Question 1

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

Accepted Answer

no asymptotes or symmetry, loc

Question 2

Find the absolute maximum and absolute minimum values of the function f(x) = 2   - 3   , -2 $\le$ x $\le$ 2.

Accepted Answer

A)  maximum 4, minimum -28 
B)  maximum 2, minimum -26 
C)  maximum 3, minimum -29 
D)  maximum 4, minimum -12 
E)  no absolute extrema 
A)  maximum 4, minimum -28 
B)  maximum 2, minimum -26 
C)  maximum 3, minimum -29 
D)  maximum 4, minimum -12 
E)  no absolute extrema

Question 3

Find and classify all the local extrema of the function f(x) = x - sin2x.

Accepted Answer

A)  local minima at x = k $\pi$   +   and local maxima at x = k $\pi$ -   , where k is an integer 
B)  local maxima at x = k $\pi$ +   and local minima at x = k $\pi$ -   , where k is an integer 
C)  local maxima at x = k $\pi$ +   and local minima at x = k $\pi$ -   , where k is an integer 
D)  local minima at x = k $\pi$ +   and local maxima at x = k $\pi$ -   , where k is an integer 
E)  local minima at x = 2k $\pi$ +   and local maxima at x = 2k $\pi$ -   , where k is an integer 
A)  local minima at x = k $\pi$   +   and local maxima at x = k $\pi$ -   , where k is an integer 
B)  local maxima at x = k $\pi$ +   and local minima at x = k $\pi$ -   , where k is an integer 
C)  local maxima at x = k $\pi$ +   and local minima at x = k $\pi$ -   , where k is an integer 
D)  local minima at x = k $\pi$ +   and local maxima at x = k $\pi$ -   , where k is an integer 
E)  local minima at x = 2k $\pi$ +   and local maxima at x = 2k $\pi$ -   , where k is an integer

Question 4

A girl is on the bank of a river that is 1 kilometre wide. She wants to travel to a town on the opposite bank but 1 kilometre upstream. She intends to row in a straight line to some point P on the opposite bank and walk the remaining distance along the (straight) bank. To what point should she row in order to reach her destination in the least possible time if she can walk 5  kilometres per hour and row 3 kilometres per hour?Refer to the figure below:

Accepted Answer

A)  x = 0 km (row directly to town) 
B)  x =1 km 
C)  x =   km 
D)  x =   km 
E)  x =   km 
A)  x = 0 km (row directly to town) 
B)  x =1 km 
C)  x =   km 
D)  x =   km 
E)  x =   km

Question 5

Newton's Method with the initial approximation x1 = 1 is used to approximate the real root of the equation x3 + 3x - 1 = 0. Determine the value of x3, the third iteration of Newton's Method.

Accepted Answer

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

Question 6

The function f(x) = 3x5 + Ax4 + Bx3 has inflection points at x = 0, x = -1, and x = 1. Find the values of the constants A and B.

Accepted Answer

A)  A = 0, B = -10 
B)  A = 0, B = 10 
C)  A has any value, B = -10 
D)  A has any value, B = 10 
E)  A = 10, B = 0 
A)  A = 0, B = -10 
B)  A = 0, B = 10 
C)  A has any value, B = -10 
D)  A has any value, B = 10 
E)  A = 10, B = 0

Question 7

The graph of f(x) = 3   (4 - x) has a cusp at

Accepted Answer

A)  x = 0 
B)  x = 1 
C)  x = -2 
D)  x = 4 
E)  The graph has no cusps. 
A)  x = 0 
B)  x = 1 
C)  x = -2 
D)  x = 4 
E)  The graph has no cusps.

Question 8

Evaluate the limit .

Accepted Answer

A)  3 
B)  0 
C)  -3 
D)  1 
E) $\infty$ 
A)  3 
B)  0 
C)  -3 
D)  1 
E) $\infty$

Question 9

Use information from the function and its first two derivatives to sketch the graph of the function

Accepted Answer

horizontal asymptote y = @#IMG-DLM& ,abs

Question 10

Find the roots of the equation x3 - 5x - 3 = 0 correct to three decimal places using Newton's Method.

Accepted Answer

A)  -1.834, -0.657, 2.491 
B)  -1.834, 0.657, 2.491 
C)  -1.824, -0.667, -2.501 
D)  -1.824, -0.667, 2.589 
E)  -1.834, -0.667, 2.491 
A)  -1.834, -0.657, 2.491 
B)  -1.834, 0.657, 2.491 
C)  -1.824, -0.667, -2.501 
D)  -1.824, -0.667, 2.589 
E)  -1.834, -0.667, 2.491

Question 11

The graph shown in the figure below is the graph of which function?

