Chapter 4 final - MR. HANSON'S MATH...

Chapter 4

1. (AB/BC, non-calculator)

For 0 < t < 9, a particle moves along the x-axis. The velocity of the particle is given by

( ) sin4

v t t

. The particle is at position x = –4 when t = 0.

(a) For 0 < t < 9, when is the particle moving to the right?

(b) Write but do not evaluate, an integral expression that gives the total distance traveled by the particle from time t = 0 to t = 9.

(c) Find the acceleration of the particle at time t = 3. Is the particle speeding up, slowing down or neither at t = 3? Justify your response.

(d) Find the position of the particle at time t = 3.


Let 3

( ) ( )x

F x f t dt . The graph of f on the interval [–3,4] consists of 2 line segments and a

semicircle, as shown in the graph above.

(a) Find F(0), F (0), and F(4).

(b) Find all relative minimum values of F(x) on the interval [–3, 4]. Justify your answer.

(c) Find the x-coordinates of the inflection points of F(x) on the interval [–3, 4]. Justify your answer.

(d) Write the equation of the line tangent to the point where x = 2.

3. (AB/BC, calculator)

t

(minutes)

0 3 5 7 8 12

C(t)

(degrees Celsius)

65 57 50 46 44 40

As a pot of coffee cools down, the temperature of the coffee is modeled by a differentiable function C, for 0 < t < 12 where time is measured in minutes and the temperature C(t) is measured in degrees Celsius. Selected values of time t are shown in the table above.

(a) Evaluate 12

0( )C t dt . Using correct units, explain the meaning of your answer in the context

of the problem.

(b) Using correct units, explain the meaning of 12

0

1( )

12C t dt in the context of the problem. Use a

trapezoidal sum with 5 subintervals indicated by the table to approximate 12

0

1( )

12C t dt .

(c) Use the data in the table to approximate the rate at which the temperature is changing at time t = 4. Use proper units and show the computations that lead to your answer.

(d) For 12 < t < 15, the rate of cooling is modeled by ( ) 2cos 0.5C t t . Based on the model,

what is the temperature of the coffee at time t = 15? Assume C(t) is continuous at t = 12.


On a typical day, the snow on Pike Mountain melts at a rate modeled by the function

( )M t , given by ( ) sin6 12

tM t

A snow maker adds snow at a rate modeled by the function S, given by 2( ) 0.006 0.12 0.87S t t t

Both M and S have units in inches per hour and t is measured in hours for 0 < t < 6.

At time t = 0, the mountain has 40 inches of snow.

(a) How much snow will melt during the 6 hour period? Indicate units of measure.

(b) Write an expression for I(t), the total number of inches of snow at any time t.

(c) Find the rate of change of the total amount of snow at time t = 3.

(d) For 0 < t < 6 , at what time t is the amount of snow a maximum? What is the maximum value? Justify your answer.


The graph of a continuous function f is shown in the figure above. The three regions between the graph of f and the x-axis are marked A, B, and C and have unsigned areas 5.5, 6, and 15.5 respectively. Let F(x) be an anti-derivative of f that is differentiable on (0, 4) and with F(1) = 9.

(a) Find F(0) and F(4).

(b)What is the minimum number of times F equals 5 on the interval [0, 4]? Show the work that leads to your conclusion.

(c). Find all intervals where F is increasing. Justify your answer.


The velocity, v(t), of a high speed rail train is positive over 0 < t < 60 seconds. The velocity of the train is recorded for selected values of t, as shown in the table below.

t

(seconds)

0 10 20 30 40 50 60

v(t)

(feet per second)

0 45 105 140 165 195 210

(a) Estimate the acceleration of the train at t = 25. Indicate units of measure.

(b) Use a left Riemann sum with three subintervals of equal length to approximate50

20( )v t dt .

Using correct units, explain the meaning of 50

20( )v t dt in the context of this problem.

