Andrew, Shelby, Randall

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 The average value of a function can be defined by taking the integral of that function from a to b and dividing it by b minus a.

 This theorem is very similar to the average velocity theorem but is expanded to include all functions not just a velocity function.

If v(t) is the velocity of a moving object as a function of time, then the average velocity from time t = a to t = b is: 𝑉 𝐴𝑉𝐺 = 𝑎 𝑏 𝑣 𝑡 𝑑𝑥 𝑏 − 𝑎 Average velocity is displacement over time.

 The area under the curve is equal to the area of the shaded region.

 The height of the rectangle is the average value.

 ◦ F(x)=x 3 -x+5 [1,5] Step 1: Use the FTC or use Numerical Integration and divide it by the interval to find the average velocity. 1 5 𝑥 3 − 𝑥 + 5 𝑑𝑥 5 − 1

◦ Step 2: Graph it

◦     Step 3: Solve for X 41 = x 3 -x+5 36 = x 3 -x 0 = x 3 -x-36 X = 3.4029

 Suppose you are driving 60 ft/s (about 40 mi/h) behind a truck. When you get the opportunity to pass, you step on the accelerator, giving the car an acceleration a = 6 / 𝑡 , where a is in (feet per seconds) per second and t is in seconds. How fast are you going 25 s later when you have passed the truck? What was your average velocity for the 25 s interval?

◦ ◦ ◦ Step 1: Integrate   A = 6*t -1/2 V = 12*t 1/2 + C Step 2: Find “C”    Start time is 0, velocity is 60, so C = 60 Step 3: Find passing speed V(25) = 12(25) 1/2 V(25) = 120 + 60

◦ Step 4: Find the displacement 0 25

(12 ∗ 𝑡

1 2

+ 60) 𝑑𝑥 25 − 0

 Disp = 100 ft/s

 Foerster, Paul A.

Calculus Concepts and applications.

Emeryville: Key Curriculum Press, 2005. Print.