Chapter 5 Problems

Thirty chapter problems: 10 from the original book and 20 added for more practice.

Read the chapter problems → · Chapter solutions →

1. Original problem · clarified

   You deposit $1000 into the stock market that you plan to keep until you retire in 45 years. Your portfolio has an annual rate of return of 10% per year. If you are 20 years old, at what rate is your wealth growing when you are 60? (Use units of dollars per year)

   Clarification: Assume continuous compounding at an annual rate of 0.10, with t in years since the deposit.

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2. Original problem · clarified

   Find $\underset{x\to 3}{lim}\frac{x-3}{\sqrt{x-3}}+x\cdot \ln (x-3)-3\ln (x-3)$

   Clarification: Take the real-domain limit as x approaches 3 from the right.

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3. Original problem

   Solve $\underset{x\to \mathrm{\infty }}{lim}{x}^{-x^x}$

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4. Original problem

   What is$\underset{x\to \mathrm{\infty }}{lim}\sin \left(\frac{\ln x}{x}\right)$

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5. Original problem

   Calculate the derivative of $f(x)={\log }_4(x^2)+3^{x\cdot {\log }_2(x)}+4\ln (x)$

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6. Original problem

   Find the derivative of $f(x)={\tan }^{-1}({\log }_{2}x)+3{\sin }^{-1}({\cos }^{-1}x)$

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7. Original problem

   Find$\underset{x\to \mathrm{\infty }}{lim}{\left(4+\frac{3}{x}\right)}^{\frac{1}{x}}$

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8. Original problem

   Find the derivative of $f(x)=5{\sec }^{-1}(4^x)+2{\cot }^{-1}({\csc }^{-1}(x))+8^{x^{\ln x}}$

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9. Original problem

   Find the slope of the tangent line at x=0 of $f(x)=2{\sin }^{-1}(x)+3^x$

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10. Original problem

    Find the slope of the tangent line at x=0.5 of $f(x)=3{\sec }^{-1}(4^{{\log }_{2}x})+2{\cos }^{-1}({\sin }^{-1}(x))+2^{x\cdot \ln (3x)}$

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11. Added problem

    Differentiate $f\left(x\right)=\ln (x^2+1)$.

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12. Added problem

    Find the derivative of $f\left(x\right)=5^{1-2x}$.

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13. Added problem

    Find the derivative and domain of $f\left(x\right)={\log }_3(x^2-4)$.

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14. Added problem

    Use logarithmic differentiation to differentiate $y=x^x$ for $x>0$.

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15. Added problem

    Differentiate $y=(x^2+1{)}^x$.

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16. Added problem

    Differentiate $f\left(x\right)=\arcsin \left(3x\right)$ and give the finite derivative domain.

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17. Added problem

    Differentiate $f\left(x\right)=\arccos \left(x^2\right)$ on its interior domain.

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18. Added problem

    Find the derivative of $f\left(x\right)=\arctan \left(e^x\right)$.

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19. Added problem

    For the one-to-one function $f\left(x\right)=x^3+x$, find $\left(f^{-1}\right)\prime \left(10\right)$ without solving for the full inverse.

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20. Added problem

    A locally invertible function has $f\left(3\right)=7$ and $f\prime \left(3\right)=-2$. Find the derivative of its inverse at 7.

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21. Added problem

    Use L’Hospital’s rule to evaluate ${\lim }_{x\to 0}(e^{2x}-1)/x$.

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22. Added problem

    Evaluate ${\lim }_{x\to 0}(e^x-1-x)/x^2$.

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23. Added problem

    Find ${\lim }_{x\to 0^{+}}x\ln x$.

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24. Added problem

    Evaluate ${\lim }_{x\to \infty }{x}^2/e^x$.

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25. Added problem

    Evaluate ${\lim }_{x\to 0}(1+2x{)}^{1/x}$.

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26. Added problem

    A student uses L’Hospital on ${\lim }_{x\to 0}(e^x+1)/(x+1)$ and gets 1. Diagnose the mistake.

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27. Added problem

    In a fictional savings model, $A\left(t\right)=2000e^{0.04t}$ dollars, with t in years. Find the instantaneous growth rate at year 5.

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28. Added problem

    A fictional substance follows $M\left(t\right)=10e^{-kt}$ grams and halves in 6 hours. Find k and its mass rate at hour 6.

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29. Added problem

    State the derivative of $\mathrm{arcsec}x$ for $\left|x\right|>1$ and explain why $\left|x\right|$ is needed.

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30. Added problem

    Show that the derivative of $\ln \left|x\right|$ is $1/x$ on both x>0 and x<0.

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