Final Exam Review Guide
Math 191, Fall 2018
Final Exam is Thursday 12/13 at 2:00PM
Directions: The exam will consist of 10 free response questions. All questions will be drawn from the questions written below. All answers on the exam must either have work shown or attached explanations. You are allowed one sheet of notes written on a 8.5x11 sheet of paper. You are also allowed the use of a calculator.
The sections covered by the exam will be: Sections 6.1,6.2,6.3,6.5
Sections 7.1,7.2,7.3,7.4,7.5,7.7,7.8 Sections 8.1,8.2
Sections 10.1,10.2,10.3,10.4
Question 1) Find the area between the following curves: (a)y =|x−3| −1, y=x2
(b)y=ex,y=xex,x= 0
(c)y = sin2(x),y= cos2(x), from−π
4 ≤x≤
π
4
Question 2) Find the volume of the following solids when they are rotated about the given axis. (a)y = 1x,y= 0,x= 1,x= 2 about the axis y= 2
(b)y= ln(x),y = 1,y= 2, x= 0 about the y-axis. (c)y = sin(x),x= 0,x= π4,y= 0 about the axisx=−1
Question 3) Write down but do not evaluate the volume of the following solids using both methods (disks/washers and cylindrical shells).
(a)y =√x,x= 0,x= 4,y = 0 around the axis,x= 5 (b)y=ex,x= 0, x= 1, y= 0 around the axis y=−1
Question 4) Letf(x) =x13
(a) Calculate the average value of f(x) on 0≤x≤8
(b) Find a value c, betweenx= 0 and x= 8 such that f(c) is the average value you calculated in part (a).
(c) Draw a diagram illustrating the results of parts (a) and (b).
Question 5) Perform the following indefinite integrals. (a)R
xsec2(x)dx
(b)R 2xx2+3+xdx
(c)R
tan4(x) sec4(x)dx (d)R √x
Question 6) Perform the following indefinite integrals. (a)R
x7ln(x)dx
(b)R x2 (1−x2)32dx
(c)R
sin3(x) cos2(x)dx
(d)R
e2xcos(ex)dx
Question 7) Consider the definite integralR13pln(x)dx
(a) Write down but do not evaluate the midpoint rule withn= 6 to approximate this integral. (b) Write down but do not evaluate the trapezoidal rule with n = 6 to approximate this inte-gral.
(c) Write down but do not evaluate Simpson’s rule with n= 6 to approximate this integral.
Question 8) Decide if the following improper integrals are convergent or divergent. Give the value of any convergent integrals.
(a)R4 1
1
xln(x)dx (b)R5∞ 1
(4x+5)32dx
(c)R∞ 1
x+1
x2+1dx
(d)R∞ 1 xe
−xdx
Question 9) Determine if the following sequences converge or diverge. If they converge, give the value they converge to.
(a)an= (
−1)nn2
n2+1
(b)an= 3nn+5+2
(c)an= e n
2
Question 10) Decide if the following series are Absolutely Convergent, Conditionally Convergent, or Divergent. (a) ∞ P n=1 (−1)n
n2
(b)
∞
P
n=1 7n5+1
n6+2n+5
(c)
∞
P
n=2 ln(n)
n
(d)
∞
P
n=0
5nn!
1×5×9×...×(4n+1)
Question 11) Decide if the following series are Absolutely Convergent, Conditionally Convergent, or Divergent.
(a)
∞
P
n=1
(47nn+5+1)3n
(b)
∞
P
n=1
tan(e−n)
(c)
∞
P
n=1
en12
(d)
∞
P
n=3
(−1)nn2+2nn+4
Question 12) Find the radius and interval of convergence for each of the following power series. (a)
∞
P
n=1
(−4)n(x−2)n
√ n (b) ∞ P n=1
x2n
9nn3
(c)
∞
P
n=1
nn(x+1)n
10n
Question 13) Use any established series from class to build Maclaurin series for each of the fol-lowing functions. Then state their Radius of Convergence.
(a)f(x) =xex3
(b)g(x) = 4+1x (c)h(x) = (4+−1x)2
Question 14) Build from scratch the Taylor Series for f(x) = sin(4x) centered at x= π8
Question 15) Find the lengths of the following curves. (a)y = 4 + 10x32 for 0≤x≤4
(b)y= ln(sec(x)) for 0≤x≤ π4
(c)y = 151x5+41x3 for 1≤x≤2
Question 16) Write down but do not evaluate an integral for the following surface areas. (a)y =√1 + 4x for 1≤x≤5 around the x-axis.
(b)y= sin(πx) for 0≤y≤1 around the y-axis. (c)x= 1−y2 for 0≤y≤1 around the y-axis.
Question 17) Consider the curve y=√x+ 2 fromx= 0 to x= 3.
Write down but do not evaulate integrals for each of the following with respect to both dx and dy (a) The area under this curve and above the x-axis.
(b) The volume of the solid created by revolving the area under this curve and above the x-axis around the x-axis
(c) The volume of the solid created by revolving the area under this curve and above the x-axis around the y-axis
(d) The arclength of this curve
(e) The surface area of the solid created by revolving this curve around the x-axis (f) The surface area of the solid created by revolving this curve around the y-axis
Question 18) Remove the parameter from each of the following and sketch the graph. Make sure to indicate how the graph is traced out as the parameter increases.
(a)x=et,y =et−1
(b)x= sin(t),y= csc(t), 0< t < π2
Question 19) Consider the following parametric curve.
x=t4−2t2,y =t3 with −2≤t≤2
(a) Find all values oft where this function has a horizontal tangent. (b) Find all values oft where this function has a veritical tangent.
(c) Carefully sketch this curve using the information you found in parts (a) and (b).
(d) Write down but do not evaluate an integral for the arclength of the curve you sketched.
Question 20) Consider the polar curve r= 3 + 2 cos(θ) (a) Carefully sketch this curve.
(b) Find the area inside this curve.
(c) Write down but do not evaluate an integral for the area outside this curve but inside the curve
r= 2.
Question 21) Consider the polar curve r= sin(2θ) (a) Carefully sketch this curve.
(b) Find the slope of the tangent line atθ= π4. Sketch this on your curve. (c) Find the area inside this curve.
(d) Write down but do not evaluate an integral for the arclength around this curve.
Question 22) For this question you will continue to use the curver = sin(2θ) (a) Find all points of intersection between this curve and the curver= cos(θ) (b) Carefully resketch this curve along on the same axes with the curve r= cos(θ) (c) Find the area located inside both curves.
Question 23) Consider the polar curve r2= 4 cos(2θ) (a) Carefully sketch this curve.
