 # Calculus for Science and Engineering III

## MATH 223

### Practice Exercises

 Hwk No. Date Assigned Assignment Date Due 1 1/15 11.1 # 44, 47, 54 11.2 # 70 1/22 2 1/17 11.2 # 7 11.3 # 12, 14, 16 1/22 3 1/22 11.4 # 35,37,40,56 1/24 4 1/24 Let u(t)=p+at and v(t)=q+bt, where p, q, a and b are vectors in three dimensional space. Let theta(u,v) be the angle between u and v. Show that in the limit as t -> + or - infinity, cos(theta(u,v)) -> cos(theta(a,b)). 11.5 # 1-4 (Explain how you match the equations with the figures.) 11.5 # 3 (show the curve lies on the given cone) 11.5 # 11, 31, and 42 1/29 5 1/29 11.6 # 6 (arc length) 11.6 # 27 (curvature, 2D) 11.6 # 34 (curvature, 3D) 11.6 # 64 (Kepler & Newton) 1/31 6 1/31 11.7 # 9 (3D graph sketching) 11.7 # 42, 43, 48 (characterizing surfaces via planar intersections) 11.7 # 51 (Intersection of two surfaces) 2/5 7 2/5 11.8 # 19 (Convert Cartesian coordinates to cylindrical and polar) 11.8 # 35 (Graph of an equation in cylindrical coords) 11.8 # 55 (Describing a surface in cylindrical & spherical coords) 11.8 # 59 (Flying from Fairbanks to St. Petersburg) 2/7 8 2/7 12.2.5, 12.2.15 (Finding a function's domain.) 12.2.30 (Drawing a graph.) 12.2.32 (Drawing level curves.) 12.2.44 (Drawing a level surface.) 12.2.53-58 (Matching. Be sure to explain how you arrive at each match.) 2/12 9 2/12 12.3 # 29 (finding limits) 12.3 # 31 (finding a region of continuity) 12.3 # 42 (finding a limit -- similar to example from class) 12.3 # 52 (a more challenging limit) 2/19 10 2/19 12.4 # 45-50 (matching practice) 12.4 # 56 (heat equation in 2D) 12.4 # 58 (Laplace's equation in 2D) 12.4 # 65 (finding the point at which a surface is horizontal) 2/21 11 2/21 12.5 # 21 (finding the highest or lowest point on a surface -- justify your answer carefully) 12.5 # 31 (minimizing the distance to a plane, within a region) 12.5 # 46 (constrained maximum volume problem -- begun in class) 12.5 # 59 (another volume maximization problem, slightly more involved) 2/26 12 2/28 12.6 # 28 (no calculator allowed for problems from section 12.6) 12.6 # 34 (estimating the maximum error of a compound measurement) 12.6 # 31 (estimating the location of a point on a curve) 12.7 # 31 (Use the chain rule to find the plane tangent to a surface) 12.7 # 38 (Dependence of resistance on three parallel resistances) 12.7 # 53 (Hint: use equation 10; see the implicit function theorem) 12.7 # 54 (Apply the implicit function theorem!) 3/4 13 3/4 12.8 # 21 (directional derivatives) 12.8 # 30 (using the gradient to find the line tangent to a curve) 12.9 # 17 (optimizing a function of 3 variables, given 2 constraints) 12.9 # 37 (inscribed circle of maximal area, using Lagrange multipliers) 3/6 14 3/6 12.8 # 27 (another directional derivative problem, for practice!) 12.10 # 1 (classifying critical points of f(x,y)) 12.10 # 4 (classifying critical points of f(x,y)) 12.10 # 23 (a case where the discriminant is zero) 3/18 - - - S P R I N G - - - B R E A K - - - 15 3/18 13.1 # 12 (practice w/ double integrals) 13.1 # 19 (practice w/ double integrals) 13.1 # 32 (check that the order of integration doesn't matter) Show that if z(x,y)=f(x)g(y), then the double integral of z(x,y)dxdy over the region a<=x<=b and c<=y<=d is equal to the product of the integral of f(x) over [a,b], times the integral of g(y) over [c,d]. 13.2 # 3 (double integral where boundary involves one of the variables) Bonus problem: Find the area under the surface z=1-(x^2+y^2) bounded by the region z >= 0. 3/25 16 3/25 13.2 # 25 13.3 # 13 13.3 # 27 13.4 # 11 13.4 # 15 13.4 # 35 (but use Pappus' Theorem instead of the hint!) 13.5 # 15 13.5 # 43 Suppose z=f(x,y) is a function of (x^2+y^2) only. Assume that f(x,y)>=0 as long as (x^2+y^2) >= a^2. Calculate the volume of the solid formed by the surface with height z=f(x,y) over the region (x^2+y^2) <= a^2, in two different ways: Integrate a system of cylindrical shells of thickness "dr", height z and circumference 2*pi*r. Use Pappus' theorem. Then show that these two approaches give the same answer. 4/1 17 4/1 13.6 #1 (triple integral over a rectangular region) 13.6 #7 (triple integral over a more complicated region) 13.6 #38 (moment of inertia of a sphere) 13.6 #40 (another moment of inertia) 13.7 # 1 (centroid of a shape with cylindrical symmetry) 13.7 # 2 (moment of inertia of same) 13.7 # 30 (volume of a solid defined in spherical coordinates) 13.7 # 31 (moment of inertia of a solid with some rotational symmetry) 4/8 18 4/8 Surface area integrals: 13.8 #1 13.8 #3 13.8 #9 13.8 #13 4/10 19 4/10 13.9 # 1 Practice finding the Jacobian 13.9 # 3 More practice finding the Jacobian 13.9 # 7 Jacobian & area 13.9 # 9 Finding an area in the plane 13.9 # 15 Finding a volume in 3 space 13.9 # 19 Using Change-of-Variables with a nonconstant function 13.9 # 20 Another look at the ellipsoid problem 13.9 # 21 Another look at the ellipsoid, continued. 4/15 20 4/17 14.1 # 7 (sketching a vector field) 14.1 # 11-14 (matching gradients with vector fields) 14.1 # 15 (finding curl and divergence) 14.1 # 32 (prove that div(curl(F)) = 0) 14.2 # 3 (2 kinds of line integrals: ds, and dx or dy) 14.2 # 11 (line integral of T dot F ds) 14.2 # 17 line integral of f(x,y,z) ds along curve C 14.2 # 19-20 (centroid & moments of inertia of a piece of wire) 14.2 # 37 work moving on a sphere in a central force field 4/24