(c) Evaluate50

20( )v t dt . Using correct units, explain the meaning of

50

20( )v t dt in the context of

this problem.

(d) Estimate the average velocity of the high speed train over the 60 second period of time using a midpoint Riemann sum with 3 subintervals.


Let g be the function defined by 2 4 for 1 2( )2 4 for 2 4

x x xg xx < x

(a) Write the equation for the line tangent to the graph of g at x = 3.

(b) Is g continuous at x = 2? Use the definition of continuity to explain your answer.

(c) Find the average value of f (x) from x = 1 to x = 4.


The graph of a piecewise linear function f is shown above. If g is the function defined by

4( ) ( )

xg x f t dt , then g(–1) equals

(a) 4

(b) – 4

(c) 0

(d) 6

(e) – 6


The velocity of a particle is given by v(t) = 4t3 – 4t for the times 0 < t < 2 in seconds. Find the average speed of the particle over that interval.

(a) 5

(b) – 4

(c) 4

(d) 10

(e) 24


Let ( )f x be a continuous function such that (1) 2f and 3( ) 6f x x . What is the value of

(5)?f

(a) 11.446

(b) 13.446

(c) 15.446

(d) 24.672

(e) 26.672

Chapter 4 (Solutions)

Question 1

For 0 < t < 9, a particle moves along the x-axis. The velocity of the particle is given by

( ) sin4

v t t

. The particle is at position x = –4 when t = 0.

(a) For 0 < t < 9, when is the particle moving to the right?

(b) Write but do not evaluate, an integral expression that gives the total distance traveled by the particle from time t = 0 to t = 9.

(c) Find the acceleration of the particle at time t = 3. Is the particle speeding up, slowing down or neither at t = 3? Justify your response.

(d) Find the position of the particle at time t = 3.

(a) ( ) 0v t for 0 4t and for 8 9t , so the

particle is moving right for those intervals.

(b) 9 9

0 0( ) sin

4v t dt t dt

(c) 23

2v

( ) ( ) cos4 4

3 2(3) (3) cos

4 4 8

a t v t t

a v

Since v(3) and a(3) have opposite signs, the particle is slowing down at t = 3.

1: answer

1: limits and integral

1: (3)3 : 1: (3)

1 : answer

va

Question 1 (cont.)

(d)

33

00

4(3) (0) ( ) 4 cos

4

4 2 2(3) 4

x x v t dt t

x

1: limits and integrand1: initial condition4 : 1 : integrates correctly1: answer

Question 2

Let 3

( ) ( )x

F x f t dt . The graph of f on the interval [–3,4] consists of 2 line segments and a

semicircle, as shown in the graph above.

(a) Find F(0), F (0), and F(4).

(b) Find all relative minimum values of F(x) on the interval [–3, 4]. Justify your answer.

(c) Find the x-coordinates of the inflection points of F(x) on the interval [–3, 4]. Justify your answer.

(d) Write the equation of the line tangent to the point where x = 2.

(a) 0 3

3 0

4

3

1 5(0) ( ) ( ) 2

2 2

(0) (0) 3

1(4) ( )

2

F f t dt f t dt

F f

F f t dt

1: (0)3 : 1 : (0)

1: (4)

FfF

Question 2 (cont.)

(b)

The only critical number of F(x) on (–3, 4) is x = 3.

However, since F = f goes from positive to negative at x = 3, the critical number x = 3 corresponds to a relative maximum of F. F(x) has no relative minima on (–3, 4).

(c) Two conditions must be met for x = c to correspond to an inflection point of f : i). there must

be a change in concavity at x = c (f must change

sign), and ii). the tangent line at f(c) must exist (f must be continuous at x = c).

Since f changes from increasing to decreasing at x = 0, and since f is defined at x = 0, then an inflection point

occurs at 5

0,2

.

