What is the relationship between slope and symmetry in differentiable functions?

In summary: In particular, if f'(p) = 5, then f'(-p) = 5 as well.In summary, for all real numbers x, f is a differentiable function such that f(-x) = f(x) and f(p) = 1 and f'(p) = 5 for some p>0. From this, we can determine that f is an even function and f'(-p) = 5. We also know that the lines tangent to the graph of f at (-p,1) and (p,1) will intersect at a point Q with x and y coordinates dependent on p.
  • #1
Nutz
1
0
:smile: 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 ß1 and ß2 are lines tangent to the graph of f at (-p,1) and (p,1) respectibely, and if ß1 and ß2 intersect at point Q, find the x and y coordinates of Q in terms of p.





SO basically the week we did this stuff I was out with pneumonia for 6 days. Sucks, but know I am stuck with this assignment sheet. I pretty much don't know where to begin on this. Any help would be greatly appreciated.
 
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  • #2
Nutz said:
:smile: 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 ß1 and ß2 are lines tangent to the graph of f at (-p,1) and (p,1) respectibely, and if ß1 and ß2 intersect at point Q, find the x and y coordinates of Q in terms of p.
SO basically the week we did this stuff I was out with pneumonia for 6 days. Sucks, but know I am stuck with this assignment sheet. I pretty much don't know where to begin on this. Any help would be greatly appreciated.

There was a very similar problem in another forum, and you might be the same person. :confused:

Go check it out, I think it's in General Math. Completely identical.
 
  • #3
You might consider:

Since f(-x) = f(x), this makes f an even function, i.e., symmetric about the y axis..

symmetry should say something about the value of the slope at equal distances from x=0.
 

1. What is a differentiable function?

A differentiable function is a mathematical function that is smooth and continuous, meaning that it has no abrupt changes or breaks. It can be differentiated, or have its slope determined, at every point on its graph.

2. How is differentiability determined?

Differentiability is determined by checking if the limit of the difference quotient exists as the change in x approaches 0. If the limit exists, the function is differentiable at that point.

3. What is the difference between differentiability and continuity?

While both differentiability and continuity involve the smoothness of a function, continuity only requires that the function is unbroken and can be drawn without lifting the pen. Differentiability goes a step further and requires that the function has a defined slope at every point.

4. Can a function be differentiable but not continuous?

No, a function cannot be differentiable but not continuous. If a function is differentiable at a point, it must also be continuous at that point. However, a function can be continuous but not differentiable at certain points.

5. What is the importance of differentiable functions in mathematics and science?

Differentiable functions are important in mathematics and science because they allow us to calculate the rate of change of a function at any point. This is crucial in understanding the behavior of many real-world phenomena, such as the growth of populations, the movement of objects, and the change of temperature over time.

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