Integrals, Exponential Functions, and Logarithms
For the following exercises (1-2), find the derivative [latex]\frac{dy}{dx}.[/latex]
- [latex]y=\text{ln}(2x)[/latex]
- [latex]y=\dfrac{1}{\text{ln}x}[/latex]
For the following exercise, find the indefinite integral.
- [latex]\displaystyle\int \frac{dx}{1+x}[/latex]
For the following exercises (4-8), find the derivative [latex]dy\text{/}dx.[/latex] (You can use a calculator to plot the function and the derivative to confirm that it is correct.)
- [latex]y=x\text{ln}(x)[/latex]
- [latex]y=\text{ln}( \sin x)[/latex]
- [latex]y=7\text{ln}(4x)[/latex]
- [latex]y=\text{ln}( \tan x)[/latex]
- [latex]y=\text{ln}({ \cos }^{2}x)[/latex]
For the following exercises (9-13), find the definite or indefinite integral.
- [latex]{\displaystyle\int }_{0}^{1}\dfrac{dt}{3+2t}[/latex]
- [latex]{\displaystyle\int }_{0}^{2}\dfrac{{x}^{3}dx}{{x}^{2}+1}[/latex]
- [latex]{\displaystyle\int }_{2}^{e}\dfrac{dx}{{(x\text{ln}(x))}^{2}}[/latex]
- [latex]{\displaystyle\int }_{0}^{\pi \text{/}4} \tan xdx[/latex]
- [latex]\displaystyle\int \frac{{(\text{ln}x)}^{2}dx}{x}[/latex]
For the following exercises (14-18), compute [latex]dy\text{/}dx[/latex] by differentiating [latex]\text{ln}y.[/latex]
- [latex]y=\sqrt{{x}^{2}+1}\sqrt{{x}^{2}-1}[/latex]
- [latex]y={x}^{-1\text{/}x}[/latex]
- [latex]y={x}^{e}[/latex]
- [latex]y=\sqrt{x}\sqrt[3]{x}\sqrt[6]{x}[/latex]
- [latex]y={e}^{\text{−}\text{ln}x}[/latex]
For the following exercises (19-20), evaluate by any method.
- [latex]{\displaystyle\int }_{1}^{{e}^{\pi }}\frac{dx}{x}+{\displaystyle\int }_{-2}^{-1}\frac{dx}{x}[/latex]
- [latex]\frac{d}{dx}{\displaystyle\int }_{x}^{{x}^{2}}\frac{dt}{t}[/latex]
For the following exercises (21-22), use the function [latex]\text{ln}x.[/latex] If you are unable to find intersection points analytically, use a calculator.
- Find the area of the region enclosed by [latex]x=1[/latex] and [latex]y=5[/latex] above [latex]y=\text{ln}x.[/latex]
- Find the area between [latex]\text{ln}x[/latex] and the [latex]x[/latex]-axis from [latex]x=1\text{ to }x=2.[/latex]
For the following exercises (23-27), verify the derivatives and antiderivatives.
- [latex]\frac{d}{dx}\text{ln}(x+\sqrt{{x}^{2}+1})=\dfrac{1}{\sqrt{1+{x}^{2}}}[/latex]
- [latex]\frac{d}{dx}\text{ln}\left(\dfrac{x-a}{x+a}\right)=\dfrac{2a}{({x}^{2}-{a}^{2})}[/latex]
- [latex]\frac{d}{dx}\text{ln}\left(\dfrac{1+\sqrt{1-{x}^{2}}}{x}\right)=-\dfrac{1}{x\sqrt{1-{x}^{2}}}[/latex]
- [latex]\frac{d}{dx}\text{ln}(x+\sqrt{{x}^{2}-{a}^{2}})=\dfrac{1}{\sqrt{{x}^{2}-{a}^{2}}}[/latex]
- [latex]\displaystyle\int \frac{dx}{x\text{ln}(x)\text{ln}(\text{ln}x)}=\text{ln}(\text{ln}(\text{ln}x))+C[/latex]
Exponential Growth and Decay
True or False? If true, prove it. If false, find the true answer (1-2).
- If you invest [latex]$500,[/latex] an annual rate of interest of [latex]3\%[/latex] yields more money in the first year than a [latex]2.5\%[/latex] continuous rate of interest.
