Question

Suppose f (x) is a differentiable function defined on the interval [0, 60]. Use the function values given in the table below to answer the following:
x: 0 15 30 45 60
f(x): -3 -1 2 6 2
a) Approximate the value of f '(20).
b) Show that f '(c1) = 1/12 for some value x = c1 in the interval (0, 60).
c) Show that f (c2) = 0 for some value x = c2 in the interval (0, 60).

Answer 1

PLEASE HELP!!! I need to know this

Answer 2

Are you really typing a reply answer for this or?

Answer 3

a) F'(20) can be approximated by finding the secant line on boundaries that hold x=20 while being as small as given. [15, 30] satisfies this. Thus F'(20) is approximated by \[F(30)-F(15) /30-15 = -1-(-3)/15 = 2/15\].
b) We prove b) using the mean value theorem for derivatives. It states that if f(x) is continuous in a finite interval [a,b] and differentiable on an open interval (a,b), then there is a point c in [a,b] such that f'(c) equals the secant lines formed between x=a and x=b. It is happenstance that the secant line formed between x= 2 and x= -3 is -1/12. It is also given that f(x) is differentiable in [-3,2]. But there's the theorem that if a function is differentiable at a point, then it is continuous at a point. Thus, we can assume that it is differentiable in [-3,2] and continuous on [-3, 2]. Thus, we've proven b.
c) Lastly, the intermediate value theorem allows us to prove c). It states that if there's two values for f(x) in the finite, continuous interval [a,b] such that one is greater than zero, and the other lesser than zero, then there is a point c2 such that f(c2) = 0. We proved in b) that there exists a finite closed interval [a,b]. Also, there's f(30) = -1 and f(45) which satisfy the other requirement for this theorem. Thus, QED.

Answer 4

I really hope this is right I don't understand this whole question so THANK YOU SO MUCH!! <3

Answer 5

What does QED mean though?