(d)

2

3

(2) (2) 1

1(2) ( )

2

F f

F f t dt

1

1( 2)2

y x

1: identifies 32 : 1 : justificationx =

1: identifies 02 : 1 : justificationx =

1: (2)2 : 1 : F (2) and tangent

line equation

F

Question 3

t

(minutes)

0 3 5 7 8 12

C(t)

(degrees Celsius)

65 57 50 46 44 40

As a pot of coffee cools down, the temperature of the coffee is modeled by a differentiable function C, for 0 < t < 12 where time is measured in minutes and the temperature C(t) is measured in degrees Celsius. Selected values of time t are shown in the table above.

(a) Evaluate 12

0( )C t dt . Using correct units, explain the meaning of your answer in the context

of the problem.

(b) Using correct units, explain the meaning of 12

0

1( )

12C t dt in the context of the problem. Use a

trapezoidal sum with 5 subintervals indicated by the table to approximate 12

0

1( )

12C t dt .

(c) Use the data in the table to approximate the rate at which the temperature is changing at time t = 4. Use proper units and show the computations that lead to your answer.

(d) For 12 < t < 15, the rate of cooling is modeled by ( ) 2cos 0.5C t t . Based on the model,

what is the temperature of the coffee at time t = 15? Assume C(t) is continuous at t = 12.

This problem has a global limits point.

(a) 12

0'( ) (12) (0)

40 65 25 C

C t dt C C

This means the temperature decreased by 25° Celsius over the first 12 minutes.

(b)

12

0

3 2 265 57 57 50 50 46

1 1 2 2 2( )1 412 12

46 44 44 402 2

C t dt

12

0

1( ) 49.9

12C t dt Celsius. This means that the average

temperature over the first 12 minutes is 49.9 Celsius.

(c) (5) (3) 50 57 7

5 3 5 3 2

C C

°C/min

(d)

15

12(15) (12) ( ) 40 4.870 35.130 CC C C t dt

1: units in a, b, c

1: answer and explanation

1: trapezoidal sum3: 1: answer

1 : explanation

1: answer

1: integrand3: 1 : uses (12)

1: answerC

Question 4

On a typical day, the snow on Pike Mountain melts at a rate modeled by the function

( )M t , given by ( ) sin6 12

tM t

A snow maker adds snow at a rate modeled by the function S, given by 2( ) 0.006 0.12 0.87S t t t

Both M and S have units in inches per hour and t is measured in hours for 0 < t < 6.

At time t = 0, the mountain has 40 inches of snow.

(a) How much snow will melt during the 6 hour period? Indicate units of measure.

(b) Write an expression for I(t), the total number of inches of snow at any time t.

(c) Find the rate of change of the total amount of snow at time t = 3.

(d) For 0 < t < 6 , at what time t is the amount of snow a maximum? What is the maximum value? Justify your answer.

(a) 6

0( ) 2"M t dt

1: integral2 : 1 : answer

Question 4 (cont.)

(b) 0

40 t

I t S t M t dt

(c) 3 3 0.194 in/hrS M

(d)

On the interval (0, 6), I(t) has one critical number at t = 4.241.

I(0) = 40 in.

I(4.241) = 41.65 in.

I(6) = 41.492 in.

The maximum snow depth is 41.65 in. at t = 4.241 hours.

1: integral2 : 1 : uses inital condition

1: answer2 : 1 : units

1: identifies critical number1: correct values of at 3 : critical point, end points1 : correctly identifies maximum value

I t

Question 5

The graph of a continuous function f is shown in the figure above. The three regions between the graph of f and the x-axis are marked A, B, and C and have unsigned areas 5.5, 6, and 15.5 respectively. Let F(x) be an anti-derivative of f that is differentiable on (0, 4) and with F(1) = 9.

(a) Find F(0) and F(4).

(b)What is the minimum number of times F equals 5 on the interval [0, 4]? Show the work that leads to your conclusion.

(c). Find all intervals where F is increasing. Justify your answer.

