Unit 5 Packet - Analytical Apps of Differentiation PDF

Preview

Summary

Analytical Apps of Differentiation...

Description

Calculus Write your questions and thoughts here!

Notes

5.1 The Mean Value Theorem We use the MVT to justify conclusions about a function over an interval.

Mean Value Theorem:

If a function 𝑓 is continuous over the interval and differentiable over the interval , then there exists a point 𝒄 within that open interval where the instantaneous rate of change equals the average rate of change over the interval.

1. Use the function 𝑓(𝑥) = −𝑥  + 3𝑥 + 10 to answer the following. a. On the interval [2, 6], what is the average rate of change?

b. On the interval (2, 6), when does the instantaneous rate of change equal the average rate of change?

MVT vs IVT 

Mean Value Theorem MVT The derivative (instantaneous rate of change) must equal the average rate of change somewhere in the interval.



Intermediate Value Theorem IVT On a given interval, you will have a 𝑦value at each of the end points of the interval. Every 𝑦-value exists between these two 𝑦-values at least once in the interval.

2. 𝑡 minutes ℎ(𝑡) feet 𝑣(𝑡) feet per minute

0

5

15

20

30

0

40

70

65

80

0

10

3

2

4

A hot air balloon is launched into the air with a human pilot. The twice-differentiable function ℎ models the balloon’s height, measured in feet, at time 𝑡, measured in minutes. The table above gives values of the ℎ(𝑡) and the vertical velocity 𝑣(𝑡) of the balloon at selected times 𝑡. a. For 5 ≤ 𝑡 ≤ 20, must there be a time 𝑡 when the balloon is 50 feet in the air? Justify your answer.

b. For 20 ≤ 𝑡 ≤ 30, must there be a time 𝑡 when the balloon’s velocity is 1.5 feet per second? Justify your answer.

5.1 The Mean Value Theorem

Practice

Calculus

1. Skater Sully is riding a skateboard back and forth on a street that runs north/south. The twice-differentiable function 𝑆 models Sully’s position on the street, measured by how many meters north he is from his starting point, at time 𝑡, measured in seconds from the start of is ride. The table below gives values of the 𝑆(𝑡) and Sully’s velocity 𝑣(𝑡) at selected times 𝑡. 𝑡 seconds 𝑆 (𝑡) meters 𝑣(𝑡) meters per second

0

20

30

60

0

−5

7

40

0

3.2

0.8

−0.9

a. For 0 ≤ 𝑡 ≤ 20, must there be a time 𝑡 when Sully is 2 meters south of his starting point? Justify your answer.

b. For 30 ≤ 𝑡 ≤ 60, must there be a time 𝑡 when Sully’s velocity is 1.1 meters per second? Justify your answer.

2. A particle is moving along the 𝑥-axis. The twice-differentiable function 𝑠 models the particles distance from the origin, measured in centimeters, at time 𝑡, measured in seconds. The table below gives values of the 𝑠(𝑡) and the velocity 𝑣(𝑡) of the particle at selected times 𝑡. 𝑡 3 10 20 25 Seconds 𝑠 (𝑡) 5 −2 −10 8 cm 𝑣(𝑡) −4 −2 3 −2 cm per second a. For 20 ≤ 𝑡 ≤ 25, must there be a time 𝑡 when the particle is at the origin? Justify your answer.

b. For 3 ≤ 𝑡 ≤ 10, must there be a time 𝑡 when the particle’s velocity is −1.5 cm per second? Justify your answer.

3. A hot air balloon is launched into the air with a human pilot. The twice-differentiable function ℎ models the balloon’s height, measured in feet, at time 𝑡, measured in minutes. The table below gives values of the ℎ(𝑡) and the vertical velocity 𝑣(𝑡) of the balloon at selected times 𝑡. 𝑡 0 6 10 40 minutes ℎ(𝑡) 0 46 35 105 feet 𝑣(𝑡) 0 6 20 5 feet per minute a. For 6 ≤ 𝑡 ≤ 10, must there be a time 𝑡 when the balloon is 50 feet in the air? Justify your answer.

b. For 10 ≤ 𝑡 ≤ 40, must there be a time 𝑡 when the balloon’s velocity is 3 feet per second? Justify your answer.

