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free/or 0.5$ |
| m862 | Show that Mx consists of the tangent vectors at t of curves in M with c (t) = x. |
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| m863 | Show that the continuity of D1 f j at a may be eliminated from the hypothesis of Theorem 2-8. |
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| m883 | Suppose C is a collection of coordinate systems for M such that (1) For each x Є M there is f Є C which is a coordinate system around ; (2) if f, g Є C, then det (f -1 0 g) 2 > 0. Show that there is a unique orientation of M such that is orientation-preserving for all f Є C. |
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| m884 | Suppose f: Rn ->Rn is differentiable and has a differentiable inverse F -1: Rn -> Rn. Show that (f-1) I (a) = (fi (f-1(a)))-1. |
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| m1244 | Two functions f , g : R -> R are equal up to nth order at if lim h -> o f(a + h) – g(a + h)/hn =0
(a). Show that f is differentiable at if and only if there is a function g of the form g(x) = a0 + a1 (x – a ) such that f and g and g are equal up to first order at a.
(b). if f1 (a),., f(n) (a) exist, show that f and the function g defined by
g(x) = f(i) (a)/ i! (x – a)i |
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| m1251 | Use Fubini s Theorem to derive an expression for the volume of a set in R3 obtained by revolving a Jordan measurable set in the yz -plane about the -axis. |
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| m1252 | Use Fubini s Theorem to give an easy proof that D1, 2f = D2, 1f if these are continuous. |
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| m1255 | Use the function f : R -> R defined by
f (x) = { x/2 + x2 sin (1/x) x ≠ 0, 0 x = 0. |
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| m1256 | Use the implicit function theorem to re-do Problem 2-15(c). Define f : R x Rn -> Rn by |
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| m1258 | Use the theorems of this section to find f1 for the following:
a. f(x, y, z) = xy
b. f(x, y) = sin (xsin (y)).
c. f(x, y, z) = sin (xsin (ysin (z))
d. f(x, y, z) = xy2
e. f(x, y, z) =xy+z
f. f(x, y, z) =(x + y)z
g. f(x, y) = sin (xy)
h. f(x, y) = sin (xy) cos(3)
i. f(x, y) =(sin(xy), sin (xsin (y)),xy) |
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| m1260 | Use Theorem 3-14 to prove Theorem 3-13 without the assumption that g1 (x) ≠ 0. |
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| m52334 | (1) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
(2) Use your calculator to evaluate the integral correct to five decimal places.
(a) y = xe-x, y = 0, x = 2; about the y-axis
(b) y = cos4x, y = - cos4x, - π/2 ≤ x ≤ π/2; about x = π
(c) x = √sin y, 0 ≤ y ≤ π, x = 0; about y = 4 |
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| m52335 | 1. Use the Law of Exponents to rewrite and simplify the expression.
(a) 4-3 / 2-8
(b) 1/ 3√x4
2. (a) b8 (2)4
(b) (6y3)4 / 2y5 |
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| m52337 | A 360-lb gorilla climbs a tree to a height of 20 ft. Find the work done if the gorilla reaches that height in
(a) 10 seconds (b) 5 seconds |
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| m52339 | (a) A company makes computer chips from square wafers of silicon. It wants to keep the side length of a wafer very close to 15 mm and it wants to know how the area A(x) of a wafer changes when the side length x changes. Find A (15) and explain its meaning in this situation.
(b) Show that the rate of change of the area of a square with respect to its side length is half its perimeter. Try to explain geometrically why this is true by drawing a square whose side length x is increased by an amount Δx. How can you approximate the resulting change in area ΔA if Δx is small? |
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| m52340 | (a) A direction field for the differential equation y = y(y - 2) (y - 4) is shown. Sketch the graphs of the solutions that satisfy the given initial conditions.
(i) y(0) = -0.3
(ii) y(0) = 1
(iii) y(0) = 3
(iv) y(0) = 4.3
(b) If the initial condition is y(0) = c, for what values of c is limt??y(t) finite? What are the equilibrium solutions?
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| m52341 | (a) A direction field for the differential equation y = x2 - y2 is shown. Sketch the solution of the initial-value problem y = x2 - y2 y(0) = 1
Use the graph to estimate the value of y(0.3).
(b) Use Euler s method with step size 0.1 to estimate y(0.3), where y(x) is the solution of the initial-value problem in part (a). Compare with your estimate from part (a).
(c) On what lines are the centers of the horizontal line segments of the direction field in part (a) located? What happens when a solution curve crosses these lines? |
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| m52343 | (a) A particle starts by moving to the right along a horizontal line; the graph of its position function is shown. When is the particle moving to the right? Moving to the left? Standing still?
(b) Draw a graph of the velocity function. |
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| m52348 | (a) Approximate f by a Taylor polynomial with degree n at the number a.
(b) Use Taylor s Inequality to estimate the accuracy of the approximation f(x) ( Tn (x) when x line in the given interval.
(c) Check your result in part (b) by graphing |Rn (x)|.
1. f(x) = (x, a = 4, n = 2, 4 ( x ( 4.2
2. f(x) = x2/3, a = 1, n = 3, 0.8 ( x ( 1.2
3. f(x) = sec x, a = 0, n = 2, - 0.2 ( x ( 0.2 |
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| m52349 | (a) Approximate f by a Taylor polynomial with degree n at the number a.
(b) Graph f and Tn on a common screen.
(c) Use Taylor s Inequality to estimate the accuracy of the approximation f(x) ( Tn(x) when lies in the given interval.
(d) Check your result in part (c) by graphing |Rn (x)|. |
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