Tanget+Lines

What is a Tangent Lin e?
 * ==== A ** tangent line ** of a curve is a line that touches one point on the curve. The word // tangent // comes from the Latin // tangere //, // to touch // . ====
 * As it passes through the point where the tangent line and the curve meet, or the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point.
 * In geometry, the tangent line (or simply the tangent) to a curve at a given point is the straight line that "just touches" the curve at that point


 * As it passes through the point where the tangent line and the curve meet, or the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point.
 * In order to find a tangent line, we need to find a ** SLOPE ** (derivative) and a ** POINT **

Steps to Finding a Tangent Line
** Step 1: Find the derivative of the curve. ** **Step 2: Evaluate the derivative at the point given.**

**Step 3: Solve for** b.
**Step 4: Write the equation.**

Here's an example that takes you through the process of finding a tangent line: Start with the equation of ** y=x^2 + 1 **

Step 1: Find the derivative, this can be done in many ways depending on your equation (chain rule, product rule, quotient rule) In this case, we will use a regular derivative by distributing the exponent and taking the derivative of each piece. When you finish the derivative you should get ** y=2x **

Step 2: Use the derivative to plug in points given. Say for this problem we are instructed to do it at x=2. The derivative at that x is y'(2)=4. This 4 is your m, or your slope

Step 3: Now we need to solve for b. Use the formula y=mx+b to plug in. We have x, y (by plugging in x=2 in the equation we get 2^2 + 1 which is 5 ), and m so now we plug those all in to find b.

You should get 5=4 x 2 + b, then use a little bit of basic algebra and you get 5=8+b, subtract 8 from both sides and you get b to equal -3.

Step 4: Now that we have all the parts, we have to put them into a new equation. Start with y=mx+b. Plug in slope (m) and y-intercept (b)

Analytical approach
====The geometric idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of [|calculus] in the 17th century. In the second book of his [|Geometry], [|René Descartes] [|said] of the problem of constructing the tangent to a curve, "And I dare say that this is not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know".====

Equations of Tangent Lines Practice Problems
**Instructions:**

Find the equations of the tangent lines of the following functions through the given points, giving your answers in slope-intercept form. Give your answers in exact terms.

 1. g ( x )= x^ 3 − 3 x <span class="mo" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">1 ; <span class="mo" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">( <span class="mn" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">1 <span class="mo" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">,− <span class="mn" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">1 <span class="mo" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">)

<span style="background-color: #ffffff; display: block; font-family: 'Times New Roman'; font-size: medium;"> <span style="font-family: Georgia,'Times New Roman',Times,serif; font-size: 1em; text-align: left;">2. <span class="mi" style="font-family: MathJax_Math; font-size: 19px; vertical-align: 0px;">f <span class="mo" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">( <span class="mi" style="font-family: MathJax_Math; font-size: 19px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">)= <span class="mn" style="font-family: MathJax_Main; font-size: 13px; vertical-align: 0px;">2/3 <span class="mi" style="font-family: MathJax_Math; font-size: 19px; vertical-align: 0px;">x^ <span class="mn" style="font-family: MathJax_Main; font-size: 13px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">+ <span class="mi" style="font-family: MathJax_Math; font-size: 19px; vertical-align: 0px;">x^ <span class="mn" style="font-family: MathJax_Main; font-size: 13px; vertical-align: 0px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">− <span class="mn" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">12 <span class="mi" style="font-family: MathJax_Math; font-size: 19px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">6 ; <span class="mi" style="font-family: MathJax_Math; font-size: 19px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">= <span class="mn" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">0 <span style="font-family: Georgia,'Times New Roman',Times,serif; font-size: 1em; text-align: left;">3. <span class="mi" style="font-family: MathJax_Math; font-size: 19px; vertical-align: 0px;">h <span class="mo" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">( <span class="mi" style="font-family: MathJax_Math; font-size: 19px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">)= <span class="mn" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">2 <span class="mi" style="font-family: MathJax_Math; font-size: 19px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 13px; vertical-align: 0px;">1/ <span style="font-family: MathJax_Main; font-size: 13px; vertical-align: 0px;">√x ; <span class="mo" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">( <span class="mn" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">1 <span class="mo" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">, <span class="mn" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 19px; vertical-align: 0px;">) <span style="background-color: #ffffff; display: block; font-family: 'Times New Roman'; font-size: medium;">

Multiple Choice
1. An equation of the tangent line to the curve y = 4x2 - 6x - 5 at the point (-1, 5) is

A) y = (8x - 6)(x - 1) + 5.

B) y = -14x - 9.

C) y = 14x + 71.

D) y = -14x + 19.

E) y = 8x - 6.

