# C what is the natural exponential function 36 graph

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Chapter 3 / Exercise 10
Applied Calculus
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(c)What is the natural exponential function? 3. , , , 4. , , , , , , 6. , , , 7–12 |||| Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 3 and 12 and, if necessary, the transformations of Section 1.3. 7. 8. 10. 12. Starting with the graph of , write the equation of the graph that results from (a) shifting 2 units downward (b) shifting 2 units to the right (c) reflecting about the -axis (d) reflecting about the -axis (e) reflecting about the -axis and then about the -axis 14. Starting with the graph of , find the equation of the graph that results from (a) reflecting about the line (b) reflecting about the line 15–16 |||| Find the domain of each function. 15. (a) (b) 16. (a) (b) t t s 1 2 t t t sin e t f x 1 1 e x f x 1 1 e x x 2 y 4 y e x y x y x y e x 13. y 2 5 1 e x y 3 e x 11. y 1 2 e x y 2 x 9. y 4 x 3 y 4 x 3 y 0.1 x y 0.3 x y 0.6 x y 0.9 x y ( 1 10 ) x y ( 1 3 ) x y 10 x y 3 x 5. y 8 x y 8 x y e x y e x y 20 x y 5 x y e x y 2 x e e 0 a 1 a 1 a 1 a 1 a 0 |||| 7.2 Exercises 5E-07(pp 422-431) 1/17/06 4:36 PM Page 431
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Chapter 3 / Exercise 10
Applied Calculus
Berresford/Rockett Expert Verified
432 ❙❙❙❙ CHAPTER 7 INVERSE FUNCTIONS 57. Under certain circumstances a rumor spreads according to the equation where is the proportion of the population that knows the rumor at time and and are positive constants. [In Sec- tion 10.5 we will see that this is a reasonable model for .] (a) Find . (b) Find the rate of spread of the rumor. ; (c) Graph for the case , with measured in hours. Use the graph to estimate how long it will take for 80% of the population to hear the rumor. ; 58. An object is attached to the end of a vibrating spring and its displacement from its equilibrium position is , where is measured in seconds and is measured in centimeters. (a) Graph the displacement function together with thefunctions and . How are thesegraphs related? Can you explain why? (b) Use the graph to estimate the maximum value of the displacement. Does it occur when the graph touches thegraph of ? (c) What is the velocity of the object when it first returns to itsequilibrium position? (d) Use the graph to estimate the time after which the displace- ment is no more than 2 cm from equilibrium. ; 59. The flash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the flash is set off. The following data describe the charge remaining on the capacitor (measured in microcoulombs, C) at time (mea- sured in seconds). (a) Use a graphing calculator or computer to find an exponen- tial model for the charge. (See Section 1.2.) (b) The derivative represents the electric current (measured in microamperes, A) flowing from the capaci- tor to the flash bulb. Use part (a) to estimate the current when s. Compare with the result of Example 2 in Section 2.1.
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