Practice Problems

Work through the odd-numbered problems 1-19. Once you have completed the problem set, check your answers.


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,
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