Question

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

Short answer

The estimated value of the cube root of 65 using linear approximation is approximately 4.0208.

Step-by-step solution

Choose an appropriate value for a

Let's choose a value of a for which the cube root is easy to calculate, and that is close to 65. Since 4^3 = 64 is very close to 65, let's choose a = 64. In this case, we know that the cube root of a is 4, i.e., $$\sqrt[3]{64} = 4$$.

Find the derivative of the cube root function

The cube root function can be written as $$f(x)=x^{\frac{1}{3}}$$. To find the derivative of this function, we'll use the power rule: $$f'(x) = \frac{d}{dx}(x^{\frac{1}{3}}) = \frac{1}{3} x^{-\frac{2}{3}}$$.

Evaluate the derivative at a

Now, we'll find the value of the derivative at the chosen point a = 64: $$f'(64) = \frac{1}{3} (64)^{-\frac{2}{3}} = \frac{1}{3}(4^{-2}) = \frac{1}{3}\cdot \frac{1}{16} = \frac{1}{48}$$.

Use the linear approximation formula

The linear approximation formula is given by: $$f(x) \approx f(a) + f'(a)(x - a)$$. Plugging in the values we've found, we get: $$\sqrt[3]{65} \approx f(64) + f'(64)(65-64)$$ $$\sqrt[3]{65} \approx 4 + \frac{1}{48}(1)$$ $$\sqrt[3]{65} \approx 4 + \frac{1}{48}$$ $$\sqrt[3]{65} \approx 4.0208$$ So, the linear approximation of the cube root of 65 is approximately 4.0208.

Key concepts

Cube Root

A cube root is a special mathematical function used to determine a number that produces a given number when multiplied by itself twice. For example, the cube root of 27 is 3 because multiplying 3 by itself twice (3 x 3 x 3) results in 27. In symbolic terms, if we say \( \sqrt[3]{x} \), it refers to the cube root of \( x \).

For instance, when dealing with the cube root of a number like 65, often a precise mental calculation is challenging. This is where approximations become valuable. By using values close to the target number, such as 64 for \( \sqrt[3]{65} \), we can estimate more manageable numbers. In this case, since \( 64 = 4^3 \), it's easier to ascertain that \( \sqrt[3]{64} = 4 \). This proximity allows for a simplified estimation, which forms the basis for more complex calculations using techniques such as linear approximation.

Derivative

A derivative fundamentally represents the rate at which a function changes at any given point. It is a core concept in calculus, assessing how a mathematical function responds to infinitesimal changes in its input. When it comes to understanding the behavior of functions like the cube root, the derivative gives essential insights.

For the cube root function, \( f(x) = x^{1/3} \), computing the derivative involves leveraging calculus tools to examine the instantaneous rate of change. The derivative of this function tells us how the cube root of a number increases or decreases as we move away from \( x \). Knowing this rate of change is especially useful when applying linear approximation to find values close to known cube roots, as it ensures our estimates remain accurate over small intervals.

Power Rule

The power rule is a straightforward method utilized in calculus to find the derivative of functions whose variables have exponents. It states that if you have a power function \( f(x) = x^n \), the derivative is \( f'(x) = nx^{n-1} \). This simple approach is extremely useful when working with polynomial expressions.

In the context of finding the derivative of the cube root function, \( f(x) = x^{1/3} \), applying the power rule involves reducing the exponent by 1 and multiplying it by the original exponent, as per the formula. So, the derivative \( f'(x) = \frac{1}{3}x^{-2/3} \). This derivative form provides the essential foundation for further calculations, such as linear approximations, allowing predictions about the function's values for inputs near a given point, like 64 for \( \sqrt[3]{65} \).