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### Math20_Fall2011_linesandplanesSolutions

Course: MATH 20, Fall 2011
School: Harvard
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University, Harvard Math 20 Fall 2011, Instructor: Rachel Epstein 1 Lines and Planes September 16, 2011 Find the following line and planes, using either parametric form or an equation. Let P = (1, 2, 3, 4), Q = (2, 3, 4, 4), and R = (0, 0, 2, 0). 1. Find the line through the points P and Q. This line is the set: {Qt + P (1 - t) | t R}. This is the same as the set 1+t 2 + t |tR 3 + t 4 There are...

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University, Harvard Math 20 Fall 2011, Instructor: Rachel Epstein 1 Lines and Planes September 16, 2011 Find the following line and planes, using either parametric form or an equation. Let P = (1, 2, 3, 4), Q = (2, 3, 4, 4), and R = (0, 0, 2, 0). 1. Find the line through the points P and Q. This line is the set: {Qt + P (1 - t) | t R}. This is the same as the set 1+t 2 + t |tR 3 + t 4 There are other right answers to this problem. To check your answer, see if you can choose t in such a way that you recover the points P and Q. For the answer above, choosing t = 0 and t = 1 works. 2. Find the plane through the points P , Q, and R. This is a bit harder, since we didn't specifically do an example like this in class. We must first convert it to a problem that says, "Find the plane through P , parallel to vectors v and w." But what should v and w be? We can use the vectors P Q and P R, or any other vector through two points on the plane. P Q = Q - P = (1, 1, 1, 0). P R = R - P = (-1, -2, -1, -4). So, the parametric form is 1 -1 1 -2 2 1 + t + u | t, u R , -1 1 3 4 0 -4 which is the same as 1+t-u 2 + t - | 2u t, u R . 3 + t - u 4 - 4u As in the previous problem, there are other right answers. See if you can recover P , Q, and R by choosing appropriate values of t and u. In the Harvard University, Math 20 Fall 2011, Instructor: Rachel Epstein 2 above solution, when t = 0 = u, we get P , when t = 1 and u = 0, we get Q, and when t = 0 and u = 1, we get R. 3. the hyperplane through P , and perpendicular to the vector R = Find 0 0 . 2 0 In this case, the hyperplane is 3-dimensional. In general, a hyperplane can be any number of dimensions. The solution is the set of all vectors x such that (x - P ) R = 0. Since R has 3 zero coordinates, the only part that really matters is the third coordinate. We get the equation (x3 - 3)2 = 0, which is the same as 2x3 = 6, or simply x3 = 3. If we convert this problem to one in only three dimensions, instead of four, you get the same equation, although you may have written it as z = 3. This plane isn't too hard to picture. It is the plane parallel to the x and y axes at height 3. In four dimensions, it is the 3-dimensional surface parallel to the x1 , x2 , and x4 axes, with x3 = 3. This is not so easy to picture.
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Harvard - MATH - 20
Harvard University, Math 20 Fall 2010, Instructor: Rachel Epstein1Review sheet 21. Vector spaces, bases, and dimension (from supplement) (a) Know the definitions of vector space, basis, and dimension. Be able to identify when something is or is not a v
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Harvard - MATH - 20
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Harvard - MATH - 20
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Harvard - MATH - 20
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Math 1a 1Limits of FunctionsFall, 2009The graph below shows the plot of a function y = f (x). Determine whether the limits shown below exist. If the given limit does exist, find this limit.. . . . . 2 . . . . . . . . . . . -2 2 4 6 8(a) lim f (x)x0
Harvard - MATH - 1a
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Harvard - MATH - 1a
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Harvard - MATH - 1a
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Harvard - MATH - 1a
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Harvard - MATH - 1a
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Harvard - MATH - 1a
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Harvard - MATH - 1a
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Harvard - MATH - 1a
Math 1a 1OptimizationFall, 2009(a) Suppose a rectangular region has fixed perimeter of 40 cm. What is the largest area the region can have?(b) Suppose now that the region was in the shape of a right triangle, not a rectangle. If the perimeter is still
Harvard - MATH - 1a
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Harvard - MATH - 1a
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Harvard - MATH - 1a
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Harvard - MATH - 1a
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CHAIN RULERecall the bottle calibration problem. If we increase the amount of water dripped into a bottle twice as much, then, no matter what the shape of the bottle is, the height of the water will raise twice as fast. This suggests that, if we have a c
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CONTINUITYWe have seen that the limit of a function as x approaches a can sometimes be found by calculating the value of the function at x = a. Functions with this property are called continuous at a. Mathematical definition is as follows. Continuity A f
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THE DEFINITE INTEGRALWe saw a limit of the formn nlim [f (x )x + f (x )x + + f (x )x] = lim 1 2 nnf (x )x ii=1Because this form arises frequently in a wide variety of situations, we give this type of limit a special name and notation. Definition 1.
