Name:
Section 2.4 Review.
Directions: For the following worksheet, please put only your answers in the spaces shown below. Please remember to
give exact answers, not decimal approximations. Attach scratch paper to show your work.
More Difference Quotients: For these difference quotients, show me your work! This time, I’m more interested in
your scratch work than in your final answer.
1. Apply the difference quotient formula to the function f (x) =
To simplify my work, I’ll use
1
h
1
h
1
h
Apply the difference quotient formula.
7
7
+3 −
+3
x+h
x
(f (x + h) − f (x)) as my difference quotient.
(f (x + h)−f (x))
7
7
+3− −3
x+h
x
1
h
1
h
1
h
7
7
−
x+h x
1
h
1
h
Cancel out the 3.
7x − 7x − 7h
(x + h)(x)
−7h
(x + h)(x)
Distribute the negative.
7x − 7(x + h)
(x + h)(x)
Note the colors, and that the −f (x) needs a parenthesis.
7x
7(x + h)
−
(x + h)x x(x + h)
1
h
7
7
+ 3. You should get −
.
x
x(x + h)
Common denominators.
Combine the fraction.
Distribute negative sign.
−7
(x + h)(x)
Cancel the 7x.
Cancel the h.
2. Apply the difference quotient formula to the function f (x) = 3x2 − 2x + 7. You should get 6x + 3h − 2.
f (x + h)−f (x)
h
3(x + h)2 − 2(x + h) + 7−(3x2 − 2x + 7)
h
3(x + h)2 − 2x − 2h + 7 − 3x2 + 2x − 7
h
Apply the difference quotient formula.
Note the colors, and that the −f (x) needs a parenthesis.
Distribute out negatives.
3(x2 + 2xh + h2 ) − 2h − 3x2
h
FOIL the (x + h)2 , cancel out the 2x and the 7.
3x2 + 6xh + 3h2 − 2h − 3x2
h
Distribute the 3.
6xh + 3h2 − 2h
h
Cancel the 3x2 .
6x + 3h − 2
Divide out by the h.
Domain and Range Review: Use the graphs below to determine the domain and range of the given functions. Give
your answer in interval notation.
f (x) =
√
√
g(x) = − 9 − x2
4−x
3. Domain of f (x): (−∞, 4]
Range of f (x): [0, ∞)
4. Domain of g(x): [−3, 3]
Range of g(x): [−3, 0]
Domain Review: Use the algebraic approach discussed in Section 2.4 to determine the domains of the following functions: Give your answer in interval notation.
r
1
5. Domain of f (x) = 5
:
Notice that we have an odd radical (non issue). Set denominator = 0.
x−7
Thus, our interval is (−∞, 7) ∪ (7, ∞).
√
9 − 4x
4
1
:
Set the radicand ≥ 0 and denominator = 0. You’ll get x ≤ and x 6= − .
6. Domain of g(x) =
1 + 8x
9
8
1
1 9
This gives you the interval −∞, −
∪ − , .
8
8 4
7. You are making a box with volume 297 cubic meters using two materials: cardboard and plywood. Cardboard costs
$0.45 per square meter while plywood costs $1.25 per square meter. The bottom of the box will be made of plywood
while the other five surfaces will be made of cardboard. Also, one side of the bottom of the box is twice as long as the
other. (See the diagram below.)
Let x be the shortest side of the bottom of the box. Answer the following:
(a) Determine the cost to build the bottom of the box as a function of x.
Cost = $1.25 (area of the bottom) = 1.25(2x2 )
(b) Use the volume to determine the height of the box as a function of x.
297
V = L W H, so 297 = (x)(2x)(h), so
= h.
2x2
(c) Determine the cost to build the front of the box as a function
of x.
297
Cost = $0.45(area of the front) = 0.45 · xh = 0.45 · x
2x2
(d) Determine the cost to build the right surface of the box as afunction
of x.
297
Cost = $0.45 (area of the right) = 0.45 · 2xh = 0.45 · 2x
2x2
(e) Determine the cost to build the entire box (all six surfaces) as a function of x.
TOTAL COST:
BOTTOM
2
1.25(2x )
+
+
TOP
+
2
0.45(2x )
FRONT
+
0.45 · x
297
2x2
+
BACK
+
0.45 · x
297
2x2
+
LEFT
+
0.45 · 2x
297
2x2
+
RIGHT
+
0.45 · 2x
297
2x2