LS30A Week 6 Review

What is the product rule?

(f x g)' = f * g' + g * f'

What is the chain rule?

(f(g(x)))' = f'(g(x))*g'(x)

What is the quotient rule?

(f/g)' = (f'(x)g(x)-f(x)g'(x))/(g(x))^2

What does the area under the graph of X'(t) really mean?

It is the net change in the value of X(t) over that particular time interval

What is an antiderivative?

A function F(t) such that F'(t) = f(t) is the antiderivative of f(t)

How do we express the definite integral with the Riemann sum expression?

S a, b f(x)dx = lim (N-->infinity) E k=0, n-1: (X'(a+k*delta x))*delta x

N going to infinity can also be though of as ___

delta t going to 0

What does the Riemann sum calculate?

Area under a derivative graph

Relate integration to distance formula

Distance = rate x time, so the rate is X'(t) and the time is delta t; we do this for many rectangles and get many distances, which we ultimately add all up together

How do we find the number of rectangles?

Do (b-a)/(delta t)

Write the Riemann sum formula when the interval is a to b and the number of rectangles is n

E k=0, n-1: (f'(a+((b-a)/n)*k))*(b-a)/n

How do we make the area we are finding exact?

Including a limit in the Riemann sum expression

Explain how integration is the opposite of differentiation

Integrating is taking the derivative function and going BACK to the original function

Name the two things that the definite integral calculates

1. The area under the graph of f(x) from a to b 2. The net change in f(x) from a to b

Are the Riemann sums an exact area?

No, it's only the approximation of the area

What happens as the number of rectangles increases?

The approximation of the area gets more accurate

If D'(t) is the rate of drug delivery, what will the integral of this function from t = 3 to 9 minutes tell you?

The cumulative amount of the drug delivered from t = 3 to 9 minutes

Explain the connection between Euler's Method and Riemann sums

The initial value of Euler's Method is like the antiderivative's initial value, while the delta t*f(Xnow) is the area of the rectangle

What is the definite integral?

The limit of the Riemann sums as N (number of rectangles) approaches infinity

What is the goal of integration?

To find the net rate of change of the antiderivative over whatever interval

What is the Fundamental Theorem of Calculus?

X(t) - X(0) = S a, b X'(t)dt