Electricity Rates in Illinois Commonwealth Edison Company supplies electricity to residential customers for a monthly customer charge of $ 15.14 plus 7.57 cents per kilowatt- hour for up to 800 kilowatt-hours (kW h). (a) Write a linear equation that relates the monthly charge C in dollars, to the number x of kilowatt-hours used in a month, 0 ≤ x ≤ 800 . (b) Graph this equation. (c) What is the monthly charge for using 200 kilowatt-hours? (d) What is the monthly charge for using 500 kilowatt-hours? (e) Interpret the slope of the line. Source: Commonwealth Edison Company, January 2015.
Electricity Rates in Illinois Commonwealth Edison Company supplies electricity to residential customers for a monthly customer charge of $ 15.14 plus 7.57 cents per kilowatt- hour for up to 800 kilowatt-hours (kW h). (a) Write a linear equation that relates the monthly charge C in dollars, to the number x of kilowatt-hours used in a month, 0 ≤ x ≤ 800 . (b) Graph this equation. (c) What is the monthly charge for using 200 kilowatt-hours? (d) What is the monthly charge for using 500 kilowatt-hours? (e) Interpret the slope of the line. Source: Commonwealth Edison Company, January 2015.
Solution Summary: The author explains the linear equation that relates the monthly charge C, in dollars, to the number x of kilowatt-hours used in a month.
Electricity Rates in Illinois Commonwealth Edison Company supplies electricity to residential customers for a monthly customer charge of
plus
cents per kilowatt- hour for up to 800 kilowatt-hours (kW h).
(a) Write a linear equation that relates the monthly charge
in dollars, to the number
of kilowatt-hours used in a month,
.
(b) Graph this equation.
(c) What is the monthly charge for using 200 kilowatt-hours?
(d) What is the monthly charge for using 500 kilowatt-hours?
(e) Interpret the slope of the line.
Source: Commonwealth Edison Company, January 2015.
A ladder 25 feet long is leaning against the wall of a building. Initially, the foot of the ladder is 7 feet from the wall. The foot of the ladder begins to slide at a rate of 2 ft/sec, causing the top of the ladder to slide down the wall. The location of the foot of the ladder, its x coordinate, at time t seconds is given by
x(t)=7+2t.
wall
y(1)
25 ft. ladder
x(1)
ground
(a) Find the formula for the location of the top of the ladder, the y coordinate, as a function of time t. The formula for y(t)= √ 25² - (7+2t)²
(b) The domain of t values for y(t) ranges from 0
(c) Calculate the average velocity of the top of the ladder on each of these time intervals (correct to three decimal places):
. (Put your cursor in the box, click and a palette will come up to help you enter your symbolic answer.)
time interval
ave velocity
[0,2]
-0.766
[6,8]
-3.225
time interval
ave velocity
-1.224
-9.798
[2,4]
[8,9]
(d) Find a time interval [a,9] so that the average velocity of the top of the ladder on this…
Total marks 15
3.
(i)
Let FRN Rm be a mapping and x = RN is a given
point. Which of the following statements are true? Construct counterex-
amples for any that are false.
(a)
If F is continuous at x then F is differentiable at x.
(b)
If F is differentiable at x then F is continuous at x.
If F is differentiable at x then F has all 1st order partial
(c)
derivatives at x.
(d) If all 1st order partial derivatives of F exist and are con-
tinuous on RN then F is differentiable at x.
[5 Marks]
(ii) Let mappings
F= (F1, F2) R³ → R² and
G=(G1, G2) R² → R²
:
be defined by
F₁ (x1, x2, x3) = x1 + x²,
G1(1, 2) = 31,
F2(x1, x2, x3) = x² + x3,
G2(1, 2)=sin(1+ y2).
By using the chain rule, calculate the Jacobian matrix of the mapping
GoF R3 R²,
i.e., JGoF(x1, x2, x3). What is JGOF(0, 0, 0)?
(iii)
[7 Marks]
Give reasons why the mapping Go F is differentiable at
(0, 0, 0) R³ and determine the derivative matrix D(GF)(0, 0, 0).
[3 Marks]
5.
(i)
Let f R2 R be defined by
f(x1, x2) = x² - 4x1x2 + 2x3.
Find all local minima of f on R².
(ii)
[10 Marks]
Give an example of a function f: R2 R which is not bounded
above and has exactly one critical point, which is a minimum. Justify briefly
Total marks 15
your answer.
[5 Marks]
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