AP Calculus 4.5 Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. Consider the function y = sin x. a) Find the equation of the tangent line when x = 0. b) Graph both equations on your calculator in a standard viewing window. Is the tangent line a good approximation for the curve? Zoom in (at the origin) a couple of times. What do you notice? **The first of the two major ideas in this section is LINEARIZATION. 2. When you see the phrase “use linearization” you are simply writing the equation of the _______________________. To do this, you need a ______________, and a _______________. The only difference between this section and previous sections is that instead of being given a point, you are going to have to pick a point that is “close” to the x-value in the original problem.

3. Suppose you wanted to use linearization to estimate 3 31 , and then determine the error in that approximation. a) First, we need an equation (with x and y). What is that equation? b) Second, you need to pick a point on the graph of the equation you wrote in part a that is “close” to the point that you are trying to estimate. What is that point? c) Write the equation of the tangent line to the equation from part a using the point from part b. Using the point you chose in part b, find the equation of the tangent line at that point. If you’d like a visual … graph both the equation of the tangent line and the equation you wrote in part a on the same screen to see how “close” the tangent line is to approximating the curve.

d) … back to the original question … What is the approximate value of 3 31 ? What is the error in that approximation?

e) How would this entire process change if you were asked to approximate 3 37 ?

4. Use linearization to approximate f (0.1) if 1

4f x

x

. Find the error for your approximation.

5. The approximate value of 4 siny x= + at x = 0.12, obtained from the tangent to the graph at x = 0, is

A 2.00

B 2.03 C 2.06 D 2.12

E 2.24

6. [With calculator] Let f be the function given by ( ) 2 2 3f x x x= - + . The tangent line to the graph of f at x = 2 is used

to approximate the values of f (x). Which of the following is the greatest value for which the error resulting from this tangent line approximation is less than 0.5?

A 2.4 B 2.5 C 2.6 D 2.7 E 2.8

The second of the two major concepts in this section is differentials. Taking a differential is no different than taking a derivative with different notation. In the case of dy/dx, we can separate dy and dx so that dy = ___________________________ and dx = __________________________________ 7. Find the differential dy when dx = –0.2 and x = 1, if 2 3xy x e= . Explain what you’ve found.

8. If 2sin 3y x , find dy if 3x and 110dx .

9. Suppose you wanted to use differentials to approximate 4 19 . … this process is similar to the “linearization” process a) Write an equation involving x and y. b) Take a differential of the equation in part a. You should have a dy, dx, and an x in your answer. c) Pick a point “close” to the point you are trying to estimate. Of the 3 variables in part b, which two do you now know what they are? … plug these in an solve for the missing variable.

d) Explain what you’ve found and how to use it to find the approximate value of 4 19 .

10. Use differentials to estimate 86 . 11. The radius of a ball bearing is measured to be 0.7 inch. If the measurement is correct to within 0.01 inch, estimate the

error in the volume of the ball bearing. [ 343

V rp= ]

12. [Calculator Required] The range R of a projectile is ( ) ( )2

0 sin 232

vR q q= ⋅ , where 2

0v is the initial velocity in feet per

second and q is the angle of elevation. Let 0 2200v = feet per second and let q change from 10 to 11. [Remember … in calculus, we never use degrees, so change the degrees into radians!]

a) Find the actual change in the range. b) Use differentials to approximate the change in the range. Are your answers “close” ?

The actual change in the amount is the difference between the actual values, whereas the approximate change is found using the differential. “Error” and “Change” will be used interchangeably in the context of the next few problems. Percent change is found by dividing the amount of change by the original amount, therefore, approximate percent change can found by dividing the differential by the original. In other words, if you want the percent error in the calculation of A,

you simply calculate dA

A.

