Question 1

Find the Maclaurin polynomial of degree 4 for sec(2x).

Accepted Answer

A)  1 +   +   
B)  1 + x +   +   +   
C)  1 - x +   -   +   
D)  1 -   +   
E)  1 -   -   
A)  1 +   +   
B)  1 + x +   +   +   
C)  1 - x +   -   +   
D)  1 -   +   
E)  1 -   -

Question 2

Use linear approximation to estimate the change of the surface area S = 4 $\pi$r2 of a sphere if its radius r is increased from 9 cm to 9.1 cm.

Accepted Answer

A)  1026.36   
B)  1024.52   
C)  1017.26   
D)  1040.62   
E)  1041.61   
A)  1026.36   
B)  1024.52   
C)  1017.26   
D)  1040.62   
E)  1041.61

Question 3

A lawnmower using a 40 m extension cord cuts grass in a circular area. It is discovered that there is a knot in the cord, and untying it lengthens the cord to 42 m. Using an appropriate linearization, estimate how much more area can be cut with the unknotted cord.

Accepted Answer

A)  160 $\pi$m2 
B)  80 $\pi$ m2 
C)  120 $\pi$ m2 
D)  320 $\pi$ m2 
E)  225 $\pi$ m2 
A)  160 $\pi$m2 
B)  80 $\pi$ m2 
C)  120 $\pi$ m2 
D)  320 $\pi$ m2 
E)  225 $\pi$ m2

Question 4

Find the local extreme and inflection points of the function f(x) =   + 4   and sketch its graph.

Accepted Answer

Local (absolute) min

Question 5

A large sheet of cardboard 2 m by 4 m will be made into a box by cutting equal-sized squares from each corner and folding up the four edges. What is the largest possible volume of such a box?

Accepted Answer

A)  1.2180 m3 
B)  1.4001 m3 
C)  1.5396 m3 
D)  1.6247 m3 
E)  1.3241 m3 
A)  1.2180 m3 
B)  1.4001 m3 
C)  1.5396 m3 
D)  1.6247 m3 
E)  1.3241 m3

Question 6

A rectangular enclosure subdivided into four pens by three parallel partitions is to be built using a total of 500 m of fencing. What dimensions will maximize the total area of the pens?

Accepted Answer

A)  28 m and 180 m 
B)  40 m and 150 m 
C)  60 m and 120 m 
D)  50 m and 125 m 
E)  45 m and 183 m 
A)  28 m and 180 m 
B)  40 m and 150 m 
C)  60 m and 120 m 
D)  50 m and 125 m 
E)  45 m and 183 m

Question 7

In a classical equation from chemistry, the pressure of a gas is proportional to the product of the volume of the gas and the temperature of the gas. In symbols, P = kVT for some constant k. How fast is the pressure changing at an instant when the volume is 20 mL and increasing at 3 mL/min and the temperature is at 21 degrees C and is decreasing at a rate of 2 degrees per minute? Assume that k = 0.8 kilograms per mL-degree.

Accepted Answer

A)  increasing at 18.4 kilograms/min 
B)  increasing at 14.4 kilograms/min 
C)  decreasing at 23.5 kilograms/min 
D)  decreasing at 12.8 kilograms/min 
E)  increasing at 12.7 kilograms/min 
A)  increasing at 18.4 kilograms/min 
B)  increasing at 14.4 kilograms/min 
C)  decreasing at 23.5 kilograms/min 
D)  decreasing at 12.8 kilograms/min 
E)  increasing at 12.7 kilograms/min

Question 8

Find the absolute maximum and absolute minimum values of the function f(x) =   - 1, 0 $\le$ x $\le$ 2.

Accepted Answer

A)  maximum 2, minimum -1 
B)  maximum 2, minimum 0 
C)  maximum 2, minimum -2 
D)  maximum 3, minimum -2 
E)  no absolute extrema 
A)  maximum 2, minimum -1 
B)  maximum 2, minimum 0 
C)  maximum 2, minimum -2 
D)  maximum 3, minimum -2 
E)  no absolute extrema

Question 9

Find two positive numbers whose sum is 16 such that the product of one number and the cube of the other number is a maximum.

Accepted Answer

A)  4 and 12 
B)  1 and 15 
C)  3 and 13 
D)  8 and 8 
E)  9 and 7 
A)  4 and 12 
B)  1 and 15 
C)  3 and 13 
D)  8 and 8 
E)  9 and 7

Question 10

Evaluate .

