Calculus Exam

Unit 1

100

Find the limit:

lim (x² - 16) / (x - 4)

x --> 4

8

200

Find the limit:

Lim (√x - 2) / (x - 4)

x --> 4

1/4

300

Find the limit

lim (x + 3) / (√9x² - 5x)

x --> infinity

1/2

400

What are the horizontal asymptotes of the graph of

y = (3 + 4x) / (1 - 4x) ?

-1

Unit 2

100

Differentiate g(x)=e^1−cos(x)

g'(x)= e^1−cos(x)(sin(x))

200

Differentiate y= (sin(3x)) / (1+x²)

y' = ((1 + x²)(3(cos(3x)) - (sin(3x))(2x)) / (1 + x⁴)

300

Find the tangent line to the graph of f(x) = (x³ + 4x- 1)(x - 2) at the point (1, -4)

y + 4 = -3(x - 1)

400

Find the tangent line to the graph of y = lnx³ at the point (1,0)

y = 3x - 3

Unit 3

100

The slope of the curve y³ - xy² = 4 at the point where y = 2 is...

1/2

200

find dy/dx by implicit differentiation x² - 5xy + 3y² = 7

dy/dx = (-2x + 5y) / (-5 + 6y)

300

find dy/dx by implicit differentiation: e^cos(x) + e^sin(y) = 1/4

dy/dx = (sin(x))(e^cos(x))) / (cos(y))(e^sin(y)))

400

The radius of a cylinder is increasing at a rate of 1 meter per hour, and the height of the cylinder is decreasing at a rate of 4 meters per hour. At a certain instant, the base radius is 5 meters and the height is 8 meters. What is the rate of change of the volume of the cylinder at the instant?

-20pi meters³/hour

Unit 4

100

A particle moves along the x-axis. The function x(t)=t³ - 3t² + 7t - 6 gives the particle's position at any time t≥0. What is the particle's acceleration a(t), at t=3?

12

200

The maximum acceleration attained on the interval o<t<3 by a particle whose velocity is given by v(t) = t³ - 3t² + 12t + 4 is...

21

300

Evaluate the limit

lim (x³ - 8) / (x² - 4)

x --> 2

3

400

Evaluate the limit

lim (sqrt(2 + x) - 2) / (x - 2)

x --> 2

1/4

Unit 5

100

Find the absolute extreme of the function f(x) = x³ - (3/2)x² on the given interval [-1, 2]

Absolute minimum: (-1, -(5/2))

Absolute maximum: (2, 2)

200

Identify the open intervals on which the function y = x - 2cos(x) (between 0 and 2 pi) is increasing or decreasing 

Increasing: (0, (7pi/6)) and ((11pi/6), 2pi)

Decreasing: ((7pi/6), (11pi/6))

300

Determine all the number(s) c which satisfy the conclusion of Mean Value Theorem for A(t)=8t+e^−3t on [−2,3] 

(calculator)

C = -1.097

400

The function f has a first derivative given by f'(x) = x(x-3)²(x+1). At what values of x does f have a relative maximum?

F has a relative max at x = -1