Download Math 132. Practice Questions From Calculus II I. Topics Covered in

Math 132.
Practice Questions From Calculus II
I. Topics Covered in Test I
0. State the following calculus rules (these are many of the key rules from Test 1 topics).
Z
• (Trapezoidal Rule)
b
f (x) dx ≈
a
• (Fundamental Theorem of Calculus I)
• (Fundamental Theorem of Calculus II)
Z
d
•
[ln |u|] =
dx
and
Z
•
Z
tan u du =
and
cot u du =
Z
•
1
du =
u
Z
sec u du =
and
csc u du =
• Suppose f and g are inverse functions and the domain of f is an interval. If f is differentiable, then g 0 (x) =
Z
d u
•
e =
dx
and
x
?
• For a > 0, a 6= 1, a = e ,
• For a > 0, a 6= 1, loga x =
eu du =
Z
d u
a =
dx
and
and
d
[loga u] =
dx
au du =
Z
1.
Evaluate
π/3
[2 cos2 x + 2 sin2 x − 3 tan x sec x − 6x2 ] dx.
0
2.
Find the area of the region that lies below y = −x2 + 20x − 64 and above the x-axis.
√
3. (a) Suppose f 0 (x) = (x + 2) x − 1, find f .
Z 5
√
(b) Evaluate
(x + 2) x − 1 dx
2
4. Use the Fundamental Theorem of Calculus to find the derivative of
Z sin(x)
(cos(t4 ) + t)dt
h(x) =
−2
5. Find an equation for the function f (x) that has the given derivative and initial value.
f 0 (x) = x(x2 + 1)5
f (−1) =
Z
13
3
π
sin x dx.
6. (a) Use the trapezoidal rule with n = 6 to estimate
0
(b) Use the trapezoidal error formula to determine the largest possible error you would expect in
your answer to (a) (you will be given the error formulas for Simpson’s Rule and the Trapezoidal
Rule if needed on the final test)?
7. Let y = (ln x)cos x for x > 1. Find
dy
and then find the tangent line at (e, 1).
dx
8. Let f (x) = x2 ln(10 − 2x2 ). Find f 0 (x) and use interval notation to give the domain of f .
9. (a) State the definition of ln x as given in the text.
(b) Show that ln(ab) = ln a + ln b when a > 0 and b > 0.
(c) Use the definition of ln x to show for integers n ≥ 2 (draw a sketch!)
1+
1
1
1 1
1
+ ... +
≥ ln n ≥ + + . . . +
2
n−1
2 3
n
(d) Verify that ln x is strictly increasing and concave down for x > 0.
10. Differentiate f (x) = x73x
2 −2x+1
and g(x) = log7 [(x2 + 7)(4x4 + x2 + 3)]
11. Find the average value of an integral function f (x) = −
Z
12. Evaluate
7 ln x
on the interval [1.1, 1.8].
x
8x3 + 2x2 + 50x + 12
dx
x2 + 6
13. Find the slope of the tangent line to the curve
cos(3x − 3y) − xe−x = −
7π −7π/2
e
2
at the point (7π/2, 5π)
14. (a) Let f (x) = 3x + 6x11 . Find a value a such that f (a) = −9, and then find (f −1 )0 (−9).
(b) Let f (x) = x2 − 17x + 76 on the interval [8.5, ∞). Find a value a such that f (a) = 10,
then find (f −1 )0 (10).
Z
15. Evaluate the integral
Z
16. Evaluate
17.
6
(4 − x)5 · 5(4−x) dx
−1
ex
dx
x2
Let f (x) =
5ex − 7
. Find f −1 (x) and find the domain of f −1 .
19ex + 11
18. Let f (x) = 8x log3 x, find f 0 (x).
II. Topics Covered on Test II
1. Practice on the new differentiation/integration formulas:
(a) Differentiate the following with respect to x:
(i) arcsin u
(ii) arctan u
(iii) arcsec u
(b) State the integral forms of the differentiation rules in (a)
(c) State the definitions of sinh x, cosh x and tanh x
(d) Differentiate sinh x, cosh x and tanh x. Then write the integral form of these differentiation
rules.