Using the Mean Value Theorem, find where the instantaneous rate of change is equivalent to the average rate of change. 5. 𝑦 = sin 3𝑥 on [0, 𝜋] 4. 𝑦 = 𝑥 − 5𝑥 + 2 on [−4, −2]

7. 𝑦 = 𝑒  on [ 0, ln 2]



6. 𝑦 = (−5𝑥 + 15)  on [1, 3]

Test Prep

5.1 The Mean Value Theorem

Calculator active problem 8. A particle moves along the 𝑥-axis so that its position at any time 𝑡 ≥ 0 is given by 𝑥(𝑡) = 𝑡  − 3𝑡 + 𝑡 + 1. For what values of 𝑡, 0 ≤ 𝑡 ≤ 2, is the particle’s instantaneous velocity the same as its average velocity on the closed interval [0, 2]?

No calculator on this problem. 9. The table below gives selected values of a function 𝑓. The function is twice differentiable with 𝑓 󰆒󰆒(𝑥) > 0. 𝑥 𝑓(𝑥 ) 3 12. 5 5 13. 9 7 16. 1 Which of the following could be the value of 𝑓 󰆒(5)?

(A)

0.5

(B)

0.7

(C)

0.9

(D) 1.1

(E) 1.3

10. Let 𝑔 be a continuous function. The graph of the piecewise-linear function 𝑔󰆒, the derivative of 𝑔, is shown above for −4 ≤ 𝑥 ≤ 4. Find the average rate of change of 𝑔󰆒 (𝑥) on the interval −4 ≤ 𝑥 ≤ 4. Does the Mean Value Theorem applied on the interval −4 ≤ 𝑥 ≤ 4 guarantee a value of 𝑐, for −4 < 𝑥 < 4, such that 𝑔󰆒󰆒(𝑐) is equal to this average rate of change? Why or why not?



y



(−4, 2)

(2, 2)

 









 

x 







(−1, 0)





Graph of 𝑔 󰆒

𝑥

𝑓 (𝑥)

𝑓 󰆒 ( 𝑥)

𝑔 (𝑥)

𝑔 󰆒(𝑥)

2

6

3

1

2

1 3 4

3 5

−2

8

−3 6

(4, −2)



Answer: No, because 𝒈′(𝒙) is not differentiable. It has several corners. The MVT only applies if the function is differentiable.

11.



2 6 3

4 3 5

The functions 𝑓 and 𝑔 are differentiable for all real numbers. The table above gives values of the functions and their first derivatives at selected values of 𝑥. The function ℎ is given by ℎ(𝑥) = 𝑓𝑔(𝑥) + 2. Must there be a value 𝑐 for 2 < 𝑐 < 4 such that ℎ󰆒 (𝑐) = 1.

12. Calculator active problem. Let 𝑓 be the function given by 𝑓(𝑥) = 2 sin 𝑥. As shown above, the graph 𝑓 crosses the origin at point 𝐴 and point 𝐵 at the  coordinate point 󰇡 , 2󰇢. Find the 𝑥-coordinate of the point on the graph of 𝑓, between points 𝐴 and 𝐵, at which the line tangent to the graph of 𝑓 is parallel to line 𝐴𝐵. Round or truncate to three decimals.

y

𝜋 𝑩 󰇡 , 2󰇢 2 𝑓(𝑥)

𝑨 (0, 0) x

13. A differentiable function 𝑔 has the property that 𝑔󰆒 (𝑥) > 2 for 1 ≤ 𝑥 ≤ 5 and 𝑔(4) = 3. Which of the following could be true? I. 𝑔(1) = −6 II. 𝑔(2) = 0 III. 𝑔(5) = 4 (A)

