Calculus Practice Limits Practice 1,0 PDF

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Just a practice on limits. Such as rate of change and everything...

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AP Calculus – Opening Section Quiz (Chapter 1a)

Name __________________________________

1. Tony Bungy has a strong cord attached to his legs and jumps off a platform towards a river below. At time t = 4 seconds, the cord becomes taut. From that time, Tony’s distance, d, in feet, above the river is given by the equation d = 100 − 75sin[1.7( t − 4)] , t in seconds. a. How far is Tony above the river when the cord becomes taut at t = 4? (2 pts)

b. Find the average rate of change of d with respect to t for the time intervals t = 4.9 to t = 5.0 and t = 5.0 to t = 5.1 (3 decimal place accuracy. Specify units. Based on these answers, is Tony going up or down at t = 5? Explain. (4 pts)

c. Estimate the instantaneous rate of change of d with respect to t at time t = 7 sec. Show how you arrive at your answer and determine if Tony is going up or down at that time. 3 decimal places. (3 pts)

d. What is the calculus name for the value you calculated in part c? (1 pt)

2. Tim has a large fish tank that has a disease in it as fish seem to be dying off. He decides not to add any more fish until he can determine the cause of the fish dying. Following is a table of dates and the number of fish in the tank on these dates. Date Fish

June 1 42

June 3 40

June 6 38

June 8 34

June 11 28

June 15 June 17 21 18

June 18 18

June 20 16

June 25 11

a. What is the average change of the number of fish in the tank from June 1 through June 25. Specify units. (2 pts)

b. Estimate the instantaneous change of the number of fish in the tank on June 15 in three ways. (3 pts)

c. Give the best estimate for the instantaneous change of the number of fish in the tank on June 25. (2 pts)

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3. Sir Mount likes to work out on a Stair-Stepping machine. This machine simulates climbing stairs in a building. The number of floors per minutes that he climbs is modeled by the function shown to the 4 right: F ( t) = 1+ 2 , where t is measured in t − 8t + 18 minutes. He does a 5-minute workout. Using 10 trapezoids of equal base, estimate the definite integral of F with respect to t from 0 to 5 and interpret its meaning. Draw the trapezoids. (Get 2-decimal place accuracy). Show how you got your answer. (8 pts)

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AP Calculus – msec mtan Practice Quiz (Chapter 2-3)

Name _____________________________

1) f ( x ) = −5x + 3 a) find the slope of the secant line between x = 2 and x = 6

b) find the slope of the tangent line at x = 3

c) find the equation of the tangent line at x = 3

b) find the slope of the tangent line at x = 2

c) find the equation of the tangent line at x = 2

b) find the slope of the tangent line at x = 1

c) find the equation of the tangent line at x = 1

2) f ( x ) = 2x 2 − 3x + 1 a) find the slope of the secant line between x = –1 and x = 2

3) f ( x ) = x 3 − x 2 a) find the slope of the secant line between x = –1 and x = 2

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4) f ( x ) = x − 2 x −5 a) find the slope of the secant line between x = 3 and x = 4

b) find the slope of the tangent line at x = 2

c) find the equation of the tangent line at x = 2

5) If an object travels a distance of s = 2t 2 − 5t + 1, where s is in feet and t is in seconds, find a) the average velocity of the object within the first 10 seconds

b) the instantaneous velocity at t = 3 seconds

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AP Calculus – msec mtan Quiz (Chapter 2-3)

Name ______________________________

SHOW ALL WORK. NO CREDIT GIVEN WITHOUT SUPPORTING WORK. 1) f ( x ) = −2x + 1 (6 points) a) find the slope of the secant line between x = 3 and x = 8

2) f ( x ) = x 2 − x − 2

x+3 x −1

c) find the equation of the tangent line at x = 3

b) find the slope of the tangent line at x = 2

c) find the equation of the tangent line at x = 2

b) find the slope of the tangent line at x = 2

c) find the equation of the tangent line at x = 2

(6 points)

a) find the slope of the secant line between x = –1 and x = 4

3) f ( x ) =

b) find the slope of the tangent line at x = 3

(6 points)

a) find the slope of the secant line between x = 3 and x = 5

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4) On earth, an object subjected to gravity covers a distance s = 16t 2 while on the moon, the distance is s = 4t 2 where s is measured in feet and t is in seconds. Find the average velocity of a falling object within the first 6 seconds on a) the earth

b) the moon

Find the instantaneous velocity of a falling object at t = 2 seconds on c) the earth

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(2 points each)

(3 points each)

d) the moon

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AP Calculus – Graphical Limits Quiz (Chapter 4)

Name __________________________________

Each problem has 6 parts. Specify the limit next to the problem. Give the answer that best describes the graph.

