Download Math 34B 2012 Quiz 2 Answers 1. Integrate the functions (a) x dx

Math 34B 2012
Quiz 2 Answers
1. Integrate the functions
(a)
Z
x
−4
x−3
+C,
dx = −
3
(5 points)
(b)
Z
cos(x)dx = sin(x) +C,
(5 points)
(c)
Z
1
1
1
(3x5 − 2x3 + x − 1)dx = x6 − x4 + x2 − x +C,
2
2
2
(5 points)
2. Compute the indefinite integrals
(a)
Z
1
(2x + 4)3 dx = (2x + 4)4 +C,
8
(5 points)
(b)
Z
x + 1/3
1p 2
√
dx =
3x + 2x + 4 +C,
3
3x2 + 2x + 4
(5 points)
(c)
Z
sin2 (x)dx =
Z
1
1
1
(1−cos(2x))dx = x− sin(2x)+C,
2
2
4
(d)
Z
2x exp(x2 )dx = exp(x2 ) +C,
1
(5 points)
(5 points)
3. Compute the definite integrals
(a) (5 points)
Z π/2
π/2
−π/2
sin(x)dx = − cos(x)|−π/2 = − cos(π/2) + cos(−π/2) = 0.
(b) (5 points)
Z 1
x + 1/3
0
3x2 + 2x + 4
dx =
1
1
ln(3x2 + 2x + 4)|10 = [ln(9) − ln(4)]
6
6
(c) (5 points)
√
1
1
1
1
[√
− e6x ]dx = [2 x + 1 − e6x ]|1−1 = 23/2 − e6 + e−6
6
6
6
x+1
−1
Z 1
(d) (5 points)
Z 1
0
1
1
(3x2 + 2)(x3 + 2x + 5)3 dx = (x3 + 2x + 5)4 |10 = [84 − 54 ]
4
4
4. (6 points) Solve the differential equation
1
dF
= e4x , F(x) = e4x +C
dx
4
Find the solution F(x) that satisfies the initial condition F(0) = 14 .
1
F(x) = e4x , C = 0
4
5. (6 points) Draw the graph of the oscillatory function
f (t) = e−t [cos(t) + sin(2t)]
over the interval 0 ≤ t ≤ π. Notice that sin and cos have different frequency
and the exponential function is decaying so the amplitude of the oscillation
decays.
2
6. (6 points) Solve the differential equation
dF
1
1
=
, F(x) = ln(2x + 1) +C
dx
2x + 1
2
Find the solution F(x) that satisfies the initial condition F(0) = 0.
F(x) =
1
ln(2x + 1), C = 0
2
7. (6 points) Compute the left Riemann integral of f (x) = x, over the interval
x ∈ [0, 1], by using 5 subintervals.
4
AL =
4
f (xk )∆x =
∑
k=1
∑ k∆x · ∆x = [1 + 2 + 3 + 4]/25 = 10/25 = 2/5
k=1
Draw your approximation to the area.
8. (7 points) Find the area of the region between the graph of the function
f (x) = x2 and the x axis, over the interval [0, 1]
Z 1
A=
x2 dx =
0
x3 1 1
| =
3 0 3
Draw the region.
9. (7 points) Find the area under the graph of f (x) = cos(x) and the x axis,
over the interval [−π/2, π/2].
Z π/2
A=
−π/2
π/2
cos dx = sin(x)|−π/2 = sin(π/2) − sin(−π/2) = 2
Draw the region.
10. (7 points) Find the area under the graph of f (t) =
over the interval [0, 1].
A=
Z 1√
0
√
t − √1t and the t axis,
1
2
4
2
[ t − √ ]dx = [ t 3/2 − 2t 1/2 ]|10 = − 2 = −
3
3
3
t
Draw the region.
3