Question 1

Let f(x) =   . Use information from f and its first two derivatives to sketch the graph of f.

Accepted Answer

Oblique asymptote y = x - 3, v

Question 2

Find the height of the circular cylinder of maximum volume that will fit inside a sphere of radius R cm.

Accepted Answer

A)    R cm 
B)    R cm 
C)    R cm 
D)    R cm 
E)    R cm 
A)    R cm 
B)    R cm 
C)    R cm 
D)    R cm 
E)    R cm

Question 3

Evaluate the limit .

Accepted Answer

A)  -2 
B)  2 
C)  0 
D)  1 
E)  -1 
A)  -2 
B)  2 
C)  0 
D)  1 
E)  -1

Question 4

Find all inflection points of the graph of f(x) = 3   - 5   + 13x.

Accepted Answer

A)  (0, 0) and (1, 11) 
B)  (0, 0) only 
C)  (1, 11) only 
D)  (-1, -21) and (1, 11) 
E)  (-1, -21), (0, 0), and (1, 11) 
A)  (0, 0) and (1, 11) 
B)  (0, 0) only 
C)  (1, 11) only 
D)  (-1, -21) and (1, 11) 
E)  (-1, -21), (0, 0), and (1, 11)

Question 5

Find all local extreme values of the function f(x) =   - 6   + 12x - 5 and their locations.

Accepted Answer

A)  local maximum 3 at x = 2, local minimum -61 at x = -2 
B)  local maximum -61 at x = -2, local minimum 3 at x = 2 
C)  local maximum 3 at x = -2, local minimum -61 at x = 2 
D)  local maximum -61 at x = 2, local minimum 3 at x = -2 
E)  no local extrema 
A)  local maximum 3 at x = 2, local minimum -61 at x = -2 
B)  local maximum -61 at x = -2, local minimum 3 at x = 2 
C)  local maximum 3 at x = -2, local minimum -61 at x = 2 
D)  local maximum -61 at x = 2, local minimum 3 at x = -2 
E)  no local extrema

Question 6

Compute the Maclaurin polynomial of degree 3 for f(x) =   .

Accepted Answer

A)  1 + 3x +   +   
B)  1 +   
C)  1 - 3x +   -   
D)  1 + 3x +   +   
E)  1 + 3x +   +   
A)  1 + 3x +   +   
B)  1 +   
C)  1 - 3x +   -   
D)  1 + 3x +   +   
E)  1 + 3x +   +

Question 7

Give the iteration formula for finding the roots of the equation sin x - x2 = 0 using Newton's Method.

Accepted Answer

A)    =   -   
B)    =   -   
C)    =   +   
D)    =   -   
E)    =   +   
A)    =   -   
B)    =   -   
C)    =   +   
D)    =   -   
E)    =   +

Question 8

Evaluate the limit .

Accepted Answer

A)  -$\infty$ 
B)  $\infty$ 
C)  1 
D)  0 
E)  -1 
A)  -$\infty$ 
B)  $\infty$ 
C)  1 
D)  0 
E)  -1

Question 9

Find the roots of the equation ln x -   = 0 to four decimal places using Newton's Method.

Accepted Answer

A)  1.2445 
B)  1.9445 
C)  1.3445 
D)  1.0445 
E)  1.0543 
A)  1.2445 
B)  1.9445 
C)  1.3445 
D)  1.0445 
E)  1.0543

Question 10

Find the inflection points of the graph of f(x) =   where c is a nonzero constant.

Accepted Answer

A)    
B)    
C)    
D)    
E)  There are no inflection points. 
A)    
B)    
C)    
D)    
E)  There are no inflection points.

Question 11

Accepted Answer

A)      , where s is some number between c and x 
B)      , where s is some number between c and x 
C)      , where s is some number between c and x 
D)      , where s is some number between c and x 
E)      , where s is some number between c and x 
A)      , where s is some number between c and x 
B)      , where s is some number between c and x 
C)      , where s is some number between c and x 
D)      , where s is some number between c and x 
E)      , where s is some number between c and x

Question 12

Given that   -4?

Accepted Answer

@#IMG-DLM& < @#IMG-DLM& , n =

Question 13

An aircraft is climbing at a 30-degree angle to the horizontal. How fast is the aircraft gaining altitude if its speed is 750 kilometres per hour?

Accepted Answer

A)  400 kilometres per hour 
B)  350 kilometres per hour 
C)  375 kilometres per hour 
D)  250 kilometres per hour 
E)  425 kilometres per hour 
A)  400 kilometres per hour 
B)  350 kilometres per hour 
C)  375 kilometres per hour 
D)  250 kilometres per hour 
E)  425 kilometres per hour

Question 14

The height of a right circular cylinder is increasing at the rate of 4 cm/s and its radius is decreasing at the rate of 2 cm/s. At what rate is the lateral surface area of the cylinder changing when the height is 3 centimetres and the radius is 1 centimetre?
The lateral surface area of a cylinder of base radius r and height h is given by S = 2 $\pi$r h

Accepted Answer

A)  decreasing at 16 $\pi$   /s 
B)  decreasing at 4 $\pi$  /s 
C)  increasing at 4 $\pi$  /s 
D)  increasing at 20 $\pi$   /s 
E)  increasing at 16 $\pi$   /s 
A)  decreasing at 16 $\pi$   /s 
B)  decreasing at 4 $\pi$  /s 
C)  increasing at 4 $\pi$  /s 
D)  increasing at 20 $\pi$   /s 
E)  increasing at 16 $\pi$   /s

Question 15

Find the equations of the vertical asymptotes of f(x) =

Accepted Answer

A)  x = 0, x = 12, and x = -2 
B)  x = 0, x = 6, and x = 4 
C)  x = 0 and x = 12 
D)  x = 12 and x = -2 
E)  x = 0, x = -8, and x = -3 
A)  x = 0, x = 12, and x = -2 
B)  x = 0, x = 6, and x = 4 
C)  x = 0 and x = 12 
D)  x = 12 and x = -2 
E)  x = 0, x = -8, and x = -3

Question 16

Evaluate .

