lesson 5 nov 3
TRANSCRIPT
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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
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change in f = rate of change * time
change of f = f(b) f(a)
f(b)
f(a)
a b
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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
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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?
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Change in price =
P(5) P(0) = 0
5
P '(t) dt =0
5
120 + 180 t( ) dt
Calculate an RSUM
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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
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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
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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
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The graph below shows the rate in gallons per hour at which oil is leaking out of a tank
y = r(t)
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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
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Shade the region whose area represents the total amount of oil that leaks out in the first hour.
y = r(t)
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Give a lower and upper estimate of the total amount of oil that leaks out in the first hour.
y = r(t)
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