Rates of change

In the following video, we illustrate this concept by considering the instantaneous velocity at time $t=2$ of a particle with position $s(t)=4t^2+3$, where $t$ is time in seconds, and $s$ is position in feet.

We now show here the computation of the instantaneous velocity at time $t=2$ seconds of a particle with position $s(t)=4t^2+3$ feet. We will need to compute either $$(1) \lim_{t\to 2}\frac{s(t)-s(2)}{t-2},\qquad\text{ or }\qquad (2) \lim_{h\to 0}\frac{s(2+h)-s(2)}{h}.$$

DO: Work through these examples carefully. Try to compute both limits before looking at the solutions below. The first one is also done on the previous video.

$$(1)\quad \lim_{t\to 2}\frac{s(t)-s(2)}{t-2}=\lim_{t\to 2}\frac{(4t^2+3)-(4(2)^2+3)}{t-2}
=\lim_{t\to 2}\frac{4t^2+3-16-3}{t-2}=\lim_{t\to 2}\frac{4t^2-16}{t-2}\\
=\lim_{t\to 2}\frac{4(t^2-4)}{t-2}=\lim_{t\to 2}\frac{4(t-2)(t+2)}{t-2}=\lim_{t\to 2}4(t+2)=4(4)=16$$

$$(2)\quad \lim_{h\to 0}\frac{s(2+h)-s(2)}{h}=\lim_{h\to 0}\frac{(4(2+h)^2+3)-(4(2^2)+3)}{h}
=\lim_{h\to 0}\frac{4(4+4h+h^2)+3-4(4)-3}{h}\\
=\lim_{h\to 0}\frac{16+16h+4h^2-16}{h}=\lim_{h\to 0}\frac{16h+4h^2}{h}=\lim_{h\to 0}(16+4h)=16$$

Notice that in both computations (until the denominator was cancelled) if we plugged $t=2$ in, we would get the indeterminate form $\frac{0}{0}$ (try it) so we had to do more work.

We see that the instantaneous velocity at $t=2$ is 16 feet/second. Notice that the units come from the units of the numerator (feet) and the denominator (seconds) of the function we are taking the limit of--the average velocity function).

Also notice that the work required in either computation is similar. Often students prefer method (1), but in the next section you will be forced to use method (2), so be sure you practice it at least as much as method (1).

The Six Pillars of Calculus

Trigonometry Review

Exponential Functions

Logarithms and Inverse functions

Intro to Limits

Limit Laws and Computations

Continuity and the Intermediate Value Theorem

Limits at Infinity

Rates of Change

The Derivative Function

Basic Differentiation Rules

Product and Quotient Rules

Derivatives of Trig Functions

The Chain Rule

Implicit Differentiation

Derivatives of Logs

Derivatives in Science

Related Rates

Linear Approximation and Differentials

Differentiation Review

Absolute and Local Extrema

The Mean Value and other Theorems

$f$ vs. $f'$

Indeterminate Forms and L'Hospital's Rule

Optimization

Newton's Method

Anti-derivatives

The Area under a curve

Definite Integrals

The Fundamental Theorem of Calculus

Instantaneous velocity