Accepted Answer

The answer of Evaluate .  
  ...

Question 17

Suppose Newton's Method applied to f(x) =   is used to "find"the root of the equation   = 0 with initial guess x0 = r $\neq$ 0. What result does the first iteration of the method yield? The second iteration? The nth iteration? Why do these not converge to the obvious root x = 0 no matter how close the initial guess r was to that root?

Accepted Answer

x₁ = -2r, x₂ = 4r, ..., x_n = @#IMG-DLM& r.

Question 18

A plane is flying horizontally at an altitude of seven kilometres and a speed of 800 kilometres per hour. At time t = 0 the plane passes over a tracking station on the ground. How fast is the angle of elevation of the plane as measured at the tracking station changing 18 seconds later?

Accepted Answer

A)  increasing at 86 rad/h 
B)  increasing at 66 rad/h 
C)  decreasing at 86 rad/h 
D)  decreasing at 66 rad/h 
E)  decreasing at 62 rad/h 
A)  increasing at 86 rad/h 
B)  increasing at 66 rad/h 
C)  decreasing at 86 rad/h 
D)  decreasing at 66 rad/h 
E)  decreasing at 62 rad/h

Question 19

If f(x) is a polynomial of degree one and L(x) is the linearization of the function f about a, then f(x) = L(x).

Accepted Answer

A) True 
 B)False

Question 20

Use the table of values below to evaluate

Accepted Answer

A)  -   
B)  - 2 
C)    
D)  -   
E)    
A)  -   
B)  - 2 
C)    
D)  -   
E)

Let f(x) = . Use information from f and its first two derivatives to sketch the graph of f.

Find the height of the circular cylinder of maximum volume that will fit inside a sphere of radius R cm.

Evaluate the limit .

Find all inflection points of the graph of f(x) = 3 - 5 + 13x.

Find all local extreme values of the function f(x) = - 6 + 12x - 5 and their locations.

Compute the Maclaurin polynomial of degree 3 for f(x) = .

Give the iteration formula for finding the roots of the equation sin x - x² = 0 using Newton's Method.

Evaluate the limit .

Find the roots of the equation ln x - = 0 to four decimal places using Newton's Method.

Find the inflection points of the graph of f(x) = where c is a nonzero constant.

Given that < 2, find an estimate for the size of the error if the Maclaurin polynomial of degree n for is used to approximate = . How large need n be to ensure that this error is less than 10^-4?

An aircraft is climbing at a 30-degree angle to the horizontal. How fast is the aircraft gaining altitude if its speed is 750 kilometres per hour?

Find the equations of the vertical asymptotes of f(x) =

Evaluate .

A plane is flying horizontally at an altitude of seven kilometres and a speed of 800 kilometres per hour. At time t = 0 the plane passes over a tracking station on the ground. How fast is the angle of elevation of the plane as measured at the tracking station changing 18 seconds later?

If f(x) is a polynomial of degree one and L(x) is the linearization of the function f about a, then f(x) = L(x).

Use the table of values below to evaluate

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

Let f(x) = . Use information from f and its first two derivatives to sketch the graph of f.

Find the height of the circular cylinder of maximum volume that will fit inside a sphere of radius R cm.

Evaluate the limit .

Find all inflection points of the graph of f(x) = 3 - 5 + 13x.

Find all local extreme values of the function f(x) = - 6 + 12x - 5 and their locations.

Compute the Maclaurin polynomial of degree 3 for f(x) = .

Give the iteration formula for finding the roots of the equation sin x - x2 = 0 using Newton's Method.

Evaluate the limit .

Find the roots of the equation ln x - = 0 to four decimal places using Newton's Method.

Find the inflection points of the graph of f(x) = where c is a nonzero constant.

Given that < 2, find an estimate for the size of the error if the Maclaurin polynomial of degree n for is used to approximate = . How large need n be to ensure that this error is less than 10-4?

An aircraft is climbing at a 30-degree angle to the horizontal. How fast is the aircraft gaining altitude if its speed is 750 kilometres per hour?

Find the equations of the vertical asymptotes of f(x) =

Evaluate .

A plane is flying horizontally at an altitude of seven kilometres and a speed of 800 kilometres per hour. At time t = 0 the plane passes over a tracking station on the ground. How fast is the angle of elevation of the plane as measured at the tracking station changing 18 seconds later?

If f(x) is a polynomial of degree one and L(x) is the linearization of the function f about a, then f(x) = L(x).

Use the table of values below to evaluate

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

Give the iteration formula for finding the roots of the equation sin x - x² = 0 using Newton's Method.

Given that < 2, find an estimate for the size of the error if the Maclaurin polynomial of degree n for is used to approximate = . How large need n be to ensure that this error is less than 10^-4?