(e) Applications of some of the differentiation rules you stated above:
Differentiate: arctan(ex + sin x)
e2x cosh(5 − x3 )
(f) Applications of some of the integration rules you stated above:
Z
Z
Z
ex
Evaluate:
cosh(5x) dx
dx
x sinh(x2 ) dx
1 + e2x
5x2 arcsin(e4x )
Z
cosh(3x) dx
1 + sinh(3x)
(g) Use the definitions of sinh x and cosh x given above to show
(i) cosh2 x − sinh2 x = 1
(ii) sinh 2x = 2 sinh x cosh x
2.
A rectangular billboard 7 feet in height stands in a field so that its bottom is 8 feet
above the ground. A cow with eye level at 4 feet above the ground stands x feet from the
billboard, as illustrated in the diagram below.
(a) Express θ, the vertical angle subtended by the billboard at her eye, in terms of x.
(b) Find the distance the cow must stand from the billboard to maximize θ.
Z
3. Evaluate the integral
Z
4. Evaluate
√
sinh 7x
√
dx
7x
8x dx
√
2x2 − x4
5. An unknown radioactive element decays into non-radioactive substances. In 660 days the
radioactivity of a sample decreases by 68 percent.
(a) What is the half-life of the element?
(b) How long will it take for a sample of 100 mg to decay to 90 mg?
6. Suppose that news spreads through a city of fixed size 200000 people at a rate proportional
to the number of people who have not heard the news.
(a) Formulate a differential equation and initial condition for y(t), the number of people who
have heard the news t days after it has happened.
( b) 5 days after a scandal in City Hall was reported, a poll showed that 100000 people have
heard the news. Using this information and the differential equation, solve for the number of
people who have heard the news after t days.
7. When an object is removed from a furnace and placed in an environment with a constant
temperature of 80◦ F, its core temperature is 1500◦ F. One hour after it is removed, the core
temperature is 1120◦ F. Use Newton’s law of cooling to find the core temperature 5 hours after
the object is removed from the furnace.
8. Solve the differential equation
satisfies y(4) = ln(4).
9. Solve the differential equation
dy
= (x − 4)e−2y and then find the particular solution that
dx
dy
= 88xy 10 subject to y(0) = 4.
dx
10. Which of the following differential equations generates the slope field given below?
(a)
dy
= xy − y
dx
(b)
dy
= y − xy
dx
(c)
dy
= x − xy
dx
(d)
dy
= xy − x
dx
11. Biologists stocked a lake with 233 fish and estimated the carrying capacity to be 9700.
The number of fish tripled in the first year. Assuming the fish population satisfies the logistic
L
equation P (t) =
, find P (t) and determine how long it will take for the population
1 + be−kt
to reach 4850.
12. Find the area between the curves x = 4 − y 2 and x = y − 2.
13. Find the area bounded by the line y = x + 2 and the curve y = (x + 2)3 .
14. The region between the graphs of y = x2 and y = 2x is revolved around the line y = 4.
Find the volume of the resulting solid.
15. A ball of radius 17 has a round hole of radius 3 drilled through its center. Find the volume
of the resulting solid.
16. (a) Use the method of volume by cross-sectional slices to find the volume of a pyramid
with a square base of 260 cubits and a height of 1000 cubits.
(b) Repeat problem (a) where the base has side b and height is h and b and h are unknown
constants.
17. Find the volume of the solid obtained by revolving the region bounded by the curves
y = 1/x3 , y = 0, x = 3 and x = 5 about the line y = 3.
18. Use the method of your choice to find the volume of the solid formed when the region
bounded by y = x2 and y = 2x is revolved about (a) the line y = −3; and (b) the line x = 7.
19. Differentiate: (a) f (x) = ex arctan(x2 )
(b) g(x) = x arcsin x +
20. Suppose arcsin(x3 y) = xy 3 . Use implicit differentiation to find
21. Evaluate the expression tan(2 cos−1 (x/3)).
22. To solve the equation (cos x)2 − 1.1 cos x − 1.26 = 0.
√
1 − x2
dy
.
dx
III. Topics Covered on Test III
1.
Find the arc length of the curve y =
1 5x
e + e−5x for 0 ≤ x ≤ 3
10
2. Find the area of the surface obtained by rotating the curve y = 9x3 for x = 0 to x = 10
about the x-axis.
