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Chapter 4: Problem 24

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error. $$\sqrt[3]{65}$$

Short Answer

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#tag_title# Check the Approximation #tag_content# To check the accuracy of our approximation, let's calculate the decimal value of $$4 + \frac{1}{48}$$: $$\sqrt[3]{65} \approx 4 + \frac{1}{48} \approx 4.0208$$ Now, let's compare this result with the actual cube root of 65 calculated using a calculator: $$\sqrt[3]{65} \approx 4.0212$$ As we can see, our linear approximation of $$\sqrt[3]{65}$$ is quite close to the actual value, with a minor difference. This confirms that our method of using a linear approximation was effective in estimating the cube root of 65.

Step by step solution

01

Choose the Suitable Value of a

Since 65 lies between two perfect cube numbers, 64 and 81 (4^3 and 5^3), we will choose the nearest cubic number a, which is 64, as it will give a smaller error.
02

Define the Function and its Derivative

We have the function $$f(x)=\sqrt[3]{x}$$. We will now find its derivative f'(x). $$f'(x) = \frac{d(\sqrt[3]{x})}{dx} = \frac{1}{3\sqrt[3]{x^2}}$$
03

Find the Values of the Function and Derivative at Point a

Next, we need to find the values of the function and its derivative at the chosen point a=64. $$f(64) = \sqrt[3]{64} = 4$$ $$f'(64) = \frac{1}{3\sqrt[3]{64^2}} = \frac{1}{3(4^2)}=\frac{1}{48}$$
04

Linear Approximation Formula

We'll use the linear approximation formula f(x) 鈮� f(a) + f'(a)(x - a) to find an estimate for $$\sqrt[3]{65}$$.
05

Calculate the Estimated Value

Now, let's plug in the values of a, x, f(a), and f'(a) into the linear approximation formula: $$f(65) \approx f(64) + f'(64)(65 - 64)$$ $$\sqrt[3]{65} \approx 4 + \frac{1}{48}(65 - 64)$$ $$\sqrt[3]{65} \approx 4 + \frac{1}{48}$$ Therefore, the cube root of 65 is approximately $$4 + \frac{1}{48}$$.

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Most popular questions from this chapter

Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation \(a(t)=v^{\prime}(t)=g,\) where \(g=-9.8 \mathrm{m} / \mathrm{s}^{2}\). a. Find the velocity of the object for all relevant times. b. Find the position of the object for all relevant times. c. Find the time when the object reaches its highest point. What is the height? d. Find the time when the object strikes the ground. A payload is dropped at an elevation of \(400 \mathrm{m}\) from a hot-air balloon that is descending at a rate of \(10 \mathrm{m} / \mathrm{s}\).

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Given the following velocity functions of an object moving along a line, find the position function with the given initial position. Then graph both the velocity and position functions. $$v(t)=2 \sqrt{t} ; s(0)=1$$

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Use analytical methods to evaluate the following limits. $$\lim _{x \rightarrow 1^{+}}\left(\frac{1}{x-1}-\frac{1}{\sqrt{x-1}}\right)$$
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