2.1 - The Tangent and Velocity Problems Calculus = The branch of mathematics dealing with change and motion / and quantites that approach other quantites… 2 Branches of Calculus Differential --> The Tangent & Velocity Problems Integral --> The Area Problem Key Idea: Limits Key people: Isaac Newton Gottfried Leibniz

The Tangent and Velocity Problems Explanation… ex: Find the equation of the tangent line to the parabola y = x2 at the point (1, 1).

Tabular Functions ex: Use the data to draw the graph of this function and estimate the slope of the tangent line at the point where t = 0.04 100 90 80 70 60 50 0.02 0.04 0.06 0.08 P

Pick table values on either side of tangent point… Triangle Method (x1, y1) (x2, y2) P x y

The Velocity Problem ex: A ball is dropped from an observation tower 450m above the ground. The ball drops according to position equation: Find the velocity of the ball after 5 seconds. Average Velocity = Distance Traveled Time Elapsed

Distance traveled = Time elapsed = 5.1 - 5 = 0.1 sec Average Velocity for t= [5 to 5.1] = 49.49 m/s We can continue to make the interval smaller and smaller and compute the average velocity… It appears we are reaching a limiting value… thus the instantaneous velocity after 5s = 49 m/s

The Velocity Problem s (position) t (time) t1 t2

SHOULDA TAKEN STATS!! HW 2.1 pg 100 #’s 1-8 all MATH MELTDOWN!!!