Math 103 Final Examination - Spring 2001

Show all work. No credit will be given for answers without proper mathematical justification. All questions are worth 10 points.

1. a) Sketch the graph of the parabola represented by the equation .

b) Find the equation of the line tangent to the parabola at the point on

the parabola where the y-coordinate = -2.

c) Find the x and y coordinates of the point of intersection of the two lines

y = and y=.

2. Given the vertex of a parabola is (1/2, 2) and another point on the parabola is (1, 3)

and the Axis of Symmetry is parallel to the Y-axis. Find the equation of the

parabola.

3. a) Sketch the graph of y = + 16 using the calculus method developed in

the course. (It is NOT necessary to find the point of inflection or concavity.)

b) Find the roots of the equation in part "a".

c) If the roots are real numbers, check that the midpoint of the line segment

joining the roots lies on the Axis of Symmetry;

OR

If the roots are complex numbers show that the average of the roots gives the

x-coordinate of points on the Axis of Symmetry.

4. a) Sketch the graph of by finding the critical points

and the x-intercept, and by investigating the behavior of the curve as and as .

b) Find both the x and y coordinates of the point on the curve at

which the tangent line has a slope of M= .

c) Find and simplify the equation of a line through the point whose

slope equals 3. Express your answer in y = mx + b form.

5.a) Find the equation of the line through the points (2, 4/3) and ( -3, 1/2). Express

your answer in y = mx + b form.

b) Reduce the system of equations and to

one equation in one unknown. Your final equation should have all non zero

terms on the left hand side and zero on the right hand side. Combine similar

terms on the left hand side.

c) If x = 5 is a root of the polynomial P(x) = x33 x2 133x + 5 use the Root

Factor Theorem and polynomial division to find the other 2 roots.

6. a) Solve for y in terms of x:.

b) Solve for y in terms of x: .

c) Solve for all values of y:.

7. a) Expand (° + °).

b) Evaluate to obtain a single numerical value. Show the

special right triangle used to find the value of the trigonometric function.

c) Evaluate the .

d) Find the area of the region bounded by the curves and

8. a) Sketch the graph of and then find the function in the form

y = f(x) that represents the bottom half of the parabola.

b) Find the equation of the parabola for which y = 3 + represents the top

half.

c) Find the domain and range of the function y = 1.

9. a) Find and simplify the derivative, dy/dx, of the function y in order to

find the x-coordinates of the critical points. DO NOT FIND THE

CORRESPONDING Y-COORDINATES.

b) If y = has x = 8/5 as the x-coordinate of a critical value of the

function, the corresponding y-coordinate can be expressed as . Find

the values of a and b.

10. a) Reduce the system of 3 equations in 3 unknowns , x, y and z, to 2 equations

containing only the 2 unknowns y and z: (SIMPLIFY THE RESULTING 2

EQUATIONS BUT DO NOT SOLVE THEM.)

-4x + 3y + = 0

- x + 2y -2z = 3

3x- y -6z = 2

b) Solve the following system of equations for all values of the "x" coordinate of

each solution set, (x,,y), that satisfies the system: (DO NOT SOLVE FOR THE

CORRESPONDING "y" value.)