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1.4 graphs1

# 1.4 graphs1 - S E C T I O N 1 3 N E W F U N C T I O N S F...

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SECTION 1.3 NEW FUNCTIONS FROM OLD FUNCTIONS |||| 45 57. The Heaviside function H is defined by It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantane- ously turned on. (a) Sketch the graph of the Heaviside function. (b) Sketch the graph of the voltage in a circuit if the switch is turned on at time and 120 volts are applied instantaneously to the circuit. Write a formula for in terms of . (c) Sketch the graph of the voltage in a circuit if the switch is turned on at time seconds and 240 volts are applied instantaneously to the circuit. Write a formula for in terms of . (Note that starting at corresponds to a translation.) 58. The Heaviside function defined in Exercise 57 can also be used to define the ramp function , which represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph of the ramp function . (b) Sketch the graph of the voltage in a circuit if the switch is turned on at time and the voltage is gradually increased to 120 volts over a 60-second time interval. Write a formula for in terms of for . (c) Sketch the graph of the voltage in a circuit if the switch is turned on at time seconds and the voltage is gradu- ally increased to 100 volts over a period of 25 seconds. Write a formula for in terms of for . 59. Let and be linear functions with equations and . Is also a linear function? If so, what is the slope of its graph? 60. If you invest dollars at 4% interest compounded annually, then the amount of the investment after one year is . Find , , and . What do these compo- sitions represent? Find a formula for the composition of copies of . 61. (a) If and , find a function such that . (Think about what operations you would have to perform on the formula for to end up with the formula for .) (b) If and , find a function such that . 62. If and , find a function such that . 63. (a) Suppose and are even functions. What can you say about and ? (b) What if and are both odd? 64. Suppose is even and is odd. What can you say about ? Suppose t is an even function and let . Is h always an even function? 66. Suppose t is an odd function and let . Is h always an odd function? What if is odd? What if is even? f f h f t h f t 65. f t t f t f f t f t t f t f h t h x 4 x 1 f x x 4 f t h t h x 3 x 2 3 x 2 f x 3 x 5 h t f t h f h x 4 x 2 4 x 7 t x 2 x 1 A n A A A A A A A A A A x 1.04 x A x x f t t x m 2 x b 2 f x m 1 x b 1 t f t 32 H t V t t 7 V t t 60 H t V t t 0 V t y tH t y ctH t t 5 H t V t t 5 V t H t V t t 0 V t H t 0 1 if t 0 if t 0 51. Use the given graphs of and to evaluate each expression, or explain why it is undefined. (a) (b) (c) (d) (e) (f) 52. Use the given graphs of and to estimate the value of for . Use these estimates to sketch a rough graph of . A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of .

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