Download Practice exam - Jessica M. Conway

Math 141H-01
PRACTICE Midterm Exam 1
February X, 2015
Name:
Instructions: Clearly answer each of the questions. Box your answers. Partial credit will be awarded
based on the clarity and correctness of your explanation of each solution. Use the backs of pages for your
scratch work. Make sure you have all 11 pages.
Trigonometric identities and useful integration formulas are given on pages 10-11.
1. (X points) Inverse Functions.
(a) Does f (x) = x2 − 1 have an inverse? Why or why not?
(b) Given g(x) = x3 − 8, what is g−1 (−8)?
(c) Given h−1 (x) = tan−1 (2x + 1), what is h(π/4)?
(d) Let f be a function such that
f (−4) = 5, f (1) = 2, f (2) = 1, f (3) = −4, f (5) = −2
1
1
f 0 (−4) = 3, f 0 (−2) = 1, f 0 (2) = , f 0 (3) = − , f 0 (5) = 2
4
4
−1
0
−1
0
Find ( f ) (−2) and ( f ) (5).
(e) w(x) =
√
x − 1. Sketch w−1 (x).
2
2. (X points)
(a) What is the domain of y = ln(x2 − 1)?
(b) What is the range of y = tan−1 (x2 )?
(c) Determine the x-intercept(s) of the
2
graph y = ex −1 − 1.
(d) Determine the y-intercept(s) of the
graph y = 21/(x+1) − 4.
(e) Compute limx→∞ (ln(1 + 2x) − ln(1 + 3x)).
(f) Sketch y = ln(x2 − 1).
3
3. (X points) Find the derivative of the given function, evaluated at the given point. Simplify completely
(no inverse trig functions in your answer!).
(a) y = tan(sin−1 (t)), t = 0.
x
(b) y = (ln x)e , x = e.
4
4. (X points) Determine the correct partial fractions expansions. DO NOT DETERMINE THE CONSTANTS AND DO NOT INTEGRATE.
(a) To integrate
ˆ
x
dx,
2
x − 7x + 12
which partial fractions expansion should you use? DO NOT DETERMINE THE CONSTANTS AND DO
NOT INTEGRATE.
(b) To integrate
ˆ
x
(x2 − 1)2 (x2 − 3x + 12)2
dx,
which partial fractions expansion should you use? DO NOT DETERMINE THE CONSTANTS AND DO
NOT INTEGRATE.
5
5. (X points) Integrate by parts
ˆ
1
x tan−1 x dx
0
6
6. (X points) Integrate using a trigonometric substitution. Simplify as much as you can.
ˆ
x3
√
dx
9 + 4x2
7
7. (X points) Integrate using whatever method you wish.
ˆ
(a)
π/4
cos2 θ sin2 θ dθ
0
ˆ
(b)
sin(t) ln(cos2 (t)) dt
8
ˆ
x
√
dx
x − x2
(c)
ˆ
(d)
2
3
x2
dx
x2 − 1
9
Reciprocal Identities
sin θ =
1
csc θ
Sum and Difference Identities
csc θ =
1
cos θ =
sec θ
tan θ =
1
cot θ
1
sin θ
1
sec θ =
cos θ
cot θ =
Quotient Identities
sin θ
= tan θ
cos θ
cos θ
= cot θ
sin θ
1
tan θ
sin (A ± B) = sin A cos B ± cos A sin B
cos (A ± B) = cos A cos B ∓ sin A sin B
tan (A ± B) =
tan A ± tan B
1 ∓ tan A tan B
Double-Angle Identities
sin 2θ = 2 sin θ cos θ
cos 2θ = cos2 θ − sin2 θ
= 2 cos2 θ − 1
= 1 − 2 sin2 θ
tan 2θ =
2 tan θ
1 − tan2 θ
Pythagorean Identities
Power reduction formulae
2
2
sin θ + cos θ = 1
tan2 θ + 1 = sec2 θ
1 − cos 2θ
2
1 + cos 2θ
cos2 θ =
2
sin2 θ =
1 + cot2 θ = csc2 θ
Half-Angle Identities
r
θ
1 − cos θ
Co-function Identities
sin = ±
2
2
π
π
r
θ
1 + cos θ
sin θ = cos
−θ
cos θ = sin
−θ
cos = ±
2
2
2
2
r
π
π
θ
1 − cos θ
−θ
cot θ = tan
−θ
tan θ = cot
tan = ±
, cos θ =
6 1
2
2
2
1 + cos θ
π
π
sec θ = csc
−θ
csc θ = sec
−θ
Product-Sum Identities
2
2
Opposite Angle Identities
sin(−θ ) = − sin θ
cos(−θ ) = cos θ
tan(−θ ) = − tan θ
1
[sin (A + B) + sin (A − B)]
2
1
cos A sin B = [sin (A + B) − sin (A − B)]
2
1
sin A sin B = [cos (A − B) − cos (A + B)]
2
1
cos A cos B = [cos (A + B) + cos (A − B)]
2
sin A cos B =
Basic Integration Formulae
ˆ
xn+1
1. xn dx =
+C (n 6= 1)
n+1
x
sin x dx = − cos x +C
sec2 x dx = tan x +C
sec x tan x dx = sec x +C
sec x dx = ln | sec x + tan x| +C
15.
csc x dx = ln | csc x − cot x| +C
12.
ˆ
ˆ
tan x dx = ln | sec x| +C
ˆ
csc x cot x dx = − csc x +C
10.
ˆ
ˆ
13.
csc2 x dx = − cot x +C
8.
ˆ
ˆ
11.
cos x dx = sin x +C
6.
ˆ
ˆ
9.
ax
+C
ln a
ˆ
ˆ
7.
1
dx = ln |x| +C
x
ax dx =
4.
x
e dx = e +C
5.
2.
ˆ
ˆ
3.
ˆ
cot x dx = ln | sin x| +C
14.
ˆ
dx
1
= tan−1
+C
2
2
x +a
a
a
x
16.
dx
−1 x
√
+C, a > 0
= sin
a
a2 − x2