Practice Problems
Work through the oddnumbered problems 155. Once you have completed the problem set, check your answers.
1. The graph of is given in Fig. 6.
(a) At which integers is continuous?
(b) At which integers is differentiable?
3. Use the values given in the table to determine the values of and .
0 
2 
3 
1 
5 




1 
–3 
2 
5 
–2 




2 
0 
–3 
2 
4 




3 
1 
–1 
0 
3 




5. Use the information in Fig. 8 to plot the values of the functions and and their derivatives at and .
7. Calculate by (a) using the product rule and (b) expanding the product and then differentiating. Verify that both methods give the same result.
13. Find values for the constants and so that the parabola has , and
15. If and are differentiable functions which always differ by a constant for all ), then what can you conclude about their graphs and their derivatives?
17. If the product of and is a constant for all , then how are and related?
27. and . (You may need to use the Bisection Algorithm or the "trace" option on a calculator to approximate where .)
29. with constants and . Can you find conditions on the constants and which will guarantee that the graph of has two distinct "vertices"? (Here a "vertex" means a place where the curve changes from increasing to decreasing or from decreasing to increasing.)
Where are the functions in problems 3037 differentiable?
39. For what values of and is differentiable at ?
41. If an arrow is shot straight up from ground level on the moon with an initial velocity of 128 feet per second, its height will be feet at seconds. Do parts (a)  (e) of problem 40 using this new equation for .
43. If an object on Earth is propelled upward from ground level with an initial velocity of feet per second, then its height at seconds will be
(a) What will be the object's velocity after seconds?
(b) What is the greatest height the object will reach?
(c) How long will the object remain aloft?
45. The best high jumpers in the world manage to lift their centers of mass approximately 6.5 feet.
(a) What is the initial vertical velocity these high jumpers attain?
(b) How long are these high jumpers in the air?
(c) With the initial velocity in part (a), how high would they lift their centers of mass on the moon?
47. (a) Find the equation of the line which is tangent to the curve at the point .
(b) Graph and and determine where intersects the axis and the axis.
(c) Determine the area of the region in the first quadrant bounded by , the axis and the axis.
49. Find values for the coefficients and so that the parabola goes through the point and is tangent to the line at the point .
51. (a) Find a function so that .
(b) Find another function so that .
(c) Can you find more functions whose derivatives are ?
53. The graph of is given in Fig. 12.
(a) Assume and sketch the graph of .
55. Assume that and are differentiable functions and that . State why each step in the following proof of the Quotient Rule is valid.
Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorgresources/wwwresources/site/wpcontent/uploads/2012/12/MA0053.3DerivativesPropertiesandFormulas.pdf
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