Math 220 (section AC2)
Practice Exam 3 answers
Spring 2012
Name
• Do not open this test until I say start.
• Turn off all electronic devices and put away all items except a pen/pencil and an eraser.
• No calculators allowed.
• You must show sufficient work to justify each answer.
• Quit working and close this test booklet when I say stop.
Expectations:
section 4.3, 4.5
• know the definitions of critical number, increasing, decreasing, concave up, concave down,
inflection point, local max/min, absolute max/min.
• be able to state and use the first derivative test
• given a formula or graph of a function f (x) (or sometimes its derivative f 0 (x)), say where
the graph is increasing, decreasing, concave up, concave down, has an inflection point, has
local max/min, or has abs max/min.
• be able to state the Extreme Value Theorem and use it to find ansolute max/min of a
continuous function on a closed interval.
• be able to sketch a graph using the information from first and second derivatives and
asymptotes.
section 4.4 Be able to use L’Hospital’s Rule when applicable.
sections 4.7 Be able to solve optimization problems. You may need to know formulas for
circumference, area (circle, rectangle, triangle), and volume (box, sphere, cyclinder, cone), the
Pythagorean theorem, similar triangles, and trig.
section 4.8
• Understand the graphical basis for Newton’s Method (that is, use the point where the
tangent line crosses the x-axis as your next estimate for a root of a function.)
• Be able to apply Newton’s Method to approximate roots, solutions, or x-intercepts.
section 4.9
• Know antiderivative formulas for 0, k (constant), x, xn (n 6= −1), x−1 , ex , ax , sin x, cos x,
1
√ 1
csc x cot x, sec2 x, csc2 x, sec x tan x, 1+x
, cf (x), f (x) ± g(x).
2,
1−x2
• Be able to solve a differential equation where values for the function or its first or second
derivative are given.
• Be able to apply these rules to problems involving acceleration, velocity, or position.
section 5.1
• Use Riemann sums (left, right, or midpoint) to estimate area and state if your estimate
is known to be an underestimate or overestimate. These sums will involve at most 8
subintervals.
• Use limits of Riemann sums to find the exact area. (You can use right Riemann sums as
done in class.)
• Understand sigma notation for sums and know
n
P
1 = n,
i=1
n
P
i =
i=1
n(n+1)
,
2
and
n
P
i2
=
i=1
n(n+1)(2n+1)
.
6
section 5.2
• Understand the definition of a definite integral as
Rb
a
f (x) dx = limn→∞
n
P
i=1
f (x∗i )∆x. Be
able to more explicitly write out the appropriate limit for a specific function on a given
interval and evaluate it.
• Know the relationship between a definite integral and area.
• Know the properties of integrals on pages 379-380. (properties 1-5)
1. Find antiderivatives of the following functions
(a) f (x) = 2ex + sec x tan x
2ex + sec x + C
(b) g(x) =
x5 − 3x
x2
1/4x4 − 3 ln |x| + C
(c) h(x) =
√
x + 4x5 + xπ
2/3x3/2 + 2/3x6 +
1
xπ+1 + C
π+1
(d) y = cos x − (1 − x2 )−1/2
sin x − sin−1 x + C
(e) g(r) = 3 sin r − 2r, g(0) = 4
−3 cos r − r2 + 7
2. Find two positive integers such that the sum of the first number and four times the second
number is 1000 and the product of the numbers is as large as possible.
x = 500, y = 125
3. What is the minimum vertical distance between the parabolas y = x2 + 1 and y = x − x2 ?
11/8
4. A cylindrical can without a top is made to contain V cm3 of liquid. Find the dimensions
that will minimize the cost of the metal to make the can.
s
r=
3
V
,h =
π
5. Evaluate the limits
3x2 − 2
x→∞ ex + 4
(a) lim
0
4x
(b) lim
e
x→0
− 1 − 4x
x2
8
3x2 − 2
x→∞ x2 + 4
(c) lim
3
(d)
lim (tan x)cos x
x→π/2−
1
cos2 x − 1
x→0
x2
(e) lim
−1
√
(f) lim+ x
x
x→0
1
s
3
V
π
6. (3 points each) Suppose that f is an odd function which is integrable on the interval [−5, 5].
If
Z 2
f (x) dx = 4 and
Z 3
0
f (x) dx = 10, then evaluate the following quantities.
2
(a)
Z 5
f (x) dx +
0
Z 3
f (x) dx
5
14
(b)
Z 2
f (x) dx
−2
0
(c)
Z 2
f (|x|) dx
−2
8
7. Fill in the missing information to show that the given definite integral can be expressed as
the limit of a Riemann sum. The only variables appearing in your limit should be n and
k. You do not need to evaluate this limit.
Z 6
5
x +8
4
dx = lim
n
X
n→∞
2
"
#
k=1
((2 +
4k 5
4
) + 8)4
n
n
8. Evaluate:
• limn→∞
n
P
3 4i 2
[( )
i=1
n
n
− 5]
1
•
n
P
[2i − 5i2 + 7]
i=1
n(n + 1) −
5n(n + 1)(2n + 1)
+ 7n
6
9. (a) State L’Hospital’s Rule.
(b) State the definition of a definite integral as a limit of sums.
(c) State the definition of an inflection point.
√
10. Find the second approximation of 5 20 using Newton’s Method.
x2 = 37/20 (x1 = 2)
11. Use Newton’s Method to find a second approximation of the root of x5 − x4 + 3x2 − 3x − 2
on the interval [1, 2].
x2 = 3/2 (x1 = 1)
12. Use Newton’s method to find second approximations to the solutions of x2 − 3x + 1 =
sin x.(you don’t have to simplify your answer)
On [0, 1] : (x1 = 0), x2 = 1/4, on [2, 3] : (x1 = 3), x2 = 3 −
1 − sin(3)
3 − cos(3)
2
x
13. Sketch a graph of f (x) = x+8
(identifying intercepts, asymptotes, intervals of increase/decrease,
intervals of concavity, local extrema, and inflection points.)
x-int: x = 0, VA: x = −8, slant asymptote: y = x − 8, increasing: (−∞, −16) ∪ (0, ∞),
dec: (−16, −8) ∪ (−8, 0), local max (−16, −32), local min (0, 0), concave up (−8, ∞), conc
down (−∞, −8)
√
14. Sketch a graph of x 2 + x(identifying intercepts, asymptotes, intervals of increase/decrease,
intervals of concavity, local extrema, and inflection points.)
x-ints x = 0, −2, domain [−2, ∞), asymptotes: none, inc (−4/3, ∞), dec (−∞, −4/3), local
(and abs) min at x = −4/3, conc up (−2, ∞)
15. Find absolute and local extrema of (x − 2)e−x .
local (and abs) max at x = 3
16. Find the local extrema of the function given by
f (x) =















x for x 6= 3, 5, 7, 9
5 for x = 3
−3 for x = 5
9 for x = 7
7 for x = 9
local maxima x = 3, 7, local minima: x = 5, 9