MATH 156, Winter 2023, Midterm Practice Questions
1. Find the following integrals.
Z
√
Z 5 Z
ln(t)
(a) ln x dx (b) dt (c) ex sin(x) dx
1 t2
2. Evaluate the following integrals.
Z ∞ Z −3 Z ∞
−5t 1
(a) e dt (b) 3/2
dx (c) xp dx, where p is a constant.
2 −∞ (−2 − x) 1
3. When the continuously compounded interest rate is r, compute the present value of
an income stream of rate R(t) = tet/20 .
4. Compute the following limits.
ln(x2 + 1)
2 3 1 1
(a) lim (b) lim+ x ln(x ) (c) lim − x
x→0 cos(x) − 1 x→0 x→0 x e −1
5. Find approximate values using a suitable Taylor polynomial of degree two to the
following expressions:
(a) ln(1.1) (b) sin(π/2 + 0.1)
6. Solve the following differential equations
dy y dy x2 ln(x)
(a) =y+ 2 (b) = , y(1) = 0
dx x dx ey
7. Find the domains of the following functions.
−1/4 x+1
(a) f (x, y) = (1+x+2y) (b) g(x, y) = ln (c) h(x, y) = ln(x) ln sin(y)
y
8. A car manufacturer produces two models whose unit prices are p and q and the de-
manded quantities are x and y, respectively. The demand equations are
p = 60,000 − 4x − y, q = 50,000 − 2x − 4y.
(a) Find the total revenue function R(x, y).
(b) Find the domain of R(x, y).
(c) When x = 5,000 and y = 2,000 cars are sold, what is the rate of change in
revenues with respect to changes in x?
1
, 9. Determine the first-order partial derivatives of the following functions:
(a) f (x, y) = (y − 2x)2 ,
y
(b) g(x, y) = x2 +y 2
for x, y 6= 0,
2
(c) h(x, y) = x1/y for x > 0, y 6= 0.
10. Determine all second-order partial derivatives of
y
(a) f (x, y) = ex + e , (b) g(x, y) = xyex+y .
∂ 100 g
Additionally, (c) what is ∂x50 ∂y 50
?
11. Use the chain rule to find the indicated partial derivatives.
(a) w = xey + ye−x , x = er , y = sr2 , ∂w ∂w
∂s
, ∂r ,
∂w ∂w
(b) w = xy, x = st, y = est , ∂s
, ∂t when s = 0, t = 1,
∂w ∂w
(c) w = cos(x − y), x = s2 sin(θ), y = s cos(θ), ∂s
, ∂θ .
12. Find all critical points of the following functions. Determine whether each critical
point gives a relative maximum, a relative minimum or a saddle point.
(a) f (x, y) = x3 + 2xy − 2y 2 − 2x,
2
(b) f (x, y) = ex sin(y),
(c) f (x, y) = 4c2 x2 + y 2 − 4cxy + 2 for a constant c.
13. A company produces two goods at unit costs $1 and $2. The demanded quantities of
the two goods are x = 12 − 2p + q and y = 15 + p − 4q, where p and q are the prices
of the two goods.
(a) What is the domain of the profit function as function of the prices p and q?
(b) Over all (p, q) in the domain, find the critical point(s) of the profit function.
14. A company produces x units of type A and y units of type B of a good. The maximal
quantities that the company can produce weekly are given by 2x2 + y 2 = b for a
positive constant b. The profits are $4 per unit of type A and $2 per unit of type B.
(a) Find x and y such that the profit is maximized.
(b) Assuming that currently b = 12 and that there is a possibility to increase b to 27
at a cost of 5. Is it worth to do it?
√ √
15. We optimize the function f (x, y) = 3x + 5y subject to constraint x + y = 4
(a) Find a constrained extremum.
2
, (b) One can show (you don’t have to do it) that the function f has another con-
strained extremum at the point (x, y) = (0, 16). Explain why the method of
Lagrange multipliers fails to find the constrained extremum at (0, 16).
16. Imagine yourself being an instructor for Math 156. You gave your students the fol-
lowing question and received one of your students’ answer:
d Compute the following integral
Z 0
xex dx.
−∞
Solution:
This is an improper integral. To solve the integral we would have to apply
integration by parts which is not allowed for improper integrals. c
Please find the mistake in student’s solution and provide your corrected version.
17. Let f be a function that has all its derivatives are defined. What conditions on f do
f (x)
you need so that the limit lim 2 equals a finite number? When this limit is finite,
x→0 x
what is its value (in terms of the function f and its derivatives)?
R∞
18. Let a be a constant. For which a, is the integral 0 x2020 eax+7 dx finite? If it is finite,
what is its value (in terms of the constant a)?
∂f
19. Let f be a function of two variables x and y such that f (0, 0) = 3, (0, 0) = 2 and
∂x
∂f 2 ∂g ∂g
(0, 0) = −1. For g(u, v) = 2f (u −3v,u cos(u)) , determine (a) ∂u (0, 0) and (b) ∂v (0, 0).
∂y
20. A company produces x units of type A and y units of type B of a good. The maximal
quantities that the company can produce weekly are given by ax2 + by 2 = 100 for
constants a > 0 and b > 0. The profits are $4 per unit of type A and $2 per unit of
type B. What is the maximal profit as a function of a and b?
