SECTION 5.3 Exponential Functions 311 124. Current in an RC Circuit The equation governing the amount of current I (in amperes) after time t (in microseconds) in a simple RC circuit consisting of a resistance R (in ohms), a capacitance C (in microfarads), and an electromotive force E (in volts) is = ( ) − I E R e t RC / E I C 1 2 R (a) If = = E R 120 volts, 2000 ohms, and = C 1.0 microfarad, how much current I1 is flowing initially ( ) = t 0? After 1000 microseconds? After 3000 microseconds? (b) What is the maximum current? (c) Graph the function ( ) = I I t , 1 measuring I along the y-axis and t along the x-axis. (d) If = = E R 120 volts, 1000 ohms, and = C 2.0 microfarads, how much current I2 is flowing initially? After 1000 microseconds? After 3000 microseconds? (e) What is the maximum current? (f) Graph the function ( ) = I I t 2 on the same coordinate axes as ( ) I t . 1 125. If f is an exponential function of the form ( ) = f x Cax with growth factor 3, and if ( ) = f 6 12, what is ( ) f 7 ? 126. Another Formula for e Use a calculator to compute the values of + + + + n 2 1 2! 1 3! 1 ! for = n 4, 6, 8, and 10. Compare each result with e. [Hint: = = ⋅ = ⋅ ⋅ 1! 1,2! 2 1,3! 3 2 1, n n n ! 1 3 2 1 .] ( ) ( )( )( ) = − ⋅ ⋅ 127. Another Formula for e Use a calculator to compute the various values of the expression. Compare the values to e. 2 1 1 1 2 2 3 3 4 4 etc. + + + + + 128. Difference Quotient If ( ) = f x a ,x show that ( ) + − = ⋅ − ≠ f x h f x h a a h h ( ) 1 0 x h 129. If ( ) = f x a ,x show that + = ⋅ f A B f A f B ( ) ( ) ( ). 130. If ( ) = f x a ,x show that ( ) − = f x f x ( ) 1 . 131. If ( ) = f x a ,x show that α [ ] ( ) = α f x f x ( ) . (a) Determine the probability that = x 5 people arrive within the next minute. (b) Determine the probability that = x 8 people arrive within the next minute. 121. Relative Humidity The relative humidity is the ratio (expressed as a percent) of the amount of water vapor in the air to the maximum amount that the air can hold at a specific temperature. The relative humidity, R, is found using the following formula: = ( ) + − + + R 10 T D 4221 459.4 4221 459.4 2 where T is the air temperature (in °F) and D is the dew point temperature (in °F). (a) Determine the relative humidity if the air temperature is ° 50 Fahrenheit and the dew point temperature is ° 41 Fahrenheit. (b) Determine the relative humidity if the air temperature is ° 68 Fahrenheit and the dew point temperature is ° 59 Fahrenheit. (c) What is the relative humidity if the air temperature and the dew point temperature are the same? 122. Learning Curve Suppose that a student has 500 vocabulary words to learn. If the student learns 15 words after 5 minutes, the function ( ) ( ) = − − L t e 500 1 t 0.0061 approximates the number of words L that the student will have learned after t minutes. (a) How many words will the student have learned after 30 minutes? (b) How many words will the student have learned after 60 minutes? 123. Current in an RL Circuit The equation governing the amount of current I (in amperes) after time t (in seconds) in a simple RL circuit consisting of a resistance R (in ohms), an inductance L (in henrys), and an electromotive force E (in volts) is [ ] = − ( ) − I E R e 1 R L t / 1 L 2 R E I (a) If = = E R 120 volts, 10 ohms, and = L 5 henrys, how much current I1 is flowing after 0.3 second? After 0.5 second? After 1 second? (b) What is the maximum current? (c) Graph this function ( ) = I I t , 1 measuring I along the y-axis and t along the x-axis. (d) If = = E R 120 volts, 5 ohms, and = L 10 henrys, how much current I2 is flowing after 0.3 second? After 0.5 second? After 1 second? (e) What is the maximum current? (f) Graph the function ( ) = I I t 2 on the same coordinate axes as ( ) I t . 1
RkJQdWJsaXNoZXIy NjM5ODQ=