FINAL EXAM PRACTICE
I. Tangent lines to parametric curves.
1. Find an equation of the tangent line to the curve at the point corresponding to the value of the
parameter:
(a)
x=e  t , y =t−ln t 9 ; t=1
(b)
x=t sin t , y =t cos t ; t=6
(c)
x=cos t t sin t , y =sin t −t cos t ; t=/3
2. Given the parametric curve x=e t cos t , y =e t sin t , find dy/dx at the point corresponding to t =/ 6 .
3. Find the points on the curve x =2cos  , y =sin 2  where the tangent is horizontal or vertical.
II. Areas of regions bounded by parametric curves.
1. Use parametric equations of an ellipse, x=a cos  , y=bsin  , 0≤≤2  , to find the area that it
encloses.
2. Find the area bounded by the curve
x =cost , y =e t , 0≤t≤

, and the lines y = 1 and x = 0.
2
III. Length of a curve given in parametric form
1. Find the length of the curve
x=3 t 2 , y =2 t 3 , 0≤t≤1 .
2. Find the exact length of the parametric curve:
t
t
x=e cos t , y =e sin t , 0≤t ≤

5
1
x= , y=ln t , 1≤t ≤2 .
t
4. The position, in feet, of a particle at time t, in seconds, is given by x=4 t 3, y=2 t 2.
Set up the integral used to find the distance the particle travels in the first 5 seconds.
3. Set up the integral to find the length of the curve
IV. Polar coordinates and cartesian coordinates of a point
1. The polar coordinates of a point are given. Find the cartesian coordinates of the point.
(a) −1,/2
(b) 3, 2/3
(c) 2  2 ,5/ 4
2. The cartesian coordinates of a point are given.
(i) Find polar coordinates r , of the point with r 0 and 0≤2 
(ii) Find polar coordinates r , of the point with r 0 and 0≤2
(a) (– 1, 1)
(b) − 3 ,−1
(c) (2, – 3)
V. Cartesian equation for a curve given in polar form and viceversa.
1. Find a cartesian equation for the curve described by the given polar equation: r =3 sin 
2. Find the polar equation for the curve represented by the given cartesian equation: x + y = 4
3. Find a cartesian equation for the curve =/ 4
6
4. Find a cartesian equation for the curve r =
sin −2cos 
VI. Area of the region bounded by polar curves
1. Find the area of the region that lies inside the first curve and outside the second curve.
r =10sin  , r =5
2. Find the area of the region that lies inside both curves: r =4 sin  , r=4 cos 
3. Find the area enclosed by one loop of the curve r =sin 2
4. Find the area of the region that lies inside the first curve and outside the second curve:
r =3 cos  , r=1cos 
VII. Slope of the tangent line to a given polar curve.
1. Find the points on the curve where the tangent line is horizontal:
r =51−cos 
2. Find the slope of the tangent to the given polar curve at the point specified by the value of  :
1
(a) r = , =

(b) r =sin 3 , =/ 6
(c) r =2−sin  , =/3
VIII. The Taylor series for
f  x  centered at a
1. Find the Taylor polynomial T3 for the function
2. Find the Taylor polynomial T3 for the function
f  x =sin x at a =1.
f  x =ln x at a =1.
3. Find the first four non-zero terms of the Taylor series centered at a = 1 for f  x =  x .
4. (a) Find the first three non-zero terms of the Taylor series of f  x =tan x about x=/ 4.
(b) Use your result from part (a) to approximate tan(.8).
IX. The radius of convergence and interval of convergence of a power series
1. Find the radius of convergence and interval of convergence of each power series:
n
∞
∞
∞
∞
n!12 x n
xn
n xn
n  x2
a. ∑ n 5
b. ∑ −1
c. ∑
d. ∑
n 2n
2n
n=1 5 n
n=1 2 n!
n=1
n=1
X. Definite integral for a volume (washers & shells).
1. Set up, but do not evaluate an integral for finding the volume of the solid obtained by rotating the region
bounded by the given curves about the specified line:
x2
, y=5−x 2 ; about the x-axis.
4
a.
y=
b.
y =sin x , y =0, x=0, x= ; about
x=−1 .
XI. The area enclosed by two curves
1. Sketch the region enclosed by the given curves, and then find the area of the region;