21. Compute the total differentials of z = f (x, y) for the following functions:
ln(2x)
(a) f (x, y) = 2x + 3, (b) f (x, y) = sin(3xy ), (c) f (x, y) = .
y2
3
1. Find the following integrals.
Z
√
Z 5 Z
ln(t)
(a) ln x dx (b) dt (c) ex sin(x) dx
1 t2
2. Evaluate the following integrals.
Z ∞ Z −3 Z ∞
−5t 1
(a) e dt (b) 3/2
dx (c) xp dx, where p is a constant.
2 −∞ (−2 − x) 1
3. When the continuously compounded interest rate is r, compute the present value of
an income stream of rate R(t) = tet/20 .
4. Compute the following limits.
ln(x2 + 1)
2 3 1 1
(a) lim (b) lim+ x ln(x ) (c) lim − x
x→0 cos(x) − 1 x→0 x→0 x e −1
5. Find approximate values using a suitable Taylor polynomial of degree two to the
following expressions:
(a) ln(1.1) (b) sin(π/2 + 0.1)
6. Solve the following differential equations
dy y dy x2 ln(x)
(a) =y+ 2 (b) = , y(1) = 0
dx x dx ey
7. Find the domains of the following functions.
−1/4 x+1
(a) f (x, y) = (1+x+2y) (b) g(x, y) = ln (c) h(x, y) = ln(x) ln sin(y)
y
8. A car manufacturer produces two models whose unit prices are p and q and the de-
manded quantities are x and y, respectively. The demand equations are
p = 60,000 − 4x − y, q = 50,000 − 2x − 4y.
(a) Find the total revenue function R(x, y).
(b) Find the domain of R(x, y).
(c) When x = 5,000 and y = 2,000 cars are sold, what is the rate of change in
revenues with respect to changes in x?
1
, 9. Determine the first-order partial derivatives of the following functions:
(a) f (x, y) = (y − 2x)2 ,
y
(b) g(x, y) = x2 +y 2
for x, y 6= 0,
2
(c) h(x, y) = x1/y for x > 0, y 6= 0.
10. Determine all second-order partial derivatives of
y
(a) f (x, y) = ex + e , (b) g(x, y) = xyex+y .
∂ 100 g
Additionally, (c) what is ∂x50 ∂y 50
?
11. Use the chain rule to find the indicated partial derivatives.
(a) w = xey + ye−x , x = er , y = sr2 , ∂w ∂w
∂s
, ∂r ,
∂w ∂w
(b) w = xy, x = st, y = est , ∂s
, ∂t when s = 0, t = 1,
∂w ∂w
(c) w = cos(x − y), x = s2 sin(θ), y = s cos(θ), ∂s
, ∂θ .
12. Find all critical points of the following functions. Determine whether each critical
point gives a relative maximum, a relative minimum or a saddle point.
(a) f (x, y) = x3 + 2xy − 2y 2 − 2x,
2
(b) f (x, y) = ex sin(y),
(c) f (x, y) = 4c2 x2 + y 2 − 4cxy + 2 for a constant c.
13. A company produces two goods at unit costs $1 and $2. The demanded quantities of
the two goods are x = 12 − 2p + q and y = 15 + p − 4q, where p and q are the prices
of the two goods.
(a) What is the domain of the profit function as function of the prices p and q?
(b) Over all (p, q) in the domain, find the critical point(s) of the profit function.
14. A company produces x units of type A and y units of type B of a good. The maximal
quantities that the company can produce weekly are given by 2x2 + y 2 = b for a
positive constant b. The profits are $4 per unit of type A and $2 per unit of type B.
(a) Find x and y such that the profit is maximized.
(b) Assuming that currently b = 12 and that there is a possibility to increase b to 27
at a cost of 5. Is it worth to do it?
√ √
15. We optimize the function f (x, y) = 3x + 5y subject to constraint x + y = 4
(a) Find a constrained extremum.
2
, (b) One can show (you don’t have to do it) that the function f has another con-
strained extremum at the point (x, y) = (0, 16). Explain why the method of
Lagrange multipliers fails to find the constrained extremum at (0, 16).
16. Imagine yourself being an instructor for Math 156. You gave your students the fol-
lowing question and received one of your students’ answer:
d Compute the following integral
Z 0
xex dx.
−∞
Solution:
This is an improper integral. To solve the integral we would have to apply
integration by parts which is not allowed for improper integrals. c
Please find the mistake in student’s solution and provide your corrected version.
17. Let f be a function that has all its derivatives are defined. What conditions on f do
f (x)
you need so that the limit lim 2 equals a finite number? When this limit is finite,
x→0 x
what is its value (in terms of the function f and its derivatives)?
R∞
18. Let a be a constant. For which a, is the integral 0 x2020 eax+7 dx finite? If it is finite,
what is its value (in terms of the constant a)?
∂f
19. Let f be a function of two variables x and y such that f (0, 0) = 3, (0, 0) = 2 and
∂x
∂f 2 ∂g ∂g
(0, 0) = −1. For g(u, v) = 2f (u −3v,u cos(u)) , determine (a) ∂u (0, 0) and (b) ∂v (0, 0).
∂y
20. A company produces x units of type A and y units of type B of a good. The maximal
quantities that the company can produce weekly are given by ax2 + by 2 = 100 for
constants a > 0 and b > 0. The profits are $4 per unit of type A and $2 per unit of
type B. What is the maximal profit as a function of a and b?
21. Compute the total differentials of z = f (x, y) for the following functions:
ln(2x)
(a) f (x, y) = 2x + 3, (b) f (x, y) = sin(3xy ), (c) f (x, y) = .
y2
3
