What are the possible values of g'(a) at a differentiable point (a, g(a))?

[Loppyfoot]

Homework Statement

If the functions f and g are defined so that f'(x) = g'(x) for all real numbers x with f(1)=2 and g(1)=3, then the graph of f ad the graph of g:

Is the answer that they do not intersect?
The other choices are:

• intersect exactly 1 time
• intersect no more than 1 time
• could intersect more than 1 time
• have a common tangent at each pt. of tangency.

How would I be able to prove this?

#2)
If the function g is differentiable at the point (a, g(a)), then which of the following are true?

g'(a) = lim g(a+h) - f(a)
h
g'(a) = lim g(a)-g(a-h)
h
g'(a) = lim g(a+h)-g(a-h)
h

I think that it is only the first one can be correct. Can any of the others be correct?

(Above, the h is on the end, but the h should be under the numerator.

 

[LCKurtz]

1. Think about the function h(x) = f(x) - g(x). What can you say about h?

 

[darkchild]

Homework Statement

If the functions f and g are defined so that f'(x) = g'(x) for all real numbers x with f(1)=2 and g(1)=3, then the graph of f ad the graph of g:

Is the answer that they do not intersect?
The other choices are:

• intersect exactly 1 time
• intersect no more than 1 time
• could intersect more than 1 time
• have a common tangent at each pt. of tangency.

How would I be able to prove this?

#2)
If the function g is differentiable at the point (a, g(a)), then which of the following are true?

g'(a) = lim g(a+h) - f(a)
h
g'(a) = lim g(a)-g(a-h)
h
g'(a) = lim g(a+h)-g(a-h)
h

I think that it is only the first one can be correct. Can any of the others be correct?

(Above, the h is on the end, but the h should be under the numerator.

I'm assuming that #2 limits are taken as h -> 0, and that the first option should have g(a) in it, not f(a). To answer this question, all you need to know is the definition of the derivative.

 

[Loppyfoot]

Yes, it is the limits as h approaches 0. I made an error however, would the 3rd equation in E2 be correct if there is a 2h in the denominator?

 

[darkchild]

Yes, it is the limits as h approaches 0. I made an error however, would the 3rd equation in E2 be correct if there is a 2h in the denominator?

What about the first option? Is that f(a) supposed to be there? Once again, you need to know the definition of the derivative.

 

[Loppyfoot]

Yea I think it is supposed to be f(a).