210 AP Calculus Exam problem tangent line to curve

In summary, the "210 AP Calculus Exam problem tangent line to curve" is a specific question on the AP Calculus exam that requires students to find the equation of the tangent line to a given curve at a specific point. The tangent line is calculated by finding the derivative of the curve at the given point and using it to write the equation in y = mx + b form. This type of problem tests a student's understanding of derivatives and their ability to apply them to real-world situations. To solve this type of problem, students should identify the given curve and point, find the derivative, and carefully follow the solution process. Common mistakes to avoid include forgetting to find the derivative, using the wrong x-coordinate, and making errors in algebra.
  • #1
karush
Gold Member
MHB
3,269
5
Find the slope of the tangent line to the graph of
$$f(x)=-x^2+4\sqrt{x}$$
at $x=4$

(A) $8-$
(B) $-10$
(C) $-9$
(D) $-5$
(E) $-7$

rewrite as
$f(x)=-x^2+4x^{1/2}$
then
$\dfrac{d}{dx}f(x)=-2x+\dfrac{2}{\sqrt{x}}$
then
$f'(4)=-2(4)++\dfrac{2}{\sqrt{4}}=-8+1=-7\quad (E)$
 
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  • #2
I edited your post to fix the spoiler. You had the opening tag embedded in a bunch of formatting tags, and the closer had a backslash instead of the forward slash.
 
  • #3
Mahalo

I'll buy you a cup of coffee 😎
 

1. What is the purpose of finding the tangent line to a curve on the AP Calculus Exam?

The purpose of finding the tangent line to a curve on the AP Calculus Exam is to test your understanding of the concept of derivatives. The tangent line represents the instantaneous rate of change of a function at a specific point, and being able to find it demonstrates your ability to apply the derivative formula.

2. How do you find the equation of the tangent line to a curve on the AP Calculus Exam?

To find the equation of the tangent line to a curve on the AP Calculus Exam, you will need to use the derivative formula. First, find the derivative of the function at the point given in the problem. Then, plug in the x-coordinate of the given point into the derivative to find the slope of the tangent line. Finally, use the point-slope formula to write the equation of the tangent line.

3. What is the difference between a tangent line and a secant line?

A tangent line touches a curve at only one point, while a secant line intersects the curve at two points. The tangent line represents the instantaneous rate of change at a specific point, while the secant line represents the average rate of change over an interval.

4. Can you use the tangent line to approximate the value of a function?

Yes, you can use the tangent line to approximate the value of a function at a specific point. This is known as linear approximation, where the tangent line is used as an approximation of the function near a given point. The closer the point is to the given point, the more accurate the approximation will be.

5. How can the tangent line be used to find the maximum or minimum value of a function?

The tangent line can be used to find the maximum or minimum value of a function by finding the point where the tangent line is horizontal, or has a slope of 0. This point represents the maximum or minimum value of the function, also known as the critical point. Additionally, the second derivative test can be used to confirm if the critical point is a maximum or minimum value.

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