Math 252 Online Graded Assignment #4

It is essential to show all steps by hand. Also, if a method is prescribed, use only that method (no credit otherwise.)

Answers must be exact. Decimal approximations may not be given unless they are asked for

1. (4 points each) Consider the parametric equations and y = 1 - t.

(i) Using a table, sketch the curve represented by the parametric equations. Write out the table that you use. Be sure to indicate the orientation of the curve.

(ii) Determine the corresponding rectangular equation, writing y as a function of x (isolating y.)

(iii) Determine the (x , y) coordinates and the value of the parameter at the point(s) where the slope of the tangent line to the curve is 4 .

(iv) What is the concavity of the graph at the point where t = -1? Your work must not rely on a graph but, rather, on calculus techniques.

2. (4 points each) Write a set of parametric equations for the each of the following figures.

(i) the line segment with endpoints at (3 , -5) and (-2 , 11), such that the parameter value t satisfies

(ii) the lower half of the circle with center (8 , -7) and radius 6

3. (4 points each) Determine the arc length of the curve given by the parametric equations and y = 6t on the interval defined by in the two different ways prescribed below. Round your answer to the nearest hundredth.

(i) Use the formula for arc length in parametric form. (See page 722 in the text.)

(ii) Write the corresponding rectangular equation and then use the arc length formula from Chapter 7, Section 4. (See page 477 in the text.)

4. (2 points each)

(i) Convert to rectangular coordinates.

(ii) Convert (-10 , 15) to polar coordinates, rounding the radius to the nearest hundredth and rounding the angle to the nearest hundredth of a degree.

5. (4 points) Convert to a rectangular equation. Use the rectangular equation to find the center and radius.

6. (4 points) Sketch a graph of , , and calculate the area of one petal, rounding your answer to the nearest hundredth.

7. (3 points) Solve the differential equation , isolating y—that is, by writing the solution as y as a function of x.

8. (3 points) Solve the differential equation, writing the solution as y as a function of x (isolating y.)