1. Find the instantaneous velocity for the following 1. Find the instantaneous velocity for the following time—distance function at time = 2 seconds. time—distance function at time = 2 seconds. Quiz Find the function that represents the slope at any location on the following function:. 2. Find the function that represents the slope at any location on the following function:.

HOMEWORK Section 10-2 Section 10-2 (page 810) (evens) 2-8, 12, 14, 18, even, even, 50 (18 problems)

10-2 Limits and Motion The Area problem.

What you’ll learn about Computing Distance Traveled (Constant Velocity) Computing Distance Traveled (Changing Velocity) Limits at Infinity The Connection to Areas The Definite Integral … and why Like the tangent line problem, the area problem has applications throughout science, engineering, economics and historicy.

Computing Distance Traveled A car travels at an average rate of 56 miles per hour for 3 hours. How far does the car travel? Distance traveled = rate (speed) * time OR: using the definition of average velocity from section 10-1:

The Velocity as a function of time plot: Time (hrs) VELOCITY Distance traveled is the area under is the area under the v(t) function the v(t) function for a specific time for a specific time interval !! interval !! (constant velocity) Area = 300 miles

What if the velocity is NOT constant (velocity is changing): Time (hrs) VELOCITY Distance traveled is the area under the v(t) function for a specific time interval. This is still pretty easy:

Velocity is NOT constant Time (sec) VELOCITYFt/sec Harder: How do you find the area you find the area under a continuously under a continuously changing curve? changing curve? Why not add the areas of a group areas of a group of rectangles (upper of rectangles (upper right corner of rectangle right corner of rectangle Is on the curve)? Right Rectangular Approximation Method (RRAM)

Velocity is NOT constant Time (sec) VELOCITY Not accurate enough! enough!(over-estimate)

Velocity is NOT constant Time (sec) VELOCITY Not accurate enough—under estimates the area. estimates the area. Left Rectangular Approximation Method (LRAM)

Velocity is NOT constant Time (sec) VELOCITY Why not make the rectangles smaller? Better, but not accurate enough. accurate enough.

Velocity is NOT constant Time (sec) VELOCITY Why not make the rectangles even smaller? Better, but still not accurate enough. accurate enough.

Velocity is NOT constant Time (sec) VELOCITY Why not make the rectangles even smaller? Even better, but still not accurate enough. still not accurate enough.

Velocity is NOT constant Time (sec) VELOCITY Why not make the rectangles infinitesmally wide? Now you’re talkin’!

Section 9-5 (We didn’t have time to cover this) Series: The sum of a sequence of numbers. Section 9-4 Sequences: A list of numbers Where (and k = 1,2,3,…) = 2, 5, 10, 17, 26 = = 65 “The summation of ‘a’ sub ‘k’ for ‘k’ = 1 to 5”

We’re going to use this idea of a summation to find out the exact area under the curve Time (sec) VELOCITY We will make the width of the rectangle infinitesimally infinitesimally small. small. We’ll call that Infinitesimally small width: small width: The height of the rectangle is just the rectangle is just the output value of the output value of the function. function. The distance traveled (area under the curve) is the sum of all of the small rectangles.

Width: (infinitesimally small) small) We’re going to use this idea of a summation to find out the exact area under the curve Time (sec) VELOCITY The velocity (area under the curve) is the sum of all of the small rectangles. Height: (function value for the left upper corner of the left upper corner of Each of these slivers (there are an infinite # of them) are an infinite # of them) Where “b” and “a” are the right and left are the right and left ends of the interval. ends of the interval.

Definite Integral

Limits at Infinity (Informal)

Calculating the Integral of a function (gives the area under the curve for any specified interval) Find: (the area under the function for the interval x = [1,5] interval x = [1,5] b = largest input value a = smalles input value n = # of rectangles (bigger is more accurate)

Calculating the Integral of a function (gives the area under the curve for any specified interval) Find: (the area under the function for the interval x = [1,5] interval x = [1,5] b = largest input value a = smalles input value n = # of rectangles (bigger is more accurate) Where does the infinite series “converge”?

Calculating the Integral of a function Find: (the area under the function for the interval x = [1,5] interval x = [1,5] Area under the line (in the interval x = [1,5] the interval x = [1,5] f(x) = 2x Area = area large triangle – area small triangle