Power and energy Energy is the capacity to do work, and power is the rate at which energy is used or consumed. Therefore, if E ( t ) is the energy function for a system, then P ( t ) = E ′( t ) is the power function. A unit of energy is the kilowatt-hour (1 kWh is the amount of energy needed to light ten 100-W lightbulbs for an hour); the corresponding units for power are kilowatts. The following figure shows the energy consumed by a small community over a 25-hour period. a. Estimate the power at t = 10 and t = 20 hr. Be sure to include units in your calculation. b. At what times on the interval [0, 25] is the power zero? c. At what times on the interval [0, 25] is the power a maximum?
Power and energy Energy is the capacity to do work, and power is the rate at which energy is used or consumed. Therefore, if E ( t ) is the energy function for a system, then P ( t ) = E ′( t ) is the power function. A unit of energy is the kilowatt-hour (1 kWh is the amount of energy needed to light ten 100-W lightbulbs for an hour); the corresponding units for power are kilowatts. The following figure shows the energy consumed by a small community over a 25-hour period. a. Estimate the power at t = 10 and t = 20 hr. Be sure to include units in your calculation. b. At what times on the interval [0, 25] is the power zero? c. At what times on the interval [0, 25] is the power a maximum?
Power and energy Energy is the capacity to do work, and power is the rate at which energy is used or consumed. Therefore, if E(t) is the energy function for a system, then P(t) = E′(t) is the power function. A unit of energy is the kilowatt-hour (1 kWh is the amount of energy needed to light ten 100-W lightbulbs for an hour); the corresponding units for power are kilowatts. The following figure shows the energy consumed by a small community over a 25-hour period.
a. Estimate the power at t = 10 and t = 20 hr. Be sure to include units in your calculation.
b. At what times on the interval [0, 25] is the power zero?
c. At what times on the interval [0, 25] is the power a maximum?
Topic 2
Evaluate S
x
dx, using u-substitution. Then find the integral using
1-x2
trigonometric substitution. Discuss the results!
Topic 3
Explain what an elementary anti-derivative is. Then consider the following
ex
integrals: fed dx
x
1
Sdx
In x
Joseph Liouville proved that the first integral does not have an elementary anti-
derivative Use this fact to prove that the second integral does not have an
elementary anti-derivative. (hint: use an appropriate u-substitution!)
1. Given the vector field F(x, y, z) = -xi, verify the relation
1
V.F(0,0,0) = lim
0+ volume inside Se
ff F• Nds
SE
where SE is the surface enclosing a cube centred at the origin and having edges of length 2€. Then,
determine if the origin is sink or source.
4
3
2
-5 4-3 -2 -1
1 2 3 4 5
12
23
-4
The function graphed above is:
Increasing on the interval(s)
Decreasing on the interval(s)
Chapter 3 Solutions
MyLab Math with Pearson eText -- Standalone Access Card -- for Calculus: Early Transcendentals (3rd Edition)
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