Math 201-DDB Final Exam May 19, 2009 Page 1 of 4
1. Approximate the given deﬁnite integral correct to 5 decimal places (i.e., within ±0.5 × 10):
1 − cosx
2 x4 x 6 X x2n
2. A function f(x) has Maclaurin series: 1 + x + + + ▯▯▯ = 1 + 2
4 9 n=1 n
ﬁnd f (k(0) for all positive integer k.
3. (a) Use the binomRal series to ﬁnd the Maclaurin series for f(x) = arcsinx and its radius of convergence
(Hint: arcsinx = x√ dt ).
(b) Use the series in Part (a) to evaluate the following limit:
x − arcsinx
4. (a) Find the third degree Taylor polynomial T3(x) for the function f(x) = ln(1+2x) centered at a = 1.
(b) Use Taylor’s inequality (or Lagrange’s remainder) to estimate the error in using3T (x) to approxi-
mate f(x) on the interval [0.5,1.5].
5. Given the curve C having parametric equations: x = 9 − t where t ∈ R
y = t − 16t
i) Find dy/dx and d y/dx .
ii) Find the x and y intercepts and coordinates of the points on C where the tangent line is vertical
iii) Sketch the graph of C showing the orientation of the curve.
iv) The curve forms a loop. Set up, but do not evaluate, the integrals needed to ﬁnd the area
enclosed by the loop and the length of the loop.
6. Given the polar curves r = 1 + 2sinθ and r = 2, do the following:
i) Sketch both graphs on the same axes.
ii) Find all the points of intersection for θ ∈ [0,2π].
iii) Find the area of the region outside the circle and inside the l¸on.
iv) Set up, but do not evaluate, the integral needed to ﬁnd the length of the inner loop of the
7. Sketch and give the name of the following surfaces:
2 2 2
i) x + y = z + 9
2 2 2
ii) z = 9 − 4x − y
8. Let r(t) = ▯sint, 2cost,sint▯.
i) Compute the velocity, speed and acceleration.
ii) Find the curvature and the tangential and normal components of acceleration.
iii) Find an equation of the quadric surface on which this space curve lies. Sketch r(t) for 0 ≤ t ≤ 2π.
(You might want to sketch the curve on the surface, to help you make a good graph.)
9. Find the limit if it exists or show that it does not exist.
x − y 3 3x y
(a) lim (b) lim
(x,y)→(0,0) + y 3 (x,y)→(0,0)+ y 2 Math 201-DDB Final Exam May 19, 2009 Page 2 of 4
10. Find the equations of the tangent plane and normal line of the surface x + y + z = 6 at the point
11. If z = f(x,y) is implicitly deﬁned by x z + sin(yz) = y secz, ﬁnd ∂x.
2 2 ∂ z
12. If z = f(x − y ,2xy) ﬁnd 2. Assume that second order partial derivatives of f are continuous.
2 3 4
13. Find and classify the critical points of f(x,y) = 6xy − 2x − 3y .
14. Use the method of Lagrange multipliers to ﬁnd the smallest and largest values of f(x,y) = xy on the
circle x + y = 1.
15. Let z = f(x,y) be a surface and f(x,y) = c a level curve on that surface. Show that the gradient of
f is always perpendicular to the level curve (Hint: You may want to represent the level curve by the
vector equation r(t) = ▯x(t),y(t)▯, and then use the chain rule.).
16. Set up, but do not evaluate, the integral needed to ﬁnd the volume of the region E lying inside
x + 4y = 4 , above the xy-plane and below z = 2 − x. Sketch the region.
Evaluate the multiple integrals in problems 17–20:
Z 9 Z 3
17. √ xy siny dydx
18. (x − y) (x + y) dA where R is the triangular region bounded by the coordinate axes and the line
x + y = 1 (Hint: Use an appropriate change of variables).
Z Z √ Z
2 4−x2 2
19. (x + y )dzdydx
−2 − √4−x2 √ x +y2
Z Z Z r