Final Exam
201-225-AB
Winter 2013
1. Given f 00 (x) = 5x − 6 and f 0 (−1) = 3 and f (−1) = −6.
(2)
(a) Find f 0 (x) =
(2)
(b) Find f (2) =
Z
(4) 2. Use the Simpson’s Rule to estimate the value of the integral
the table and give your answer to 4 decimals. Hint: n = 6.
x
5
6
7
8
9
10
11
11
f (x) dx. Use all the data in
5
f(x)
2
5
3
2
0
-3
-5
(4) 3. Determine the equation of the tangent line to the curve at the specified point: y =
−ex
, (1, −e)
x
dy
for the following. Do not simplify your answer.
dx
(a) arctan(xy) = 1 + x3 y
4. Find
(4)
(4)
(4)
(4)
(4)
(b) y = arcsin(2x + 1) + sec−1 (5x)
√
(c) y = ln(sin x) + x3 cos x
(d) y = etan 3x + cot3 x
log7 (sec x)
(e) y =
3x
(4) 5. Use Newton’s method on the following function f (x) and initial approximation x1 to find x3 ,
the third approximation to the root of the equation. Given,
f (x) = x3 + 2x − 4,
x1 = 1
(4) 6. When two electric resistors R1 and R2 are in series, their total resistance (the sum) is 32 Ω . If
the same resistors are in parallel, their total resistance (the reciprocal of which equals the sum
of the reciprocals of the individual resistances) is the maximum posible for two such resistors.
What is the resistance of each?
(4) 7. Use the Washer Method to setup the integral needed to find the volume generated by rey2
volving the region bounded by x =
, x = y + 4 and y = 0 about the y-axis. DO NOT
2
SOLVE THE INTEGRAL.
(4) 8. Sketch the region enclosed by the given curves y1 = x2 , y2 = 3x + 4 and find its area.
dy
(4) 9. Solve the given differential equation, cos2 x + y csc x dx
= 0.
Page 1
Final Exam
201-225-AB
Winter 2013
(4) 10. Find the Fourier series for the square wave function
(
−1, if −π ≤ x < 0
f (x) =
1, if 0 ≤ x < π
(4) 11. Solve the given first order differential equation, y 0 + 2y = sin x.
t
(in A) is sent through an electric dryer circuit containing a previously
+1
uncharged 2.0µF capacitor. How long does it take for the capacitor voltage to reach 120 V ?
(4) 12. A current i = √
(4)
(4)
(4)
(4)
(4)
(4)
(4)
(4)
(4)
t2
13. Integrate the following integrals.
Z
(a) y = tan9 x sec2 x dx
Z e
cos(ln x)
(b) y =
dx
x
Z1
(c) y = esin x cos x dx
Z
x
√
dx
(d) y =
x2 + 2x + 5
Z
1
(e) y =
dx
x2 + 7x + 10
Z
(f) y = sin3 x cos x dx
Z
(g) y = ln x dx
Z
x+4
dx
(h) y =
x(x2 + 1)
Z
(i) y = sec3 x dx
Answers
11
5
1. (a) x2 − 6x −
2
2
(b) = −24
2.
19
3
3. y = −e
3x2 y + 3x4 y 3 − y
x
2
5
(b) √
+ √
2
−4x − 4x 5x 25x2 − 1
√
1
(c) √ cot x + 3x2 cos x − x3 sin x
2 x
4. (a)
(d) 3etan 3x · sec2 3x − 3 csc2 x cot2 x
Page 2
Final Exam
(e)
201-225-AB
tan x
log7 (sec x) ln 3
−
3x ln x
3x
5. x3 = 1.18
6. R1 = 16Ω; R2 = 16Ω
Z 4 y2
7. v =
π (y + 4)2 −
2
0
125
6
r
2
9.
cos3 x + 2c
3
π
· sin[(2n + 1)x] Let n start at 0.
10.
4(2n + 1)
8.
11.
2
1
c
sin x − cos x + 2x
5
5
e
12. 0.022s
tan10 x
+c
10
(b) sin(1)
13. (a)
(c) esin x + c
1
x+1
+c
(d) (x2 + 2x + 5) − arctan
2
2
1 x + 2 (e) ln +c
3
x + 5
sin4 x
+c
4
(g) x ln x − x + c
(f)
(h) ln
(i)
(x2
x4
+ arctan x + c
+ 1)2
1
[sec x tan x + ln | sec x + tan x|] + c
2
Page 3
Winter 2013