I only

(B)

II only

(C)

I and II only

(D)

I and III only

(E)

II and III only



14. Calculator active problem. Let 𝑓 be the function with 𝑓(1) = 𝑒 , 𝑓(4) = , and derivative given by 𝑓 󰆒 (𝑥) = (𝑥 − 1) sin(𝑒𝑥). How many values of 𝑥 in the open interval (1, 4) satisfy the conclusion of the Mean Value Theorem for the function 𝑓 on the closed interval [1, 4]?

(A) None (B) One (C) Two (D) More than two

Calculus

Notes

5.2 Critical Points

Write your questions and thoughts here!

Extreme Value Theorem: If a function 𝑓 is continuous over the interval [𝑎, 𝑏], then 𝑓 has at least one minimum value and at least one maximum value on [𝑎, 𝑏]. 𝒚

Global vs. Local Extrema or

Absolute vs. Relative Extrema 𝒙

Find all extreme values. Identify the type and where they occur. 1. 2.

Write your questions and thoughts here!

Critical Point: A point that has a possibility of being an extrema (max or min).

How do you find a critical point? 1. 𝑓 󰆒 (𝑥) does not exist 2. 𝑓 󰆒 (𝑥) = 0 Find all critical points  3. 𝑓 (𝑥 ) = 𝑥 − 9𝑥 + 24 

5.2 Critical Points Calculus

4. 𝑔 (𝑥) =



√ 

Practice

Find all extreme values. Identify the type and where they occur. For example, an answer could be written as “absolute max of 𝟑 at 𝒙 = 𝟏.” 1. 2. 3.

Find the critical points. 4. 𝑓(𝑥) = 4𝑥 − 9 𝑥 − 12𝑥 + 3

7. 𝑓 ( 𝑥) = ( ln𝑥 )

5. 𝑔( 𝑡 ) =

  

 6. ℎ( 𝑥) = √𝑥 − 2

9. 𝑔(𝑥) = 𝑒  − 𝑥



8. ℎ (𝑥 ) = 2 sin 󰇡  󰇢 where −2𝜋 ≤ 𝑥 ≤ 2𝜋

Test Prep

5.2 Critical Points

10. Calculator active problem. The first derivative of the function 𝑓 is given by 𝑓 󰆒 (𝑥) = How many critical values does 𝑓 have on the open interval (0, 10)? A) One

(B) Two

(C) Three

(D) Four

  

(E) Six

11. If 𝑓 is a continuous, decreasing function on [0,10] with a critical point at (4, 2), which of the following statements must be false? (A) 𝑓(10) is an absolute minimum of f on [0,10]. (B) 𝑓(4) is neither a relative maximum nor a relative minimum. (C) 𝑓′(4) does not exist (D) 𝑓′(4) = 0 (E) 𝑓 󰆒 (4) < 0



− . 

Calculus Write your questions and thoughts here!

5.3IncreasingandDecreasingIntervals

Notes

Find the critical points of the graph of 𝑓.

𝒇󰇛𝒙󰇜

When the slope of a function is positive, the function is increasing. When the slope of a function is negative, the function is decreasing.

𝒙 Sign of 𝒇 󰆒 󰇛𝒙 󰇜

󰇛∞, 3󰇜

3

󰇛3, 5󰇜

5

󰇛5, 6󰇜

6

󰇛6, ∞󰇜

Positive

0

Negative

0

Positive

DNE

Negative

1. Find the intervals on which the function 𝑓󰇛𝑥󰇜  𝑥   4𝑥  1 is increasing and decreasing and justify your answers. a. First find the critical points.

b. In between the 𝑥-values, the derivative must be positive or negative. c. We can use a chart to help keep track of the information. Write the critical points of the derivative first. 𝑥

󰇛∞, 2󰇜

2

󰇛2, ∞󰇜

Sign of 𝑓 󰆒 󰇛𝑥󰇜

Negative

0

Positive

d. Answer statements with justification: 𝒇 is decreasing on 󰇛∞, 𝟐󰇜 because 𝒇󰆒󰇛𝒙󰇜  𝟎. 𝒇 is increasing on 󰇛𝟐, ∞󰇜 because 𝒇󰆒󰇛𝒙󰇜  𝟎.