Accepted Answer

A)  1 
B)  e 
C)    
D)  $\infty$ 
E)  -6 
A)  1 
B)  e 
C)    
D)  $\infty$ 
E)  -6

Question 11

Find the local extrema and inflection points and sketch the graph of the given function f(x) = x(x - 2)2.

Accepted Answer

Local max (2/3, 32/2

Question 12

Find any extreme values of the function f(x) = x   .

Accepted Answer

A)  absolute minimum 0 at x = 1, no maxima 
B)  absolute minimum 0 at x = 0, local maximum 2 at x = 2 
C)  absolute minimum 0 at x = 0, no local maxima 
D)  absolute minimum 0 at x = 1, local maximum 2 at x = 2 
E)  no absolute or local extrema 
A)  absolute minimum 0 at x = 1, no maxima 
B)  absolute minimum 0 at x = 0, local maximum 2 at x = 2 
C)  absolute minimum 0 at x = 0, no local maxima 
D)  absolute minimum 0 at x = 1, local maximum 2 at x = 2 
E)  no absolute or local extrema

Question 13

Find the extreme values of f(x) = x +   cos x on [0,$\pi$].

Accepted Answer

A)  maximum   + 1, minimum   - 1 
B)  maximum   , minimum $\pi$ -   
C)  maximum $\pi$ -   , minimum   
D)  maximum   +   , minimum   -   
E)  no extreme values 
A)  maximum   + 1, minimum   - 1 
B)  maximum   , minimum $\pi$ -   
C)  maximum $\pi$ -   , minimum   
D)  maximum   +   , minimum   -   
E)  no extreme values

Question 14

Find any extreme values of the function f(x) =   - 7 and their locations.

Accepted Answer

A)  local (and absolute) minimum -7 at x = 5/2, no local maxima 
B)  local (and absolute) maximum -7 at x = 5/2, no local minima 
C)  local (and absolute) minimum -7 at x = 5/2, local maximum -2 at x = 0 
D)  local (and absolute) maximum -7 at x = 5/2, local minimum -2 at x = 0 
E)  no absolute or local extrema 
A)  local (and absolute) minimum -7 at x = 5/2, no local maxima 
B)  local (and absolute) maximum -7 at x = 5/2, no local minima 
C)  local (and absolute) minimum -7 at x = 5/2, local maximum -2 at x = 0 
D)  local (and absolute) maximum -7 at x = 5/2, local minimum -2 at x = 0 
E)  no absolute or local extrema

Question 15

Water is leaking out of an inverted conical tank at a rate of 0.1 cubic metres per minute at the same time the water is being pumped into the tank at a constant rate. The tank has height 6 metres and the diameter at the top is 4 metres. If the water level is rising at a rate of 20 centimetres per minute, find the rate at which water is being pumped into the tank at the instant when the water is 2 metres deep.

Accepted Answer

A)  0.179   /min 
B)  0.279   /min 
C)  0.379   /min 
D)  0.479   /min 
E)  0.365   /min 
A)  0.179   /min 
B)  0.279   /min 
C)  0.379   /min 
D)  0.479   /min 
E)  0.365   /min

Question 16

The graph shown below is the graph of which function?

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 17

There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree planted in the orchard, the output per tree drops by 10 apples. How many trees should be added to the existing orchard to maximize the total output of trees?

Accepted Answer

A)  25 additional trees 
B)  20 additional trees 
C)  15 additional trees 
D)  10 additional trees 
E)  5 additional trees 
A)  25 additional trees 
B)  20 additional trees 
C)  15 additional trees 
D)  10 additional trees 
E)  5 additional trees

Question 18

Use Newton's Method to find the roots of the equation cos x = x.

Accepted Answer

A)  0.739335 
B)  0.739161 
C)  0.739085 
D)  0.739204 
E)  0.739185 
A)  0.739335 
B)  0.739161 
C)  0.739085 
D)  0.739204 
E)  0.739185

Question 19

The power P (watts) in an electric circuit is related to the circuit's resistance R (ohms) and the current I (amperes) by the equation P = R I2. If the power of a circuit is a constant 40 watts and the resistance R is decreasing at the constant rate of 4 ohms /s, at what rate is the current I changing at the instant the current is 2 amperes?

Accepted Answer

A)  The current is decreasing at the rate of   amperes / s. 
B)  The current is increasing at the rate of   amperes / s. 
C)  The current is decreasing at the rate of   amperes / s. 
D)  The current is increasing at the rate of   amperes / s. 
E)  The current is decreasing at the rate of   amperes / s. 
A)  The current is decreasing at the rate of   amperes / s. 
B)  The current is increasing at the rate of   amperes / s. 
C)  The current is decreasing at the rate of   amperes / s. 
D)  The current is increasing at the rate of   amperes / s. 
E)  The current is decreasing at the rate of   amperes / s.

