Question 1

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

Accepted Answer

Local max at (0, 1), local min at (±1, 0), inflections at (±1/   , 4/9)

Question 2

At what value(s) of x does the graph of f(x) = x   have inflections?

Accepted Answer

A)  ±   
B)  0 and ±   
C)  ±   
D)  0 and ±   
E)  0 
A)  ±   
B)  0 and ±   
C)  ±   
D)  0 and ±   
E)  0 D

Question 3

Using the second derivative test, classify the critical points of the function f(t) = t3 - t2 - t + 2 and locate any points of inflection.

Accepted Answer

A)  local max at t = -1/3, local min at t = 1, inflection at t = 1/3 
B)  local min at t = -1/3, local max at t = 1, inflection at t = 1/3 
C)  local max at t = 1/3, local min at t = -1, inflection at t = -1/3 
D)  local min at t = 1/3, local max at t = -1, inflection at t = -1/3 
E)  none of the above 
A)  local max at t = -1/3, local min at t = 1, inflection at t = 1/3 
B)  local min at t = -1/3, local max at t = 1, inflection at t = 1/3 
C)  local max at t = 1/3, local min at t = -1, inflection at t = -1/3 
D)  local min at t = 1/3, local max at t = -1, inflection at t = -1/3 
E)  none of the above A

Question 4

A function  f(x)  satisfies the following conditions:

f(1) = 0,	$~~~~~~~~$  fS1U1P1(n)S1S1P0 (1) = n  for  n = 1, 2, and 3,
	3x  ≤  fS1U1P1(4)S1S1P0 (x)  ≤  9x  provided  x  ≥  1.

(a)	What is the Taylor polynomial pS1U1B13S1U1B0  (x)  of degree 3  for  f(x)  about  x = 1?
(b)	What is the approximate value for  f(3/2)  supplied by   (x)?
(c)	Based on the bounds for  fS1U1P1(4)S1S1P0 (x)  given above, what is the smallest interval that you can be sure
	contains the value  f(3/2)?
(d)	Based on your answer to (c), what is the best approximation you can give for  f(3/2)?

Accepted Answer

The answer of A function  f(x)  satisfies the...

Question 5

A plane flying horizontally at an altitude of 1 kilometre and a speed of 500 kilometres per hour passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing at the instant when the plane is 2 kilometres away from the station.

Accepted Answer

A)  350 kilometres per hour 
B)  462 kilometres per hour 
C)  402 kilometres per hour 
D)  433 kilometres per hour 
E)  382 kilometres per hour 
A)  350 kilometres per hour 
B)  462 kilometres per hour 
C)  402 kilometres per hour 
D)  433 kilometres per hour 
E)  382 kilometres per hour

Question 6

At what values of t does the function g(t) =   have extreme values?

Accepted Answer

A)  absolute maximum at t = 1, absolute minimum at t = -1, no other local extrema 
B)  absolute maximum at t = 1, absolute minimum at t = 0, no other local extrema 
C)  absolute maximum at t = 2, absolute minimum at t = 0, no other local extrema 
D)  absolute maximum at t = 2, absolute minimum at t = -2, no other local extrema 
E)  no absolute or local extrema 
A)  absolute maximum at t = 1, absolute minimum at t = -1, no other local extrema 
B)  absolute maximum at t = 1, absolute minimum at t = 0, no other local extrema 
C)  absolute maximum at t = 2, absolute minimum at t = 0, no other local extrema 
D)  absolute maximum at t = 2, absolute minimum at t = -2, no other local extrema 
E)  no absolute or local extrema

Question 7

Find a suitable linearization for sin(x°) useful near x = 60°.

Accepted Answer

A)  L(x) =   +   (x - 60) 
B)  L(x) =   +   (x - 60) 
C)  L(x) =   -   (x - 60) 
D)  L(x) =   -   (x - 60) 
E)  L(x) =   +   (x - 60) 
A)  L(x) =   +   (x - 60) 
B)  L(x) =   +   (x - 60) 
C)  L(x) =   -   (x - 60) 
D)  L(x) =   -   (x - 60) 
E)  L(x) =   +   (x - 60)

Question 8

Let f(x) = 18   + 9   . The first and the second order derivatives of f are given by   (x) =   and   (x) =   , respectively. Determine:(a) intervals of concavity.(b) the x and y coordinates of the inflection points.

Accepted Answer

(a) concave down on

Question 9

For what value of k will f(x) = 2x - 3k   , have a local minimum at x = 1?