a. y =cos x , y=sin 2 x  with 0≤ x≤
2
2
b. y = x −2 x , y = x4
c.
y = x 2 , y =4 x− x 2
d.
x=2 y , x y =1
2
XII. Use Partial Fraction Decomposition to evaluate the indefinite integral
x−1
dx
1. ∫ 2
x 2 x
dx
dx
2. ∫ 2
x 3 x−4
x 2 −2 x −1
dx
3. ∫
 x −12  x 21
XIII. Evaluate the definite integral (no calculators)
/ 2
1.
∫ sin3  cos2  d 
0
4
2.
∫ t e−t dt
0
1
3.
∫ 2  x13 dx
0
XIV. Evaluate the indefinite integral
1.
2.
∫ 20 x e5 x dx
∫ x cos7 x  dx
3.
sin x
dx
∫ 1cos
2
x
ANSWERS:
I.
16
 x−e 1
e
  31/  3−1
1. (a)
y=−
(b)
y=
x
6 
6
(c)
y −.342=  3  x−1.407
2.
3. horizontal tangent at   2 , 1 ,− 2 ,1 ,  2 ,−1 ,− 2 ,−1 ; vertical tangent at (2, 0), (0,2)
II.

1. a b 
2.
1−e 2
2
III.
2
1. 4  2−2
 2  e −1 
 /5
2.
3.
∫
1
IV.


3
3
− ,3 
2
2


2. (a)(i)  2 , 3
(a)(ii) − 2 , 7
4
4


(b)(i) 2, 7
(b)(ii) −2,
6
6
(c) (i)   13 ,arctan −3/22
1(a) ( 0, – 1)
1(b)







1 1
 dt
t4 t 2
5
4.
∫  4 2 4 t 2 dt
0
1(c) −2,−2 


(c)(ii) − 13 ,arctan −3/ 2
V.
2
1.
  
x 2 y −
3
3
=
2
2
2
2. r =
4
cos sin 
3. y = x
4. y = 2x + 6
VI.
1.
25 25  3

3
2
2. 2 −4 3. /8
4. 
VII.

1. 0,0 , −

15 15  3
15 15  3
,−
, − ,
4
4
4
4

2. (a) −
(b) − 3
VIII.
1. T 3  x =sin 1cos1⋅ x −1−
1
1
2
3
2.  x−1−  x−1   x−1
2
3
sin 1
cos 1
⋅ x −12−
⋅ x−13
2
6
(c)
2−  3
1−2  3
1
1
1
2
3
3. T 4  x =1  x−1−  x−1   x−1
2
8
16
4. (a)
2
   
T 3  x =12 x −


2 x−
4
4
T 3 0.8=1.0296301
(b)
IX.
I =−5,5
1 (a) R = 5,
(b) R = 2,
I =−4, 0
(c) R=0,
{ 12 }
I= −
(d)
R=∞ , I =−∞ , ∞
X.
2
[
 ]
2 2
1. (a) ∫  5− x  − x
4
−2
XI.
1. 1/2
2 2

dx
2. 125/6
(using washers)
∫ 2  x1sin x dx
(b)
(using shells)
0
3. 8/3
4. 9/8
XII.
1.
3 ∣
1
ln x2∣− ln∣x∣C
2
2
XIII.
1. 2/15
2.
2. 1−5 e−4
1 x−1
ln
C
5 x 4
∣ ∣
3. ln∣x −1∣
1
1
− ln  x 21arctan  xC
x−1 2
3. 15/2
XIV.
1.
4
4 x e5x − e 5 x C
5
2.
1
x
cos 7 x  sin7 xC
49
7
3. −arctan cos x C