calculus 2

Math 126 Name (Print): Takehome 4 4/21/2017

You may not use your books, notes, or cell phones on this exam.

You are required to show your work on each problem on this exam. A correct answer, unsupported by calcu- lations, explainations, or algebraic work will recieve no credit; an incorrect answer supported substantially with correct calculations and explanations might still receive partial credit.

Do not write graded material on this page.

Math 126 Takehome 4 - Page 2 of 6 4/21/2017

1. Consider the parametric curve x = t2 − 1 , y = t4 − t2 , −2 ≤ t ≤ 2

(a) Approximate the graph of the function by plotting ordered pairs obtained from integer t values on an xy−plane (label).

(b) Solve for y = f (x) or x = f (y).

Math 126 Takehome 4 - Page 3 of 6 4/21/2017

2. Consider the parametric curve x = t2 − 1 , y = t4 − t2 , −∞≤ t ≤∞

Use formulas given to find the following:

(a) Find the equation of the tangent line to the curve at t = 2.

(b) Find d 2y

d x2 .

Math 126 Takehome 4 - Page 4 of 6 4/21/2017

3. (a) Find the arc length of the the curve x = t2 , y = 1

30 (20t + 25)

3 2 , 0 ≤ t ≤ 1

(b) Find the length of the polar curve r = θ2 , 0 ≤ θ ≤ 3

Math 126 Takehome 4 - Page 5 of 6 4/21/2017

4. (a) Convert (4, π3 )P to rectangular.

(b) Convert (4,−2)R to polar.

(c) Convert (x − 5)2 + y2 = 25 to polar.

(d) Convert r = 1 + 3r cos(θ) to rectangular.

(e) Find all polar coordinates equivalent to (−7, 2π19 )P.

Math 126 Takehome 4 - Page 6 of 6 4/21/2017

5. Sketch a picture describing the regions and use them to set up the integrals that give the area of the region that is described. MAKE SURE TO LABEL YOUR PICTURES. YOU DO NOT HAVE TO EVALUATE THE INTEGRALS.

(a) The area inside the circle of radius 2 centered at (0, 0), above the line y = x, and below the line y = −x √

3

(b) The area inside the circle r = 2 cos(θ) (convert to rectangular to graph) and outside the unit circle.