MCV4U Exam Review

Rates of Change and Derivatives

1. Use the following graph of the piecewise function to fill in the following table.

2. Evaluate the following limits:

a) if evaluate

b)

c)

d)

2. Determine the slope of the tangent at to the function

3. Determine the equation of the tangent at to the function

4. Analyze the continuity of the piecewise function :

5. Use First Principles to determine the derivative of the function

.

Applications of Derivatives

1. Determine given

2. Given , determine the derivatives of the following functions.

Simplify your solutions.

a)

b)

c)

d)

3. A piece of sheet metal, by is to be used to make a rectangular box with an open top.

Determine the dimensions that will give the box with the largest volume.

Curve Sketching

1. Identify the type of point for each of the following and then sketch a graph of what such a function

could look like

continued...

SOLUTION:

2. Determine the points of inflection and the intervals of concavity of the function

0

Points of Inflection:

3. Graph the function . Label and intercepts (when

possible), any asymptotes, local maxima, local minima, and points of inflection.

decreasing local min increasing local max decreasing local min increasing

concave up point of

inflection concave down point of

inflection concave up

4. Graph the function

. Label and intercepts (when possible), any asymptotes,

local maxima, local minima, and points of inflection.

(work to the right)

decr. VA decr. VA decr.

concave

down VA concave

up POI concave

down VA concave

up

Exponential and Trig Derivatives

1. Find the equation of the tangent to the function at

s

2. Determine the maximum length of a rod that can be taken from a hallway that is wide around a

corner to a perpendicular hallway that is wide.

Introduction to Vectors

1. Determine the resultant of the following vectors:

a)

b) a vector long and the vector long at an angle of clockwise from the first vector.

2. Determine the Northern and Western components of the boat going at a bearing of

3. Is on the plane determined by the span of ?

4. What is a linear combination and how does it relate to a spanning set?

5. Sketch the parallelepiped formed by the vectors .

Applications of Vectors

1. Determine the angle between the vectors and .

2. How much horizontal work is done if Mrs. Bethany is pulling a trolley on a flat surface for and

she is pulling with a force of along the handle, which is at an angle of above the horizontal?

3. Determine the normal of the vectors and

4. Determine the magnitude of the torque on a nut if the wrench is 21cm long and if the force is

applied at an angle of to the wrench.

5. Find the scalar projection of the vector on .

6. Determine the vector projection of the vector on .

7. At what bearing should a ship aim in order to travel at a bearing of if the wind is

pushing the boat at a bearing of and the ship's propellers are pushing it at a speed of ?

.

8. What is an equilibrant?

9. A mass of is suspended from a ceiling by two lengths of rope that make angles of and

with the ceiling. Determine the tension in each of the ropes.

area of a parallelogram

Equations of Lines and Planes

1. a) Consider the line going through the points and . Fill in the table:

Vector Equation: Parametric Equations:

Cartesian Equation:

b) Consider the line going through the points and . Fill in the table:

Vector Equation: Parametric Equations:

Symmetric Equation:

c) Consider the plane containing the lines and

. Fill in the table:

Vector Equation:

Parametric Equations:

Cartesian Equation:

2. Sketch the plane .

3. Show that the line is contained in the plane

Points, Lines, and Planes

1. What is a consistent system? An inconsistent system?

2. What is the intersection between the lines and

.

Therefore, the lines intersect at a point:

3. Determine the intersection between the planes

.

3 7 -2 13 4 -1 3 24 5 -2 -4 6

Reduce to: 0 0 1 3 4 -1 3 24 3 0 10 42

.

4. Determine the distance between the point and the plane

.