MAT1300 FInal
MAT1300D
Solution to Final Examination
Fall 2007
Solution to Final Examination
MAT1300D, Fall 2007
Part I. Multiple-Choice Questions (30 marks)
1.
D
2.
3.
E
A
4.
5.
B
C
6.
A
7.
8.
C
E
9.
D
10.
C
1. Find the equation of the tangent line to the graph of y = 3 x − 1 when x = 4.
(A) y =
1
1
3
3
3 x + 1 ; (B) y = x + 3 ; (C) y = x + 7 ; (D) y = x + 2 ; (E) y = x − 1 .
2
2
4
4
4
Solution. y' =
3
. When x = 4, y = 5, and y' =
2 x
3
3 line is y = (x − 4) + 5, or y = x + 2.
4
4
3
. Hence, the equation of the tangent
4
x 2 − x − 12
.
x →−3 x+3 2. Calculate lim
(A)
1
5
(B) 3
(C)
7
4
(D)
4
9
(E) −7.
Solution. x2 − x − 12 = (x + 3)(x − 4). x 2 − x − 12
( x + 3)( x − 4)
= lim
= lim ( x − 4) = −7. x →−3 x →−3 x →−3 x+3 x+3 lim 3. On what interval is the function g(x) = −2x3 + 12x2 − 36x + 3 concave down?
(A) (2, ∞)
(B) (2, 3)
(C) (−1, ∞)
(D) (−∞, −1)
(E) (−2, 4).
Solution. g' = −6x2 + 24x − 36, g" = −12x + 24. g'' = 0 implies x = 2. When x < 2, g" <
0, the graph of g(x) is concave down.
4. Which of the following statements is true for the function g(x) = 2x3 + 3x2 − 36x + 2?
(A) x = −1 is a local minimum.
(C) x = −3 is a local minimum.
(E) x = 1 is a local maximum.
(B) x = −3 is a local maximum.
(D) x = 1 is a local minimum.
1
MAT1300D
Solution to Final Examination
Fall 2007
Solution. g' = 6x2 + 6x − 36. Critical points are the roots of x2 + x − 6 = 0, x = −3, 2.
When −∞ < x < −3, g' > 0; when −3 < x < 2, g' < 0; when 2 < x < ∞, g' > 0. g attains a local maximum at x = −3.
∫
5. Calculate
(A) 10
Solution.
4
0
(3 x + 1)dx
(C) 20
(B) 15
∫
(D) 25
(E) 30
4
4
2
(3 x + 1)dx = 3∫ x1/ 2 dx + ∫ dx = 3 43 / 2 + 4 = 16 + 4 = 20 .
0
0
0
3
4
6a. Suppose that for a certain product, the demand function is given by D(x) = 11 − x2 and the
Solution to Final Examination
Fall 2007
Solution to Final Examination
MAT1300D, Fall 2007
Part I. Multiple-Choice Questions (30 marks)
1.
D
2.
3.
E
A
4.
5.
B
C
6.
A
7.
8.
C
E
9.
D
10.
C
1. Find the equation of the tangent line to the graph of y = 3 x − 1 when x = 4.
(A) y =
1
1
3
3
3 x + 1 ; (B) y = x + 3 ; (C) y = x + 7 ; (D) y = x + 2 ; (E) y = x − 1 .
2
2
4
4
4
Solution. y' =
3
. When x = 4, y = 5, and y' =
2 x
3
3 line is y = (x − 4) + 5, or y = x + 2.
4
4
3
. Hence, the equation of the tangent
4
x 2 − x − 12
.
x →−3 x+3 2. Calculate lim
(A)
1
5
(B) 3
(C)
7
4
(D)
4
9
(E) −7.
Solution. x2 − x − 12 = (x + 3)(x − 4). x 2 − x − 12
( x + 3)( x − 4)
= lim
= lim ( x − 4) = −7. x →−3 x →−3 x →−3 x+3 x+3 lim 3. On what interval is the function g(x) = −2x3 + 12x2 − 36x + 3 concave down?
(A) (2, ∞)
(B) (2, 3)
(C) (−1, ∞)
(D) (−∞, −1)
(E) (−2, 4).
Solution. g' = −6x2 + 24x − 36, g" = −12x + 24. g'' = 0 implies x = 2. When x < 2, g" <
0, the graph of g(x) is concave down.
4. Which of the following statements is true for the function g(x) = 2x3 + 3x2 − 36x + 2?
(A) x = −1 is a local minimum.
(C) x = −3 is a local minimum.
(E) x = 1 is a local maximum.
(B) x = −3 is a local maximum.
(D) x = 1 is a local minimum.
1
MAT1300D
Solution to Final Examination
Fall 2007
Solution. g' = 6x2 + 6x − 36. Critical points are the roots of x2 + x − 6 = 0, x = −3, 2.
When −∞ < x < −3, g' > 0; when −3 < x < 2, g' < 0; when 2 < x < ∞, g' > 0. g attains a local maximum at x = −3.
∫
5. Calculate
(A) 10
Solution.
4
0
(3 x + 1)dx
(C) 20
(B) 15
∫
(D) 25
(E) 30
4
4
2
(3 x + 1)dx = 3∫ x1/ 2 dx + ∫ dx = 3 43 / 2 + 4 = 16 + 4 = 20 .
0
0
0
3
4
6a. Suppose that for a certain product, the demand function is given by D(x) = 11 − x2 and the