MATH 1210-6
Spring 2003
Midterm exam III
Student Name:
Student ID Number:
Course Abbreviation and Number:
Course Title:
Instructor:
Math 1210
Calculus I
Vladimir Vinogradov
Date of Exam:
Time Period:
Duration of Exam:
Number of Exam Pages:
(including this cover sheet)
Exam Type:
Additional Materials Allowed:
April 7, 2003
Start time: 12:55 pm
1 hours
6
Closed Book
Calculator
QUESTION
VALUE
1
2
3
4
5
20
20
20
20
20
TOTAL
100
*) The bonus question counts for 10 points maximum.
SCORE
End Time: 1:55 pm
1. (20 points) Find the indefinite integral of:
Z
24x(x2 − 1)3 dx
ANSWER:
2. (20 points) Find the function whose value at 1 is 0 and whose derivative is given:
dy
3(x2 + 1)
= 3
dx
(x + 3x)2
ANSWER:
2
3. (20 points) Calculate the definite integrals:
π/3
Z
³
´
4 sin(4x) − cos(4x) dx
π/6
ANSWER:
4. (20 points) Find the solution to the following differential equation such that y(0) = 1
√
dy
= xy 2 + 2x2 y 2
dx
ANSWER:
3
5. (20 points) What is the area of the region bounded by the curves y1 = 5 − (x − 2)2 and y2 = 1.
ANSWER:
4
Bonus question (10 points). Evaluate
d
dx
Zx2
f (t)dt
−x2
ANSWER:
5
Useful formulae
Differentiation
Product rule: (f (x)g(x))0 = f 0 (x)g(x) + f (x)g 0 (x)
³
´
f (x) 0
g(x)
=
f 0 (x)g(x)−f (x)g 0 (x)
g 2 (x)
d
dx f (g(x))
=
df
dg
Quotient rule:
Chain rule:
·
dg
dx
Power rule (xα )0 = αxα−1
Trigonometric functions: (cos x)0 = − sin x, (sin x)0 = cos x
Integration
Z
Z
0
f (g(x))g (x)dx =
g(b)
Z
Zb
f (g(x))g 0 (x)dx =
a
f (u)du = F (g(x)) + C
f (u)du = F (g(b)) − F (g(a))
g(a)
cos(0) = 1, sin(0) = 0
√
π
π
1
3
cos( ) =
, sin( ) =
6
2
6
2
√
√
π
2
π
2
cos( ) =
, sin( ) =
4
2
4
2
√
π
1
π
3
cos( ) = , sin( ) =
3
2
3
2
π
π
cos( ) = 0, sin( ) = 1
2
2
cos(π) = −1, sin(π) = 0
6