(a)

1

0

1

0

1 0

0 1

9 5.5

14.5

F F f x dx

F F f x dx

2 : evaluates each definite integral 3 : 1 : uses initial condition for both cases

Question 5 (cont.)

4

14 1

9 6 15.5

18.5

F F f x dx

(b) We know (0) 14.5F , 1 9F , and

(4) 18.5F . We calculate 3

1(3) (1) ( ) 9 ( 6) 3F F f x dx . Since

1 5 (3)F F , ( ) 5F c for some c in

1,3 . Since 3 5 (4)F F , ( ) 5F c for

some c in 3, 4 .Thus, the function F equals

5 at least two times on the interval 0, 4

because of the Intermediate Value Theorem.

(c) F is increasing on the interval 3, 4 ,

since 0f on (3,4).

1: (3) 34 : 1 : conclusion

2: analysis

F

1: answer2 : 1 : justification

Question 6

The velocity, v(t), of a high-speed-rail train is positive over 0 < t < 60 seconds. The velocity of the train is recorded for selected values of t, as shown in the table below.

t

(seconds)

0 10 20 30 40 50 60

v(t)

(feet per second)

0 45 105 140 165 195 210

(a) Estimate the acceleration of the train at t = 25. Indicate units of measure.

(b) Use a left Riemann sum with three subintervals of equal length to approximate50

20( )v t dt .

Using correct units, explain the meaning of 50

20( )v t dt in the context of this problem.

(c) Evaluate50

20( )v t dt . Using correct units, explain the meaning of

50

20( )v t dt in the context of

this problem.

(d) Estimate the average velocity of the high-speed train over the 60 second period of time using a midpoint Riemann sum with 3 subintervals.

(a)

2

(30) (20)(25) (25)

30 20140 105

(25) 3.5 ft / sec30 20

v va v

a

1: difference quotient with answer2 : 1 : units

Question 6 (cont.)

(b) 50

20( ) 10(105 140 165) 4100 ftv t dt

This represents the total distance, in feet, traveled by the train from time 20t seconds to time 50t seconds.

(c) 50

20( ) (50) (20) 90v t dt v v . This means

that the velocity changed by 90 ft/sec from 20t seconds to time 50t seconds.

(d)

60

0

1 1( ) 20 (45 140 195)

60 60126.7 ft / sec

v t dt

1: left Riemann sum3: 1: interprets as total distance

1 : units

1: Fundamental Theorem of Calculus2 : with answer

1 : explanation

1: expression for average velocity2 : 1 : midpoint Riemann sum

Question 7

Let g be the function defined by 2 4 for 1 2( )2 4 for 2 4

x x xg xx < x

(a) Write the equation for the line tangent to the graph of g at x = 3.

(b) Is g continuous at x = 2? Use the definition of continuity to explain your answer.

(c) Find the average value of f (x) from x = 1 to x = 4.

(a)

12 4

2 41

(3)2

3 2

12 ( 3)

2

dx

dx x

g

g

y x

(b)

2

2

2

lim ( ) 2 8 4,

lim ( ) 4 4 0

x

x

g x

g x

Since the left and right limits differ, g(x) is discontinuous at x = 2.

1: (3)3 : 1: (3)

1 : equation of tangent

gg

1: answer2 : 1 :explantion involving limits

Question 7 (cont.)

(c)

4 2 4

1 1 2

33 2 2 42

1 2

1 1( ) ( ) ( )

4 1 3

1 1 12 ] (2 4) ]

3 3 3

1 11 8 19

3 3 3 9

g x dx g x dx g x dx

x x x

1: antiderivative of polynomial

4 : 2: antiderivative of 2 41 : average value form and answer

x

Questions 8-10

8. a 4

1( 1) ( ) 5 1 4g f t dt

9. a

2 3

04 4 10

52 0 2

t t dt

10. e

5

1

5 3

1

5 1

5 2 6

5 2 24.672 26.672

f f f x dx

f x dx

f

Chapter 4 final - MR. HANSON'S MATH...

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