Calculus Question

Question

Using the table below, estimate the instantaneous rate of change when x=3

x: 1, 2, 3, 4, 5

y: 0, -2, -6, -12, -20

Answer 1

There are a couple of ways of doing this problem. Please note that this problems asks to ESTIMATE the instantaneous velocity. There is no hard right or wrong answer.

Method 1) Calculate the average velocity just below the point in question.

We use the point (3, -6) and (2, -2) and essentially use m = (y₂ - y₁)/(x₂ - x₁) except most calculus books identify it as (f(x₂) - f(x₁)/(x₂ - x₁) but, at any rate, its (-2 - -6)/(2 - 3) = -4 length/time (you don't specify the units).

Method 2: Calculate the average velocity just above the point in question.

We use the point (3, -6) and (4, -12) and m = (y₂ - y₁)/(x₂ - x₁) again to get (-12 - -6)/(4 - 3) = -6 length/time.

Method 3: Average the two above

(-4 + -6)/2 = -5 length/time

Method 4: Curve Fit the data to get a Least Squares equation and then, using the equation, calculate instantaneous velocity.

Using your TI-84 calculator, put the data in a list and use the Least Squares curve-fitting to get various possible equations. A line has an r² = 0.95. A quadratic has an r² = 1. Can't beat that so let's use it. The equation that fits the data is y = -x² + x. You can either take the derivative (if you've been instructed in that) or you can use the limit definition. I'll assume you have not learned to take a derivative but have learned the limit definition. The instantaneous velocity = the limit as h approaches zero of [f(x+h) - f(x)]/h

f(x+h) = -(x + h)² + (x + h) = -(x² + 2xh + h²) + x + h = -x² - 2xh + h² + x + h

f(x) = -x² + x

f(x + h) - f(x) = -x² - 2xh + h² + x + h -(-x² + x) = -x² - 2xh + h² + x + h + x² - x = -2xh + h² + h = h(-2x + h + 1)

[f(x + h) - f(x)]/h = h(-2x + h + 1)/h = -2x + h + 1

The limit as h approaches zero is -2x + 1

So the instantaneous velocity, based on curve fitting, is -2(3) + 1 = -5 (same solution as method 3)