Calculus 1 : How to find slope by graphing functions

Study concepts, example questions & explanations for Calculus 1

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Example Questions

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Example Question #1 : How To Find Slope By Graphing Functions

What is the slope of the tangent line of f(x) = 3x4 – 5x3 – 4x at x = 40?

Possible Answers:

None of the other answers

743,996

331,841

684,910

768,000

Correct answer:

743,996

Explanation:

The first derivative is easy:

f'(x) = 12x3 – 15x2 – 4

The slope of the tangent line is found by calculating f'(40) = 12 * 403 – 15 * 402 – 4 = 768,000 – 24,000 – 4 = 743,996

Example Question #2 : How To Find Slope By Graphing Functions

Find the slope of the line tangent to  when  is equal to .

Possible Answers:

Correct answer:

Explanation:

To find the slope of a tangent line, we need to find the first derivative of the function at that point. In other words, we need y'(6).

Taking the first derivative using the Power Rule  we get the following.

Substituting in 6 for b and solving we get:

.

So our answer is 320160

Example Question #3 : How To Find Slope By Graphing Functions

Find function which gives the slope of the line tangent to .

Possible Answers:

Correct answer:

Explanation:

To find the slope of a tangent line, we need the first derivative.

Recall that to find the first derivative of a polynomial, we need to decrease each exponent by one and multiply by the original number.

Example Question #4 : How To Find Slope By Graphing Functions

Find the slope of the line tangent to  at .

Possible Answers:

Correct answer:

Explanation:

The slope of the tangent line can be found easily via derivatives. To find the slope of the tangent line at s=16, find b'(16) using the power rule on each term which states:

Applying this rule we get:

Therefore, the slope we are looking for is 454.

 

Example Question #5 : How To Find Slope By Graphing Functions

Find the slope of  at .

Possible Answers:

Correct answer:

Explanation:

To find the slope of the line at that point, find the derivative of f(x) and plug in that point. 

Remember that the derivative of  and the derivative of   

Now plug in  

Example Question #6 : How To Find Slope By Graphing Functions

Find the slope of at  given . Assume the integration constant is zero.

Possible Answers:

Correct answer:

Explanation:

The first step here is to integrate  in order to get .

Here the problem tells us that the integration constant , so

Plug in  here

Example Question #7 : How To Find Slope By Graphing Functions

Consider the curve

.

What is the slope of this curve at ?

Possible Answers:

Correct answer:

Explanation:

The slope of a curve at any point is equal to the derivative of the curve at that point.

Remembering that the derivative of  and using the power rule on the second term we find the derivative to be:

.

Pluggin in  we find that the slope is .

Example Question #8 : How To Find Slope By Graphing Functions

Find the line tangent to  at .

Possible Answers:

Correct answer:

Explanation:

Find the line tangent to  at .

 

First, we find :

Next, we find the derivative:

Therefore, the slope at  is:

.

Using point-slope form, we can write the tangent line:

Simplifying this gives us:

Example Question #9 : How To Find Slope By Graphing Functions

An isosceles triangle has one point at , one point at  and one point on the -axis. What is the slope of the line between the point on the -axis and ?

Possible Answers:

Correct answer:

Explanation:

The other point of the triangle must be at  as it must be equidistant from the other two points of the triangle. Since all points on the y-axis are  units away from the other points in the  direction, the third point must be equidistant in the  direction from both  and . The distance between these points is , so the third point must have a y-value of . The third point is now at  so the slope of the line from  to  is as follows.

Example Question #10 : How To Find Slope By Graphing Functions

What is the slope of the line tangent to the graph of  at ?

Possible Answers:

Correct answer:

Explanation:

We must take the derivative of the function using the chain rule yielding .

The chain rule is .

Also remember that the derivative of  is .

Applying these rules we get the following.

Plugging in the value for  we get  which is .

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