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    Mathematics IA

    Worked Examples

    CALCULUS: THE LOGARITHM ANDEXPONENTIAL FUNCTIONS

    Produced by the Maths Learning Service,CLPD, The University of Adelaide.

    April 18, 2011

    The questions on this page have worked solutions and links to videos onthe following pages. Click on the link with each question to go straight tothe relevant page.

    Questions

    1. See Page 3 for worked solutions.

    (a) Let a >0. Find the derivative of the function ln(ax).

    (b) Using the definition of the function lnx

    and the Fundamental Theo-rem of Calculus, show that ln a+ ln b= ln(ab), for all a, b >0.

    2. See Page 4 for worked solutions.

    Use the upper sum U6 for

    2.5

    1

    1

    t dt to show that e >2.5.

    3. See Page 6 for worked solutions.Using the properties of ln x, show that (ea)b = eab.

    4. See Page 7 for worked solutions.Solve the following equations for x R:

    (a) ln(x+ 3) + ln(x 3) = 3 (b) 3ex = 2 + 5ex .

    5. See Page 9 for worked solutions.Find the derivatives of the following functions:

    (a) ln(5x4) (b) x2ex

    6. See Page 10 for worked solutions.Find the derivatives of the following functions:

    (a) log10(x+ 2) (b) 3 23x (c) xlnx

    1

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    7. See Page 12 for worked solutions.Use the definitions of sinh xand cosh xto show thatsinh2 x= 1

    2

    cosh(2x) 1

    .

    8. See Page 13 for worked solutions.Let arctanh xbe the inverse function of tanh x. Write down the domainand range of arctanh xand find a formula for arctanh x.

    9. See Page 15 for worked solutions.Find the derivatives of the following functions:

    (a) sinh(x2) (b) tanh3(x) (c) cosh x cos x

    2

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    1. Click here to go to question list.

    (a) Let a >0. Find the derivative of the function ln(ax).

    (b) Using the definition of the function ln xand the Fundamental Theo-rem of Calculus, show that ln a+ ln b= ln(ab), for all a, b >0.

    Link to video on YouTube

    3

    http://www.youtube.com/watch?v=ZuPfFGspx8khttp://www.youtube.com/watch?v=ZuPfFGspx8k

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    2. Click here to go to question list.

    Use the upper sum U6 for

    2.5

    1

    1

    t dt to show that e >2.5.

    Link to video on YouTube

    4

    http://www.youtube.com/watch?v=Lwq6QjxenE8http://www.youtube.com/watch?v=Lwq6QjxenE8

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    5

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    3. Click here to go to question list.Using the properties of ln x, show that (ea)b = eab.

    Link to video on YouTube

    6

    http://www.youtube.com/watch?v=kaoD9JKNIZYhttp://www.youtube.com/watch?v=kaoD9JKNIZY

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    4. Click here to go to question list.Solve the following equations for x R:

    (a) ln(x+ 3) + ln(x 3) = 3 (b) 3ex = 2 + 5ex .

    Link to video on YouTube

    7

    http://www.youtube.com/watch?v=32A5o5rCe48http://www.youtube.com/watch?v=32A5o5rCe48

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    8

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    5. Click here to go to question list.Find the derivatives of the following functions:

    (a) ln(5x4) (b) x2ex

    Link to video on YouTube

    9

    http://www.youtube.com/watch?v=qgHvcI76LoEhttp://www.youtube.com/watch?v=qgHvcI76LoE

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    6. Click here to go to question list.Find the derivatives of the following functions:

    (a) log10(x+ 2) (b) 3 23x (c) xlnx

    Link to video on YouTube

    10

    http://www.youtube.com/watch?v=BSjD6GXDBFAhttp://www.youtube.com/watch?v=BSjD6GXDBFA

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    11

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    7. Click here to go to question list.Use the definitions of sinh xand cosh xto show thatsinh2 x= 1

    2

    cosh(2x) 1

    .

    Link to video on YouTube

    12

    http://www.youtube.com/watch?v=atL-GBidt1ghttp://www.youtube.com/watch?v=atL-GBidt1g

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    8. Click here to go to question list.Let arctanh xbe the inverse function of tanh x. Write down the domainand range of arctanh xand find a formula for arctanh x.

    Link to video on YouTube

    13

    http://www.youtube.com/watch?v=tTP8PcjYpKghttp://www.youtube.com/watch?v=tTP8PcjYpKg

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    14

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    9. Click here to go to question list.Find the derivatives of the following functions:

    (a) sinh(x2) (b) tanh3(x) (c) cosh x cos x

    Link to video on YouTube

    15

    http://www.youtube.com/watch?v=20r2a6xBoCwhttp://www.youtube.com/watch?v=20r2a6xBoCw

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