Rates of Change and Limits Notes

Rates of Change and Limits

Learning Goals

• Describe rates of change in a variety of ways.
• Describe connections between average rate of change and secant lines.
• Make connections between points on a graph and instantaneous rate of change.

Calculus Definition

• CALCULUS is the study of continuous change.

The Two Fundamental Problems of Calculus

• The Area Problem: Finding the area under a curve.
• The Tangent Problem: Finding the tangent to a curve at a point.

Tangent Definition

• The tangent to a curve at a point $P$ is the straight line through $P$ that most resembles the curve at $P$.
• The goal is to find the rate of change (slope of the tangent) at different given points.
• To find the tangent, we search for a point $Q$ close to point $P$.

What is a Limit?

• Imagine trying to find the circumference and area of a circle in 300 BC.
• Approximations improve with more sides added to the polygon.
• Let:
  • $b$ = side lengths of the polygons
  • $h$ = the height of the triangle
  • $n$ = the number of sides
• As $n$ gets large:
  • Perimeter approaches the circumference.
• Area of the circle:
  • None of the individual calculations for the area of the polygon will equal the area of the circle.

Average Rate of Change

• Average Rate of Change: The rate of change that takes place over an interval.
• Secant Line: The line connecting two points on a curve.

Example 1

• A rock is tossed upward from a cliff that is $120$ m above the water.
• The height of the rock above the water is modeled by $H(t) = -5t^2 + 10t + 120$, where $H(t)$ is the height in metres and $t$ is time in seconds.
  • a) Find the average rate of change during the first second.
  • b) Find the average rate of change between 1 and 6 seconds.

Estimated Instantaneous Rate of Change:

• To calculate the rate of change at a specific point, we use the estimated instantaneous rate of change.
• Tangent Line: A line that touches a curve only at one point.

Example 2

• Estimate the slope of the function $f(x) = x^2$ at $P(2, 4)$.
• Choose two points (point $P$ and point $Q$) on either side of $x = 2$.
• The closer we get to 2, the better the approximation will be.
• Choose $P(1.99, …)$ and $Q(2.01, …)$

Instantaneous Rate of Change

• For the function $f(x)$, we will use two points $P$ and $Q$, to find the instantaneous rate of change.
• Let’s say that we want the tangent to point $P(a, f(a))$.
• We will express $Q$ as a generic point very close to $P$.
• We add a very small, non-zero amount called $h$ to the $x$-value of $P$: $Q(a + h, f(a + h))$
• The smaller $h$ gets, the more accurate our tangent is.
• Ultimately, we want $h \rightarrow 0$ ($h$ to approach zero).

Example 3

• Find the slope of the tangent at $P(3, 9)$ for $f(x) = x^2$.

Limit of a Function

• In general, the slope of a tangent to a curve $y = f(x)$ at any point of tangency $P(x, f(x))$ is:

Example 4

• Find the slope of the tangent of the curve $f(x) = \frac{7x-1}{x}$ at $P(1, 6)$.

Example 5

• Find the slope of the tangent to the curve $f(x) = \sqrt{x}$ at $x = 16$.

Example 6

• Find the slope of the tangent to $y = 2x^2 - 4x$ at the point $(2, 0)$.

Example 7

• Find the slope of the tangent to $f(x) = \sqrt{x}$ when $x = 4$.

Example 8

• A toy rocket is launched in the air so that its height, $s$, at time $t$ is measured by $s(t) = -5t^2 + 30t + 2$. What is the instantaneous velocity at 4 seconds?