1)

2)

a) lim− f ( x )

b) lim+ f ( x )

c) lim f ( x)

d) f (1)

e) lim f (x )

f) lim f ( x )

x →1

x →1

x →−∞

x →1

x →∞

3)

a) lim− f (x )

b) lim+ f ( x)

c) lim f ( x)

d) f ( 2)

e) lim f (x )

f) lim f ( x)

x →2

x →2

x →−∞

x →2

x →∞

4)

a) lim− f (x )

b) lim+ f ( x)

c) lim f ( x)

a) lim− f (x )

b) lim+ f ( x)

c) lim f ( x)

d) f ( 3)

e) lim f (x )

f) lim f ( x)

d) f ( 0)

e) lim f (x )

f) lim f ( x)

a) lim− f ( x )

b) lim+ f ( x )

c) lim f ( x)

d) f (1)

e) lim f (x )

f) lim f ( x )

x →3

x →3

x →−∞

x →3

x →∞

5)

a) lim− f (x ) x→1

d) f ( 1) !

x →0

x →0

x →−∞

x →0

x →∞

6)

!b) lim+ f (x ) x→1

e) lim f ( x ) x→−∞

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c) lim f (x ) x→1

f) lim f ( x ) x→∞

x →1

x →1

x →−∞

x →1

x →∞

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7)

8)

a) lim− f ( x )

b) lim+ f ( x )

c) lim f ( x)

a) lim− f (x )

b) lim+ f ( x)

c) lim f ( x)

d) f (1)

e) lim f (x )

f) lim f ( x )

d) f ( 3)

e) lim f (x )

f) lim f ( x)

x →1

x →1

x →−∞

x →1

x →∞

x →3

x →3

x →−∞

x →3

x →∞

10)

9)

a) lim− f (x )

b) lim+ f ( x)

c) lim f ( x)

a) lim− f (x )

b) lim+ f ( x)

c) lim f ( x)

d) f ( 2)

e) lim f (x )

f) lim f ( x)

d) f ( 0)

e) lim f (x )

f) lim f ( x)

x →2

x →2

x →−∞

x →2

x →∞

x →0

x →0

x →−∞

x →0

x →∞

12)

11)

a) lim− f (x )

b) lim+ f ( x)

c) lim f ( x)

a) lim− f (x )

b) lim+ f ( x)

c) lim f ( x)

d) f ( 0)

e) lim f (x )

f) lim f ( x)

d) f ( 0)

e) lim f (x )

f) lim f ( x)

x →0

x →0

x →−∞

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x →0

x →∞

x →0

x →0

x →−∞

x →0

x →∞

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AP Calculus – Algebraic Limits Quiz (Chapter 5)

Name __________________________________

Find the following limits. If no work is necessary, you may just write the answer. However, problems that cannot be found by inspection must work to justify your answers. Proper notation MUST be used or points will be taken off.

1. lim 3

x→−∞

x→−4

(

4. lim y 2 − 3y −1 y→−1

3. lim ( 5x +1)

2. lim 4x

x→4

)

4x +1 x→∞ 5x − 2

7. lim

x8 x →−∞1000 x 5 + 1

10. lim

13. lim x →5

x x −5

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x 2 − x −12 2 x →4 x − 9x + 20

x 2 − 3x −1 x→3 x+4

6. lim

1 x→−∞ x −15

2 9. lim 3x − 5x +17 x→∞ 0.01x 3

5. lim

8. lim

x 4 −16 x →−2 x + 2

−3x 4 − 2 x →−∞ 5x − 6x 4

11. lim

12. lim

14. lim x →3

x−4 x2 − 9

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15. lim

⎧ 2 16. f ( x ) ⎨ x − 6, x ≥ 4 = 3x − 2, x < 4 ⎩

⎧ −5 , x > 0 ⎪ 17. f ( x ) = ⎨ x ⎪⎩ 2 x , x < 0

18. lim

x 3 − 27 19. lim 2 x →3 x − 6x + 9

⎛ 1 1⎞ − ⎟ ⎜ 20. lim⎜ x + 5 5 ⎟ x →0 x ⎜ ⎟ ⎝ ⎠

x +1 x →0 x 2

21. lim x →−∞

4x + 3 2x 2 − 50

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x →4

x x−4

⎛ e− x ⎞ 22. lim ⎜ x→∞ ⎝ csc x ⎟ ⎠

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AP Calculus – Algebraic Limits AP Type Quiz (Chapter 5) 1) Find lim x →0

