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The final examination questions for calculus i, mathematics 105, from december 2003. The questions cover various topics including differentiation, integration, and differential equations.
Typology: Exams
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December 9 Mathematics 105 2003 Calculus I Final Examination (18 points) I. Calculate:
A. If f ( x )= x ln( x^2 + 1 ), then f' ( x )=
B. If t
t f t cos ( )= , then f' ( t )=
cx + d
a dx
dx x
3 ( t^2 1 ) dt
sin lim x (^0) ex
x
(10 points) II. Using this table of values for the function f ,
x 0.0 0.5 1.0 1. f (x) (^) -2 -5 -2 0
0
f ( x ) dx
B. estimate f ' (1.0)
(22 points) V. Here is a graph of f ' , the derivative of the function f. y
3
f '
1
x
2 4 6 8
Warning: These questions are about f, f ', and f ". The graph you see above is the graph of f '.
A. The values of x (if any) at which f ' is not differentiable:
B. The values of x (if any) at which f ' is not continuous:
C. The interval(s) where f increases:
D. The interval(s) where f is concave down:
E. The inflection points (if any) of f :
6
0
f ' ( t ) dt
G. The average value of f ' over the interval [0, 6] :
H. If f(0) = 5, then f(4) =
I. f " (3) =
J. The critical points (if any) for f :
K. The local minima (if any) for f.
(6 points) VI. A snowstorm lasts 4 hours. Suppose f ( t ) is the rate of snowfall (in inches per hour) t hours after the start of the storm.
A. Write a sentence (including units) describing the quantity measured by
4
2
f ( t ) dt.
B. Write a sentence (including units) describing the quantity measured by f ' (3).
(9 points) IX. Given the equation x^2 y + xy^2 =^6 ,
A. what is the value of dx
dy at the point where x = 2 and y = 1?
B. Give the local linearization of this curve at the point (2, 1).
C. Use your answer to part B to approximate y when x = 1..
