Approximate - Calculus - Solved Exam, Exams of Calculus

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Math 112 (Calculus I)

Final Exam Form A KEY

Multiple Choice. Fill in the answer to each problem on your scantron. Make sure your name, section and instructor is on your scantron.

Short Answer. Fill in the blank with the appropriate answer. 1 point each

1. (10 points)

(a) If f (x) = e^3 x^ then f ′′(x) =

Solution: 9 e^3 x (b)

d dx

(a^3 + cos^3 x) =

Solution: −3 cos^2 x sin x

(c)

d dx

ex

2

Solution: 2 xex

2

(d)

d dx

(tan−^1 (x^2 )) =

Solution:

2 x 1 + x^4

(e) lim x→ 0 +

ln(1 + x) x

Solution: 0 /0 so L’Hˆopital gives lim x→ 0 +

1 /(1 + x) 1

(f)

d dx

ln(sinh(x)) =

Solution:

cosh(x) sinh(x)

(g)

d dx

sin(π^2 + e^3 ) =

Solution: 0 (h) Use the linearization of f (x) =

x at a = 9 to approximate

Solution: Linearization at a = 9 is y =

(x − 9) + 3 so

(i) lim x→ 0

x^2 + 3 ex^

Solution: 3 (j)

3 x^2 + 2x + 1 dx =

Solution: x^3 + x^2 + x + C

1. (6 points) If a car is initially moving at 14 meters per second, and the driver begins braking so that the car slows down at the constant rate of 2 meters per second per second until it stops, how far will the car travel from the time that the driver began braking until the time the car stops? Solution:

a(t) = − 2 m/s^2 v(t) = − 2 t + v 0 = − 2 t + 14 Stops when v(t) = 0 t = 7

Dist =

0

v(t) dt = −t^2 + 14t

7

0

= 49 m

1. (6 points) lim x→ 0 +^

x csc(x) =

1. (6 points) Find the derivative of f (x) = xsin^ x.
2. (6 points) Sketch the graph of a function f (x) which is twice differentiable on the interval (−∞, ∞) and which has the following properties:
   • The second derivative f ′′(x) is positive on the interval (0, ∞) and negative on the interval (−∞, 0).
   • lim x→∞ f (x) = 2
   • The first derivative is zero only at x = − 3

Solution: Graph must be

• concave down on the interval (−∞, 0)
• have a max (absolute) at x = − 3
• have a point of inflection at x = 0
• concave up on the interval (0, ∞)
• Right hand horizontal asymptote (approached from above) at y = 2