Accepted Answer

A)  y =   
B)  y =   
C)  y =   
D)  y =   
E)  y =   
A)  y =   
B)  y =   
C)  y =   
D)  y =   
E)  y =

Question 12

A poster has an area of 800 cm2. An image is to be printed on the poster so that there are 4 cm margins at the top and bottom and 2 cm margins on either side. Find the maximum area such an image can have.

Accepted Answer

A)  792   
B)  800   
C)  512   
D)  648   
E)  625   
A)  792   
B)  800   
C)  512   
D)  648   
E)  625

Question 13

At a certain instant the length of one leg of a right triangle is 6 m and is increasing at 2 m/min while the length of the other leg is 8 m and is decreasing at 3 m/min. How fast is the area of the triangle changing at that instant?

Accepted Answer

A)  decreasing at 1   /min 
B)  decreasing at 2   /min 
C)  unchanging, holding steady 
D)  increasing at 2   /min 
E)  increasing at 1   /min 
A)  decreasing at 1   /min 
B)  decreasing at 2   /min 
C)  unchanging, holding steady 
D)  increasing at 2   /min 
E)  increasing at 1   /min

Question 14

Determine the concavity of f(x) = cos x + sin x on [0, 2π] and identify any points of inflection.

Accepted Answer

f(x) is concave downwards on [

Question 15

Find the ratio of the height h to the radius r of a cylindrical container having given volume V and having the least possible total surface area.

Accepted Answer

A)    = 2 
B)    =   
C)    = 3 
D)    =   
E)    =   
A)    = 2 
B)    =   
C)    = 3 
D)    =   
E)    =

Question 16

A balloon is 100 metres off the ground and rising vertically at the constant rate of 3 metres per second just as an automobile passes beneath it travelling along a straight, level road at the constant rate of 72 kilometres per hour. How fast is the distance between them changing one second later? (rounded to the nearest hundredth of a metre)

Accepted Answer

A)  6.76 metres per second 
B)  43.71 metres per second 
C)  20.22 metres per second 
D)  10.00 metres per second 
E)  1.28 metres per second 
A)  6.76 metres per second 
B)  43.71 metres per second 
C)  20.22 metres per second 
D)  10.00 metres per second 
E)  1.28 metres per second

Question 17

Determine the concavity of f(x) =   - 24   + 6x + 18 and identify any points of inflection.

Accepted Answer

A)  concave downwards on (-$\infty$, 8), upwards on (8, $\infty$); inflection at x = 8 
B)  concave downwards on (-$\infty$, -8), upwards on (-8, $\infty$); inflection at x = -8 
C)  concave upwards on (- $\infty$, 8), downwards on (8, $\infty$); inflection at x = 8 
D)  concave upwards on (- $\infty$, -8), downwards on (-8, $\infty$); inflection at x = -8 
E)  concave upwards on (-$\infty$, $\infty$); no inflection points 
A)  concave downwards on (-$\infty$, 8), upwards on (8, $\infty$); inflection at x = 8 
B)  concave downwards on (-$\infty$, -8), upwards on (-8, $\infty$); inflection at x = -8 
C)  concave upwards on (- $\infty$, 8), downwards on (8, $\infty$); inflection at x = 8 
D)  concave upwards on (- $\infty$, -8), downwards on (-8, $\infty$); inflection at x = -8 
E)  concave upwards on (-$\infty$, $\infty$); no inflection points

Question 18

Evaluate .

Accepted Answer

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

Question 19

Let P(x) be a polynomial in x, let k be a positive integer, and let   be a number such that   is a factor of   (x) but   is not a factor of   (x). For what values of k is it possible that P(x) has a local maximum or minimum value at x =  ?

Accepted Answer

A)  even values of k 
B)  k = 1 only 
C)  k = 2 only 
D)  odd values of k 
E)  k = 0 only 
A)  even values of k 
B)  k = 1 only 
C)  k = 2 only 
D)  odd values of k 
E)  k = 0 only

Question 20

Find the point (x, y) on the graph of y =   nearest the point (4, 0).

