1. 3.1.Tangent Lines and Rates of Change. Average and instantenious velocity. Rita Korsunsky

2. not a tangent, 2 pts of intersection.. c Is it a tangent to the curve at c?. c Is it a tangent to the curve at c? The tangent line to the graph of f is a line that touches the graph at one isolated point and could possibly intersect it again at another point. Tangent Lines Yes! Slope of the graph of f at point c is the slope of the tangent line at point c.

3. Slope Slope of secant line = x y.. Q (x, f(x)) P (c, f(c)) O Let’s first find the Slope of secant line:

4. Finding the slope of the tangent line at pt C Let’s pick 2 nd point closer and closer to C and calculate the slopes of Secant Lines. Watch the animation: Slope of tangent line at pt C  Slope of Secant Line when x is approaching to C

5. Slope Slope of tangent line = Let x - c = h x = c + h Slope of tangent line at pt C  Slope of Secant Line when x is approaching to C

6. To Find the Equation of the Tangent Line at x=c: 1.Find the slope of tangent at any point x 2. Plug in x = c into m x to find the slope m c at the point (c, f(c)). 3. Substitute coordinates (c,f(c)) and slope m c into the point-slope equation of a line:

7. Example 1 Solution:

8. Average Velocity and rate of change

9. Instantaneous Velocity and rate of change

10. Example 2 A sandbag is dropped from a hot-air balloon that is hovering at a height of 512 feet above the ground. If air resistance is disregarded, then the distance s(t) from the ground to the sand bag after t second is given by: Find the velocity of the sandbag at: (a) t = a sec (b) t = 2 sec (c) the instant it strikes the ground (a) Find Velocity at t = a

11. Example 2 continued (b) Find velocity at t = 2 sec (c) Find velocity at instant it hits the ground