13. The circumference of the equator of a sphere is measured as 10 cm with a possible error of 0.4 cm. This measurement is then used to calculate the radius. The radius is then used to calculate the surface area and volume of the sphere. Estimate the percentage errors in the calculated values of the following: a) percent error in the measurement of the radius

b) percent error in the calculation of the surface area [ 24S rp= ]

c) percent error in the calculation of the the volume [ 343

V rp= ]

14. How accurately should you measure the radius of a sphere in order to be reasonably sure the volume of the sphere is

within 2% of its actual value? [Volume of a sphere: 343

V rp= ]

15. A right circular cone has a radius that is one-third of the height. How accurately must the radius be measured so that

error in calculating the volume is no more than 3% ? [Volume of a cone: 213

V r hp= ]

AP Calculus 4.6 Worksheet All work must OK … I couldI’d just throw since I found ibeen working

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7. A spherical container is deflated such that its radius decreases at a constant rate of 10 cm/min. At what rate must air be removed when the radius is 5 cm? [The volume of a sphere is 34

3V rp= ]

8. A pebble is dropped into a still pool and sends out a circular ripple whose radius increases at a constant rate of 4 ft/s. How fast is the area of the region enclosed by the ripple increasing at the end of 8 s? 9. Liquid is pouring through a cone shaped filter at a rate of 3 in3/min. Assume that the height of the cone is 12 inches and the radius of the base of the cone is 3 inches. How rapidly is the depth of the liquid in the filter decreasing when the level is 6 inches deep? 10. Sand pours out of a chute into a conical pile whose height is always one half its diameter. If the height increases at a constant rate of 4 ft/min, at what rate is sand pouring from the chute when the pile is 15 ft high?

11. The radius r, height h, and volume V of a right circular cylinder are related by the equation 2V r hp= .

a) How is dV

dt related to

dh

dt if r is constant?

b) How is dV

dt related to

dr

dt if h is constant?

c) How is dV

dt related to

dr

dt and

dh

dt if neither r nor h is constant?

12. Complete #42 on page 254 of your textbook.

AP Calculus 5.1 Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. Suppose an oil pump is producing 800 gallons per hour for the first 5 hours of operation. For the next 4 hours, the pumps production is increased to 900 gallons per hour, and then for the next 3 hours, the production is cut to 600 gallons per hour. a) Make a graph modeling this situation. b) The term “area under a graph” is the area between the graph and the horizontal axis. Find the area under the graph from 0 to 5 hours. What does this value represent? c) Find the total area under the graph for the entire 12 hours. What does this value represent? 2. The function f is continuous on the closed interval [2, 8] and has values that are given in the table below.

x 2 5 7 8 f (x) 10 30 40 20

Using the subintervals [2, 5], [5, 7], and [7, 8], what are the following approximations of the area under the curve? A) LRAM B) RRAM C) Trapezoid Approximation D) Write an algebraic expression (you don’t have enough information to simplify it) that would give an approximation of the area under the curve using MRAM.

hours

Gallons/hour

3. Let R be the region enclosed between the graphs of 22y x x= - and the x-axis for 0 < x < 2.

a) Sketch the region R. b) Partition [0, 2] into 4 subintervals and find the following: i) LRAM ii) RRAM iii) MRAM iv) Trapezoidal Approximation 3. Oil is leaking out of a tanker damaged at sea. The damage to the tanker is worsening as evidenced by the increased leakage each hour, recorded in the table. a) Find an estimate using a Midpoint Sum for the total quantity of oil that has escaped in the first 8 hours using 4 intervals of equal width. b) Without calculating them, will LRAM or RRAM yield a “higher” estimate in this case? Why?

Time (h) Leakage (gal/h) 0 50 1 70 2 97 3 136 4 190 5 265 6 369 7 516 8 720

4. Sylvie’s Old World Cheeses has found that the cost, in dollars per kilogram, of the cheese it produces is

( ) 0.012 6.50c x x=- + ,

where x is the number of kilograms of cheese produced and 0 < x < 300. a) Draw a sketch of the cost function. Label each axes with the correct units. b) If you wanted to find the total cost of producing 200 kg of cheese how is this represented on your sketch? c) Find the total cost of producing 200 kg of cheese.

5. A particle is moving along the x-axis with velocity given by ( ) 2 1v t t= + , where velocity is measured in feet/sec.

a) Draw a sketch of the velocity function for 0 < t < 5. b) What does the area under the graph of velocity represent? c) If the object originally began at x = 3, where is the object located at t = 2? … What about t = 5?

