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2-4 Rates of change & tangent lines

Learn about average rate of change, slope of a curve at a point, and normal lines to the curve. This concept is essential for finding the equations of tangent lines and normal lines in calculus.

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2-4 Rates of change & tangent lines

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  1. 2-4 Rates of change & tangent lines

  2. Average Rate of Change = • This is the slope of the secant line through 2 points on f (x) • (x1, f (x1)) and (x2, f (x2)) • As the intervals get closer together we approach the slope • of a tangent line Def: Slope of a Curve at a Point If y = f (x) and P(a, f (a)) is a point on the curve, then (as long as the limit exists)

  3. We can use the definition above to find the equation of the tangent line to the curve. We need two things: (1) a slope (2) a point Ex 4) Let a) Find the slope of the curve at x = a. b) Where does the slope equal ?

  4. Difference Quotient of f at a Two Notes: (1) secant slope  its limit is the slope of the curve & the tangent at x = a (2) average rate of change  its limit is the function rate of change at x = a

  5. Normal to a Curve Def: Normal Line: the line  to the tangent at a point Ex 5) Write an equation for the normal line to the curve First find slope of tangent: (tangent)  = ½ (1, 3) y – 3 = ½(x – 1)

  6. homework Pg. 84 #6 – 54 (mult of 6) Pg. 92 #1 – 6, 8 – 9

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