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Chapter 3: Problem 46

Evaluate the derivative of the following functions at the given point. $$A=\pi r^{2} ; r=3$$

Short Answer

Expert verified
Answer: The rate at which the area of the circle changes with respect to its radius when r = 3 is $$6\pi$$ square units per unit increase in the radius.

Step by step solution

01

We are given a function $$A = \pi r^2$$, which represents the area of a circle with respect to its radius (r). We need to find the derivative of A with respect to r, and then evaluate it at r = 3. #Step 2: Differentiate A with respect to r#

To find the derivative of A with respect to r, we'll apply the power rule. The power rule states that if $$f(x) = x^n$$, where n is a constant, then $$f'(x) = nx^{n-1}$$. Using the power rule, we get $$\frac{dA}{dr} = \frac{d}{dr}(\pi r^2) = 2\pi r^{2-1} = 2\pi r$$. #Step 3: Evaluate the derivative at r = 3#
02

Now that we have the derivative, we need to evaluate it at r = 3. Substitute r = 3 into the derivative: $$\frac{dA}{dr}|_{r=3} = 2\pi (3) = 6 \pi$$. #Step 4: Interpret the result#

The derivative we found, $$\frac{dA}{dr} = 6 \pi$$, represents the rate at which the area of the circle changes with respect to its radius when r = 3. In other words, when the radius of the circle is 3 units, the area is increasing at a rate of $$6\pi$$ square units per unit increase in the radius.

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