Accepted Answer

A)  k =   
B)  k = -   
C)  k = 3 
D)  k = 0 
E)  k = -3 
A)  k =   
B)  k = -   
C)  k = 3 
D)  k = 0 
E)  k = -3

Question 10

A hemispherical bowl of radius 10 inches is filled with water to a depth of x inches. The volume V of water in the bowl (in cubic centimetres) is given by the formulaV =   (30   -   ). Suppose that you measure the depth of water in the bowl to be 5 cm with a maximum possible measured error of 0.5 cm. Estimate the maximum error in the calculated volume of water in the bowl using a linear approximation.

Accepted Answer

A)  117.81 cm3 
B)  114.63 cm3 
C)  112.62 cm3 
D)  121.61 cm3 
E)  81.12 cm3 
A)  117.81 cm3 
B)  114.63 cm3 
C)  112.62 cm3 
D)  121.61 cm3 
E)  81.12 cm3

Question 11

A closed rectangular container with a square base is to have a volume of 12 cubic metres. The material for the top and bottom of the container will cost $6 per square metre and the material for the sides will cost $4 per square metre. Let x be the length of a side of the base (in metres) and C(x) be the total cost of the container (in dollars). Determine
 (a) the total cost C(x).
(b) the dimensions of the most economical box.

Accepted Answer

(a) total cost C(x)

Question 12

Water is leaking out of an inverted conical cup at a rate of 2 cubic centimetres per second. The radius of the cone is six centimetres and the height is 10 centimetres. Find the rate at which the water depth is changing at time t when the depth of the water is 3 centimetres.

Accepted Answer

A)  decreasing at   cm/sec 
B)  decreasing at   cm/sec 
C)  decreasing at   cm/sec 
D)  decreasing at   cm/sec 
E)  decreasing at   cm/sec 
A)  decreasing at   cm/sec 
B)  decreasing at   cm/sec 
C)  decreasing at   cm/sec 
D)  decreasing at   cm/sec 
E)  decreasing at   cm/sec

Question 13

Evaluate   x ( $\pi$ - 2   (7x)).

Accepted Answer

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

Question 14

Find   correct to four decimal places using Newton's Method.

Accepted Answer

A)  2.6672 
B)  2.6627 
C)  2.6698 
D)  2.6684 
E)  2.6667 
A)  2.6672 
B)  2.6627 
C)  2.6698 
D)  2.6684 
E)  2.6667

Question 15

Find the solutions of the equation cos x - x4 = 0 to 4 decimal places using Newton's Method.

Accepted Answer

A)  0.8241, -0.6421 
B)  0.8906, -0.8906 
C)  0.8241, -0.8241 
D)  0.6421, -0.6421 
E)  0.8954, -0.8954 
A)  0.8241, -0.6421 
B)  0.8906, -0.8906 
C)  0.8241, -0.8241 
D)  0.6421, -0.6421 
E)  0.8954, -0.8954

Question 16

If the linear approximation of f(x) =   at x = -1 is used to estimate the value of   , then the estimated value is $\textbf{      smaller than   }$  the exact value.

Accepted Answer

A) True 
 B)False

Question 17

Find the smallest possible area of a triangle formed by the coordinate axes and a line tangent to the ellipse   +   = 1, where a > 0 and b > 0.

Accepted Answer

A)  ab square units 
B)  2ab square units 
C)    +   square units 
D)    square units 
E)    square units 
A)  ab square units 
B)  2ab square units 
C)    +   square units 
D)    square units 
E)    square units

Question 18

Find the best linear approximation to the function p(x) = -x3 + 3x at the point (2, -2).

Accepted Answer

A)  L(x) = 16 - 9x 
B)  L(x) = 9x - 20 
C)  L(x) = 14 - 8x 
D)  L(x) = 8x - 18 
E)  L(x) = 20 - 9x 
A)  L(x) = 16 - 9x 
B)  L(x) = 9x - 20 
C)  L(x) = 14 - 8x 
D)  L(x) = 8x - 18 
E)  L(x) = 20 - 9x

Question 19

For what values of the nonnegative constant c is the minimum distance from the point(0,c) to the parabola y = x2 equal to c?

Accepted Answer

A)  0 $\le$c $\le$  
B)  0 $\le$c $\le$1 
C)  only c = 0 
D)  0 $\le$ c < $\infty$ 
E)  only c =   
A)  0 $\le$c $\le$  
B)  0 $\le$c $\le$1 
C)  only c = 0 
D)  0 $\le$ c < $\infty$ 
E)  only c =

Question 20

Find the absolute maximum and minimum values (if any) of f(x) =   .

