Chapter 3 Problems

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

Read the chapter problems → · Chapter solutions →

1. Original problem

   Find the slope of the tangent line at x = –1 of the piecewise function:​
   $f(x)=\{\begin{array}{rlr}{x}^2+1& \phantom{p}if& x<1\\ -x+2& \phantom{p}if& x>1\end{array}$

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2. Original problem

   Find the derivative of the functions in #2 – 5 using only the power rule:​
   $f(x)=\frac{2x^3+4x^2+3x+\frac{1}{x}}{x}$

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3. Original problem · clarified

   $f(x)=3e^x+\frac{4}{x}+\sqrt{x}$

   Clarification: Use the exponential derivative rule as well as the power rule.

   Show your work for Problem 3

   Shared source instruction or preceding problem:

   Find the derivative of the functions in #2 – 5 using only the power rule:​
   

   The source instruction in Problem 2 governs Problems 2–5. Each task’s own existing clarification stays attached and takes precedence where supplied.

   

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4. Original problem

   $f(x)=\sec x+\tan x(1-\sin x)$

   Show your work for Problem 4

   Shared source instruction or preceding problem:

   Find the derivative of the functions in #2 – 5 using only the power rule:​
   

   The source instruction in Problem 2 governs Problems 2–5. Each task’s own existing clarification stays attached and takes precedence where supplied.

   Instruction clarification for Problems 4 and 5: the original shared direction says to use only the power rule. These two problems contain trigonometric functions and require trigonometric derivative rules; the power-rule-only restriction cannot be followed for them. Use the appropriate trigonometric derivative rules. The original direction, task text and corrected answers are retained.

   

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5. Original problem

   $f(x)=\frac{\cos x}{\sqrt{1-{\cos }^{2}x}}+\cos x\cot x+\sin x$

   Show your work for Problem 5

   Shared source instruction or preceding problem:

   Find the derivative of the functions in #2 – 5 using only the power rule:​
   

   The source instruction in Problem 2 governs Problems 2–5. Each task’s own existing clarification stays attached and takes precedence where supplied.

   Instruction clarification for Problems 4 and 5: the original shared direction says to use only the power rule. These two problems contain trigonometric functions and require trigonometric derivative rules; the power-rule-only restriction cannot be followed for them. Use the appropriate trigonometric derivative rules. The original direction, task text and corrected answers are retained.

   

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6. Original problem

   Carry out three iterations of the Newton-Raphson approximation method for the function $f(x)=x^3-3x^2+3x+3$. Use x₀= 0.

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7. Original problem

   Carry out two iterations of Newton Raphson approximation, given that:​
   $f(x)=\ln (x),f^{\prime }(x)=\frac{1}{x}$​
   Use x_o= e

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   Source discrepancy: this image gives initial value 0, but the existing readable problem gives initial value e. Both source versions are retained. This overlay does not choose a replacement starting value or change an answer.

   

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8. Original problem

   Calculate the second derivative of the functions in #8 – 10: $f(x)=2x^3+4x^2+3x+\frac{1}{x}$

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9. Original problem

   $f(x)=\sin x-\cos x$

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   Shared source instruction or preceding problem:

   Calculate the second derivative of the functions in #8 – 10:

   The source instruction in Problem 8 governs Problems 8–10.

   

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10. Original problem

    $f(x)=3e^x+\frac{x^2-1}{x+1}+{\left(4\sqrt{x}\right)}^3$

    Show your work for Problem 10

    Shared source instruction or preceding problem:

    Calculate the second derivative of the functions in #8 – 10:

    The source instruction in Problem 8 governs Problems 8–10.

    

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11. Added problem

    Find the derivative of $f\left(x\right)=5x^4-3x^2+2x-7$.

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12. Added problem

    Differentiate $f\left(x\right)=1/\sqrt{x}+x\sqrt{x}$ for $x>0$.

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13. Added problem

    Differentiate $f\left(x\right)=2\sin x-3\cos x$.

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14. Added problem

    Find $f\prime \left(x\right)$ for $f\left(x\right)=4e^x-\tan x$, and state its exclusions.

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15. Added problem

    Find the tangent line to $y=x^3$ at $x=2$.

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16. Added problem

    Let $f\left(x\right)=x^2$ for $x\le 2$ and $f\left(x\right)=ax+b$ for $x>2$. Find a and b so f is differentiable at 2.

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17. Added problem

    Where is $f\left(x\right)=|x-3|$ differentiable, and what is its derivative?

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18. Added problem

    Use Newton's method for $x^2-3=0$, starting at $x_0=2$, and compute two iterations.

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19. Added problem

    Why does Newton's method fail immediately for $f\left(x\right)=x^3-1$ with $x_0=0$?

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20. Added problem

    For $f\left(x\right)=2x^3-3x^2$, find f″ and the intervals of concavity.

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21. Added problem

    A position is $s\left(t\right)=t^3-6t^2+9t$ meters, with t in seconds. Find velocity and acceleration, then evaluate both at $t=1$.

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22. Added problem

    Simplify first, then differentiate $f\left(x\right)=(1+\cos x)/\sin x-\cot x$.

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23. Added problem

    Use the difference quotient to derive $f\prime \left(a\right)$ for $f\left(x\right)=x^2+3x$.

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24. Added problem

    A derivative table gives $f\prime \left(a\right)=2$, $f\prime \left(b\right)=-1$, and $f\prime \left(c\right)=0$. What can you conclude at these three inputs? Can you declare c a maximum?

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25. Added problem

    Use the tangent line at $x=4$ to estimate $\sqrt{4.1}$.

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26. Added problem

    Find the third derivative of $f\left(x\right)=6e^x-\sin x$.

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27. Added problem

    Use the difference quotient to examine $f\left(x\right)=x^{1/3}$ at zero.

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28. Added problem

    Show that $f\left(x\right)=1/x^2$ is decreasing and concave up on $x>0$.

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29. Added problem

    For $f\left(x\right)=x^3-2x+2$, compute two Newton steps from $x_0=0$. What problem do you notice?

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30. Added problem

    A student writes $d\left(x^{-3}\right)/dx=3x^{-2}$. Diagnose and fix the error.

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