PowerPoint Format
Download
Report
Transcript PowerPoint Format
Math 1241, Spring 2014
Section 3.3
Rates of Change
Average vs. Instantaneous Rates
Average Speed
β’ The concept of speed (distance traveled
divided by time traveled) is a familiar instance
of a rate of change.
β’ Example: To drive the 15.5 miles from Clayton
State to Turner field, it takes 18 minutes.
β’
15.5 πππππ
18 πππ.
= 0.88 mi/ min = 52.7 mph
β’ Question: If you drove to Turner Field, would
your speedometer always read 52.7 mph?
Average Speed
β’ The 52.7 mph is an average speed. Your
speedometer measures something else.
β’ In pre-Calculus courses, you solve problems
assuming that speed (or some other rate of
change) is constant: it does not change.
β’ One of the main features of Calculus: we can
solve problems where speed (or some other
rate or change) is not necessarily constant.
Average Rate of Change
β’ For a function y = f(x), we define the average
rate of change from x = a to x = b as:
π π β π(π)
πβπ
β’ In the case of speed:
β We often use t instead of x (for obvious reasons).
β f(t) is the total distance weβve travelled at time t.
β Numerator = Change/difference in distance
β Denominator = Change/difference in time
Average Rate of Change = Slope
β’ The blue curve is the
graph of y = f(x).
β’ The two blue dots show
the functionβs values at
x = a and x = b.
β’ The red line is called a
secant line. Its slope
equals the average rate
of change of f(x) from a
to b.
Instantaneous Speed
β’ Your speedometer measures your speed βat a
given time.β What does this mean?
β’ Average speed: Change in distance divided by
change in time. We canβt do this βat a given
time,β because the change in time is zero (in
the denominator).
β’ Solution: Take the limit as change in time
approaches zero!
Instantaneous Rates of Change
β’ Take the limit of (average rate of change), as
the change in the independent (x) variable
approaches zero. There are two ways to do so:
π π β π(π)
lim
πβπ
πβπ
π π + β β π(π)
lim
ββ0
β
β’ Although these appear to be different
formulas, note that h = b β a (thus b = a + h).
Graphical Demonstration
β’ Itβs somewhat difficult to do dynamic graphs in
Graph, so weβll use the following link:
https://www.desmos.com/calculator/irip8pnpdf
β’ Left-click on one of the dots and hold down the
button. Drag your mouse to see how the secant line
(in red) changes.
β’ As the dots get closer together, the slope of the
secant line approaches the instantaneous rate of
change.
Tangent Lines
Consider what happens to the secant line as
π β π (or as β β 0).
β’ Any secant line contains the point (π, π π ).
β’ The slope of the secant line approaches the
instantaneous rate of change (at x = a).
The line through (π, π π ) with slope equal to
the instantaneous rate of change (at x = a) is
called the tangent line (at x = a).
β’ The tangent line of a circle is a special case.
Graphical Demonstration
β’ Using the link from earlier:
https://www.desmos.com/calculator/irip8pnpdf
β’ Change the function definition to sqrt(4-x^2).
This is the upper half of the circle centered at
(0,0) with radius 2.
β’ Drag the two points close together. The secant
line is very close to a tangent line of the circle.
Tangent Lines in Graph
β’ Fortunately, Graph will draw a tangent line.
1
π₯
β Start by graphing a function. Iβll use π₯ + .
β Select Function -> Insert Tangent/Normal from the
menu (or press F2, or use the toolbar button).
β In the βx = β field, type the x-value where you
want the tangent line (Iβll use x = 1). Use a
different color than the original function.
β’ Zoom in on the point where the function
touches the tangent line. What do you see?
An important note
β’ The instantaneous rate of change of the
function f(x) at x = a is a limit.
β’ To actually compute it, we need to know the
function value for x values closer and closer to
x = a. This would mean infinitely many values!
β’ We can avoid this if we have a formula for f(x)
that is valid near x = a. Weβll usually take this
approach.
Algebraic Example
Find the instantaneous rate of change of the
function π π₯ = π₯ 2 at the point a = 1.
β’ Before computing the limit, use Graph to draw
the tangent line. What is the slope?
β’ We need to evaluate one of the following:
π₯2 β 1 2
1 + β 2 β (1)2
lim
lim
π₯β1 π₯ β 1
ββ0
β
β’ Try both forms; which one is easier?