2.4 rates of change and tangent lines calculus. finding average rate of change
TRANSCRIPT
![Page 1: 2.4 Rates of Change and Tangent Lines Calculus. Finding average rate of change](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f1c5503460f94c3205e/html5/thumbnails/1.jpg)
2.4 Rates of Change and Tangent Lines
Calculus
![Page 2: 2.4 Rates of Change and Tangent Lines Calculus. Finding average rate of change](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f1c5503460f94c3205e/html5/thumbnails/2.jpg)
Finding average rate of change
• Find the average rate of change of over the interval [1, 3].
•12
![Page 3: 2.4 Rates of Change and Tangent Lines Calculus. Finding average rate of change](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f1c5503460f94c3205e/html5/thumbnails/3.jpg)
Slope of a secant line
• Use points P(23, 150) and Q(45, 340) to compute the average rate of change and the slope of the secant line PQ. • 8.6 flies/day•We can always think about average rate of change as the slope of a secant line.
![Page 4: 2.4 Rates of Change and Tangent Lines Calculus. Finding average rate of change](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f1c5503460f94c3205e/html5/thumbnails/4.jpg)
Instantaneous rate of change
• What about the growth of the population on day 23? We move point Q closer to point P to get a better estimate.
• Notice the secant line appears to be approaching the tangent line.
• So we could use the slope of the tangent line as the instantaneous rate of change at
![Page 5: 2.4 Rates of Change and Tangent Lines Calculus. Finding average rate of change](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f1c5503460f94c3205e/html5/thumbnails/5.jpg)
Steps for finding the slope of the tangent
1. Start with what we can calculate- the slope of the secant through a point P and a point nearby (Q) on the curve.
2. Find the limiting value of the secant slope (if it exists) as Q approaches P along the curve.
3. Define the slope of the curve at P to be this number and define the tangent to the curve at P to be the line through P with this slope.
![Page 6: 2.4 Rates of Change and Tangent Lines Calculus. Finding average rate of change](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f1c5503460f94c3205e/html5/thumbnails/6.jpg)
Definition: Slope of a curve at a point
• The expression is the difference quotient of f at a.
![Page 7: 2.4 Rates of Change and Tangent Lines Calculus. Finding average rate of change](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f1c5503460f94c3205e/html5/thumbnails/7.jpg)
Example: Finding slope and tangent line
• Find the slope of the parabola at the point P(2, 4). Write an equation for the tangent to the parabola at this point.
![Page 8: 2.4 Rates of Change and Tangent Lines Calculus. Finding average rate of change](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f1c5503460f94c3205e/html5/thumbnails/8.jpg)
Example:
• Find the slope of the curve at .
• Where does the slope equal -1/4?
![Page 9: 2.4 Rates of Change and Tangent Lines Calculus. Finding average rate of change](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f1c5503460f94c3205e/html5/thumbnails/9.jpg)
Lines normal to a curve
• The normal line to a curve at a point is the line perpendicular to the tangent at that point.•Write an equation for the normal to the curve at
•
![Page 10: 2.4 Rates of Change and Tangent Lines Calculus. Finding average rate of change](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f1c5503460f94c3205e/html5/thumbnails/10.jpg)
Free fall…again
• Find the speed of the falling rock (discussed earlier in this chapter) at sec. • Remember:
• 32 ft/sec