Question 20

Find the minimum values of the function f(x) =   -   .

Accepted Answer

A)  -   
B)  0 
C)  -   
D)  -   
E)  no local or absolute minima 
A)  -   
B)  0 
C)  -   
D)  -   
E)  no local or absolute minima

Find the Maclaurin polynomial of degree 4 for sec(2x).

Use linear approximation to estimate the change of the surface area S = 4 $\pi$ r² of a sphere if its radius r is increased from 9 cm to 9.1 cm.

A lawnmower using a 40 m extension cord cuts grass in a circular area. It is discovered that there is a knot in the cord, and untying it lengthens the cord to 42 m. Using an appropriate linearization, estimate how much more area can be cut with the unknotted cord.

Find the local extreme and inflection points of the function f(x) = + 4 and sketch its graph.

A large sheet of cardboard 2 m by 4 m will be made into a box by cutting equal-sized squares from each corner and folding up the four edges. What is the largest possible volume of such a box?

A rectangular enclosure subdivided into four pens by three parallel partitions is to be built using a total of 500 m of fencing. What dimensions will maximize the total area of the pens?

Find the absolute maximum and absolute minimum values of the function f(x) = $Find the absolute maximum and absolute minimum values of the function f(x) = - 1, 0 \le x \le 2.$ - 1, 0 $\le$ x $\le$ 2.

Find two positive numbers whose sum is 16 such that the product of one number and the cube of the other number is a maximum.

Evaluate .

Find the local extrema and inflection points and sketch the graph of the given function f(x) = x(x - 2)².

Find any extreme values of the function f(x) = x .

Find the extreme values of f(x) = x + $Find the extreme values of f(x) = x + cos x on [0, \pi ].$ cos x on [0, $\pi$ ].

Find any extreme values of the function f(x) = - 7 and their locations.

The graph shown below is the graph of which function?

There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree planted in the orchard, the output per tree drops by 10 apples. How many trees should be added to the existing orchard to maximize the total output of trees?

Use Newton's Method to find the roots of the equation cos x = x.

Find the minimum values of the function f(x) = - .

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Exam 5: More Applications of Differentiation

Find the Maclaurin polynomial of degree 4 for sec(2x).

Use linear approximation to estimate the change of the surface area S = 4 π\piπ r2 of a sphere if its radius r is increased from 9 cm to 9.1 cm.

A lawnmower using a 40 m extension cord cuts grass in a circular area. It is discovered that there is a knot in the cord, and untying it lengthens the cord to 42 m. Using an appropriate linearization, estimate how much more area can be cut with the unknotted cord.

Find the local extreme and inflection points of the function f(x) = + 4 and sketch its graph.

A large sheet of cardboard 2 m by 4 m will be made into a box by cutting equal-sized squares from each corner and folding up the four edges. What is the largest possible volume of such a box?

A rectangular enclosure subdivided into four pens by three parallel partitions is to be built using a total of 500 m of fencing. What dimensions will maximize the total area of the pens?

Find the absolute maximum and absolute minimum values of the function f(x) = - 1, 0 ≤\le≤ x ≤\le≤ 2.

Find two positive numbers whose sum is 16 such that the product of one number and the cube of the other number is a maximum.

Evaluate .

Find the local extrema and inflection points and sketch the graph of the given function f(x) = x(x - 2)2.

Find any extreme values of the function f(x) = x .

Find the extreme values of f(x) = x + cos x on [0, π\piπ ].

Find any extreme values of the function f(x) = - 7 and their locations.

The graph shown below is the graph of which function?

There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree planted in the orchard, the output per tree drops by 10 apples. How many trees should be added to the existing orchard to maximize the total output of trees?

Use Newton's Method to find the roots of the equation cos x = x.

Find the minimum values of the function f(x) = - .

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Use linear approximation to estimate the change of the surface area S = 4 $\pi$ r² of a sphere if its radius r is increased from 9 cm to 9.1 cm.

Find the absolute maximum and absolute minimum values of the function f(x) = $Find the absolute maximum and absolute minimum values of the function f(x) = - 1, 0 \le x \le 2.$ - 1, 0 $\le$ x $\le$ 2.

Find the local extrema and inflection points and sketch the graph of the given function f(x) = x(x - 2)².

Find the extreme values of f(x) = x + $Find the extreme values of f(x) = x + cos x on [0, \pi ].$ cos x on [0, $\pi$ ].