Question

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Answer 1

Answer:

(B)  $\displaystyle \frac{W(3.1) - W(2.9)}{0.2}$

General Formulas and Concepts:

Calculus

Limits

Derivatives

• The definition of a derivative is the slope of the tangent line.

Derivative Notation

Instantaneous Rates

• Tangent Line: $\displaystyle f'(x) = \frac{f(b) - f(a)}{b - a}$

Step-by-step explanation:

Since we are trying to find a rate at which W(t) changes, we must find the derivative at t = 3.

We are given 2 close answer choices that would have the same numerical answer but different meanings:

1. (A)  $\displaystyle \lim_{t \to 3} W(t)$
2. (B)  $\displaystyle \frac{W(3.1) - W(2.9)}{0.2}$

If we look at answer choice (A), we see that our units would simply just be volume. It would not have the units of a rate of change. Yes, it may be the closest numerically correct answer, but it does not tell us the rate at which the volume would be changing and it is not a derivative.

If we look at answer choice (B), we see that our units would be cm³/s, and that is most certainly a rate of change. Answer choice (B) is also a derivative at t = 3, and a derivative tells us what rate something is changing.

∴ Answer choice (B) will give us the best estimate for the value of the instantaneous rate of change of W(t) when t = 3.

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

Book: College Calculus 10e