MATH 222 Study Guide - Maxima And Minima, Nappe, Cylindrical Coordinate System
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201-DDB Final Exam — May 2011
Calculus III Page 1
1. Let f(x) = Zx
tcos √t dt:(6)
(a) ﬁnd a power series representation for f(x);
(b) use this series to approximate f(x) = Z1/2
tcos √t dt correctly to 4 decimal places.
2. Find the power series representation for each of the following functions, and state the radius of(6)
(a) f(x) = 1
4−3x, centered at x= 2.
(b) f(x) = 3
2 + x−x2, centered at x= 0.
3. Let f(x) = 3
√8 + x:(8)
(a) use the Binomial theorem to ﬁnd the ﬁrst 5 terms of the Maclaurin series for f(x), and its radius
(b) approximate 3
√8.2 correctly to 4 decimal places.
4. Let Cbe the plane curve deﬁned by parametric equations (x= 3t−t3
(a) Show the orientation of C.
(b) Find and simplify dy
dx and d2y
(c) At what points does Chave a vertical tangent line?
(d) Set up (but do not evaluate) the integral needed to ﬁnd the area
of the region enclosed by the loop.
5. (a) Sketch the graph of r= 2 sin(3θ).(6)
(b) Find the area of the region enclosed by the curve.
(c) Set up (but do not evaluate) the integral needed to ﬁnd the length of one loop of the curve.
6. Let Cbe the space curve deﬁned by the vector equation r(t) = het,etsin t, etcos ti.(10)
(a) Find the equation of a quadric surface on which Clies. Sketch both the surface and the curve.
(b) Find the unit tangent vector Tand the unit normal vector N.
(c) Find the length of Con the interval 0 ≤t≤1.
(d) Find the curvature κof C.
(e) Find the parametric equations of the tangent line to Cat the point where t= 0.
7. Sketch and describe the following. Show all your work.(9)
(a) The surface f(x, y) = px2+ 2y2+ 1.
(b) The level curve of z=y
x2+y2corresponding to z=1
(c) The surface ρ= csc ϕcot ϕ.
201-DDB Final Exam — May 2011
Calculus III Page 2
8. Let rbe a three-times-diﬀerentiable function of t. Simplify: [r·(r′×r′′)]′.(2)
9. Find the limit (or if appropriate, show that it does not exist):(4)
10. Show that if f(t) is diﬀerentiable, then z=f(x/y) is a solution of the partial diﬀerentiable equation(3)
∂y = 0.
11. Let Cbe the curve formed by the intersection of the level surface x2y+yz +z2+ 1 = 0 and the plane(3)
x+y+z= 1. Let P0(1,−1,1) be a point on C. Find a tangent vector to Cat P0.
12. Let z=f(x, y) be implicitly deﬁned by sin(xy) + xz4+y3z= 2, and let P0(0,1,2) be a point on this(6)
(a) Find the equation of the tangent plane to the surface at P0.
(b) Find ∇f(0,1).
(c) Find an approximation of f(−0.05,1.10).
13. Find and classify the critical points of f(x, y) = y2+x2y+x2−2y.(5)
14. Use Lagrange Multipliers to ﬁnd the points on the sphere x2+y2+z2= 3 where the maximum and(5)
minimum values of the product xyz are found.
−√1−y2ln(x2+y2+ 1) dx dy (b) Z4
√zdx dy dz
16. Sketch the solid region Sbounded below by z=px2+y2, and bounded above by ρ= 2 cos φ.(5)
Find the volume of S.
17. Sketch the solid region Sbounded below by the plane z= 0, laterally by the surface x2+(y−1)2= 1,(6)
and above by the surface z=x2+y2.
Set up the triple integrals representing the volume of Sin
(a) cartesian coordinates (b) cylindrical coordinates
201-DDB Final Exam–May 2011 ANSWERS Page 1 of 2
1. (a) f(x) = ∞
2. (a) f(x) = ∞
2n+1 , and R=2
(b) f(x) = ∞
2n+1 + (−1)nxn, and R= 1
3. (a) f(x) = 2 + x
248832 +···and R= 8
(b) f(0.2) = 2 + 0.2
with absolute value of error less than 5(0.2)3
20736 = 0.19 ×10−5
4. (a) Counterclockwise orientation
dx2=2(1 + t2)
(c) Vertical tangents at (±2,3)
(d) A= 2 Z√3
xdy = 2 Z√3
(3t−t3)(6t)dt = 12 Z√3
2(2 sin(3θ))2dθ =π
0p4 + 32 cos2(3θ)dθ = 2 Zπ/3
0p1 + 8 cos2(3θ)dθ
6. (a) The curve lies on the cone x2=y2+z2. Note that x=etso x > 0 implying that the curve
spirals around x=py2+z2, the upper nappe of the cone.
(b) T(t) = 1
√3h1,sin t+ cos t, cos t−sin ti
and N(t) = 1
√2h0,cos t−sin t, −sin t−cos ti
Page 1 (marks) (6: let f (x) = z x. 0 t cos t dt: (a) nd a power series representation for f (x); (b) use this series to approximate f (x) = z 1/2. 0 t cos t dt correctly to 4 decimal places. (6: find the power series representation for each of the following functions, and state the radius of convergence. (a) f (x) = (b) f (x) = Show all your work. (a) the surface f (x, y) = px2 + 2y2 + 1. (b) the level curve of z = y x2 + y2 corresponding to z = 1. 4 . (c) the surface = csc cot . = 0. (3: let c be the curve formed by the intersection of the level surface x2y + yz + z2 + 1 = 0 and the plane x + y + z = 1. 1z ln(x2 + y2 + 1) dx dy (b) z 4.