NAME Rec. Instructor: Signature Rec. Time #1 #2 #3 #4 #5 ...
NAME Signature
Rec. Instructor: Rec. Time
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7 15 20 8 7 7 8 8 6 7 7 100
CALCULUS I - EXAM 2 - SPRING 2019 March 7, 2019
Show all work for full credit. No books, notes or calculators are permitted. The point value of each problem is given in the left-hand margin. No need to simplify derivatives.
(7) 1. Find the equation of the line tangent to y = x3 - 2x2 + 1 at x = 2 .
(15) 2. Differentiate the following ( arctan x is the inverse tangent tan-1 x ).
d a)
dx
2x
+
5 x2
+
7
=
b) d x3 arctan x = dx
d x2 c) dx x3 + ex =
(20) 3. Differentiate using the chain rule
d a)
1 + e2x =
dx
d b)
ln x2 + x + 1
=
dx
d c)
ex3+2x-1 =
dx
d) d tan3 (2t) = dt
page 2 of 6
page 3 of 6
(8) 4. A 10 foot ladder is leaning against a wall. The top of the ladder is sliding down the wall at 2 feet per minute. How fast is the bottom of the ladder moving away from the wall when it is 6 feet from the wall?
(7) 5. a) Find the linear approximation to f (x) = 3 x near x = 8 .
b) Use part (a) to approximate 3 8.12
.
page 4 of 6
(7) 6. Use implicit differentiation to find dy for y5 + xy3 = 5x + 7. dx
dy (8) 7. Use logarithmic differentiation to find the derivative for
dx x2x y = (x2 + 1)5
page 5 of 6
(8) 8. An object moves in a straight line with position after t seconds given by s(t) = 9t2 - 2t3.
Here distance is measured in feet. a) Find the velocity at time t = 2
b) Find the acceleration at time t = 2
c) Is the object speeding up or slowing down at time t = 2?
(6) 9. Given the graph of y = f (x) sketch the graph of the derivative y = f (x).
4 3 2 1
-6 -5 -4 -3 -2 -1 -1 -2 -3 -4
12345678 y = f (x)
(7) 10. Find the second derivative y for y = x3 sin x.
page 6 of 6
(7) 11 a) Give the limit definition of the derivative function for f (x) f (x) = lim
b) Use the limit definition to find f (x) for f (x) = x2 + x .
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