W
<span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; text-align: left;">e need to determine if the point <span style="display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;"><span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">,− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">) <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;">is on the graph of g(x). We could graph it or plug in the point <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 and see if we get <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">y <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">=− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;">. Plugging in the point is more exact (since our graph my LOOK like the point is on the curve when it may not be). So let's evaluate <span style="display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;"><span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">g <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">) at <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;">.<span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">g <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mn" style="font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)+ <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">=− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;">Step 1: Calculate the derivative and slope. <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;"> <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">g <span class="mo" style="font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">′ <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mn" style="font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3  <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;">This is the equation that gives the slope of <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">g <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)  at every point on the curve. To get the slope at the specific point <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1, we evaluate <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">g <span class="mo" style="font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">′ <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mn" style="font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">0. <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;">Step 2:Using the slope from step 1 and the point given in the problem statement, find the equation of the tangent line. <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;">Okay, so the slope <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">m <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">0, which means the point-slope form of the tangent line is: <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;"> <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">y <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">−(− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">0 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">) → <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">y <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">=− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1. <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;">This equation is in slope-intercept form ( <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">y <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">mx <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">+ <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">b ), so we are done. <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;"><span class="minHdr" style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 1em;">Final Answer <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;">The equation of the tangent line to <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">g <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)= <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mn" style="font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1  through the point <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">,− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)  is

** y=−1. **
<span style="background-color: #ffffff; display: block; font-family: 'Times New Roman'; font-size: 14px; text-align: left;"> <span style="background-color: #ffffff; display: block; font-family: 'Times New Roman'; font-size: 14px; text-align: left;"> <span style="background-color: #ffffff; display: block; font-family: 'Times New Roman'; font-size: 14px; text-align: left;">2. <span style="background-color: #ffffff; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;">Step 1: Find the derivative of <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">f <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">). <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">f <span class="mo" style="font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">′ <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">23 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mn" style="font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)+ <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">12 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mn" style="font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">12 Now use the value <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">0, to calculate the slope. <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">f <span class="mo" style="font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">′ <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">0 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">0 <span class="mn" style="font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)+ <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">0 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">12 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">=− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">12 Step 2: Find the equation of the tangent line. We know that <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">m <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">=− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">12 and <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">0  but, we need the y-value of the function, <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">f <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">0 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">0 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">0 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">0 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">6 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">6. So our point is <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">0 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">, <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">6 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">). Now we can use the point-slope form of the equation of a line to get the tangent line. <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">y <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">6 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">=− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">12 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">0 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">) → <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">y <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">=− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">12 <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">6 To verify the correctness of this solution, it's a good idea to graph both the function and the tangent line on the same set of axes. <span class="minHdr" style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 1em;">Final Answer The equation of the tangent line to <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">f <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">23 <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mn" style="font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">+ <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mn" style="font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">12 <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">6 at <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">0  is =** y=−12x+6. **= <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;">3. <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;">We need to verify that the point <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">, <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">) is on the graph of <span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">h <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">). <span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">h <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)= <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)+ <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">/ <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">√ <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span style="font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px;"> Since <span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">h <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)= <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 we know that the point <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">, <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">) is on the curve. So we can use the [|technique outlined above] <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;">.Step 1: Find the derivative and calculate the slope. <span style="font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px;"> First, we rewrite the square root term to be able to use the <span style="background-color: #ffffff; color: #004276; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;">power rule <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;">.<span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">h <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)= <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">+ <span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">− <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">1 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">/ <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">2 <span style="font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px;"> In this step, we rewrote the square root as a <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">/ <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;"> power and then moved the term to the numerator, which entailed changing the sign of the exponent. Now we can take the derivative. <span style="font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px;"><span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">h <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">′ <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)= <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">+(− <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">/ <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">) <span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">− <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">3 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">/ <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">2 To find the slope at the point <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">, <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">), we need to calculate <span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">h <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">′ <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">) <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;">.<span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">h <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">′ <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)= <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">+(− <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">/ <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)( <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">) <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">− <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">3 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">/ <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 10px; vertical-align: 0px;">2 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">− <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">/ <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">/ <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 Step 2: Find the equation of the tangent line. <span style="font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px;">We have the slope <span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">m <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">/ <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 and the point <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">, <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">) <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;">.Let's use the slope-intercept form this time. <span style="background-color: #ffffff; color: #888888; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;"><span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">y <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">mx <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">+ <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">b <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">→ <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">=( <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">/ <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)( <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)+ <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">b <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">→ <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">b <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">− <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">/ <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">= <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">/ <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">→ <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">y <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">=( <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">/ <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">) <span class="mi" style="font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">/ <span class="mn" style="font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span style="background-color: #ffffff; display: block; font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px; text-align: left;"> To check the answer, it would be a good idea to graph the function and the tangent line on the same set of axes. <span style="font-family: Georgia,'Times New Roman',Times,serif; font-size: 14px;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif;">Final Answer The equation of the tangent line to the curve <span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">h <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">)= <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">2 <span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">+ <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mi" style="color: #888888; font-family: MathJax_Math; font-size: 15px; vertical-align: 0px;">x <span style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">√ at the point <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">( <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">1 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">, <span class="mn" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">3 <span class="mo" style="color: #888888; font-family: MathJax_Main; font-size: 15px; vertical-align: 0px;">) is

Multiple Choice Answers:
1. B.