Accepted Answer

A)  minimum -1, maximum 1 
B)  minimum -1, no maximum 
C)  no minimum, maximum 1 
D)  minimum -1, maximum 0 
E)  no absolute maximum or minimum 
A)  minimum -1, maximum 1 
B)  minimum -1, no maximum 
C)  no minimum, maximum 1 
D)  minimum -1, maximum 0 
E)  no absolute maximum or minimum

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

At what value(s) of x does the graph of f(x) = x have inflections?

Using the second derivative test, classify the critical points of the function f(t) = t³ - t² - t + 2 and locate any points of inflection.

A plane flying horizontally at an altitude of 1 kilometre and a speed of 500 kilometres per hour passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing at the instant when the plane is 2 kilometres away from the station.

At what values of t does the function g(t) = have extreme values?

Find a suitable linearization for sin(x°) useful near x = 60°.

Let f(x) = 18 + 9 . The first and the second order derivatives of f are given by (x) = and (x) = , respectively. Determine:(a) intervals of concavity.(b) the x and y coordinates of the inflection points.

For what value of k will f(x) = 2x - 3k , have a local minimum at x = 1?

Water is leaking out of an inverted conical cup at a rate of 2 cubic centimetres per second. The radius of the cone is six centimetres and the height is 10 centimetres. Find the rate at which the water depth is changing at time t when the depth of the water is 3 centimetres.

Evaluate $Evaluate x ( \pi - 2 (7x)).$ x ( $\pi$ - 2 $Evaluate x ( \pi - 2 (7x)).$ (7x)).

Find correct to four decimal places using Newton's Method.

Find the solutions of the equation cos x - x⁴ = 0 to 4 decimal places using Newton's Method.

Find the smallest possible area of a triangle formed by the coordinate axes and a line tangent to the ellipse + = 1, where a > 0 and b > 0.

Find the best linear approximation to the function p(x) = -x³ + 3x at the point (2, -2).

For what values of the nonnegative constant c is the minimum distance from the point(0,c) to the parabola y = x2 equal to c?

Find the absolute maximum and minimum values (if any) of 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 local extrema and inflection points of the function f(x) = and sketch its graph.

At what value(s) of x does the graph of f(x) = x have inflections?

Using the second derivative test, classify the critical points of the function f(t) = t3 - t2 - t + 2 and locate any points of inflection.

A plane flying horizontally at an altitude of 1 kilometre and a speed of 500 kilometres per hour passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing at the instant when the plane is 2 kilometres away from the station.

At what values of t does the function g(t) = have extreme values?

Find a suitable linearization for sin(x°) useful near x = 60°.

Let f(x) = 18 + 9 . The first and the second order derivatives of f are given by (x) = and (x) = , respectively. Determine:(a) intervals of concavity.(b) the x and y coordinates of the inflection points.

For what value of k will f(x) = 2x - 3k , have a local minimum at x = 1?

Water is leaking out of an inverted conical cup at a rate of 2 cubic centimetres per second. The radius of the cone is six centimetres and the height is 10 centimetres. Find the rate at which the water depth is changing at time t when the depth of the water is 3 centimetres.

Evaluate x ( π\piπ - 2 (7x)).

Find correct to four decimal places using Newton's Method.

Find the solutions of the equation cos x - x4 = 0 to 4 decimal places using Newton's Method.

If the linear approximation of f(x) = at x = -1 is used to estimate the value of , then the estimated value is smaller than \textbf{ smaller than } smaller than the exact value.

Find the smallest possible area of a triangle formed by the coordinate axes and a line tangent to the ellipse + = 1, where a > 0 and b > 0.

Find the best linear approximation to the function p(x) = -x3 + 3x at the point (2, -2).

For what values of the nonnegative constant c is the minimum distance from the point(0,c) to the parabola y = x2 equal to c?

Find the absolute maximum and minimum values (if any) of 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

Using the second derivative test, classify the critical points of the function f(t) = t³ - t² - t + 2 and locate any points of inflection.

Evaluate $Evaluate x ( \pi - 2 (7x)).$ x ( $\pi$ - 2 $Evaluate x ( \pi - 2 (7x)).$ (7x)).

Find the solutions of the equation cos x - x⁴ = 0 to 4 decimal places using Newton's Method.

Find the best linear approximation to the function p(x) = -x³ + 3x at the point (2, -2).