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Math 2413
Lab 2
Name ____________________________________
(Dr. Khoury)
Show all work
Due as scheduled
1. If the graphs of the differentiable piecewise functions f and g are given below, use formulas learned in class
to evaluate each of the following: (Use Rise and Run to find the slope)
a.
g(2) 
g(2) 
f (1) 
f (1) 
g(1) 
g(1) 
f (5) 
f (5) 
g(5) 
g(5) 
f (0) 
f (0) 
d

 f (x)g(x) 
dx
x 1
b.
d  f (x) 


dx  g(x) 

d

 f (x) 20
dx
x 5
c.
x 5
2. Find f ( x) .

a. f (x)  1  2x  1  2x  .
5
6
b. f (x)  2x  1
2x  1
3. For f (x)  4  3x  x 3
a. Find the slope of the tangent line at x  1

10
.
b. Find an equation of the tangent line to the
graph of f (x)  4  3x  x 3 at the
point (1, 0) .
4. For what values a & b is the line 2x  y  b tangent to the parabola y  ax 2 when x  2 ?
5. Find a parabola with equation y  ax 2  bx  c that has slope 4 at x  1 , slope 8 at x  1 , and passes
through the point (2,15).
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6. Find a cubic function f (x)  ax3  bx 2  cx  d whose graph has horizontal tangents at the points (2,6)
and (2,0) .
7. Find all the points on the graph of the function f (x)  2sin x  sin 2 x at which the tangent line is
horizontal, 0  x  2 .
8. Find the 100th derivative of f (x)  xex . Write your answer in simplest form.
9. A curve C is defined by the parametric equations x  t 2 , y  t3  3t .
a. Show that C has two tangents at the point
c. Use your calculator to illustrate parts a and b
(3,0) and find their equations.
by graphing C and the tangent lines.
b. Find the points on C where the tangent is
horizontal or vertical.
10. Find equations of the tangent lines to the curve C with equation y 
ln x
at the points x  1 and x  e .
x
11. Suppose that the only information we have about a function f is that f (1)  5 and the graph of the derivative
is as shown below. Based on this information, Find the equation of the tangent line at (1,5) .
a. Use linear approximation to estimate f (0.99)
b. Use linear approximation to estimate f (1.01)
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12. Use Differentials to approximate 1.01 . Start by graphing the tangent line to the graph of f (x)  x
at the point (1,1), then show on the graph both the actual value and the approximate value.
13. Use implicit differentiation to find y and y in terms of x and y if x 2  6xy  y 2  8 . Write your answer
in simplest form.
14. Find a formula for the nth derivative of y  ln(1  x) . Write your answer in simplest form.
15. Use Logarithmic Differentiation only to find f (x) for f (x) 
(3x 2  1)3 (2x  3) 4
.
5
(1  6x)
16. Suppose f is a one-to-one differentiable function and its inverse function f 1 is also differentiable.
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a. Use implicit differentiation to show that (f 1)(x) 
provided that the denominator is not 0.
f (f 1(x))
b. If f (4)  5 and f (4) 
2
, find (f 1)(5) .
3
17. If f (x)  ln(3x 2  x  5)4 , find f (x). Write your answer in simplest form.
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4
2
18. If f (x)  e x  x  4 , find f (x). Write your answer in simplest form.
19. If f (x)  3x(x3  4)6 , find f (x) . Write your answer in simplest form.
20. If f (x)  (1  x 2 )3 (2  x 2 )5 , find f (x) . Write your answer in simplest form.
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 2  x2 
21. If f (x)  
, find f (x) . Write your answer in simplest form.
 3  x 3 


22. If f (x)  tan 2 (4x3) , find f (x) . Write your answer in simplest form.
23. If f (x)  tan 1  x  1  x 2  , find f (x) . Write your answer in simplest form.


