201-DDB Final Exam — May 2011 Calculus III Page 1
(Marks) Z x √
(6) 1. Let f(x) = tcos tdt:
(a) ﬁnd a power series representation for f(x);
Z 1/2 √
(b) use this series to approximate f(x) = tcos tdt correctly to 4 decimal places.
(6) 2. Find the power series representation for each of the following functions, and state the radius of
(a) f(x) = , centered at x = 2.
4 − 3x
(b) f(x) = 2 , centered at x = 0.
2 + x − x
(8) 3. Let f(x) = 38 + x:
(a) use the Binomial theorem to ﬁnd the ﬁrst 5 terms of the Maclaurin series for f(x), and its radius
(b) approximate 8.2 correctly to 4 decimal places.
x = 3t − t3
(8) 4. Let C be the plane curve deﬁned by parametric equations
y = 3t2
(a) Show the orientation of C.
dy d y
(b) Find and simplify and 2.
(c) At what points does C have 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.
(6) 5. (a) Sketch the graph of r = 2sin(3θ).
(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.
(10) 6. Let C be the space curve deﬁned by the vector equation r(t) = ▯ e , tsint,etcost▯.
(a) Find the equation of a quadric surface on which C lies. Sketch both the surface and the curve.
(b) Find the unit tangent vector T and the unit normal vector N.
(c) Find the length of C on the interval 0 ≤ t ≤ 1.
(d) Find the curvature κ of C.
(e) Find the parametric equations of the tangent line to C at the point where t = 0.
(9) 7. Sketch and describe the following. Show all your work.
(a) The surface f(x,y) = x + 2y + 1.
(b) The level curve of z = y corresponding to z = .1
x + y2 4
(c) The surface ρ = cscϕcot ϕ. 201-DDB Final Exam — May 2011 Calculus III Page 2
′ ′′ ′
(2) 8. Let r be a three-times-diﬀerentiable function of t. Simplify: [r · (r × r )] .
(4) 9. Find the limit (or if appropriate, show that it does not exist):
(a) lim p (b) lim (x + y )ln(x + y ) 2
(x,y)→(0,0) x + y 2 (x,y)→(0,0)
(3) 10. Show that if f(t) is diﬀerentiable, then z = f(x/y) is a solution of the partial diﬀerentiable equation
x + y = 0.
(3) 11. Let C be the curve formed by the intersection of the level surface x y +yz +z +1 = 0 and the plane
x + y + z = 1. Let P0(1,−1,1) be a point on C. Find a tangent vector to C at P .0
(6) 12. Let z = f(x,y) be implicitly deﬁned by sin(xy)+xz +y z = 2, and let P (0,102) be a point on this
(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).
2 2 2
(5) 13. Find and classify the critical points of f(x,y) = y + x y + x − 2y.
(5) 14. Use Lagrange Multipliers to ﬁnd the points on the sphere x + y + z = 3 where the maximum and
minimum values of the product xyz are found.
(8) 15. Evaluate: √
Z 1 Z 1−y2 Z 4Z 1 Z 2 cos(x )
(a) √ ln(x + y + 1)dxdy (b) √ dxdy dz
−1 − 1−y2 0 0 2y z
(5) 16. Sketch the solid region S bound