MATH 155A FALL 13
SOLUTIONS TO THE PRACTICE MIDTERM 2.
Question 1. Find y .
1
1
(a) y = .
5
x
x3
(b) y =
tan x
.
1 + cos x
(c) y = x
(d) y =
21+
cos x.
1
.
sin(x sin x)
(e) y = sin2 (cos( sin(x).
Solution. Direct application of the dierentiation rules yie
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MATH 155A FALL 13
EXAMPLES APPENDIX D
If 0 < <
2
is such that
2(cos )2 + (8
3) cos 4 3 = 0,
nd .
Solution. Rewrite the equation as
8 3
cos 2 3 = 0.
(cos ) +
(1)
2
Using the quadratic formula, we see that the equation
8 3
2
x2 3=0
x +
2
has solutions
3
x=
MATH 155A FALL 13
EXAMPLES OF SECTIONS 1.1 AND 1.2
Find the domain of the following functions.
(a) f (x) = x3 + 2x 7.
(b) g(x) =
3x
.
x3 4x
Solutions.
(a) This function is a polynomial, hence its domain is R. Alternatively, notice that given any
real num
MATH 155A FALL 13
EXAMPLES OF SECTIONS 1.5 AND 1.6
Evaluate the following limits, when possible.
(a)
lim 3x
x2
(b)
x
.
x0 |x|
lim
(c)
x2 2x 3
.
x3 x2 4x + 3
lim
Solutions.
(a) When x approaches 2, 3x approaches 6, with no undened expressions arising. Henc
MATH 155A FALL 13
PRACTICE MIDTERM 4.
Question 1. Find the derivative of the following functions.
x
(a) f (x) =
t+
t dt.
0
Solution. By the FTC,
f (x) =
x
(b) f (x) =
0
x+
x.
z2
dz.
z4 + 1
Solution. By the FTC and the chain rule with u = x,
x
.
f (x) =
2
MATH 155A FALL 13
PRACTICE FINAL SOLUTIONS.
Question 1. Find an expression for the function whose graph consists of the line segment from
the point (2, 2) to the point (1, 0) together with the top half of the circle with center the
origin and radius 1.
So
MATH 155A FALL 13
PRACTICE MIDTERM 1.
Question 1. Find the domain of the following functions.
(a) f (x) =
2x3 5
.
x2 +x6
(b) g(x) =
x+1
1 .
1+ x+1
(c) f (x) =
5x+
1
.
x10
Question 2. An electricity company charges its customers a base rate of $10 a month
MATH 155A FALL 13
PRACTICE FINAL.
Question 1. Find an expression for the function whose graph consists of the line segment from
the point (2, 2) to the point (1, 0) together with the top half of the circle with center the
origin and radius 1.
Question 2.
MATH 155A FALL 13
PRACTICE MIDTERM 3, SOLUTIONS.
Question 1. Find all the maxima and minima of the functions on the given intervals.
(a) f (x) = cos2 x 2 sin x, on [0, 2].
(b) f (x) = x 6 x, on [10, 6].
2
5
(c) f (x) = 5x 3 2x 3 , on (, ).
Solution. (a) a
MATH 155A FALL 13
PRACTICE MIDTERM 2.
Question 1. Find y .
1
1
(a) y = .
5
x
x3
(b) y =
tan x
.
1 + cos x
(c) y = x
(d) y =
21+
cos x.
1
.
sin(x sin x)
(e) y = sin2 (cos( sin(x).
Question 2. Find an equation for the tangent line and normal line to the cur
MATH 155A FALL 13
PRACTICE MIDTERM 3.
Question 1. Find all the maxima and minima of the functions on the given intervals.
(a) f (x) = cos2 x 2 sin x, on [0, 2].
(b) f (x) = x 6 x, on [10, 6].
2
5
(c) f (x) = 5x 3 2x 3 , on (, ).
Question 2. Show that the
MATH 155A FALL 13
EXAMPLES SECTIONS 1.7 AND 1.8.
Question 1. What value should A have in order to make the function
f (x) =
x2 7,
x 5,
A cos( 15 x) 3, x > 5,
continuous?
Question 2. Show that the equation
cos2 x = x3 ,
has at least one solution.
Question
MATH 155A FALL 13
PRACTICE MIDTERM 4.
Question 1. Find the derivative of the following functions.
x
(a) f (x) =
t+
t dt.
0
x
(b) f (x) =
0
x2
(c) f (x) =
tan x
z2
dz.
z4 + 1
1
dt.
2 + t4
Question 2. Evaluate the following indenite integrals.
sin x
dx.
x
(
MATH 155A FALL 13
MORE EXAMPLES WITH LIMITS
Evaluate the following limits, when possible.
(a)
lim
x 2
(x )4
2
.
tan x
(b)
lim | csc |.
Solutions.
(a) We cannot plug in x = because tan is undened. Notice, however, that when x
2
2
approaches from the left,
NAME (131121an
155 A: Accelerated Single Variable Calculus
Test 2, Fall 2011
Problem Points TYour Score
1 15 _l
2 30
_____L___.l___7
3 15
___J_ g
4 20
F
5 20
6 3 extra credit
7 3 extra credit
100 + 6