Download Math1325: Test 3 Review

Math1325: Test 3 Review
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the derivative.
1) y = 5x2 e3x
1)
A) 10ex3x(3x + 2)
2) y =
B) 5xe 3x(2x + 3)
C) 10xe3x(2x + 3)
D) 5xe 3x(3x + 2)
e-x + 1
ex
A)
2)
-ex + 2
e2x
B)
ex - 2
e2x
C)
-ex - 2
e2x
D)
ex + 2
e2x
Find the derivative of the function.
3) y = ln (4x3 - x2 )
12x - 2
A)
4x2
3)
12x - 2
B)
4x2 - x
4x - 2
C)
4x2 - x
12x - 2
D)
4x3 - x
ex
B)
x
ex(x ln x + 1)
C)
x
ex(ln x + x)
D)
x
Find the derivative.
4) f(x) = ex ln x, x > 0
4)
A) ex ln x
5) f(x) =
ln x
ex
A)
ln x
xex
5)
B)
1 - x ln x
ex
C)
1
xex
D)
1 - x ln x
xex
Find all points where the function is discontinuous.
6)
A) None
6)
B) x = 1
C) x = -2
D) x = -2, x = 1
Find the largest open interval where the function is changing as requested.
7) Increasing y = (x2 - 9)2
A) (-3, 0)
8) Decreasing f(x) = A) (-∞, 3)
7)
B) (-∞, 0)
C) (3, ∞)
D) (-3, 3)
B) (-3, ∞)
C) (-∞, -3)
D) (3, ∞)
x+3
8)
1
Find the indicated absolute extremum as well as all values of x where it occurs on the specified domain.
9) f(x) = x3 - 3x2 ; [0, 4]
Minimum
A) 16 at x = 4
C) No absolute minimum
9)
B) -4 at x = 2
D) 0 at x = 0
Determine the location of each local extremum of the function.
10) f(x) = x3 - 6x2 + 12x - 1
10)
A) Local maximum at 2; local minimum at -2
B) Local maximum at 2
C) Local minimum at 2
D) No local extrema
Use the first derivative test to determine the location of each local extremum and the value of the function at that
extremum.
x2
11) f(x) =
11)
x2+ 5
A) No local extrema
C) Local minimum at (5, 0.83333333)
B) Local minimum at (0, 0)
D) Local maximum at (0, 0)
Evaluate f''(c) at the point.
12) f(x) = (x2 - 4)(x3 - 3), c = 1
A) f''(1) = 36
B) f''(1) = -4
13) f(x) = ln (3x2 - 2), c = -1
A) f''(-1) = -30
12)
C) f''(1) = -10
D) f''(1) = 4
C) f''(-1) = -6
D) f''(-1) = -1
13)
B) f''(-1) = 30
Find the coordinates of the points of inflection for the function.
14) f(x) = x2 + 8x + 18
A) (-5, 1)
C) (-3, 3)
14)
B) (-4, 2)
D) There are no points of inflection.
Find the largest open intervals where the function is concave upward.
15) f(x) = 4x3 - 45x2 + 150x
15
A) ,∞
4
15
B) -∞,
4
15
C)
,∞
4
15)
15
D) -∞, 4
Find the indicated absolute extremum as well as all values of x where it occurs on the specified domain.
16) f(x) = (x + 1)2 (x - 2); [-2, 1]
Maximum
A) -4 at x = -2
C) 0 at x = -1
B) -2 at x = 0
D) No absolute maximum
2
16)
Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither.
17) f(x) = -x3 - 3x2 + 24x - 4
17)
A) Local maximum at 4; local minimum at -2
B) Local maximum at -4; local minimum at 2
C) Local maximum at -2; local minimum at 4
D) Local maximum at 2; local minimum at -4
Find the absolute extremum within the specified domain.
1
18) Minimum of f(x) = x3 - 2x2 + 3x - 4; [-2, 5]
3
A) 2, -
62
3
B) 2, -
18)
61
3
C) -2, -
61
3
D) -2, -
62
3
Solve the problem.
19) Find the dimensions that produce the maximum floor area for a one-story house that is rectangular
in shape and has a perimeter of 163 ft.
