i got a big final exam tomorow, and im trying my best to understand all of the practice problems so i at least have a prayer. there will most likely be more questions from me throughout the night, and i appreciate anything that anybody can help me with
1. Let
f (x) be the function defined by
f
(x) = (integral from 0 to x) sin(t^2) dt
Answer the following questions, and mention any theorems or results that you are using to justify your answers:
(a) Is
g(t) = sin(t^2) a continuous function?
(b) Is g(t) integrable? Explain.
(c) Use the graph of sin(t) to sketch a rough graph for g(t) = sin(t^2).
(d) Is f (x) continuous? Why?
(e) Is f (x) differentiable? Why?
(f) Find f '(x) and f ''(x).
(g) Find all the critical points of f (x) (i.e. those values of x such that f 0(x) = 0), and find all the values where
there may be a change of concavity (i.e. those points such that f ''(x) = 0).
(h) Determine the intervals where f (x) is increasing/decreasing, and where f (x) is convex/concave.
(i) Sketch the graph of f (x). Notice that f (0) = 0.
(j) Use Taylor polynomials to approximate the value f (1). Make sure that the error (in absolute value) of your
approximation is less than 0:001.
first off, would i be correct in saying that f(x)=1/2 - 1/2 cos(x^2)....i kind of forgot how to evaluate these. heres my initial thoughts
a) yes, i dont really know why, other than that sin is continuous
b)yes, i dont know why
c)would that be my above f(x) function?
d)yes, i dont know why
e)
f)wouldnt f'(x)=sin(t^2) and then f''(x)=2t cos(t^2)?
g)
hmm. after doing this myself, it seems im getting more and more confsed, so i dont want to learn this the wrong way. these seem like easy questions, just im having trouble understanding them. the whole f(x) and g(t) thing is throwing me off
once again, any help is extremely appreciated