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?