Instantaneous Rate of Change

1 / 50

Instantaneous Rate of Change - PowerPoint PPT Presentation

Instantaneous Rate of Change. Sections 3.1-3.3 Section 4.1. Average Change. Defined equations in terms of their changes e.g., exponential  constant percentage change Will use this concept motivate derivatives. Average Change. Rate of change Difference between two values

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

PowerPoint Slideshow about 'Instantaneous Rate of Change' - winka

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

Instantaneous Rate of Change

Sections 3.1-3.3

Section 4.1

Average Change
• Defined equations in terms of their changes
• e.g., exponential  constant percentage change
• Will use this concept motivate derivatives
Average Change
• Rate of change
• Difference between two values
• Percentage change
• Difference between two values as a percentage of the original value
• Average change
• Change per unit of time
Average Change
• Ex: r(t) = pool sales
Average Change
• Rate of change
• Percent change
• Average change
Average Change

Rate of change from April to August

Rate of change

Average Change

Rate of change from April to August

r(8) - r(4)

Average Change

Average rate of change from April to August

r(8) - r(4)

Average Change

Average rate of change from April to August

r(8) - r(4)

8-4

Average Change

Average rate of change from April to August

r(8) - r(4)

8-4

Average Change
• In-Class
• Pg 167: 1, 2, 4, 6, 7, 8, 10
Average Change

Average rate of change from April to August

r(8) - r(4)

8-4

Average Change

Average rate of change from April to August

rise

run

Average Change

Average change between two points

Slope of the secant line between the two points

=

Instantaneous Rate of Change
• Rate of change at this instant
• Average rate of change over an infinitesimally small range
Instantaneous Rates of Change
• Tangent line
• Secant line that touches the graph at the point evaluated

The instantaneous rate of change is the slope of the tangent line at the point evaluated

Instantaneous Rates of Change
• Local linearity
• Zoom in enough and anything looks like a line.
Instantaneous Rates of Change
• Tangent lines don’t intersect graph at the point of tangency, but
• Tangent lines can intersect graph at other points
Instantaneous Rates of Change
• Exists only where you have a continuous function
• Does not exist at breakpoints
Instantaneous Rates of Change
• In-Class
• Pg 185: 7, 8, 9, 10

Derivatives

Section 3.3

Section 4.1

Derivatives
• Another phrase for instantaneous rate of change

Instantaneous rate of change

Rate of change

=

=

Slope of curve

Slope of tangent line

=

=

Derivative

Derivatives
• Notation

“Derivative of f with respect to x”

Derivatives
• Ex: Profit versus number of employees
• p(t) = 20*ln(t)
Derivatives
• Notation
• Interpretation
Derivatives
• In-Class
• Pg 203: 1, 2, 4, 6, 10