Question

I know it's hard to draw a graph here, but I just need an explanation of both if possible. I appreciate any help I can get! :)

"Determine whether the following statements are true and give an explanation or counterexample. Explain why or why not, using complete sentences and appropriate terminology correctly. For part a, include a nice graph to support your claim.

a. Consider the graph of the parabola f(x)=x^2. For a>0 and h>0, the secant line through (a,f(a)) and (a+h,f(a+h)) and always has greater slope than the tangent line at (a+f(a)).

b. d/dx (π^5)=5π^4 "

Answer 1

The secant line has slope
((a+h)^2 - a^2)/h = (a^2+2ah+h^2-a^2)/h = (2a+h)
The slope of the tangent line is 2a
I'm sure you can graph y=x^2, right?

d/dx π^5 = 0
because π is a constant
However,
d/dx x^5 = 5x^4
so at x=π, the tangent line has slope 5π^4

Answer 2

Thanks for your help! I mostly struggled with the explanations, but the graphing was fine.