Use substitution or elimination method to predict when developing countries’ carbon dioxide emissions equal developed countries’ carbon dioxide emissions.

MATH2412 Exam II Review

Show All Necessary Working

1. Find the exact value of the expression, if it is defined.


b) d)

2. Verify that each equation is an identity.



3. a) Write the given expression in terms of x and y only.

cos(sin−1 x − tan−1 y)

b) Evaluate sin 2, where cos = -2/5 with in Quadrant II.

4. Prove that each equation is an identity.



5) i) Use rotation of axes to show that the following equation represents a hyperbola. (Write the equation in XY-coordinates. Use a rotation angle that satisfies and that eliminates the -term.)

ii) Consider the following.

a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola.

b) Use a rotation of axes to eliminate the -term. (Write an equation in -coordinates. Use a rotation angle that satisfies.)

6) i) Find the rectangular coordinates of the point with the following polar coordinates:



ii) Find polar coordinates of the point whose rectangular coordinates are



7) Sketch a graph of the polar equation, and express the equation in rectangular coordinates.



8) Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions.

a) Ellipse, eccentricity = , directrix y=

b) Hyperbola, eccentricity = , directrix x=

c) Parabola, directrix

9) A polar equation of a conic is given.

a) Show that the conic is a hyperbola, and sketch the graph.

b) Find the vertices (in polar coordinates) and directrix (as an equation in rectangular coordinates).

c) Find the center of the hyperbola (in polar coordinates), and sketch the asymptotes.

10) a) Find parametric equations for the line with the given properties.

, passing through the point

b) Find parametric equations for the line with the given properties. Passing through (6, 7) and (7, 8).

Question 11.

a) Find the component form of the vector with initial point (-3 , 5) and terminal point (3 , 8).

b) If the vector v = is sketched with initial point (2 , 5), what is its terminal point?

c) Sketch representations of the vector w = with initial points at (0 , 0), (2 , 2), (-2 , -1), and (1 , 4).