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MAT235Y1 Lecture 17: Mat235 note17

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1 Nov 2018

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MAT235Y1 Lecture 16: Mat235 note16

MAT235Y1 Lecture 17: Mat235 note17

MAT235Y1 Lecture Notes - Lecture 20: Itz

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14. 2limitsandc. org (cid:1334) (cid:3085)(cid:3289) l if fix gets arbitrary close to. Therefore there is no limit of 9(cid:1886) as im hoo since approachiugoolfromdihutdirationsleadtoditf. De. fi tip arbitrary close to l when 1(cid:2782) are arbitrary close to cab. Es hm las cmhlab alongany curve that an. N find him (cid:5294) (cid:1266) x444 if it exists. 1 thelimit (cid:6997) (cid:1356) (cid:19779) (cid:7170) e (cid:1252) weguess equalto 0 (cid:1601)(cid:2191) 1(cid:3565)1 it. D inorder to prove that (cid:5333)(cid:3289)(cid:1382) dne we need to find two ways in which lxnn sa. li such that fun approaches to diff valuesalong each direction example iii. 2 if we want to show hat (cid:5333)(cid:3289)(cid:2781)(cid:1785) we willtry to bound forsimpleq hs. tlim9lm (cid:5294) sagsblaampleiuwlthlro. no. 9(cid:2698)(cid:2781) 1(cid:3565)1 (cid:1323) 0 g hlx. de flm lkglxy limhlxnx. Answer and (cid:1323)0 flag 0 flx. de (cid:19783)2 i__________i. Although these goes to 0 if y ii fix (cid:1685) (cid:1323) (cid:4547)(cid:2191) (cid:1323) (cid:1286) (cid:1286) no limit.

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Related Questions

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Please do the following question by yourself and check which arethe wrong answers,please post your answers and compare to it.

Let r in and u =(1 1 0), v = (1 4 1), w = (r2 1 r2), b = (3+2r 5+12r 2r). For which values r in is the set {u. v, w} linearly independent? For which values r in is the vector b a linear combination of u. v and w? For which of these values of r can b be written as a linear combination of u. v and w in more than one way? The set 3 of all column vectors of length three, with real entries, is a vector space. Is the subset B = {(x y z) in 3 | xy + yz = 0} a subspace of 3? Justify your answer. Show that the set of all twice differentiable functions f : satisfying the differential equation sin(x) f"(x) + x2f(x) = 0 is a vector space with respect to the usual operations of addition of functions and multiplication by scalars. Here, f" denotes the second derivative of f. Let S be the following subset of the vector space 3 of all real polynomials p of degree at most 3: S = {p in 3 | p(l) = 0, p'(l) = 0}, where p' is the derivative of p. Determine whether S is a subspace of 3. Determine whether the polynomial q(x) = x - 2x2 + x3 is an element of S. u = (1 1 0), v = (1 4 1), w = (r2 1 r2) if {u, v, w} is linearly independent set 1(4r2 - 1)-1(r2 - r2) ne 0 4r2 ne 1 r ne plusmn1/2 this set is linearly independent set for values of e ne plusmn1/2 - - - - - - - (3+2r 5+12r 2r) = a1 (1 1 0) + a2 (1 4 1) + a3 (r2 1 r2) this forms system of equations in a1, a2, a3 if 4r2 - 1 ne 0 then system has unique solution if 4r2 - 1 = 0 and -2 - 4r = 0 rArr r = -1/2, the system has more than one solution if 4r2 - 1 ne 0 b can be written as linear combination of u,v,w as uniquely if r = -1/2 then b can be written as linear combination of u,v,w more than one way - - - - - - - - - (1 1 -1) in B and (1 0 1) in B (1 1 -1) + (1 0 1) = (2 1 0) not in B since 2times1+1times0 ne 0 so addition of vector is failed, this set is not forms subspace consider the following differential equation (sinx)f"(x) + x2f (x) = 0 each solution is continuous function the set of all solutions is subset of space of set of all continuous functions n from R to R let f,g are two solutions (sinx)f"(x) + x2f (x) = 0 (sinx)g"(x)+x2g(x) = 0 ccheck whether af+bg is solution or not (sinx)f"(x) + x2f(x) = 0 rArr (sinx) af"(x) + x2?f (x) = 0 (sinx)g"(x) + x2g (x) = 0 rArr (sinx)bg" (x) + x2bg (x) = 0 (sinx)(af"(x)+bg"(x)) +x2 (af (x) + bg(x)) = 0 af+bg is also solution to differential equation the set of all solutions to differential equation forms subspace - - - - - - - - - - consider the following subset of P3 {p in P3/.p(l)=0, p'(l) = 0} let p, q in S rArr p(l)= 0, p'(l) = 0 and q(l) = 0, q'(l) = 0 (ap + bq) (l) = ap (l) +b q(l) = a (0) + b (0) = 0 (ap + bq)'(1) = ap'(l) +bq'(1) = a (0) + b(0) = 0 p, q in S rArr ap +bq in S then S is subspace q(x) = x - 2x2 + x3 q(1) = 1 - 2 + 1 = 0 q'(x)=1 - 4x + 3x2 q'(l) = 1 - 4 + 3 = 0 q is element of S

Need detailed help/answers/advice.1. A picture framer has a piece of wood that measures one inch wide by 50 inches long. She would like to make a picture frame as shown below, and uses all 50 inches of wood she has. Notice that x and y in our sketch below are the outside dimensions of the frame. The picture looks like a rectangle with the measurements 1 inch around all sides and it says that the length of the rectangle is Y and the width is X. A. Write an equation that expresses the required relationship between variables x, y, and the total length of the wood. I got 2(x+y)=50 B. Solve your equation from part (A) for y in terms of x. Got y=25-x C. Write an expression (i.e. a function) that gives the interior area of the frame as a function of its exterior height x. x^2+25x-50?? Confused on everything from here on out. D. Find the values of x that make the interior area of the picture frame equal to zero. Explain why this is the case in the context of the picture frame problem. Draw the frames that correspond to these two extreme conditions. E. Use your function to find the inside dimensions that will make this area a maximum. State also the maximum internal area of the picture frame. The second part of it goes like this: Now the picture framer would like to rework this problem for a piece of wood of arbitrary length L. [In PART I, L was 50 inches. Now L becomes a new variable so that we can generalize what we did in PART I for a piece of wood of any arbitrary length, L] A. Write an equation that expresses the required relationship between variables x, y, and the total length of the wood L. In PART I, you had a linear equation of the form ax+by = 50 where a and b were numbers. Now you will have a slightly more complicated equation involving x, y and L, i.e. your equation will have three variables, x, y, and L in it. B. Solve your equation from part (A) for y in terms of x and L. C. Use your answer to part (B) to write an expression (i.e. a function) that gives the interior area of the frame as a function of its exterior height x, and the wood length L. [The variable y will not appear in this equation] D. Use your function to find the inside dimensions that will make this area a maximum. Now each of your inside dimensions will depend on L, i.e. they will be functions of L.

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