Calculus Answers

If f(x)=∫x0(t3+7t2+4)dt
then
f′′(x)=

List corresponding features of the graphs of a function f, its first derivative f′, and (an) antiderivative F. Describe a strategy whereby, given a plot showing the graphs of f, f′, and F, you can determine which is which. Apply your strategy to identify the graphs A (blue), B (red) and C (green) as the graphs of a function f, its first derivative f′, and (an) antiderivative F:

is the graph of the function f.

is the graph of the function's first derivative f′.

is the graph of an antiderivative of the function F.

Consider the function f(x)=5x2−4x7. Let F(x) be the antiderivative of f(x) with F(1)=0.

Then F(x)=

This question summarizes a few simple facts about polynomials. Recall that a polynomial p can be written in the form
p(x)=∑i=0nαixi
where we assume that αn≠0. The degree of a polynomial is the largest exponent present, and so the degree of p is n. Fill in the blanks in the following questions:

The degree of p′(x) is
.

The degree of ∫p(x)dx is
.

The degree of p2(x) (i.e., the square of p) is
.

The (n+1)-th derivative of p is

A plane leaves an airport traveling at 400 mph in the direction N 45° E. A wind is blowing at 40 mph in the direction N 45° W. What is the ground speed of the plane?

In an opera house, the base of a chandelier is 160 ft above the floor. Suppose the Phantom of
the Opera dislocates the chandelier and causes it to fall from rest and crash on the floor.
a. What is the equation of motion of the chandelier?
b. Find the instantaneous velocity of the chandelier at 1 sec and 1.5 sec.
c. Find how long it takes the chandelier to hit the floor.
d. What is the speed of the chandelier when it hits the floor?

: A toy rocket rises vertically in such a way that t seconds after its liftoff, it is
s(t)= -16t^2 + 200t feet above the ground.
(a) How high is the rocket after 6 seconds?
(b) What is the average velocity of the rocket over the first 6 seconds of flight (between t=0 and t=6)?
(c) What is the instantaneous velocity of the rocket at t=2 sec?

Jun throws a stone straight upward alongside a tree with an initial velocity of 32 ft/sec. The stone rises until it is even with the top of the tree and then falls back to the ground. How tall is the tree?

A car is travelling at 100 ft/sec when the driver suddenly applies the brakes. The position
function of the skidding car is

A water tank in the form of an inverted right-circular cone is being emptied at the rate of
6 m^3 /min. The altitude of the cone is 24m, and the base radius is 12m. Find how fast the water level is lowering when the water is 10m deep?