6. Suppose that in a memory experiment, the rate of memorizing is given by ( ) 20.009 0.2M t t t=- + , where ( )M t is the

memory rate, in words per minute. The graph is shown below. a) Explain what the shaded area represents in the context of this problem. b) Use MRAM with 5 rectangles of equal width to approximate the area of the shaded region. 7. Complete each sentence with ALWAYS, SOMETIMES, or NEVER. a) If a function is concave up, then LRAM will ____________ overestimate the actual area under the curve. b) If a function is decreasing, then RRAM will _____________ overestimate the actual area under the curve. 8. Which rule is simply the average of LRAM and RRAM? 9. If f is a positive, continuous function on an interval [a, b], which of the following rectangular approximation methods has a limit equal to the actual area under the curve from a to b as the number of rectangles approaches infinity?

I. LRAM II. RRAM

III. MRAM A I and II only B III only C I and III only D I, II, and III E None of these 10. A truck moves with positive velocity v (t) from time t = 3 to time t = 15. The area under the graph of v (t) between t = 3 and t = 15 gives A the velocity of the truck at t = 15 B the acceleration of the truck at t = 15 C the position of the truck at t = 15 D the distance traveled by the truck from t = 3 to t = 15 E The average position of ht e truck in the interval t = 3 and t = 15. Want more practice? … Complete the following problems from the textbook:

pages 270 – 273: #3, 4, 11 (use 4 rectangles and any RAM), 17, 23, 30, 35

Time (in minutes)

Mem

ory

Rat

e (i

n w

ords

per

min

ute)

10

AP Calculus 5.2 Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit. An Activity: Instead of using LRAM and RRAM, let’s introduce a “lower” and “upper” estimate to use for this example. A lower estimate, uses the lowest y-value in an interval regardless of whether this point is on the left side or the right side. Similarly, an upper estimate uses the highest y-value in an interval. A car is traveling so that its speed is never decreasing during a 10 – second interval. The speed at various moments in time is listed in the table below.

Time (sec) 0 2 4 6 8 10 Speed (ft/sec) 30 36 40 48 54 60

a) Explain why the best lower estimate for the distance traveled in the first 2 seconds is 60 feet. b) Explain why the best upper estimate for the distance traveled in the first 2 seconds is 72 feet. c) Find the best lower estimate for the distance traveled in the first 10 seconds. : An answer of 300 feet (which ignores some of the data) is not correct. d) Find the best upper estimate for the distance traveled in the first 10 seconds. : An answer of 600 feet (which ignores some of the data) is not correct. … These sums of products that you have found in c and d are called __________________. e) If you choose the lower estimate for your approximation of how far the car travels, what is the maximum amount your approximation could differ from the exact distance? … In other words, how far apart are your estimates? f) Choose speeds to correspond with t = 1, 3, 5, 7, and 9 seconds. Keep the nondecreasing nature of the above table and do not select the average of the consecutive speeds. Find new best upper and lower estimates for the distance traveled for these 10 seconds. … how far apart are your estimates?

Time (sec) 0 1 2 3 4 5 6 7 8 9 10 Speed (ft/sec) 30 36 40 48 54 60

New Upper Estimate: _______ New Lower Estimate: _________ g) Compare how far apart your upper and lower estimates are with at least two other groups/individuals who didn’t use the same numbers as you did. What do you notice? h) Make a prediction … How far apart do you think the estimates would be if we extended the last table to include speeds that correspond with t = 0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7,5, 8.5, and 9.5 seconds? i) If we continue to introduce more entries into our charts, what happens to the upper and lower estimates? h) Write an expression giving the ACTUAL distance this car traveled in 10 seconds, if it's velocity was v (t).

1. Graphically speaking, if ( )f x is always above the x-axis, what does ( )b

a

f x dxò mean?

2. Given the graph of f (x) below, answer the following questions:

a) Is ( )b

a

f x dxò positive, negative, or zero? Why?

b) Is ( )c

b

f x dxò positive, negative, or zero? Why?

c) Is ( )c

a

f x dxò positive, negative, or zero? Why?