A) 13.58 ft x 40.75 ft
B) 40.75 ft x 163 ft
C) 40.75 ft x 40.75 ft
D) 81.5 ft x 81.5 ft
19)
20) An architect needs to design a rectangular room with an area of 72 ft2 . What dimensions should he
use in order to minimize the perimeter?
A) 8.49 ft × 8.49 ft
B) 14.4 ft × 72 ft
C) 8.49 ft × 18 ft
D) 18 ft × 18 ft
20)
21) A piece of molding 152 cm long is to be cut to form a rectangular picture frame. What dimensions
will enclose the largest area?
A) 38 cm × 38 cm
B) 12.33 cm × 38 cm
C) 30.4 cm × 30.4 cm
D) 12.33 cm × 12.33 cm
21)
22) Of all numbers whose difference is 12, find the two that have the minimum product.
A) 0 and 12
B) 1 and 13
C) 6 and -6
D) 24 and 12
22)
23) Find two numbers whose sum is 490 and whose product is as large as possible.
A) 10 and 480
B) 1 and 489
C) 244 and 246
D) 245 and 245
23)
24) Find two numbers x and y such that their sum is 120 and x2y is maximized.
A) x = 90, y = 30
B) x = 40, y = 80
C) x = 80, y = 40
24)
D) x = 30, y = 90
Find the open interval(s) where the function is changing as requested.
25) Increasing; f(x) = x2 - 2x + 1
A) (-∞, 1)
B) (1, ∞)
C) (-∞, 0)
25)
D) (0, ∞)
Find the x-value of all points where the function has relative extrema. Find the value(s) of any relative extrema.
26) f(x) = 3x4 + 16x3 + 24x2 + 32
26)
A) Relative minimum of 30 at -1.
B) Relative maximum of 48 at -2; Relative minimum of 32 at 0.
C) No relative extrema.
D) Relative minimum of 32 at 0.
Sketch the graph and show all extrema, inflection points, and asymptotes where applicable.
3
27) f(x) = 2x3 - 12x2 + 18x
27)
y
24
12
-8
-4
4
8
x
-12
-24
A) Rel min: 2,10
No inflection points
B) Rel:max 1,8 ; Rel min: 3,0
Inflection point: 2,4
y
-8
y
24
24
12
12
-4
4
8
x
-8
-4
4
-12
-12
-24
-24
C) No extrema
Inflection point: 0, 0
8
x
8
x
D) Rel:max: 0, 0
Rel min: 1,-1
Inflection point: 0.5,-0.5
y
24
y
24
12
12
-8
-4
4
8
x
-8
-4
4
-12
-12
-24
-24
4
28) f(x) =
x2
2
x +2
28)
y
1
-3
-2
-1
1
2
3 x
-1
A) Rel min: 0,
1
2
B) Rel min: (0,0)
Inflection points: -
No inflection points
6 1
,
,
3 4
6 1
,
3 4
y
y
1
1
-3
-2
-1
1
2
3 x
-3
-2
-1
1
2
3 x
1
2
3 x
-1
-1
C) Rel min: (0,0)
No inflection points
D) Rel min: 0,-
1
2
No inflection points
y
y
1
1
-3
-2
-1
1
2
3 x
-3
-2
-1
-1
-1
5
Solve the problem.
29) A private shipping company will accept a box for domestic shipment only if the sum of its length
and girth (distance around) does not exceed 120 in. What dimensions will give a box with a square
end the largest possible volume?
A) 20 in. × 20 in. × 100 in.
C) 20 in. × 20 in. × 40 in.
B) 20 in. × 40 in. × 40 in.
D) 40 in. × 40 in. × 40 in.
Find the open interval(s) where the function is changing as requested.
x+2
30) Decreasing; f(x) =
x-3
A) (-∞, -3)
29)
B) (-∞, 3), (3, ∞)
C) (-∞, 2), (2, ∞)
6
30)
D) none
Answer Key
Testname: 1325-3
1) D
2) C
3) B
4) C
5) D
6) B
7) C
8) B
9) B
10) D
11) B
12) C
13) A
14) D
15) C
16) C
17) D
18) D
19) C
20) A
21) A
22) C
23) D
24) C
25) B
26) D
27) B
28) B
29) C
30) B
7