Unformatted text preview: 2.1 Slope of the Tangent Line and the Derivative of a Function Study definitions: Slope of the tangent line, p. 97 Derivative, p. 99** p. 103104: 5, 17 Consider the slope of the secant line as an approximation to the slope of the tangent line. How good an approximation is it? Prof. G. Battaly, Westchester Community College Title: Approximation (2 of 11) Consider the slope of the secant line as an approximation to the slope of the tangent line. Getting better. Prof. G. Battaly, Westchester Community College Title: Delta X smaller (3 of 11) Consider the slope of the secant line as an approximation to the slope of the tangent line. and better Prof. G. Battaly, Westchester Community College Title: Delta X even smaller (4 of 11) Slope of the secant line:
Calc Web Pages secant_tangent_slopes_dx.swf Slope of the tangent line at a point (c, f(c)): Calc Web Pages Prof. G. Battaly, Westchester Community College Title: slopes (5 of 11) Ex: Finding the slope of a tangent line at a particular point. Use equation derived above. Prof. G. Battaly, Westchester Community College Calc Web Pages Title: ex: slope at point (6 of 11) Calc Web Pages This generalizes from a specific point (c,f(c)) to any point on the curve (x,f(x)) It can be interpreted as the slope of the tangent line to the curve at any point on the curve. Prof. G. Battaly, Westchester Community College Calc Web Pages Title: derivative (7 of 11) Ex: Finding the equation of a tangent line at a particular point requires first finding the slope of the tangent at that point. We use the definition of the derivative as an example here. The tangent line would be of the form: y y 1 = m (xx1) using (3,4) y 4 = (4)(x (3)) = (4)(x+3) = 4x 12 Therefore, y = 4x 8 Prof. G. Battaly, Westchester Community College Calc Web Pages Title: Ex: using derivative (8 of 11) Calc Web Pages Prof. G. Battaly, Westchester Community College Title: Alternate Form (9 of 11) Calc Web Pages Prof. G. Battaly, Westchester Community College Title: example (10 of 11) Calc Web Pages Prof. G. Battaly, Westchester Community College Title: continuity& differentiation (11 of 11) ...
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 Approximation, Derivative, Slope, Westchester Community College

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