- If given a half-life of [latex]t[/latex] years, the constant [latex]k[/latex] for [latex]y={e}^{kt}[/latex] is calculated by [latex]k=\text{ln}(1\text{/}2)\text{/}t.[/latex]
For the following exercises (3-11), use [latex]y={y}_{0}{e}^{kt}.[/latex]
- If bacteria increase by a factor of [latex]10[/latex] in [latex]10[/latex] hours, how many hours does it take to increase by [latex]100?[/latex]
- If a relic contains [latex]90\%[/latex] as much radiocarbon as new material, can it have come from the time of Christ (approximately 2000 years ago)? Note that the half-life of radiocarbon is [latex]5730[/latex] years.
- The populations of New York and Los Angeles are growing at [latex]1\%[/latex] and [latex]1.4\%[/latex] a year, respectively. Starting from [latex]8[/latex] million (New York) and [latex]6[/latex] million (Los Angeles), when are the populations equal?
- The effect of advertising decays exponentially. If [latex]40\%[/latex] of the population remembers a new product after [latex]3[/latex] days, how long will [latex]20\%[/latex] remember it?
- If [latex]y=100[/latex] at [latex]t=4[/latex] and [latex]y=10[/latex] at [latex]t=8,[/latex] when does [latex]y=1?[/latex]
- What continuous interest rate has the same yield as an annual rate of [latex]9\text{%}?[/latex]
- You are trying to save [latex]$50,000[/latex] in [latex]20[/latex] years for college tuition for your child. If interest is a continuous [latex]10\text{%},[/latex] how much do you need to invest initially?
- You are trying to thaw some vegetables that are at a temperature of [latex]1\text{°}\text{F}\text{.}[/latex] To thaw vegetables safely, you must put them in the refrigerator, which has an ambient temperature of [latex]44\text{°}\text{F}.[/latex] You check on your vegetables [latex]2[/latex] hours after putting them in the refrigerator to find that they are now [latex]12\text{°}\text{F}\text{.}[/latex] Plot the resulting temperature curve and use it to determine when the vegetables reach [latex]33\text{°}\text{F}\text{.}[/latex]
- The spent fuel of a nuclear reactor contains plutonium-239, which has a half-life of [latex]24,000[/latex] years. If [latex]1[/latex] barrel containing [latex]10[/latex]kg of plutonium-239 is sealed, how many years must pass until only [latex]10g[/latex] of plutonium-239 is left?
For the next set of exercises (12-15), use the following table, which features the world population by decade.
| Years since 1950 | Population (millions) |
|---|---|
| [latex]0[/latex] | [latex]2,556[/latex] |
| [latex]10[/latex] | [latex]3,039[/latex] |
| [latex]20[/latex] | [latex]3,706[/latex] |
| [latex]30[/latex] | [latex]4,453[/latex] |
| [latex]40[/latex] | [latex]5,279[/latex] |
| [latex]50[/latex] | [latex]6,083[/latex] |
| [latex]60[/latex] | [latex]6,849[/latex] |
- The best-fit exponential curve to the data of the form [latex]P(t)=a{e}^{bt}[/latex] is given by [latex]P(t)=2686{e}^{0.01604t}.[/latex] Use a graphing calculator to graph the data and the exponential curve together.
- Find and graph the derivative [latex]{y}^{\prime }[/latex] of your equation. Where is it increasing and what is the meaning of this increase?
- Find and graph the second derivative of your equation. Where is it increasing and what is the meaning of this increase?
- Find the predicted date when the population reaches [latex]10[/latex] billion. Using your previous answers about the first and second derivatives, explain why exponential growth is unsuccessful in predicting the future.
For the next set of exercises (16-18), use the following table, which shows the population of San Francisco during the 19th century.
| Years since 1850 | Population (thousands) |
|---|---|
| [latex]0[/latex] | [latex]21.00[/latex] |
| [latex]10[/latex] | [latex]56.80[/latex] |
| [latex]20[/latex] | [latex]149.5[/latex] |
| [latex]30[/latex] | [latex]234.0[/latex] |
- The best-fit exponential curve to the data of the form [latex]P(t)=a{e}^{bt}[/latex] is given by [latex]P(t)=35.26{e}^{0.06407t}.[/latex] Use a graphing calculator to graph the data and the exponential curve together.
