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State Standard – 4.1 Students demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function. Objective – To be able to find the tangent line.

Definition of a Tangent Line: –1 –5–4–3–2– P Q Tangent Line

Slope: ax P (a,f(a)) Q (x,f(x)) x – a f(x) – f(a)

Definition The tangent line to the curve y = f(x) at the point P(a,f(a)) is the line through P with the slope: Provided that this limit exists.

Example 1 Find an equation of the tangent line to the parabola y = x 2 at the point (2,4). Use Point Slope y – y 1 = m (x – x 1 ) y – 4 = 4(x – 2) y – 4 = 4x – 8 +4 y = 4x – 4

Provided that this limit exists. For many purposes it is desirable to rewrite this expression in an alternative form by letting: h = x – a Then x = a + h

Example 2 Find an equation of the tangent line to the hyperbola at the point (3,1). y – y 1 = m (x – x 1 ) y – 1 = - 1 / 3 (x – 3) y – 1 = - 1 / 3 x y = - 1 / 3 x + 2

Example 3 Find an equation of the tangent line to the parabola y = x 2 at the point (3,9). y – y 1 = m (x – x 1 ) y – 9 = 6(x – 3) y – 9 = 6x – y = 6x – 9

Example 4 Find an equation of the tangent line to the parabola y = x 2 –4 at the point (1,-3). y – y 1 = m (x – x 1 ) y – -3 = 2(x – 1) y + 3 = 2x – 2 -3 y = 2x – 5

Pg a, 5b, 6a, 6b, 7 – 10, 11a, 12a, 13b, and 14b