AP Calculus BC Test Prep: Solving f(-x)=f(x) Problem | Ryan

In summary, the conversation discusses a problem involving a differentiable function that is symmetric about the y-axis. The problem asks for the derivative of the function at certain points and the coordinates of a point of intersection with two tangent lines. The solution involves taking the derivative of the given function and using the concept of symmetry to find the answers.
  • #1
Ryush806
2
0
To review for the AP Calculus BC test coming up in May, my teacher has been giving us problems from past AP tests to help us review. However, I have absolutely no idea how to do on of the problems that I was assigned:

For all real numbers x, f is a differentiable function such that f(-x)=f(x). Let f(p)=1 and f'(p)=5 for some p>0.
a) Find f'(-p).
b) Find f'(0).
c) If line 1 and line 2 are lines tangent to the graph of f at (-p,1) and (p,1), respectively, and if line 1 and line 2 intersect at point Q, find the x- and y-coordinates of Q in terms of p.

I'm sure this problem is not incredibly hard but I'm very much confused. Please help me get started on it.

Ryan
 
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  • #2
All of these are based on the notion that f is symetric about x=0.
 
  • #3
f is symmetric in the sense that f is even - reflective symmetry in the y axis.

take the derivative on both sides of f(x)=f(-x) what do you get?

if you've done that properly what you've got defines another form of symmetry. f' is an odd function. odd functions have rotational symmetry about the origin.

you shoudl be able to find the answers now. remember z=-z when and only when z=0
 
  • #4
Originally posted by matt grime
take the derivative on both sides of f(x)=f(-x) what do you get?

Thank you for your help so far. I'm confused about taking the derivative of f(x)=f(-x). I'm not the best at this theory stuff.

Ryan
 
  • #5
f(x)=f(-x)

so d/dx(f(x)) = d/dx(f(-x))

use the chain rule on the rhs if you need to to get

f'(x) = -f'(-x)

or better I reckon -f'(x) = f'(-x)

so f'(-p) = -f'(p)

and -f'(0) = f'(-0) = f'(0)

so it must be that f'(0)=0
 

FAQ: AP Calculus BC Test Prep: Solving f(-x)=f(x) Problem | Ryan

1. What is the purpose of solving f(-x)=f(x) problems in AP Calculus BC?

Solving f(-x)=f(x) problems is an important skill in AP Calculus BC as it allows us to determine whether a given function is even, odd, or neither. This information can help us simplify the function and make it easier to integrate or differentiate.

2. How do I solve f(-x)=f(x) problems in AP Calculus BC?

To solve f(-x)=f(x) problems, we can use three methods: substitution, graphing, and algebraic manipulation. Substitution involves replacing x with -x in the original function and simplifying. Graphing involves graphing the original function and its reflection across the y-axis to see if they overlap. Algebraic manipulation involves setting f(-x)=f(x) and solving for x.

3. Can I use the same approach for all f(-x)=f(x) problems in AP Calculus BC?

No, the approach may vary depending on the given function. Some functions may be easier to solve using substitution, while others may require graphing or algebraic manipulation. It is important to be familiar with all three methods and determine which one is most suitable for the given problem.

4. Are there any tips for solving f(-x)=f(x) problems in AP Calculus BC?

One helpful tip is to remember that even functions are symmetric about the y-axis, while odd functions are symmetric about the origin. This can help us visualize the function and determine its symmetry without having to graph it. Additionally, it is important to carefully check your work and make sure that the function remains unchanged after substituting -x for x.

5. How can I practice solving f(-x)=f(x) problems for the AP Calculus BC test?

There are a variety of resources available for practicing f(-x)=f(x) problems, such as textbooks, online practice problems, and past AP Calculus BC exams. It is also helpful to work on problems with a study group or to seek help from a tutor if needed. Consistent practice and understanding of the concepts will help you feel confident and prepared for the AP Calculus BC test.

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