1. Math 1304 Calculus I 2.7 – Derivatives, Tangents, and Rates

2. Tangent Line Definition: The tangent line to a curve y = f(x) at the point P(a, f(a)) is the line through P with slope

3. Alternate formula for tangent The tangent line to a curve y = f(x) at the point P(a, f(a)) is the line through P with slope

4. Derivative The following limits are the same. Either of the above formulas is called the derivative of the function f(x) at x = a.

5. Examples Find an equation of the tangent line to the parabola y = x 2 at the point (1, 1). Find an equation of the tangent line to the hyperbola y = 2/x at the point (2, 1). Find slope of tangent to the function f(x)=sqrt(x) at the points (1,1), (4,2), and (9,3).

6. Instantaneous Rate of Change is given by the derivative x1x1 x2x2 y1y1 y2y2 xx yy

7. Physics: Velocity Velocity is the derivative of position Velocity at time t is the limit of the average velocity as the change in time approaches zero

8. Example A ball is thrown up into the air with a velocity of 40 ft/sec. Its height after t seconds is given by: y = 40t - 16 t 2 Find a formula for the velocity: v What is the velocity after 2 seconds (when t=2)?

9. Economics: Marginal Cost The cost of producing x units of a certain commodity is given by C(x) = 5000 +10x+0.05 x 2. Find the instantaneous rate of change, when x =100. This is called the marginal cost and is the derivative of the cost.