Name ______________________________

x+2− 2 . Show all work. (3 points) x

⎧ x 2 − 5x − 6 ⎪ , x ≠6 2) Let F ( x) = ⎨ x − 6 ⎪3k + 2 , x = 6 ⎩

a) Find lim F(x) . Show all proper steps. (1 points) x →6

b) Find the value of k such that lim F(x) = F( 6) . Show all work. (2 points) x →6

3) If it exists, find lim x →∞

x −3 x −3 . If it does not, explain why. (3 points) − lim x − 3 x →3 x − 3

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AP Calculus – Derivatives Exam (Chapter 6, 7, 8)

Name ______________________________

Find the derivative of the following functions. Be sure to use proper notation to show the derivative. Simplify according to the rules established in class. (1-12 - 2 pts, 13 - 18 - 3 pts) 1. f ( x ) = −12

2. g( x ) = 7x

3. y = 7 − 8x

4. f ( x ) = 8x 2 − 3

5. h( x) = −12x 2 − 4 x + 5

6. y = ax 2 − bx − ab + 4, a,b are constants

7. f ( x ) = π 2 − 3π + 1

8. y = 12x 4 −17x 3 −

9. y =

9x 3 15x 2 7x 4 − + − 3 2 4 11

11. y =

7 5 − x3 x

13. y = (2x 5 − 8x 3 − 7x −1)(11x 3 − x 2 + 1)

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10. y = −

x + 20.4 2

2 (3x 5 − 5x 2 −12 ) 5

12. y =

13x 3 − 7x 2 + x . 2x

14. y =

x 2 + 5x − 2 5x − 6

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15. y =

1 3x − 5

17. y = 4 x

16. y = 3 x −

2

(

x+3

5

1 x

2 18. y = x − 2x − 3 x

)

19. Using the calculator, find the equation of the tangent line to y =

cos x at x = 0. (3 pts) e2 x

Using the chart below, find f ′(5) if f ( x ) is given by: Show necessary work. (4 pts each)

g(5) 2

20. f ( x ) =

h( x) 2g( x)

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g′(5 ) 8

h( 5) 1 − 2

h′( 5) 3

21. f ( x ) = g ( x )[ g( x ) + h ( x )]

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22. Given f ( x ) = 2x 2 + x − 3 , find f (x ) by using the definition of the derivative. (4 pts)

23. Find the equation of the tangent line to f (x )=

1 2 x + 3x + 3 at the point (–2,–1). Sketch both curves. Show 2

all work. (6 pts)

24. Find the equation of the line that is tangent to f (x )= x 2 − 4 x − 7 and parallel to 2x + y = 4. Show all work. (5 pts)

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AP Calculus – Chain Rule Quiz (Chapter 9)

Name ______________________________

x 2 f ( x) π 3 f ′( x)

7

g( x)

5

g ′( x )

−4

3 3

4 9

9 4

π 4 π 2

−3

4

−3

2

9

π

π 2

For each problem, find the requested information. Show your steps. Answers should first be calculated exactly and then found accurate to 3 decimal places. (3 pts each) 1.

d [ f ( x) + g (x )] at x = 2 dx

2.

d [ f ( x) ⋅ g(x )] at x = 4 dx

3.

d ⎡ f (x )⎤ ⎢ ⎥ at x = 9 dx ⎣ g( x) ⎦

4.

3 d f ( x) ] at x = 9 [ dx

5.

d 2 [ x ⋅ g(x )] at x = 4 dx

6.

