Berkeley City College
Practice Problems
Math 3B - Calculus II - Chapters 10 - Parametric & Polar Curves
Name__________________________________________
Graph the curve whose parametric representation is given.
1) x = 2 t - 3 , y = 3 7 - t , 3 ≤ t ≤ 7
1)
2) x = 2t - 1, y = t2 + 3; -4 ≤ t ≤ 4
2)
y
10
10 x
-10
-10
Obtain the Cartesian equation of the curve by eliminating the parameter.
3) x = cos t , y = 8 sin t
3)
4) x = 9 sec t , y = 7 tan t
4)
5) x = t3 - 5t, y = t2 - 5
5)
Find dy/dx without eliminating the parameter.
6) x = 1 - 9 cos t, y = 1 + 6 sin t, t ≠ n!
6)
7) x = 1/t6, y = -3 + ln t
7)
8) x = 3t2 + t, y = t2 - 8t + 4
8)
9) x = ln(2t), y = e2t
9)
Find d2y/dx2 without eliminating the parameter.
10) x = ln(2t), y = ln(8t)3, t > 0
10)
2
2
11) x = t + 8t, y = t - 2t, t ≠ -8
2
2
11)
12) x =
12)
t + 7, y =
Instructor: K Pernell
5t, t > 0
1
Find an equation for the line tangent to the curve at the point defined by the given value of t.
13) x = csc t, y = 12 cot t, t = !
3
14) x = t, y =
2t, t = 2
13)
14)
Obtain the Cartesian equation of the curve by eliminating the parameter.
15) x = 5 tan θ, y = 4 cot θ
15)
16) x = 9 sin2 t, y = 9 cos2 t; 0 ≤ t ≤ 2"
16)
17) x = 7 cos t , y = 4 sin t
17)
Find the length of the parametric curve defined over the given interval.
18) x = 5 sin3 t, y = 5 cos3 t, 0 ≤ t ≤ !/2
18)
Find the area of the surface generated by revolving the curves about the indicated axis.
19) x = 2 + cos t, y = sin t, 0 ≤ t ≤ !; x-axis
19)
For the given rectangular equation, write an equivalent polar equation.
20) x2 + y 2 = 64
20)
21) xy = 7
21)
For the given polar equation, write an equivalent rectangular equation.
22) r cos θ = 11
23) r = -9 csc θ
22)
23)
Find the slope of the polar curve at the indicated point.
"
24) r = 1 - sin θ, θ =
3
24)
25) r = -9 - 8 cos θ, θ = "
2
25)
26) r = 4 , θ = 4"
θ
26)
Find the length of the curve.
27) The spiral r = 5θ2, 0 ≤ θ ≤ 2
3
27)
θ
28) The curve r = a cos2
, 0 ≤ θ ≤ ", a > 0
2
28)
2
29) The spiral r = e4θ, 0 ≤ θ ≤ "
29)
30) The cardioid r = 9(1 - cos θ)
30)
3
Answer Key
Testname: MATH3B_CH10_PRACTICE
1)
y
6
4
2
-6
-4
-2
2
4
6
x
-2
-4
-6
2)
y
10
10 x
-10
-10
3) 64x2 + y 2 = 64
x2 - y 2 = 1
4)
81
49
5) x2 = y 3 + 5y 2
2
6) cot t
3
6
7) - t
6
8)
2t - 8
6t + 1
9) 2te2t
10) 0
10
11)
(t + 8)3
12) - 7 5
t3/2
13) y = 24x - 12
3
4
Answer Key
Testname: MATH3B_CH10_PRACTICE
14) y = 1 x + 1
2
15) xy = 20
16) x + y = 9
x2 + y 2 = 1
17)
49
16
18) 15
19) 4"
20) r = 8
21) r2 =
7
sin θ cos θ
22) x = 11
23) y = -9
24) 1
8
25)
9
26) -4"
280
27)
3
28) 2a
17 (e4" - 1)
4
29)
30) 72
5