Math 1337 Review for Final Exam Chapters 1 through 5
Please show all work on this page. Use correct mathematical notation at all times. No graphing calculators are allowed on this exam.
1. Use L’Hospital’s Rule (if appropriate) to find the limit:
2. Find each of the following derivatives:
3. Find the first derivative:
f(x) = ln (sec x)
f(x) = cos (ln x)
f(x) = ln[cos (2x2 + 3)]
4. Find the first derivative:
f(x) = cos (ex + 5)
f(x) = 2x3∙ esin (4x+1)
f(x) = sin−1(e3x)
5. Use implicit differentiation to find the first derivative:
3x – xy + 1 = 0
x3 + y4 + xy + x + y = 1
6. Find the equation of the line tangent to the graph of f(x) at the indicated point.
y = 4 cos x at
y = 3 tan4x at
y = (1 + x) cos x at (0, 1)
at (1, 1)
f(x) = (2 + x)e-x at (0, 2)
y = x ln x at (e, e)
7. Use logarithmic differentiation to find the derivative of the function.
y = xcos x
y = (sin x)ln x
8. Determine intervals where f(x) is increasing and decreasing and find all local extrema.
9. Determine the intervals where f(x) is concave up and concave down and find all inflection points.
10. Use the Second Derivative Test to find all local extrema of the given function:
f(x) = x4 – 32x + 64
11. Find all absolute extrema of the given function over the indicated interval.
over [1, 16]
over [1, 4]
12. Given:
a. Find all values of x for which f(x) is discontinuous.
b. Find the coordinates of any hole in the graph (if there is one).
c. Find the equation of any vertical asymptote.
d. Find the equation of any horizontal asymptote.
13. Related Rates Problems
a. A balloon is being filled with helium at the rate of 4 ft3/min. Find the rate, in square feet per minute,
at which the surface area is increasing when the volume is ft3. ; SA = 4r2
b. A circular conical reservoir, vertex down, has depth 20 ft and radius of the top 10 ft. Water is leaking out so that the surface is falling at the rate of ½ ft/hr. Find the rate, in cubic feet per hour,
at which the water is leaving the reservoir when the water is 8 ft deep.
c. Suppose two bikers leave the same place but not at the same time. Biker X is traveling east at a rate
of 8 miles per hour and Biker Y is traveling north at a rate of 14 miles per hour. At a certain time
Biker X is 6 miles east of the starting point while Biker Y is 8 miles north of the starting place. At
this instant, at what rate is the distance between the two bikers changing?
14. Exponential Growth and Decay
a. . A bacteria population has 300 cells initially and grows at a rate proportional to its size. After 3 hours
the population has increased to 1200 cells.
i. Write an equation to represent the number of cells present in the population after t hours.
ii. Find the rate of growth after 6 hours.
b. Suppose that 10 grams of the plutonium isotope Pu-239 was released in the Chernobyl nuclear
accident in 1986. If the half-life of Pu-239 is 24,360 years, how long will it take for the 10 grams to
decay to 1 gram?
15. Optimization
a. A rectangular page is to contain 24 square inches of print. The margins at the top and bottom of the
page are to be 1.5 inches. The margins on the left and right sides of the page are to be 1 inch. What
should the dimensions of the page be so that the least amount of paper is used?
b. A farmer plans to fence a rectangular pasture adjacent to a river. He plans to use the river as one side
of the pasture and fence the other 3 sides. The pasture must contain 90,000 square meters in order to
provide enough grass for the herd of horses that will be contained in the pasture. Find the minimum
cost of fencing this pasture if the side opposite the river costs $6 per meter and the other two sides
cost $3 per meter.
16. Evaluate each of the following indefinite integrals:
17. Evaluate each of the following definite integrals:
18. Use the Substitution Rule to evaluate each indefinite integral:
19. Use the Substitution Rule to evaluate each definite integral:
20. Find the particular position function s(t), if
a. if the velocity of a particle at time t is given by v(t) = 4t + 4 (where t > 0) and s(1) = 2
b. if the object’s velocity at time t be given by (where t > 0)
and s(0) = 15.
c. if the velocity of the object at time t is given by v(t) = et + 12t2 + 3 (where t > 0) and s(0) = 5.
21. Find the average value of the function:
a. f(x) = x2 – 3 over [1, 3].
b. f(x) = 2x2 + 3 on the interval [0. 2].
c. f(x) = cos x over
22. Find the area between the x-axis and the graph of y over the indicated interval:
a. from x = 1 to x = 16.
b. y = sin x from x = 0 to x =