AP Calculus – Fall 2017

Meaning of a Derivative- Rate of Change

E. Q. : What is a derivative?

Day / Topic / Assignment
Aug 21&Aug 22
Mon &Tues / 2.7Tangents, Velocities, and Other Rates of Change
EQ: What is a derivative? / Definition of a Derivative
Meaning of the Defn.of a derivative
Interpreting the Derivative
Discuss relationship between position & velocity
Ticket out the door: compare and contrast
Pg. 151: #17-20, 42, 43ab, 44ab, 45, 47-52
Aug 23
Wed / 2.8 The Derivative as a Function
What is the derivative of a function at a point and how is it related to the tangent line? / Homework skills mini quiz interpreting the deriv.
Pg. 151: #27, 33-37 odd
Pg. 163: #21, 23-27, 35a
Using the Defn. of a Derivative
Do the problem at the bottom of this sheet: Derivatives Numerically** (know!)
Ticket out the Door: Using the definition of derivative and derivatives numerically
Aug 24
Thur / 2.8 The Derivative as a Function
How do you find the derivative as a function and what does the function tell us about the derivative? What is a second derivative? / Homework Skills check mini quiz
Chapter 2—The Derivative
Major Curve Pieces
The Derivative Function
Intro to Curve Sketching
Pg. 164: #37-40, 43-45, 53
Aug 25
Fri / Sec. 2.8 What does say about ?
What does the first derivative and the second derivative tell us about the function? What are the units of f ‘ ? / In class: Definition of Derivative Quiz
What does say about f
Curve Sketching Given f(x)
AB Calculus: Curve Sketching #1
Ticket out the door: curve sketching
Pg. 162 #1-11, 13-15
Pg. 168 #35, 42-44, 47-50
Mon 8/28 / Review / Review Sheet—Summary Beginning Derivatives
Tues 8/29 / Test / Good Luck!

Answers to even numbered problems:

p. 150 #18) y+3 = 4(x-5) #20) f(4) = 3 and #42) The slope of the tangent at t = 1 hours seems to be about #44) a) i) [2006, 2008] 2120 locations per year

ii) [2006, 2007] 2571 locations per year iii) [2005, 2006] 2199 locations per year

(b) (2571+2199)/2 = 2385 locations per year

#48) (a) f ′ (5) is the rate of growth of the bacteria population when t = 5 hours. Its units are bacteria per hour. (b) With unlimited space and nutrients, f ′ should increase as t increases; so

f ′ (5) < f ′ (10). If the supply of nutrients is limited, the growth rate slows down at some point in time, and the opposite may be true. #50 (a) f ′ (8) is the rate of change of the quantity of coffee sold with respect to the price per pound when the price is $8 per pound. The units for f ′ (8) are pounds/(dollars/pound). (b) f ′ (8) is negative since the quantity of coffee sold will decrease as the price charged for it increases. People are generally less willing to buy a product when its price increases. #52 a) S ′ (T) is the rate of change of the maximum sustainable speed of Coho salmon with respect to the temperature. Its units are (cm/s)/°C. b) For T = 15°C, it appears the tangent line of the curve goes through the points (10, 25) and (20, 32). So S ′ (15) ≈ (cm/s)/°C. This tells us that at T=15°C, the maximum sustainable speed of Coho salmon is changing at a rate of 0.7 (cm/s)/°C. In a similar fashion for T = 25°C, we can use the points (20, 35) and (25, 25) to obtain S ′ (25)≈ − 2 (cm/s)/°C. . As it gets warmer than 20°C, the maximum sustainable speed decreases rapidly.

P. 162 # 2 a) f ′ (0) ≈ 6 b) f ′ (1) ≈ 0 © f ′ (2) ≈ − 1.5 d) f ′ (3) ≈ − 1.3 e) f ′ (4) ≈ −0.8

f) f ′ (5) ≈ −0.3 (g) f ′ (6) ≈ 0 h) f ′ (7) ≈ 0.2

#4 #6 #8 #10

#14) (a) F ′ (v) is the instantaneous rate of change of fuel economy with respect to speed. (b) Graphs will vary depending on estimates of F ′ but will change from positive to negative at about v = 50.

© to save gas, drive at the speed where F is a maximum and F ′ is 0 which is about 50 mi/h

#24 f ′ (x) = 3x -1 #26 . #38 f is not differentiable at x = 0 because there is a discontinuity there and at x = 3, because the graph has a vertical tangent there. #40 f is not differentiable at x = −1 because there is a discontinuity there and at x = 2, because the graph has a corner there. #44 Where d has horizontal tangents only c is 0, so d′ = c. c has negative tangents for x < 0 and b is the only graph that is negative for x<0, so c′=b. b has positive tangents everywhere except at x = 0. The only graph that is positive on the same domain is a, so b′=a. We conclude that

d = f, c = f ′, b = f ″ and a = f ‴.

P. 168 #42 #44

#48 a is the graph of f, c is the graph of f ′ and b is the graph of f ″. #50) a) F ′ (1950) ≈ 0.11,

F ′ (1965) ≈ −0.16 and F ′ (1987) ≈ 0.02. (b) The rate of change of the average number of children born to each woman was increasing by 0.11 in 1950, decreasing by 0.16 in 1965 and increasing by 0.02 in 1987. (c) There are many possible reasons: In the baby boom era, there was optimism about the economy and family size was rising. In the baby bust ear, there was less economic optimism, and it was considered less socially responsible to have a large family. In the baby-boomlet era, there was increased economic optimism and a return to more conservative attitudes.