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2.4 Rates of Change & Tangent Lines

2.4 Rates of Change & Tangent Lines. Average Rate of Change. The average rate of change of a quantity over a period of time is the slope on that interval of time. Ex.: Find the average rate of change of f(x) = x 3 – x over the interval [1, 3]. Secant & Tangent Lines.

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2.4 Rates of Change & Tangent Lines

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  1. 2.4Rates of Change & Tangent Lines

  2. Average Rate of Change • The average rate of change of a quantity over a period of time is the slope on that interval of time. Ex.: Find the average rate of change of f(x) = x3 – x over the interval [1, 3].

  3. Secant & Tangent Lines • Secant lines touch a graph at two points. • The slope of a secant line represents the AVERAGE RATE OF CHANGE of a function over a given interval. (5, 7) (1, 1)

  4. Secant & Tangent Lines • A tangent line touches a graph at one point only. • Tangent lines determine the direction of a body’s (graph’s) motion at every point along its path. • Tangent lines represent the INSTANTANEOUS RATE OF CHANGE. (the slope at an actual point, not over an interval)

  5. Tangent Lines • The more secant lines you draw, the closer you are getting to a tangent line. • SOUND FAMILIAR TO SOME CONCEPT WE’VE DONE???

  6. Tangent Lines • Example: Consider the function f(x) = x2 for x ≥ 0. What is the slope of the curve at (1, 1)? The slope at (1, 1) can be approximated by a secant line through (4, 16).

  7. Tangent Lines • Example: Consider the function f(x) = x2 for x ≥ 0. What is the slope of the curve at (1, 1)? The slope at (1, 1) can be better approximated by a secant line through (3, 9).

  8. Tangent Lines • Example: Consider the function f(x) = x2 for x ≥ 0. What is the slope of the curve at (1, 1)? The slope at (1, 1) can be even better approximated by a secant line through (2, 4).

  9. Tangent Lines • Example: Consider the function f(x) = x2 for x ≥ 0. What is the slope of the curve at (1, 1)? An even better approximation for the slope at (1, 1) would be to use a secant line through (1.1, 1.21). How long could we continue to do this?

  10. Tangent Lines • Example: Consider the function f(x) = x2 for x ≥ 0. What is the slope of the curve at (1, 1)? What about using the point (1+h, (1+h)2) to find the slope at (1, 1)? (where h is a small change) If h is a small change, I can say h 0. Therefore the slope of a tangent line at (1, 1) is 2.

  11. Tangent Lines The slope of the curve at the point is: • The slope of a curve at a point is the same as the slope of the tangent line to the curve at that point.

  12. Tangent Lines • Example: Find the slope of the parabola y = x2 at the point where x = 2. Then, write an equation of the tangent line at this point.

  13. Tangent Lines • Example: Find the slope of the parabola at the point (2, 4). Then, write an equation of the tangent line at this point. The slope of a line tangent to the parabola at (2, 4) is m = 4. To find the equation of the tangent line, use y = mx + b Since m = 4 and b = -4, the equation of the tangent line is y = 4x – 4

  14. Tangent Lines • Example: Let . Find the slope of the curve at x = a. (get common denominator) 0

  15. Tangent Lines • Example: Let . Where does the slope equal ? We just found that the slope at any point a of f(x) is Therefore, when does ? Substituting in these a values into x in the original function, we see the graph has a slope of -1/4 at (2, 1/2) and (2, -1/2)

  16. Tangent Lines • The following statements mean the same thing: • The slope of y = f(x) at x = a • The slope of the tangent line to y = f(x) at x = a • The Instantaneous rate of change of f(x) with respect to x at x = a

  17. Normal Lines • The normal line to a curve at a point is the line that is perpendicular to the tangent line at that point. Example: Write an equation for the normal to the curve f(x) = 4 – x2 at x = 1. Slope of tangent line: (Slope of Tangent line)

  18. Normal Lines • The normal line to a curve at a point is the line that is perpendicular to the tangent line at that point. Example: Write an equation for the normal to the curve f(x) = 4 – x2 at x = 1. Slope of tangent line Normal Line: Slope of normal line Normal Line:

  19. Wrapping it Back Together • Problem at the beginning of Chapter 2: A rock breaks loose from the top of a tall cliff. What is the speed of the rock at 2 seconds? Speed/Velocity is an INSTANTANEOUS RATE OF CHANGE. Free fall equation: y = 16t2

  20. Wrapping it Back Together 0

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