MCV4U Final Examination

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MCV4U

Final Examination

A. Knowledge and Understanding

Question 1: (5 Marks)

Simplify and then evaluate the following limit

a) b)

Question 2: (10 Marks)

Find the first and second derivatives of the following functions. For a, find the first derivative from first principles. Find the first and second  derivative of b. Find the second derivative any way you choose. For c only find the first derivative.

a. f(x) = x2 + 1

b. f(x) = cos(x) ex

c. f(x) = x sin(x2)

Question 3: (8 Marks)

Using first principles, find the first derivative of the following functions

a) f(x)= 7x+2

b) f(x)= 3x2 + 7x+14

Question 4: (12 Marks)

Let A = [1,5,2] and B = [2,9, - 1]

a) Find A×B, and verify that A×B is perpendicular to both A and B

b) Find A.B

c) Find A-B

d) Find Proj(A on to B)

e) Find 3A-2B

f) Find if a linear combination of A and B can be made to make vector C= [5,23,1]

B. Thinking

Question 5: (8 Marks)

Find the area of the triangle with the given verities; (7,3,4),(1,0,6),(4,5,-2), calculate the parallelogram with sides; A=[6,2,1], B=[2,-4,1]

Question 6: (10 Marks)

The position, at time t, in seconds, of an object moving along a line is given bys(t) = 3t3 - 40.5t2 + 162t for O <= t <= 8.

a. Determine the velocity and the acceleration at any time t.

b. When is the object stationary? When is it advancing? When is it retreating?

c. At what time, t, is the velocity not changing?

d. At what time, t, is the velocity decreasing?

e. At what time, t, is the velocity increasing?

C. Communication

Question 7: (7 Marks)

What does the following limit indicate about the graph of the function f(x) = x2? Explain with the use of a graph.

Question 8: (6 Marks)

Find the general equation of a line passing through the point (3, 0) with slope -6

D. Application

Question 9: (7 Marks)

Find the intersection of the three planes:

x+y+2z=-2

3x-y+14z=6

x+2y=-5

Question 10: (6 Marks)

show in a diagram the ways that three non-parallel planes can intersect.

Question 11: (11 Marks)

A fence 4 feet tall runs parallel to a tall building at a distance of 4 feet from the building.

What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?

Question 12: (10 Marks)

Find two positive numbers whose product is 750 and for which the sum of one and 10 times the other is a minimum.


2024-01-02