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Harvard - MATH - 1a
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Harvard - MATH - 1a
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Harvard - MATH - 1a
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THE FUNDAMENTAL THEOREM OF CALCULUSThe first part of the Fundamental Theorem of Calculus describes functions defined by an equation of the formxg(x) =af (t)dtwhere f is a continuous function on [a, b] and x varies between a and b.xExample 1. Let g
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Harvard - MATH - 1a
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Harvard - MATH - 1a
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Harvard - MATH - 1a
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Harvard - MATH - 1a
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Harvard - MATH - 1a
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Harvard - MATH - 1a
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Harvard - MATH - 1a
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Harvard - MATH - 1a
Math 1a: Derivatives of trigonometric functionsOctober 14, 2009d d sin x = cos x and cos x = - sin x to compute dx dx c. d sec x dx1. Use a.d tan x dxb.d cot x dxd.d csc x dx2. Find the following limits. a. limh0tan h hb. limh0sin 2h sin h3.
Harvard - MATH - 1a
Math 1a: The chain ruleOctober 16, 20091. Differentiate following functions. 1. (x - 1)(x + 3)112. ex23. f (x) = 2x tan x4.x+x+x2. Some liquid is poured in a (conical) glass. Denote the volume of the liquid in the glass at time t by V (t). At wh
Harvard - MATH - 1a
Math 1a: Implicit differentiationOctober 19, 20091. Find y if y 5 + x2 y 3 = 1 + yex .22. The equation 2(x2 + y 2 )2 = 25(x2 - y 2 ) gives a curve know as a lemniscate. Find the equation of the tangent line to the lemniscate at (-3, 1).13. Find the
Harvard - MATH - 1a
Math 1a: Derivatives of logarithmic functionsOctober 21, 20091. Consider the function f (x) = x - e log x for x &gt; 0. a. Find f (x) c. For which x is f (x) minimum?b. On which intervals is f (x) increasing/decreasing? d. Which one is bigger: e or e ?2.
Harvard - MATH - 1a
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Harvard - MATH - 1a
Math 1a: Related ratesOctober 26, 20091. You are driving along a straight highway where the speed limit is 65 miles per hour. You pass a highway patrol car sitting 1000 feet from the highway on your left. Thirty-five seconds later, the highway patrolman
Harvard - MATH - 1a
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Harvard - MATH - 1a
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Harvard - MATH - 1a
Math 1a: Curve sketchingNovember 2, 2009Sketch each of the following functions, showing all important details. Use some of the following to extract as much information about the plot as you can: 1. the domain of definition of the function, vertical and
Harvard - MATH - 1a
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Harvard - MATH - 1a
MATH 1A: OPTIMIZATION (PART II)1. The Statue of Liberty is 32 m tall, and rests on a base 49 m tall. How far away from the base should you stand to get the best view of the statue? (In other words, to maximize the angle subtended by the statue at your ey
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Harvard - MATH - 1a
MATH 1A: ANTIDERIVATIVES1. Water is leaking out of a tank at the rate of (10 - t) liters per minute, where t is time in minutes. The leaking starts at t = 0. How much water has leaked after 4 minutes? What if the leak had started at t = 1?2. For each of
Harvard - MATH - 1a
AREAS AND DISTANCES1. Express the following sums using the notation: (1) 13 + 23 + + 1003(2) 1 + 3 + 5 + + 101(3) 1 -1 2+1 3-1 4 +1 292. Write down the definition of the area under the given function on the given interval. Approximate the area