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⎡ ⎤ d⎢ 1 ⎥ at x = 4 dx ⎢⎣ f (x ) ⎥⎦

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f ( x)

2 π 3

f ′( x)

7

g( x)

5

g ′( x )

−4

x

3 3

4 9

9 4

π 4 π 2

−3

4

−3

2

9

π

π 2

7.

d f ( g( x)) at x = 3 dx

8.

d g( f ( x) ) at x = 9 dx

9.

d ⎡ 2x ⎤ ⎢ ⎥ at x = 4 dx ⎣ f (x )⎦

10.

d dx

[

]

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[

[

]

]

f (2x ) at x = 2

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AP Calculus – Chain Rule Practice Exam (Chapter 9)

Name ______________________________

For problems 1-6, find the derivative. Simplify according to the rules established in class. 1. f (x ) = ( x 3 − 3x 2 + 2x −1)

⎛ 3x − 2 ⎞ 3. y = ⎜ ⎟ ⎝ 4 x −1 ⎠

3

2. y = 3 1+ 2x − x 2

8

5. f ( x ) = x 3 5 − x

4

2

4. f (x ) = ( x 2 + x + 3) (1− x )

6. y = x 2 + 1, find

d2y dx 2

7. Find the equation of the lines relative to the curve y = (3x −1) 3 at x = 1. Show necessary work. a) tangent

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b) perpendicular

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Find the point(s) of horizontal tangency to the following functions. Show all work.

(

8. y = x 2 − 2x + 5

2

)

9. y = x 4 − 2x 2 + 2

Given that f (5 ) = 3, f ′( 5) = −1, g (5 ) = 2, g′(5 ) = π , f ′( 2) = 6 , find the derivatives of the following at x = 5. g( x ) 10. y = f ( x ) ⋅ g( x) 11. y = 2 f (x )

12. y = 3 f ( x ) + 5

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13. y = f ( g( x))

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AP Calculus – Chain Rule Exam (Chapter 9)

Name ______________________________

For problems 1-8, find the derivative. Simplify according to the rules established in class. (4 points each) 5

4

1. f ( x ) = ( x 2 + 3x − 4 )

2. f (x ) = 2x (x 3 − 6 )

3. y = x 2 + 6x − 8

4. f ( x ) =

4 ⎛ x −6 ⎞ 5. f ( x ) = ⎜ ⎟ ⎝ 2x + 1⎠

6. f (x ) = ( x 2 + 3x − 8 ) (1− x 4 )

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−8 2

( 7 − 2x)

3

2

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7. f (x ) = x 2 16 − x 2

2

8. y = 5x 2 + 3, find d y2 dx

9. Find the equation of the lines relative to the curve y = 3 x 2 + x − 4 at x = 3. Show necessary work. (6 pts) a) tangent

b) perpendicular

Find the point(s) of horizontal tangency to the following functions. Show all work. (4 points each) x − 3⎞ 10. f (x ) = ⎛⎜ ⎝ x + 5 ⎟⎠

2

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11. f (x ) =

2 3

x 2 + 4x − 4

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Given that f (4 ) = 2, f ′( 4) = −3, g (4 ) = 5, g′( 4) = 6, f ′(5) = −3 , find the derivatives of the following at x = 4. (3 points each) g x 12. y = f ( x ) ⋅ g( x) 13. y = ( ) 2 f (x )

14. y =

f ( x)

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15. y = f ( g( x))

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AP Calculus – 5 AP Type problems (Chapter 1-9)

Name ______________________________

For each problem do the work in the area provided. It is suggested that you do the work on scrap paper first and then transfer to these sheets when you are confident of your answers. These problems are due in classroom _________ by _________ on Date: ______________________ 3 1) Given the function defined as f (x ) = x 3 − x 2 − 6x + 10 2 a) Find the point(s) of horizontal tangency of f.

b) Write an equation of the line normal (perpendicular) to the graph of f at x = 0.

c) Find the x and y-coordinates of all points on the graph of f where the line tangent to the graph is parallel to the x-axis.

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2) Selected values of x and f (x ) are given in the table below. x f (x )

0.9 1.226

1.0 1

1.2 0.754

a) Using the table, find the largest possible approximation of the rate of change of f at x = 1. Justify your answer.

b) Using the answer in part (a), write an equation of the line tangent to the graph at the point where x = 1.

c) If the function f is modeled by the function f ( x ) = x 4 − 3 x 2 + 3, find the equation of the line tangent to the graph at the point where x = 1.

d) Write, but do not solve, two equations to find the x-coordinate of each point at which the line tangent to the graph of f is i) parallel t...