MCV4U Unit 3 Test Curve Sketching (V1). Name: Knowledge: /25 ... Use the
second derivative test to classify the critical points as local maxima, local minima,
...
MCV4U Unit 3 Test Curve Sketching (V1) Name: Knowledge:
/25
Application:
/19
Inquiry:
/8
Comm.:
KNOWLEDGE Identify the letter of the choice that best completes the statement or answers the question.
____ 1.
The function a. b.
____ 2.
When does the function a. (0, 0) b. (1, 1)
____ 3.
When does the function a. (1, 2) b. (–1, –20)
____ 4.
Evaluate
____ 5.
____ 6.
have a local minimum? c. (1, 0) d. (–1, 0)
have a point of inflection? c. (2, 1) d. (0, –3)
.
____ 7.
c. d.
Find the absolute maximum of the function a. 6 b. 0
Find the critical point for the function a. (–1, 0) b. (1, 0.5)
If a. b.
(1 mark each)
increases on which interval? c. d.
a. 5 b. 0
on the interval c. –1 d. 8
. c. (0, 0) d. no critical point
, where a, b, c, and d are constants, find c. d.
.
/7
.
8. Analyze and graph f ( x) 2 x 6 x 8 . Show all work including intercepts, critical points, and inflection points. (11 marks) 3
Intercepts:
2
Critical Points:
Inflection points:
y
9. Find the critical points of the function f ( x) x 12 x 1 . Use the second derivative test to classify the critical points as local maxima, local minima, or neither. (4 marks) 3
10. State the possible point(s) of inflection and the intervals of concavity of a function with f ( x) x 8x 3x 5 . (3 marks) 4
3
Application 11. The height of a motorcycle stunt man is given by h(t ) 4.9t 24.5t 2 , where h is the height in metres, t 2
seconds after the motorcycle flies off an inclined ramp. Determine the maximum and minimum height in the interval of 0 t 4 seconds. (4 marks)
x2 2x 6x 2 2 12. Given: f ( x) , f ' ( x) , f ' ' ( x) . Sketch and completely label the graph using the 1 x2 1 x 2 2 1 x 2 3 following headings.
(15 marks)
x-intercept(s)
y-intercept
Vertical Asymptotes
Horizontal Asymptotes
Coordinates of Local Maxima/Minima
Intervals of Increase/Decrease
Points of inflection
Intervals of Concavity
y
x
Inquiry 13. Sketch a possible function that satisfies the following criteria:
(4 marks)
f ' ' ( x) 0 when x 4
y
f ' ' ( x) 0 when x 4
f ' ( x) 0 for all x
lim f ( x)
x 4
lim f ( x)
x 4
x
lim f ( x) 2
x
f (2) 0
14. The function f ( x) 2kx 3x px 3 has a local minimum at x = -1 and a point of inflection at x = 1. Determine the values of k and p. (4 marks) 3
2
Communication ANSWER ONLY 15 OR 16 AND 17 15. Sketch a graph of f (x) based on the information from the table:
,3
f ' ( x) f ' ' ( x)
3,0
3
0
0, 2
0
0
2,
2 0
(4 marks)
y x
16. Consider the function f ( x)
1 . Use limits to determine the horizontal asymptote. x 1 2
(2 marks)
17. If f ' (a) 0 , then there will always be a local extreme at x = a . Is this statement true or false? If this is true, explain. If it is false, give a counter example to disprove it. (2 marks)
3 marks for form
