MATH 1210-6
Spring 2003
Midterm exam I
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:
February 7, 2003
Start time: 12:55 pm
1 hours
6
Closed Book
Calculator
QUESTION VALUE SCORE
1
40
2
40
3
20
TOTAL
100
*) The bonus question counts for 10 points maximum.
End Time: 1:55 pm
1. (40 points) Evaluate the derivative of the function
y(x) = sin(π sin(π sin(πx)))
at x = 0.
ANSWER:
2
2. (40 points) The relation
2x2 + y 2 = 9
determines a curve in the x − y plane (an ellipse).
a) Find the equation of the line which goes through the point (2,-1) on the curve and is perpendicular
to the tangent line to the curve at this point.
b) At what points (x0 , y0 ) does this curve have a horizontal tangent line.
ANSWER:
3
3. (20 points) Differentiate
s
x−1
x+1
ANSWER:
4
Bonus question (10 points). As you know (sin x)0 = cos x and (cos x)0 = − sin x. Find the
hundredth derivative of
y(x) = sin(2x)
y (100) (x) =?
ANSWER:
5
Useful formulae
Constant multiple rule:
(k f (x))0 = k f 0 (x)
Sum rule:
(f (x) + g(x))0 = f 0 (x) + g 0 (x)
Product rule:
(f (x)g(x))0 = f 0 (x)g(x) + f (x)g 0 (x)
Quotient rule:
Chain rule:
Ã
f (x)
g(x)
!0
=
f 0 (x)g(x) − f (x)g 0 (x)
g 2 (x)
d
df dg
f (g(x)) =
·
dx
dg dx
d
df dg dh
f (g(h(x))) =
·
·
dx
dg dh dx
d
df dg dh du
f (g(h(u(x)))) =
·
·
·
dx
dg dh du dx
Power rule
(xα )0 = αxα−1
Trigonometric functions:
(cos x)0 = − sin x
(sin x)0 = cos x
cos(0) = 1, sin(0) = 0
π
π
cos( ) = 0, sin( ) = 1
2
2
cos(π) = −1, sin(π) = 0
6