CALCULUS CHRISTMAS

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-­‐1   20000   1  

3   0   9   15   -­‐10  

-­‐15   4   5  

78   2   11  −25

−150

43

12

225

22

−49

−22

133

−12

lim x→3x2 − 6x + 9x2 − 5x + 6

1.

lim x→3x2 − 6x + 9x2 − 5x + 6

1.

ANSWER:  0      

f (x) = 2x +3( )2Find the value of f’(x) at x = -1.

2.

f (x) = 2x +3( )2Find the value of f’(x) at x = -1.

2.

ANSWER:    4  

f (x) = 15cos(2x)

What is the value of the derivative at ?

3. x = π

4

f (x) = 15cos(2x)

What is the value of the derivative at ?

3. x = π

4

ANSWER:    

−25

f (x) =2x + b, x ≤ 5x2, x > 5

"#$

%$

What value of b will make the function continuous?

4.

f (x) =2x + b, x ≤ 5x2, x > 5

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%$

What value of b will make the function continuous?

4.

ANSWER:    15  

A snowball is melting at a rate of 2π cm3/min. At what rate is the radius changing when the radius is 5?

5.

A snowball is melting at a rate of 2π cm3/min. At what rate is the radius changing when the radius is 5?

5.

ANSWER:  −150

f (x) = x3 − 2x2 + 5

Where does the graph of the derivative of f(x) have a minimum?

6.

f (x) = x3 − 2x2 + 5

Where does the graph of the derivative of f(x) have a maximum or minimum?

6.

ANSWER:   43

f (x) = x3 − 4x +1

What is the y-intercept of the tangent line to f(x) at x = 2?

7.

f (x) = x3 − 4x +1

What is the y-intercept of the tangent line to f(x) at x = 2?

7.

ANSWER:    -­‐15  

A reindeer farmer wants to use 400 feet of fence to make a pen of maximum area to hold his reindeer. He will use a side of the barn as one side of the pen. What is the largest area that can be enclosed?

8.

A reindeer farmer wants to use 400 feet of fence to make a pen of maximum area to hold his reindeer. He will use a side of the barn as one side of the pen. What is the largest area that can be enclosed?

8.

ANSWER:  20000  

What is the value of the derivative of f(x) at x = 3?

9.

f (x) = 2x −34x − 2

What is the value of the derivative of f(x) at x = 3?

9.

f (x) = 2x −34x − 2

225

ANSWER:  

What is the value of the limit?

10.

lim x→∞

2x2 −3x + 54x2 − 2x − 7

What is the value of the limit?

10.

lim x→∞

2x2 −3x + 54x2 − 2x − 7

ANSWER:  12

What is the value of ?

11.

lim x→1 f (x)

What is the value of ?

11.

lim x→1 f (x)

ANSWER:    1  

What is the value of ?

12.

f (2)

What is the value of ?

12.

f (2)

ANSWER:    2  

What is the value of the trig expression?

13.

cos 7π4

What is the value of the trig expression?

13.

cos 7π4

ANSWER:     22

A particle moves along a straight line so that its position is represented by s(t) = t2-4t+3.

14.

How far did the particle travel from t = 1 to t = 4?

A particle moves along a straight line so that its position is represented by s(t) = t2-4t+3.

14.

How far did the particle travel from t = 1 to t = 4?

ANSWER:  5  

At what x-value is there a relative minimum value for the function f(x) = 5 + 15x + 6x2 –x3

15.

At what x-value is there a relative minimum value for the function f(x) = 5 + 15x + 6x2 –x3

15.

ANSWER:  -­‐1  

What is h’(2) if h(x) = f(g(x)) and g(2) = 3, f’(3) = 5, and g’(2) = -2?

16.

What is h’(2) if h(x) = f(g(x)) and g(2) = 3, f’(3) = 5, and g’(2) = -2?

16.

ANSWER:  -­‐10  

17.

limh→0

cos π4+ h

"

#$

%

&'− cos

π4

h=

17.

limh→0

cos π4+ h

"

#$

%

&'− cos

π4

h=

ANSWER:  −22

18. lim x→0

3sin3xx

=

18. lim x→0

3sin3xx

=

ANSWER:  9  

What is the slope of the tangent line to the curve 3x2 + y3 = 39 when x = 2?

19.

What is the slope of the tangent line to the curve 3x2 + y3 = 39 when x = 2? 19.

ANSWER:    −49

It is a little known fact that Santa actually sends out two sleighs on Christmas Eve. One travels west at a constant velocity of 50 miles per hour and a second sleigh travels south at a constant velocity of 60 miles per hour. How fast is the distance between them changing after 30 minutes?

20.

It is a little known fact that Santa actually sends out two sleighs on Christmas Eve. One travels west at a constant velocity of 50 miles per hour and a second sleigh travels south at a constant velocity of 60 miles per hour. How fast is the distance between them changing after 30 minutes?

20.

ANSWER:  78    

At what positive value of x does f(x) have a removable discontinuity?

21.

f (x) = 2x3 + x2 − 25x +122x3 +3x2 − 23x −12

At what positive value of x does f(x) have a removable discontinuity? 21.

f (x) = 2x3 + x2 − 25x +122x3 +3x2 − 23x −12

ANSWER:  3  

What is the slope of the normal line to y = tan (2x) at x = π ?

22.

What is the slope of the normal line to y = tan (2x) at x = π ? 22.

ANSWER:    

−12

If f(3) = 5 and f’(3) = -2, what is the y-intercept of the tangent line to y = f(x) at x = 3?

23.

If f(3) = 5 and f’(3) = -2, what is the y-intercept of the tangent line to y = f(x) at x = 3?

23.

ANSWER:  11  

If f(x) = 8x – x3, find the value of c that is guaranteed by the Mean Value Theorem over the interval (1, 3).

24.

If f(x) = 8x – x3, find the value of c that is guaranteed by the Mean Value Theorem over the interval (1, 3).

24.

ANSWER:     133

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