- Find and graph the derivative [latex]{y}^{\prime }[/latex] of your equation. Where is it increasing? What is the meaning of this increase? Is there a value where the increase is maximal?
- Find and graph the second derivative of your equation. Where is it increasing? What is the meaning of this increase?
Calculus of the Hyperbolic Functions
- Find expressions for [latex]\text{cosh}x+\text{sinh}x[/latex] and [latex]\text{cosh}x-\text{sinh}x.[/latex] Use a calculator to graph these functions and ensure your expression is correct.
- Show that [latex]\text{cosh}(x)[/latex] and [latex]\text{sinh}(x)[/latex] satisfy [latex]y\text{″}=y.[/latex]
- Derive [latex]{\text{cosh}}^{2}(x)+{\text{sinh}}^{2}(x)=\text{cosh}(2x)[/latex] from the definition.
- Prove [latex]\text{sinh}(x+y)=\text{sinh}(x)\text{cosh}(y)+\text{cosh}(x)\text{sinh}(y)[/latex] by changing the expression to exponentials.
For the following exercises (5-9), find the derivatives of the given functions and graph along with the function to ensure your answer is correct.
- [latex]\text{cosh}(3x+1)[/latex]
- [latex]\frac{1}{\text{cosh}(x)}[/latex]
- [latex]{\text{cosh}}^{2}(x)+{\text{sinh}}^{2}(x)[/latex]
- [latex]\text{tanh}(\sqrt{{x}^{2}+1})[/latex]
- [latex]{\text{sinh}}^{6}(x)[/latex]
For the following exercises (10-14), find the antiderivatives for the given functions.
- [latex]\text{cosh}(2x+1)[/latex]
- [latex]x\text{cosh}({x}^{2})[/latex]
- [latex]{\text{cosh}}^{2}(x)\text{sinh}(x)[/latex]
- [latex]\frac{\text{sinh}(x)}{1+\text{cosh}(x)}[/latex]
- [latex]\text{cosh}(x)+\text{sinh}(x)[/latex]
For the following exercises (15-18), find the derivatives for the functions.
- [latex]{\text{tanh}}^{-1}(4x)[/latex]
- [latex]{\text{sinh}}^{-1}(\text{cosh}(x))[/latex]
- [latex]{\text{tanh}}^{-1}( \cos (x))[/latex]
- [latex]\text{ln}({\text{tanh}}^{-1}(x))[/latex]
For the following exercises (19-21), find the antiderivatives for the functions.
- [latex]\displaystyle\int \frac{dx}{{a}^{2}-{x}^{2}}[/latex]
- [latex]\displaystyle\int \frac{xdx}{\sqrt{{x}^{2}+1}}[/latex]
- [latex]\displaystyle\int \frac{{e}^{x}}{\sqrt{{e}^{2x}-1}}[/latex]
For the following exercises (22-26), solve each problem.
- A chain hangs from two posts [latex]2[/latex] m apart to form a catenary described by the equation [latex]y=2\text{cosh}(x\text{/}2)-1.[/latex] Find the slope of the catenary at the left fence post.
- A high-voltage power line is a catenary described by [latex]y=10\text{cosh}(x\text{/}10).[/latex] Find the ratio of the area under the catenary to its arc length. What do you notice?
- Prove the formula for the derivative of [latex]y={\text{sinh}}^{-1}(x)[/latex] by differentiating [latex]x=\text{sinh}(y).[/latex] (Hint: Use hyperbolic trigonometric identities.)
- Prove the formula for the derivative of [latex]y={\text{sech}}^{-1}(x)[/latex] by differentiating [latex]x=\text{sech}(y).[/latex] (Hint: Use hyperbolic trigonometric identities.)
- Prove the expression for [latex]{\text{sinh}}^{-1}(x).[/latex] Multiply [latex]x=\text{sinh}(y)=(1\text{/}2)({e}^{y}-{e}^{\text{−}y})[/latex] by [latex]2{e}^{y}[/latex] and solve for [latex]y.[/latex] Does your expression match the text?