Practice Problems

1. Fig. 10 shows the tangent line to a function $\mathrm{g}$ at the point $(2,2)$ and a line segment $\Delta \mathrm{x}$ units long. On the figure, label the locations of 

(a) $2+\Delta x$ on the $x$-axis, (b) the point $(2+\Delta x, g(2+\Delta x))$, and

(c) the point $\left(2+\Delta x, g(2)+g^{\prime}(2) \cdot \Delta x\right)$.

(d) How large is the "error", $\left(g(2)+g^{\prime}(2) \cdot \Delta x\right)-(g(2+\Delta x))?$

In problem 3, find the equation of the tangent line $L$ to the given function $f$ at the given point $(\mathrm{a}, \mathrm{f}(\mathrm{a}))$. Use the value $\mathrm{L}(\mathrm{a}+\Delta \mathrm{x})$ to approximate the value of $\mathrm{f}(\mathrm{a}+\Delta \mathrm{x})$

3. (a) $\mathrm{f}(\mathrm{x})=\sqrt{\mathrm{x}}, \mathrm{a}=4, \Delta \mathrm{x}=0.2$

(b) $\mathrm{f}(\mathrm{x})=\sqrt{\mathrm{x}}, \mathrm{a}=81, \Delta \mathrm{x}=-1$

(c) $f(x)=\sin (x), a=0, \Delta x=0.3$

In problem 7, use the Linear Approximation Process to derive each approximation formula for $\mathrm{x}$ "close to" $\mathbf{0}$.

7. (a) $\ln (1+x) \approx x$

(b) $\cos (\mathrm{x}) \approx 1$

(c) $\tan (\mathrm{x}) \approx \mathrm{x}$

(d) $\sin (\pi / 2+x) \approx 1$

9. A rectangle has one side on the $\mathrm{x}$-axis, one side on the y-axis, and a corner on the graph of $y=x^{2}+1$ (Fig. 13).

(a) Use Linear Approximation of the area formula to estimate the increase in the area of the rectangle if the base grows from 2 to 2.3 inches.

(b) Calculate exactly the increase in the area of the rectangle as the base grows from 2 to 2.3 inches.

11. You are minting gold coins which must have a volume of $47.3 \pm 0.1 \mathrm{~cm}^{3}$. If you can manufacture the coins to be exactly $2 \mathrm{~cm}$ high, how much variation can you allow for the radius?

13. Your company is making dice (cubes) and the specifications require that their volume be $87 \pm 2 \mathrm{~cm}^{3}$. How long should each side be and how much variance can a side have in order to meet the specifications?

15. The period $\mathrm{P}$, in seconds, for a pendulum to make one complete swing and return to the release point is $\mathrm{P}=2 \pi$ $\sqrt{\mathrm{L} / \mathrm{g}}$ where $\mathrm{L}$ is the length of the pendulum in feet and $\mathrm{g}$ is $32 \mathrm{feet} / \mathrm{sec}^{2}$.

(a) If $L=2$ feet, what is the period of the pendulum?

(b) If $P=1$ second, how long is the pendulum?

(c) Estimate the change in $\mathrm{P}$ if $\mathrm{L}$ increases from $\mathrm{2}$ feet to $\mathrm{2.1}$ feet.

(d) The length of a $\mathrm{24}$ foot pendulum is increasing $\mathrm{2}$ inches per hour. Is the period getting longer or shorter? How fast is the period changing?

17. For the function in Fig. 14, estimate the value of $\mathbf{d f}$ when

(a) $\mathrm{x}=2$ and $\mathrm{dx}=1$

(b) $\mathrm{x}=4$ and $\mathrm{dx}=-1$

(c) $x=3$ and $d x=2$

19. Calculate the differentials df of the following functions:

(a) $f(x)=x^{2}-3 x$

(b) $f(x)=e^{x}$

(c) $f(x)=\sin (5 x)$

(d) $f(x)=x^{3}+2 x$ with $x=1$ and $d x=0.2$

(e) $\mathrm{f}(\mathrm{x})=\ln (\mathrm{x})$ with $\mathrm{x}=\mathrm{e}$ and $\mathrm{dx}=-0.1$

(f) $\mathrm{f}(\mathrm{x})=\sqrt{2 \mathrm{x}+5}$ with $\mathrm{x}=22$ and $\mathrm{dx}=3$.

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-3.9-Linear-Approximation.pdf
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