lesson 5 nov 3
TRANSCRIPT
Given a positiontime function, s = f(t) we differentiate to find
the velocity, v = dsdt
Given a velocitytime function, we integrate to find the total
net distance traveled, s(b) s(a) = s '(t) dta
b
change in f = rate of change * time
change of f = f(b) f(a)
f(b)
f(a)
a b
Fundamental Theorem of Calculus
Let the function f be continuous on [a, b] with derivative f '.Then
total change in f = f(b) f(a) = f ' (t) dta
b
* If we know f(a), the Fundamental Theorem enables us to reconstruct the function from a knowledge of its derivative
The price of a new car is $24 500. The price of a new car is
changing at a rate of 120 + 180 t dollars per year. =dPdt
How much will the car cost 5 years from now?
Change in price =
P(5) P(0) = 0
5
P '(t) dt =0
5
120 + 180 t( ) dt
Calculate an RSUM
Given the function f(x) = x3
Compute
a) left and right sums with 50 subdivisions
b) the Fundamental Theorem of Calculus
the derivative is:
1
2
The velocity of a car in km per hour is given by
v(t) = 3t2 + 2t for t 0
Calculate the distance traveled from t = 1 to t = 4 hours
a) using the left and right sums with n = 100
b) using the Fundamental Theorem and the fact that if
f(t) = t3 + t2 f '(t) = 3t2 + 2t
The function f(x) = sin(2x) has the derivative f '(x) = 2 cos(2x)
Compute 0
/2
2cos(2x) dx
a) left and right sums with 50 subdivisions
b) the Fundamental Theorem of Calculus
using
The graph below shows the rate in gallons per hour at which oil is leaking out of a tank
y = r(t)
Write a definite integral that represents the total amount of oil that leaks out in the first hour.
y = r(t)
0
1
r(t) dt
Shade the region whose area represents the total amount of oil that leaks out in the first hour.
y = r(t)
Give a lower and upper estimate of the total amount of oil that leaks out in the first hour.
y = r(t)