Mathematics IA Worked Examples CALCULUS: THE … IA Worked Examples CALCULUS: THE LOGARITHM AND...
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Mathematics IA Worked Examples CALCULUS: THE LOGARITHM AND EXPONENTIAL FUNCTIONS Produced by the Maths Learning Centre, The University of Adelaide. May 3, 2013 The questions on this page have worked solutions and links to videos on the following pages. Click on the link with each question to go straight to the 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 ln x 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 U 6 for Z 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 (e a ) b = e ab . 4. See Page 7 for worked solutions. Solve the following equations for x ∈ R: (a) ln(x + 3) + ln(x - 3) = 3 (b) 3e x =2+5e -x . 5. See Page 9 for worked solutions. Find the derivatives of the following functions: (a) ln(5x 4 ) (b) x 2 e x 6. See Page 10 for worked solutions. Find the derivatives of the following functions: (a) log 10 (x + 2) (b) 3 · 2 3x (c) x ln x 1
Transcript of Mathematics IA Worked Examples CALCULUS: THE … IA Worked Examples CALCULUS: THE LOGARITHM AND...
Mathematics IAWorked Examples
CALCULUS: THE LOGARITHM ANDEXPONENTIAL FUNCTIONS
Produced by the Maths Learning Centre,The University of Adelaide.
May 3, 2013
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
tdt 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 + 5e−x .
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
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7. See Page 12 for worked solutions.Use the definitions of sinhx and cosh x to show thatsinh2 x = 1
2
(cosh(2x)− 1
).
8. See Page 13 for worked solutions.Let arctanhx be the inverse function of tanhx. Write down the domainand range of arctanhx and find a formula for arctanhx.
9. See Page 15 for worked solutions.Find the derivatives of the following functions:
(a) sinh(x2) (b) tanh3(x) (c) cosh x cosx
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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 lnx and the Fundamental Theo-rem of Calculus, show that ln a + ln b = ln(ab), for all a, b > 0.
Link to video on YouTube
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2. Click here to go to question list.
Use the upper sum U6 for
∫ 2.5
1
1
tdt to show that e > 2.5.
Link to video on YouTube
4
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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
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 + 5e−x .
Link to video on YouTube
7
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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
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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
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7. Click here to go to question list.Use the definitions of sinhx and cosh x to show thatsinh2 x = 1
2
(cosh(2x)− 1
).
Link to video on YouTube
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8. Click here to go to question list.Let arctanhx be the inverse function of tanhx. Write down the domainand range of arctanhx and find a formula for arctanhx.
Link to video on YouTube
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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 cosx
Link to video on YouTube
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