 



2. Find the intervals on which the function 𝑓󰇛𝑥󰇜   𝑥  𝑥  15𝑥  2 is increasing and decreasing and justify your answers.

𝑥

󰇛∞, 5󰇜

5

󰇛5, 3󰇜

3

󰇛3, ∞󰇜

Sign of 𝑓 󰆒󰇛𝑥 󰇜

Positive

0

Negative

0

Positive

Answer statements with justification: 𝒇 is increasing on 󰇛∞, 𝟓󰇜 and 󰇛𝟑, ∞󰇜 because 𝒇󰆒 󰇛𝒙󰇜  𝟎. 𝒇 is decreasing on 󰇛𝟓, 𝟑󰇜 because 𝒇󰆒 󰇛𝒙󰇜  𝟎.

Graph of 𝒇 󰆒 . Is 𝒇 increasing or decreasing? 3. Determine the intervals where 𝑓 is increasing and decreasing based on the graph of 𝑓 󰆒 .

𝒇 󰆒 󰇛𝒙󰇜

Increasing: 󰇛∞, 𝟑󰇜 and 󰇛𝟏, 𝟑󰇜 because 𝑓 󰆒  0. Decreasing: 󰇛𝟑, 𝟏󰇜 and 󰇛𝟑, ∞󰇜 because 𝑓 󰆒  0.

Application of rate of change If you want to know if something is increasing or decreasing, you look at the sign of its rate of change. The sign of a rate of change can tell you if the DEPENDENT variable is increasing or decreasing. Interpret the following: virus cases miles students mastery checks 0 0 0 0 year month  hour week The number of students is increasing.

The number of miles is decreasing.

The number of mastery checks is increasing.

The rate of virus cases per month is decreasing.

4. The rate of change of fruit flies in Mr. Kelly’s kitchen at time 𝑡 days is modeled by 𝑅󰇛𝑡󰇜  2𝑡 cos󰇛𝑡 󰇜 flies per day. Show that the number of flies is decreasing at time 𝑡  3. 

5.3IncreasingandDecreasingIntervals



Practice

Calculus

The following graphs show the derivative of 𝒇, 𝒇 󰆒 . Identify the intervals when 𝒇 is increasing and decreasing. Include a justification statement. 1. 2. y

𝒇󰆒

   







   

x 







𝒇 󰆒 󰇛𝒙󰇜



Increasing:

Increasing:

Decreasing:

Decreasing:

For each function, find the intervals where it is increasing and decreasing, and JUSTIFY your conclusion. Construct a sign chart to help you organize the information, but do not use a calculator. 3. 𝑓 󰇛𝑥󰇜  𝑥   12𝑥  1 4. 𝑔󰇛𝑥󰇜  𝑥  󰇛𝑥  3󰇜

5. 𝑓󰇛𝑥󰇜  𝑥  𝑒 

6. 𝑔󰇛𝑡󰇜  12󰇛1  cos 𝑡󰇜 on the interval 󰇛0, 2𝜋󰇜

The derivative 𝒇 󰆒 is given for each problem. Use a calculator to help you answer each question about 𝒇.    8. 𝑓 󰆒 󰇛𝑥󰇜   sin 𝑥  𝑥 cos 𝑥 for 9. 𝑓󰆒 󰇛𝑥󰇜   𝑒  sin 𝑥 for 0  7. 𝑓 󰆒 󰇛𝑥󰇜   . . On what  0  𝑥  𝜋. On which interval(s) 𝑥  4. On what intervals is 𝑓 intervals is 𝑓 increasing? is 𝑓 decreasing? decreasing?