3. Complete the following questions from you textbook: page 283 #47 – 51, 54, 55, and 56 4. Complete the following questions from your textbook by graphing the function and evaluating the integral using geometry: Pages 282 – 283 #8, 14, 16, 19, 20, 25, and 26 5. Complete the following questions from your textbook using your calculator.: Page 283 #33 – 36 6. Complete the following questions from your textbook: page 283 #29 – 32 (write an integral for each situation) and 43 and 46

a b c

AP Calculus 5.3 Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The graph of f shown below consists of line segments and a semicircle. Evaluate each definite integral.

a) ( )2

0

f x dxò b) ( )6

2

f x dxò

c) ( )2

4

f x dx-ò d) ( )

6

4

f x dx-ò

e) ( )6

4

f x dx-ò f) ( )

6

4

2f x dx-

é ù+ë ûò

2. Part e above, gives a way to find the total Area between the x – axis and the function between x = –4 and x = 6. Without using absolute value signs, write two different expressions that can be used to find the total area between the x – axis and the function between x = –4 and x = 6. 3. Complete the following questions from your textbook: pages 290 – 292 #1, 4, 5, 47, and 48.

4. [Calculator Required … for now] What is the average value of 2 3 1y x x= + on [0, 2] ?

5. Complete the following questions from your textbook: pages 291 – 292 #11 – 16, 40, 41, 49, and 50 6. Complete the following questions from your textbook: page 293 #1 – 4.

x

y

7. [Calculator Required] … Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function F defined by

( ) 82 4sin2

tF t

æ ö÷ç= + ÷ç ÷çè ø for 0 30t£ £ ,

where F (t) is measured in cars per minute and t is measured in minutes. a) Is traffic flow increasing or decreasing at t = 7 ? Give a reason for your answer. b) What is the average value of the traffic flow over the time interval 10 15t£ £ ? Indicate units of measure.

c) What is the average rate of change of the traffic flow over the time interval 10 15t£ £ ? Indicate units of measure 8. [Calculator Required] At different altitudes in Earth’s atmosphere, sound travels at different speeds. The speed of sound s (x) (in meters per second) can be modeled by

( ) 34

32

32

4 341 if 0 11.5

295 if 11.5 22

278.5 if 22 32

254.5 if 32 50

404.5 if 50 80

x x

x

x xs x

x x

x x

ì - + £ <ïïïï £ <ïïï + £ <=íïï + £ <ïïïï- + £ £ïî

where x is measured in kilometers. What is the average speed of sound over the interval [0. 80]?

9. A blood vessel is 360 millimeters (mm) long with circular cross sections of varying diameter. The table below gives the measurements of the diameter of the blook vessel at selected points along the length of the blood vessel, where x represents the distance from one end of the blood vessel, and B (x) is a twice differentiable function that represents the diameter at that point.

Distance x (mm)

0 60 120 180 240 300 360

Diameter B (x) (mm)

24 30 28 30 26 24 26

a) Write an integral expression in terms of B (x) that represents the average radius, in mm, of the blood vessel between x = 0 and x = 360. b) Approximate the value of your answer from part a using the data from the table and a midpoint Riemann Sum with three subintervals of equal length. Show the computations that lead to your answer.

c) Using correct units, explain the meaning of ( )

2275

1252

B xdxp

æ ö÷ç ÷ç ÷ç ÷÷çè øò in terms of the blood vessel.

AP Calculus 5.4 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. For questions 1 – 10, use the Fundamental Theorem of Calculus (Evaluation Part) to evaluate each definite integral. Use your memory of derivative rules and/or the chart from your notes. You should start making a list of all the rules on ONE page!

1. ( )4

3 2

2

5x x dx-

+ò 2. 5

3

dx

xò

3.

32

12

2

1

1dx

x-ò 4.

3

2

1

1

1dx

x-

+ò

5. 2

0

5x dxò 6. 12

5

7x dx-ò

7. 5

2

6 dx-ò 8. ( )

2

sin 5x dxp

p

ò

9. ( )4

2

0

sec x dx

p

ò 10. 3

2

1

xe dx-ò

For questions 11 and 12, setup and evaluate an expression involving definite integrals in order to find the total AREA of the region between the curve and the x-axis. [No Calculator!]

11. 23 3y x= - on the interval –2 < x < 2 12. y x= on the interval 0 < x < 9

13. What are all the values of k for which 2

2

0k

x dx =ò ?

A –2 B 0 C 2 D –2 and 2 E –2, 0, and 2 For questions 14 and 15, find the average value of the function on the specified interval without a calculator. 14. ( ) 29 3g x x= - on the interval [0, 4] 15. ( ) ( ) ( )csc coth x x x= on the interval [ ]2 4,p p

16. Including start-up costs, it costs a printer $50 to print 25 copies of a newsletter, after which the

marginal cost (in dollars per copy) at x copies is given by ( ) 2'C x

x= . Find the total cost of printing

2500 newsletters.

