Sec 3.1. lim and lim e 0. Exponential Functions. f x 9, write the equation of the graph that results from: A. Limit Rules

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1 Sec 3. Eponential Functions A. Limit Rules. r lim a a r. I a, then lim a and lim a 0 3. I 0 a, then lim a 0 and lim a 4. lim e 0 5. e lim and lim e 0 Eamples:. Starting with the graph o a.) Shiting 9 units upward. 9, write the equation o the graph that results rom: b.) Shiting 7 units to the right. c.) Relecting about the -ais. 6 e. The domain o the unction 3. Find the eponential unction Ca whose graph goes through the points (0, 5) and (, 0). Desiré Taylor Math 4

2 4. Evaluate the ollowing limit lim Evaluate the ollowing limit lim e e e 5 6. Evaluate the ollowing limit lim e e Evaluate the ollowing limit lime cos Desiré Taylor Math 4

3 Sec 3. Logarithmic Functions A. Limit Rules. lim ln 0. lim ln 3. lim log 0 4. lim log a a Eamples:. lim log lim logsin 0 3. limlog log Desiré Taylor Math 4

4 4. I is one-to-one and 3, then a.) b.) 3 5. Find the inverse or each o the ollowing: 4 a.) 9 5 h e 9 3 b.) c.) ln3 0 Desiré Taylor Math 4

5 , ind For Suppose, is the inverse unction o a dierentiable unction and then 4 8. I lna, lnb 3, and ln c, evaluate ln b c a 9. Solve each equation or : a.) b.) Desiré Taylor Math 4 3

6 Sec 3.3 Derivatives o Eponential and Logarithmic Functions A. Derivatives d d.) log = a ln a d 3.) a d = a ln a d ln d =.) d 4.) e d = e Eamples: Find the derivative o each.) ln 3 0.) y e 5 cos ) 6 ln 4.) 3 e 5.) log ) 3 7 Desiré Taylor Math 4

7 B. Logarithmic Dierentiation Eamples: Find the derivative o each.) y.) y Desiré Taylor Math 4

8 Sec 3.4 Eponential Growth and Decay A. Population Growth Where: P(t) = Population ater t years P(0) = Initial Population K = Growth constant T = Time P kt t P0 e A bacteria culture initially contains 600 cells and grows at a rate proportional to its size. Ater 5 hours the population has increased to 60. a.) Find an epression or the number o bacteria ater hours. b.) Find the number o bacteria ater 7 hours. c.) Find the rate o growth ater 7 hours. (Remember: Rate = Derivative) d.) When will the population reach 4000?

9 B. Hal Lie Where: P(t) = Population ater t years P(0) = Initial Population K = Growth constant T = Time P kt t P0 e The hal-lie o cesium-37 is 30 years. Suppose we have a 900-mg sample. a.) Find the mass that remains ater years. (Find an epression or the mass that remains ater years.) b.) How much o the sample remains ater 50 years? c.) Ater how long will only 4 mg remain?

10 C. Newton s Law o Cooling T kt t T 0 Ts e Ts Where: T(t) = Temperature ater time t T s = Temperature o surrounding area T 0 = Initial temperature o object K = Growth constant T = Time T t Alternatively C e kt T Where: T(t) = Temperature ater time t T s = Temperature o surrounding area C = Initial temp - surrounding temp K = Growth constant T = Time s A roast turkey is taken rom an oven when its temperature has reached 75 Fahrenheit and is placed on a table in a room where the temperature is 65 Fahrenheit. a.) I the temperature o the turkey is 55 Fahrenheit ater hal an hour, what is its temperature ater 45 minutes? b.) When will the turkey have cooled to 0 Fahrenheit?

11 D. Interest Compound Interest A P Where: A = Future Value P = Initial Value r = Interest rate n = Number o times per year compounded t = Time in years r n nt Where: A = Future Value P = Initial Value r = Interest rate t = Time in years Compound Interest A Pe rt I 8000 dollars is invested at interest, ind the value o the investment at the end o 5 years i interest is compounded a.) annually b.) quarterly c.) monthly d.) continuously

12 Sec 3.5 Inverse Trigonometric Functions A. Unit Circle and Common Values

13 B. Derivatives o Inverse Trigonometric Functions (You must know these!) d d sin.) d d cos.) d d tan 3.) d 4.) csc d d 5.) sec d d d cot 6.) Eamples.) Find the eact value o each epression. Your answer should be either a raction or an integer..) Let 7 tan. Find.. Find 8 3.) Let cos e. 4.) Find the limit: 5.) Find the limit:

14 Sec 3.7 Indeterminate orms and L Hospital s Rule A. Indeterminate orms I we have a limit o the orm orm o type 0 0 I we have a limit o the orm orm o type lim where both 0 and g 0 a g lim where both and g a g, then we have the in determinant then we have the in determinant B. L Hospital s Rule Suppose that and g are dierentiable, g 0 and that lim a g 0 0 or that lim a g (i.e. we have an in determinant orm o the type 0 0 or ), then lim a g lim a g Eamples:.).) 9 lim ) ln lim

15 4.) lim 0 sin cos 5.) a lim e lim ) 7.) 8.) 9.)

16 So the idea is to be able to get your limit problem into the orm: lim so you can use L Hospital s Rule a g I you have () g() and you check to make sure you get either then you will need to rewrite it irst. you could either rewrite it as lim a or lim a g( ) g ( ) 0 or 0 *always put the EASY unction on the bottom! 0.) lim cot sin 6 0.) =.)

17 C. Other Indeterminate Forms (you will need to rewrite this as either 0 0 or ) *Try using ractions or actoring 3.).).) 3.) For each o these orms you will need to start by rewriting the problems as y = lim. Then you will need to take the natural log (LN) o both sides in order to get your eponential unction into a multiplication problem using the property o logs. You can then change that into one unction divided by another so you can use LH Rule. And lastly, once you get that answer you must set it equal to the LN y that you started with on the let hand side. (Phew.It s tough, but you can do it!) 4.) lim 0

18 5.) lim 3 6.) lim 7 0