For #10-12, calculator use is encouraged.



10. The rate of money brought in by a particular mutual fund is represented by 𝑚󰇛𝑡󰇜  󰇡󰇢 thousand dollars per

year where 𝑡 is measured in years. Is the amount of money from this mutual fund increasing or decreasing at time 𝑡  5 years? Justify your answer.

11. The number of hair follicles on Mr. Sullivan’s scalp is measured by the function ℎ󰇛𝑡󰇜  500𝑒  where 𝑡 is measured in years. Is the amount of hair increasing or decreasing at 𝑡  7 years? Justify your answer.



12. The rate at which rainwater flows into a street gutter is modeled by the function 𝐺󰇛𝑡󰇜  10 sin 󰇡󰇢 cubic

feet per hour where 𝑡 is measured in hours and 0  𝑡  8. The gutter’s drainage system allows water to flow out of the gutter at a rate modeled by 𝐷󰇛𝑡󰇜  0.02𝑥   0.05𝑥   0.87𝑥 for 0  𝑡  8. Is the amount of water in the gutter increasing or decreasing at time 𝑡  4 hours? Give a reason for your answer.

Test Prep

5.3IncreasingandDecreasingIntervals 13.

𝑥 𝑓󰇛𝑥󰇜

1 6

2 1

3 3

4 6

5 8

The table above gives values of a function 𝑓 at selected values of 𝑥. If 𝑓 is twice-differentiable on the interval 1  𝑥  5, which of the following statements could be true? (A)

𝑓 󰆒 is negative and decreasing for 1  𝑥  5.

(B)

𝑓 󰆒 is negative and increasing for 1  𝑥  5.

(C)

𝑓 󰆒 is positive and decreasing for 1  𝑥  5.

(D)

𝑓 󰆒 is positive and increasing for 1  𝑥  5.

14. Let 𝑓 be the function given by 𝑓󰇛𝑥󰇜  4  𝑥. 𝑔 is a function with derivative given by 𝑔󰆒 󰇛𝑥󰇜  𝑓󰇛𝑥󰇜𝑓 󰆒 󰇛𝑥󰇜󰇛𝑥  2󰇜 On what intervals is 𝑔 decreasing?

(A)

󰇛∞, 2 󰇠 and 󰇟2, ∞󰇜

(B)

󰇛∞, 2 󰇠 only

(D)

󰇟2, ∞󰇜 only

(E)

󰇟4, ∞󰇜 only

(C)

󰇟2, 4󰇠 only

15. Particle 𝑋 moves along the positive 𝑥-axis so that its position at time 𝑡  0 is given by 𝑥󰇛𝑡󰇜  2𝑡   7𝑡   4. (a) Is particle 𝑋 moving toward the left or toward the right at time 𝑡  2? Give a reason for your answer.

(b) At what time 𝑡  0 is particle 𝑋 farthest to the left? Justify your answer.

(c) A second particle, 𝑌, moves along the positive 𝑦-axis so that its position at time 𝑡 is given by 𝑦󰇛𝑡󰇜  4𝑡  5. At any time 𝑡, 𝑡  0, the origin and the positions of the particles 𝑋 and 𝑌 are the vertices of a rectangle in the first quadrant. Find the rate of change of the area of the rectangle at time 𝑡  2. Show the work that leads to your answer.

5.4TheFirstDerivativeTest

Calculus Write your questions and thoughts here!