17. Suppose ( ) 2 3 1f x x x= - + . Find K so that ( ) ( )x x

a b

f t dt K f t dt+ =ò ò if a = –1 and b = 2.

18. If you know ( )9

7

' 15f x dx-

=ò , and you know ( )7 4f - = , what does ( )9f = ?

19. The graph of ( )'h x is given below. If h (–2) = 6, what does h (3) = ?

Need more practice? … try the following from your textbook: FTOC (Evaluation part) Page 291 #19 – 30 Page 303 #27 – 40 Total Area (no calculator) Page 303 #45 – 48 Average Value (by hand) Page 291 #31 – 36

x

( )'h x

AP Calculus 5.4 Worksheet Day 2

1. If a is a constant and ( ) ( )x

a

g x w t dt=ò , what is ( )'g x ?

2. If a is a constant and ( ) ( )a

x

g x w t dt=ò , what is ( )'g x ?

3. Find 5

3

1x

tde dt

dx-

é ùê ú+ê úê úë ûò .

4. [Optional] Want more practice with the “simple” version of the FTOC? … Complete the following questions from the book: p302 #1 – 8, 13, and 14.

5. If q (x) and p (x) are differential functions of x and ( ) ( )( )

( )p x

q x

g x w t dt= ò , what is ( )'g x ?

6. Find sin

3

1

1x

dt dt

dx

é ùê ú+ê úê úë ûò .

7. Find ( )3

sin

x

x

df t dt

dx

é ùê úê úê úë ûò

8. Find 3

2

sin

xt

x

de dt

dx

é ùê úê úê úë ûò

9. [Optional] Want more practice with the “extended” version of the FTOC? … Complete the following questions from the book: p302 #9 – 12, 15 – 20.

All the questions this page come from your book … They begin that process of “putting it all together” that the last video introduced. Don’t SKIP these just because they are in your book! 10. Complete #57 on page 303 of your textbook. 11. Complete #58 and 59 on page 304 of your textbook. 12. Complete #60 and 61 on page 304 of your textbook.

13. The graph of a differentiable function f on the interval [–2, 10] is shown in the figure below.

The graph of f has a horizontal tangent line at x = 4. Let ( ) ( )4

9x

h x f t dt= +ò for –2 < x < 10.

a) Find h (4), ( )' 4h , and ( )'' 4h

b) On what intervals is h increasing? Justify your answer. c) On what intervals is h concave downward? Justify your answer.

d) Find the Trapezoidal Sum to approximate ( )10

2

f x dx-ò using 6 subintervals of length = 2.

14. The function g is define and differentiable on the closed interval [–7, 5] and satisfies g (0) = 5. The graph of ( )'g x , the derivative of g, consists of

a semicircle and three line segments as shown in the figure. a) Write an expression for g (x). b) Use your expression to find g (3) and g (–2). c) Find the x-coordinate of each point of inflection of the graph of g (x) on the interval (–7, 5). Explain your reasoning.

–2 2 4 6 8 10

–1

2

1

3

Graph of f

Graph of g¢ (x)

15. The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function r of time t, where t is measured in minutes. For 0 < t < 12, the graph of r is concave down. The table below gives selected values of the rate of change, ( )'r t , of the radius of the balloon over the time interval 0 < t < 12. The radius of the balloon is

30 feet when t = 5. (The Volume of a sphere is given by 34

3V rp= )

t

(minutes) 0 2 5 7 11 12

( )'r t

(feet per minute)

5.7 4.0 2.0 1.2 0.6 0.5

a) Estimate the radius of the balloon when t = 5.4 using the tangent line approximation at t = 5. Is your estimate greater than or less than the true value? Give a reason for your answer. b) Find the rate of change of the volume of the balloon with respect to time when t = 5. Indicate units of measure. c) Use a right Riemann Sum with the five subintervals indicated by the data in the table to

approximate ( )12

0

'r t dtò . Using correct units, explain the meaning of ( )12

0

'r t dtò in terms of the radius of

the balloon.

d) Is your approximation in part c greater than or less than ( )12

0

'r t dtò ? Give a reason for your

answer.