Download Math 150 Practice Exam 1. Someone ties a hat to a large rocket and

Math 150 Practice Exam
1. Someone ties a hat to a large rocket and lights the rocket. After a few seconds, the rocket is
travelling straight up with a constant velocity of 50 m/s. A group of math students gather 300
meters away from the launch pad to view the magnificent sight. At what rate is the distance
between the rocket and the group changing when the rocket is 400 meters above the
launching pad?
(Please show all steps as given in class.)
2. Find the critical numbers of the following function
f (=
x) x 5/3 − x 2/3
3. Find the absolute maximum and minimum values of f ( x) =x 3 − 6 x 2 − 3 on the interval
[−2,3] .
4. Mean Value Theorem
a. State the two conditions that must be verified to use the mean value theorem for a
function f on an interval [a, b] .
b. Find all numbers c that satisfy the conclusion of the Mean Value Theorem for the
function f ( x=
) x 3 + 2 x on the interval [1,3] .
5. Use the stated function and its derivatives to find the critical numbers, intervals of
increase/decrease, relative extrema, concavity, and inflection points.
f ( x) =
−2 x 4 − 8 x3 + 5
f '( x) =
−8 x 2 ( x + 3)
f ''( x) =
−24 x ( x + 2 )
a. Create the number line and chart for the first derivative.
b. Create the number line and chart for the second derivative.
6. Derivative Tests
a. A continuous function has a critical number at x = 2 and it is known that f '(1) = +4
and f '(3) = −2 . Can we conclude that there is a relative extrema at x = 2 ? Why or
why not?
b. A function has a critical number at x = 5 and it is known that f '(5) = 0 and
f ''(5) = −2 . Can we conclude that there is a relative extrema at x = 5 ? Why or why
not?
7. The graph of the derivative of a function is shown below.
Sketch a possible graph of the original function f .
8. Evaluate the following limit:
lim
x →∞
(
x2 + 5x − x2 + x
)
3
9. State a function with a horizontal asymptote of y = . (No work is needed.)
5
10. A rectangular storage container with an open top is to have a volume of 8 ft3. The length of
its base is twice its width. Material for the base costs $4 per square foot while material for
the sides costs $2 per square foot. Find the WIDTH of the box that will maximize the
volume. You do NOT need to compute the other dimensions or minimal cost.
11. For each part, find AN antiderivative of the given function.
a. f ( x) = x 3 + 6 x + 2
b. f=
( x) x ( x3 + 4 )
12. A tuba is dropped from a height of 100 meters. Its acceleration due to gravity is −32 ft/s2.
Find the height of the tuba ONE second after it is released.