Notes

The First Derivative Test is when we use the first derivative to “test” whether or not a function has a maximum or minimum. Start with something we know. A quadratic function’s graph is a parabola. We know 𝑓󰇛𝑥󰇜  𝑥   4𝑥  2 opens up, so 𝑓 will have a minimum. Examine the graph of this parabola and describe the behavior of 𝑓 󰆒󰇛𝑥 󰇜 around the minimum.



y

     

x 







    

Justification statements

Assume 𝑐 and 𝑑 are critical numbers of a function 𝑓. There is a minimum value at 𝑥  𝑐 because 𝑓 󰆒 changes signs from negative to positive. There is a maximum value at 𝑥  𝑑 because 𝑓 󰆒 changes signs from positive to negative. 1. Use the First Derivative Test to find the 𝑥 -values of any relative extrema of 𝑓󰇛𝑥󰇜  

󰇛𝑥   4󰇜 .

    

If 𝒉󰇛𝒄󰇜 does not exist, then 𝒙  𝒄 cannot be a critical point.

2. Find the relative max/min of the function ℎ󰇛𝑥󰇜 





“What is the maximum value” is not the same as “where is the maximum”.



5.4TheFirstDerivativeTest



Practice

Calculus

1. Assume 𝑓󰇛𝑥󰇜 is continuous for all real numbers. The sign of its derivative is given in the table below for the domain of 𝑓. Identify all relative extrema and justify your answers.

Interval 𝒇 󰆒 󰇛𝒙 󰇜

󰇛∞, 𝟐󰇜

󰇛𝟐, 𝟎󰇜

󰇛𝟎, 𝟑󰇜

󰇛𝟑, ∞󰇜

Positive

Negative

Negative

Positive

For each problem, the graph of 𝒇󰆒 , the derivative of 𝒇, is shown. Find all relative max/min of 𝒇 and justify. 3. 2.

For each problem, the derivative of a function 𝒈 is given. Find all relative max/min of 𝒈 and justify. 5. 𝑔󰆒 󰇛𝑥󰇜  𝑥   5𝑥  4 4. 𝑔󰆒 󰇛𝑥󰇜  󰇛𝑥  4󰇜𝑒 

Use a calculator to help find all x-values of relative max/min of 𝒇. No justification necessary.   6. 𝑓 󰆒 󰇛𝑥󰇜  𝑥   6 cos 󰇛𝑥  󰇜  2 8. 𝑓󰆒 󰇛𝑥󰇜  √𝑥   2  𝑥   5𝑥 7. 𝑓 󰆒 󰇛𝑥󰇜  

Use the First Derivative Test to locate the 𝒙-value of all extrema. Classify if it is a relative max or min and justify your answer. 10. 𝑔󰇛𝑥󰇜  𝑥𝑒  9. 𝑓󰇛𝑥󰇜  𝑥   12𝑥  1

11. ℎ󰇛𝑥󰇜 

 

13. What is the maximum value of 𝑔󰇛𝑥󰇜  2 cos 𝑥 on the open interval 󰇛𝜋, 𝜋󰇜?



12. 𝑓󰇛𝑥󰇜  󰇛𝑥  5󰇜 

14. What is the relative minimum value of ℎ󰇛𝑥󰇜  𝑥   6𝑥   3?

Test Prep

5.4TheFirstDerivativeTest

15. If 𝑔 is a differentiable function such that 𝑔󰇛𝑥󰇜  0 for all real numbers 𝑥 and if 𝑓 󰆒 󰇛𝑥󰇜  󰇛𝑥   𝑥  12󰇜𝑔󰇛𝑥󰇜, which of the following is true?

(A)

𝑓 has a relative maximum at 𝑥  3 and a relative minimum at 𝑥  4.

(B)

𝑓 has a relative minimum at 𝑥  3 and a relative maximum at 𝑥  4.

(C)

𝑓 has a relative maximum at 𝑥  3 and a relative minimum at 𝑥  4.

(D)

𝑓 has a relative minimum at 𝑥  3 and a relative maximum at 𝑥  4.

(E)

It cannot be determined if 𝑓 has any relative extrema.

16. Let 𝑓 be a twice-differentiable function defined on the interval 2.1  𝑥  2.1 with 𝑓󰇛1󰇜  2. The graph of 𝑓 󰆒 , t...