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  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. (1, 1) (5, 7)

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)

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???

Tangent Lines  Example: Consider the function f(x) = x 2 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).

Tangent Lines  Example: Consider the function f(x) = x 2 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).

Tangent Lines  Example: Consider the function f(x) = x 2 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).

Tangent Lines  Example: Consider the function f(x) = x 2 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?

Tangent Lines  Example: Consider the function f(x) = x 2 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.

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.

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

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

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

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)

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

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 – x 2 at x = 1. Slope of tangent line: (Slope of Tangent line)

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 – x 2 at x = 1. Slope of tangent line Slope of normal line Normal Line:

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 = 16t 2

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