AP Calculus AB and BC

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Clinic # 1: Writing Justifications

 Example 1:

Given y = 2x2 – 8x + 9 explain how f(x) possesses extreme values at the determined points on [-1, 3].

 

 

 

Example 2:

Given the following table of values:

x

1

3

6

7

8

12

16

17

h(x)

-9

4

2

5

-4

-4

3

-1

 

Determine the least number of zeros the function h(x) could possess; justify your answer.

 

 

 

 

 

Student assignment:

  1. For what values of m is the function

 

a.)    Is the function continuous?

 

 

 

b.)    Is the function differentiable?

 

 

 

  1. For y =  determine all the extreme values of the function.

 

 

 

 

  1. On what intervals is the graph of y = 2x + 3 concave up? Justify your answer.

 

 

 

 

 

  1. Given the following table of values

t

1

2

3

4

5

6

v(t)

9

0

-3

9

2

-7

 

Explain why there must be at least one time on 1  6 where the acceleration of the object is zero.

 

 

 

 

 

 

  1. If E(t) represents the rate of decline of the elephant population in sub-Saharan Africa from 2000 (t=20) to 2010 (t=30), in elephants per year, explain the meaning of

 

 

 

 

 

 

 

 

 

 

 

  1. A student is given the following separable differential equation:  and proceeds to solve using the following steps:

 

 

 

ln

 

2-y = A

y = - A

Without re-working the problem…(AT ALL) explain all the student’s error(s).

 

 

 

 

 

  1. Given the graph of f’(x), the derivative of f(x) pictured below, justify or refute the claim(s) below:

 

 

 

 

 

 

 

  1. F(x) possesses a local maximum at x = -2

 

 

 

  1. F(x) possesses a point of inflection at x = 0

 

 

 

  1. F(x) possesses a local maximum at x = 1

 

 

 

 

  1. F’’(x) is concave down on (-2, 1)

 

 

 

  1. F(x) is decreasing on the interval (

 

 

 

  1. (answer this question, with justification) – on what interval(s) is f(x) concave down?

 

 

 

  1.  (answer this question, with justification) – on what interval(s) is f(x) increasing?

 

 

 

  1. If (0, 4) is on the graph of f, the equation of the tangent line at that point would be:

y – 4 = -2x

 

 

 

  1.  

x

2

5

7

9

13

15

f(x)

-5

6

3

0

-4

2

 

For the given function values listed above, must there be a point, c, on (5, 13) where f(c) = 7. Justify your answer.

 

 

 

 

 

  1. Given that R(t) represents the number of rats in the city of New York for any year after 1987 (t = 0), explain the meaning  in the context of the problem, provide units.

 

 

 

  1. Given v(t) is the velocity of any object, explain why the expression  cannot provide the average velocity of an object on the interval (3, 6).

 

 

 

 

  1. Provided S’(t) represents the rate at which sand is added to a beach the morning after a hurricane beginning at 9 A.M. (t = 9) in cubic meters per second, explain the meaning of

 

 

 

  1. The table of values below represent selected values of a twice-differentiable function, strictly decreasing function, P(x)

x

-2

3

5

8

9

12

p(x)

23

15

7

6

0

-4

 

Compute a left-hand rectangular approximation for  and explain whether the approximation is an over or underestimate of the actual integral.

 

 

 

  1. Given the function, f(x) = 2xex – 4x, determine all points of inflection on the graph of f(x). Justify your answer.

 

 

 

 

  1. Determine the absolute maximum of the function y = e-x on [-1, 1]. Justify your answer.

 

 

 

 

  1. Given the following table of values for a function, g(t)

t

2

3

6

8

9

11

12

15

19

g(t)

0

1

8

9

-4

-8

-12

1

9

 

Explain why there must be at least two times in the interval (2, 19) where g’(t)=0.

 

 

 

 

 

 

 

  1. Given the following table of values for the function f’(x)

x

-4

-3

-2

-1

0

1

2

3

4

f'(x)

8

12

14

8

9

0

-3

8

5

 

Answer the following questions:

  1. Must there be a time on (2, 4) where f(x) is concave down? Justify your answer.

 

 

 

 

  1. Must f(x) possess more than three critical points? Justify your answer.

 

 

  1. F(x) possesses a critical point at x = 1

 

 

 

 

  1. Is it possible for  and the f(0) fails to exist? Explain your answer. If possible, create a function that satisfies the given conditions.

 

 

 

  1. Does f(x) =  on [-1, 4] satisfy the mean value theorem for some value, c, on [-1 4]:

F’(c) =

 

 

 

 

 

  1. Given f(x) represents the rate at which leaves fall off a tree in autumn from September 24 (t = 1) to October 19, in pounds per hour. Explain, in the context of the problem, the meaning of

 

 

 Past AP AB Calculus Examples with Solutions and Scoring Rubrics

Specific Example of Writing Justifications for Behavior of Functions