24. If f (x)  ln(sec x  tan x) , find f (x) . Write your answer in simplest form.


 sin x, for 0  x  3
2
25. Let f (x)  
. Determine a and b so the f is differentiable at x 
.
3
3a

 x+b, for <x  2
 
3
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

1  a cos x, for x  3

26. Let f (x)  
. Determine a and b so the f is differentiable at x  .
3
 b+sin  x  , for x> 

3
 2
27. Find the equations of the tangent lines to the curve y 
x 1
that are parallel to the line x  2y  2 .
x 1
28. Two cars start moving from the same point. One travels south at 60 mph and the other travels west at 25
mph. At what rate is the distance between the cars increasing two hours later?
29. Gravel is being dumped from a conveyor belt at a rate of 30 cubic feet per minute and its coarseness is
such that it forms a pile in a shape of a cone whose base diameter is always double its height.
a. How fast is the height of the pile
increasing when the pile is 10 ft
high?
b. How fast is the diameter of the
pile increasing when the pile is
10 ft high?
30. A conical paper cup 8 inches across the top and 6 inches deep is full of water. The cup springs a leak at
the bottom and loses water at the rate of 2 cubic inches per minute. How fast is the water level dropping
at the instant when the water is exactly 3 inches deep? (Hint: the volume of the cone is v   r 2h ).
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31. A man starts walking north at 90 feet per minute from a point P. Five minutes later a women starts walking
south at 120 feet per minute from a point 500 feet due east of P. Use calculus to find at what rate are the
people moving apart 30 minutes after the woman starts walking?
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32. Use the definition of the derivative to evaluate each of the following limits.
100
(x  1)
tan x  3
a. lim
b. lim


x 1 x  1
x
x
3
3
33. Find a formula for the 99th derivative of y  cos(2x) . Write your answer in simplest form.
34. Find parametric equations of the line segment from (1,3) to (5,8) . If these parametric equations
represent the function f(x), find parametric equations of f 1 (x) . Draw f(x), f 1 (x) , and y  x , on
the same graph and find the coordinates of the point common to all three lines.
35. An airplane is flying on a flight that will take it directly over
a radar tracking station. If the distance between the plane and
the radar is s miles and is decreasing at a rate of 400 mph
dx
when s is 10 miles, what is the speed of the plane
?
dt
36. Find the rate of change in the angle of elevation  of the camera
at 10 seconds after lift-off. Assume the distance between the
camera and the point of lift-off is 2000 ft and the height s of the
2
rocket is s  50t , where s is measured in feet and t is measured
in seconds.
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37. Prove the following identities:
a. sinh(x  y)  sinh x cosh y  cosh x sinh y
c. tanh(x  y) 
b. cosh(x  y)  cosh x cosh y  sinh x sinh y
tanh x  tanh y
1  tanh x tanh y
d. sinh(2x)  2sinh x cosh x
e. cosh(2x)  cosh 2 x  sinh 2 x
g.
f. tanh(ln x) 
1  tanh x
 e 2x
1  tanh x
2
x 1
2
x 1
h. (cosh x  sinh x)3  cosh 3x  sinh 3x
38. Assume tanh x 
12
. Find the values of each of the following:
13
a. sinh x 
b. cosh x 
39. Assume cosh x 
5
, x  0 . Find the values of each of the following:
3
a. sinh x 
b. tanh x 
c. coth x 
c. coth x 
d. sech x 
d. sech x 
e. csch x 
e. csch x 
40. Find the derivative of each of the following: Simplify your answer.

a. y  tanh 1  e
2x

2
d. y  x tanh x  ln 1  x
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b. y  x sinh x  cosh x
c. y  x 2 sinh 1(2x)
1  x 
2
e. y  x sinh    9  x
3
f. y 
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1  cosh x
1  cosh x
 x 
41. A telephone line hangs between two poles 14 m apart in the shape of the catenary y  20 cosh    15 ,
 20 
where x and y are measured in meters.
a. Find the slope of this
b. Find the angle
curve where it meets the
 between the line and
right pole.
the pole.
42. A curve passes through the point (0,5) and has the property that the slope of the curve at every point P is
twice the y-coordinate of P. What is the equation of the curve?
43. A roast turkey is taken from an oven when its temperature has reached 185 F and is placed on a table in a
room where the temperature is 75 F.
a. If the temperature of the turkey is 150 F after 30 minutes, what is the temperature after 45 minutes?
b. When will the turkey have cooled to 100 F?
44. If $3000 is invested at 5%, find the value of the investment at the end of 5 years if the interest is
compounded
a. Daily
b. Weekly
c. Monthly
d. Annually
45. The half-life of Cesium-137 is 30 years. Suppose we have a 100 mg sample.
a. Find the mass that remains after t years.
b. How much of the sample remains after 100 years?
c. After how long will only 1 mg remain?
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e. Continuously