Accepted Answer

A)  (2, 2) 
B)  (3, )   
C)  (4, )   
D)  (1, )   
E)  (5, )   
A)  (2, 2) 
B)  (3, )   
C)  (4, )   
D)  (1, )   
E)  (5, )

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

Find and classify all the local extrema of the function f(x) = x - sin2x.

Newton's Method with the initial approximation x₁ = 1 is used to approximate the real root of the equation x³ + 3x - 1 = 0. Determine the value of x₃, the third iteration of Newton's Method.

The function f(x) = 3x⁵ + Ax⁴ + Bx³ has inflection points at x = 0, x = -1, and x = 1. Find the values of the constants A and B.

The graph of f(x) = 3 (4 - x) has a cusp at

Evaluate the limit .

Use information from the function and its first two derivatives to sketch the graph of the function

Find the roots of the equation x³ - 5x - 3 = 0 correct to three decimal places using Newton's Method.

The graph shown in the figure below is the graph of which function?

A poster has an area of 800 cm². An image is to be printed on the poster so that there are 4 cm margins at the top and bottom and 2 cm margins on either side. Find the maximum area such an image can have.

At a certain instant the length of one leg of a right triangle is 6 m and is increasing at 2 m/min while the length of the other leg is 8 m and is decreasing at 3 m/min. How fast is the area of the triangle changing at that instant?

Determine the concavity of f(x) = cos x + sin x on [0, 2π] and identify any points of inflection.

Find the ratio of the height h to the radius r of a cylindrical container having given volume V and having the least possible total surface area.

Determine the concavity of f(x) = - 24 + 6x + 18 and identify any points of inflection.

Evaluate .

Let P(x) be a polynomial in x, let k be a positive integer, and let be a number such that is a factor of (x) but is not a factor of (x). For what values of k is it possible that P(x) has a local maximum or minimum value at x = 11ee7b18_881f_7ad5_ae82_ef6a0704a9e3_TB9661_11?

Find the point (x, y) on the graph of y = nearest the point (4, 0).

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

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

Find the absolute maximum and absolute minimum values of the function f(x) = 2 - 3 , -2 ≤\le≤ x ≤\le≤ 2.

Find and classify all the local extrema of the function f(x) = x - sin2x.

Newton's Method with the initial approximation x1 = 1 is used to approximate the real root of the equation x3 + 3x - 1 = 0. Determine the value of x3, the third iteration of Newton's Method.

The function f(x) = 3x5 + Ax4 + Bx3 has inflection points at x = 0, x = -1, and x = 1. Find the values of the constants A and B.

The graph of f(x) = 3 (4 - x) has a cusp at

Evaluate the limit .

Use information from the function and its first two derivatives to sketch the graph of the function

Find the roots of the equation x3 - 5x - 3 = 0 correct to three decimal places using Newton's Method.

The graph shown in the figure below is the graph of which function?

A poster has an area of 800 cm2. An image is to be printed on the poster so that there are 4 cm margins at the top and bottom and 2 cm margins on either side. Find the maximum area such an image can have.

At a certain instant the length of one leg of a right triangle is 6 m and is increasing at 2 m/min while the length of the other leg is 8 m and is decreasing at 3 m/min. How fast is the area of the triangle changing at that instant?

Determine the concavity of f(x) = cos x + sin x on [0, 2π] and identify any points of inflection.

Find the ratio of the height h to the radius r of a cylindrical container having given volume V and having the least possible total surface area.

Determine the concavity of f(x) = - 24 + 6x + 18 and identify any points of inflection.

Evaluate .

Let P(x) be a polynomial in x, let k be a positive integer, and let be a number such that is a factor of (x) but is not a factor of (x). For what values of k is it possible that P(x) has a local maximum or minimum value at x = 11ee7b18_881f_7ad5_ae82_ef6a0704a9e3_TB9661_11?

Find the point (x, y) on the graph of y = nearest the point (4, 0).

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

Newton's Method with the initial approximation x₁ = 1 is used to approximate the real root of the equation x³ + 3x - 1 = 0. Determine the value of x₃, the third iteration of Newton's Method.

The function f(x) = 3x⁵ + Ax⁴ + Bx³ has inflection points at x = 0, x = -1, and x = 1. Find the values of the constants A and B.

Find the roots of the equation x³ - 5x - 3 = 0 correct to three decimal places using Newton's Method.

A poster has an area of 800 cm². An image is to be printed on the poster so that there are 4 cm margins at the top and bottom and 2 cm margins on either side. Find the maximum area such an image can have.