Practice Exam Problems
Integration
• What is the best change of variables to make to evaluate
R
x2
√dx
x2 −4
R √3
• Make the change of variables x = tan t in the integral 1 x3 √dx
. What is the resulting t integral
1+x2
(including limits)? You need not evaluate the t integral.
R
• Evaluate the integral sec3 xdx.
R
• Evaluate the integral cos3 x sin2 xdx
R
• Evaluate the integral x2 sin(2x)dx
R
• Evaluate the integral x arctan xdx
R
• Evaluate the integral sec5 x tan3 xdx
R
√
• HARD! Evaluate the integral arctan xdx. (Hint: Substitution then integrate by parts.
Fundamental Theorem, MVT, IV Theorem
• State the fundamental theorem (in either form)
R cos x t2
d
√
• Evaluate dx
dt
sin x
1−t2
p
• Does the function f (x) = x(1 − x) + |x − 1/2| on [0, 1] satisfy the hypotheses of the mean value
theorem? The intermediate value theorem?
• Suppose that f (x) is continous on [0, 1] and differentiable on (0, 1), and that f (0) = −5, f (1) = −4.
Is it true that there exists a point c ∈ (0, 1) such that f (c) = −3? If it is true explain why using the
appropriate theorem. If not how would you change the problem to make it true.
• Suppose that f (x) is continous on [0, 1] and differentiable on (0, 1), and that f (0) = −3, f (1) = −4.
Is it true that there exists a point c ∈ (0, 1) such that f 0 (c) = 0? If it is true explain why using the
appropriate theorem. If not how would you change the problem to make it true.
• Use the intermediate value theorem to prove that there is a time each day between the hours of 1 : 00
and 2 : 00 when the hands of a mechanical clock point in the same direction. Approximately what
time is this?
Riemann Sums
• Give the lower and upper Riemann sums for
picture
Rπ
0
sin2 (x)dx with n = 4, along with an appropriate
• Give the left and right Riemann sums L3 and R3 for
from largest to smallest.
R3
2
x2 − 5 with n = 3. Order L3 , R3 and
Related Rates, Minimization and Maximization
R3
2
x2 − 5
• A open topped rectangular box with a square cross-section (i.e. the box is x by x by L) has volume
1 m3 . What is the minimum surface area of the box?
• The Batmobile is 1 km north of an intersection and is headed south at 120 kph. The Joker is 2 km east
of the intersection and is headed west at 150 kph. What is the rate of change of the distance (as the
bat flies) between Batman and the Joker.
• An ice rink takes the form of a rectangle with a semi-circle on each end. Suppose that the perimeter
of the rink is constrained to be 314 m. What is the maximum area of the rink? Do you think that ice
rinks are designed to maximize area?
Misc.
• Compute the limit limn→0 e
sin x
x
• Find the coefficient of the x17 term of Taylor series for x2 sin(2x3 ).
• Find the Taylor series for
sin(x)−x
cos(x)−1
to order 3.