3. Set-up and evaluate an integral to find:
(a) The arc length of the curve y =
x3
1
+ , 1 ≤ x ≤ 2.
6
2x
(b) The surface area obtained by revolving the curve in (a) about the x-axis.
4.
A tank containing water has a trapezoidal cross section. The width of the edge of the
top of the trapezoid is b2 = 2 meters, the width of the base of the trapezoid is b1 = 1.5 meters
and the height of the trapezoid is h = 1.8 meters. The depth of the water in the trough is
d = 1.4 meters (see the diagram below that is not drawn to scale), and the length of the trough
is 5 meters. Find the work required to pump the water out over the top edge of the trough.
Use the fact that the water weighs 1000 kg per cubic meter.
5. Consider the planar lamina bounded by y = −x2 +4x+2 and y = x+2 with uniform density
ρ. Find M , Mx , My and (x̄, ȳ). Set-up the integrals and then use technology to evaluate the
integrals.
6. The cross-section of the end of a trough is described by x4 ≤ y ≤ 1 and where −1 ≤ x ≤ 1.
Then the width of the top of the cross-section of the trough is 2 feet, and the length of the
trough is 10 feet. The trough is full of pig swill that weights 50 pounds per cubic foot. (The
cross-section of the end of the trough is shown in the figure below).
(a) Find the fluid force of the swill on one end of the trough.
(b) Find the amount of work required to pump all of the swill over the top edge of the trough.
18x3 + 5x2 + 128x + 35
dx.
x2 + 7
Z
7. Evaluate the integral
Z
8. Use integration by parts to evaluate
Z
ln x dx and
xn ln x dx where n 6= −1. What
happens when n = −1?
Z
9. Use tabular integration by parts to evaluate
Z
x3 e−5x dx.
π/5
10. Evaluate the trigonometric integral
sin(2x) sin x dx.
0
Z
11. Evaluate the integral
x2
√
1
dx.
121 − x2
12. For each of the following integrals, find an appropriate trigonometric substitution of the
form x = f (t) to simplify the integral.
Z
(a) (8x2 − 4)3/2 dx.
Z
(b)
Z
(c)
Z
(d)
√
x2
dx.
5x2 + 2
√
x 7x2 + 42x + 60 dx.
√
x
dx.
−50 − 6x2 − 36x
13. The velocity of a particle moving along a line is given by v(t) = t sin2 (4t) for 0 ≤ t ≤ 3.
Find the distance travelled by the particle.
Z
14. Evaluate the integral
Z
15. Evaluate the integral
Z
16. Evaluate the integral
sin3 (18x) cos8 (18x) dx.
tan3 (9x) sec(9x) dx.
105x6 sec4 (x7 ) dx.
17. Find the arc length for the curve y = ln(4x) where 1 ≤ x ≤ 9.
Z
18. Evaluate the integral
41
√
dx.
45 − 96x − 64x2
IV. Questions from Section 8.5 to 8.8
3
Z
1. Evaluate the integral
−2
Z
2. Evaluate the integral
x3 − 2
dx.
(x + 8)(x + 7)
9x3 + 7x2 + 100x + 125
dx.
x4 + 25x2
3. Use the table of integrals in your text to evaluate
Z √
Z
4. Use the table of integrals in your text to evaluate
5 − 4x − 4x2 dx
3
√
1
dt
.
9t2 − 1
e−8x − 1
x→0 sin(13x)
5. Evaluate the limit lim
6. Evaluate the limit lim
x→∞
6
1+
x
15x
.
5
5
−
.
ln x x − 1
13
13
√
dx converges or diverges. If it converges
3
x−3
7. Use L’Hopital’s rule to evaluate lim+
x→1
Z
8. Determine whether the integral
3
determine its value.
∞
Z
xe−3x dx converges or diverges. If it converges, deter-
9. Determine whether the integral
mine its value.
3
Z
10. Determine whether the integral
mine its value.
4
∞
ln x
dx converges or diverges. If it converges, deterx
Z
3
11. Determine whether the improper integral
−3
determine its value.
1
dx converges or diverges. If it converges,
x4