MATH 151 Midterm: MATH 151 TAMU Y2009 2009c Exam 1a Solutions

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turquoisegnu128

31 Jan 2019

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F (x) = 1, lim x 1 only to check continuity for x = 1 and x = 2. lim lim x 1 x 1+ f (x) = 2. f (x) does not exist, hence f (x) is not. Thus continuous at x = 1. lim lim x 2+ x 2 is continuous at x = 2. The only place of discontu- nity is x = 1. f (x) = 1, f (2) = 1, thus f (x) f (x) = 1: b: lim x 3+ x 1 x2(x + 3) cal asymptote and. = since x = 3 is a verti- x 1. < 0 for x > 3. x2(x + 3: d: first, note f (x) = x3 x2 + 3x 1 is continuous, so we will apply the intermediate value theorem. 3x denominstor by x: x x x2 + 3x.

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4. (21 pts) Let g be the function given by ( 6 - X 2 + x if x < 2, x + -2 g(x) = x2 – x-1 if x>2. (a) (6 pts) Use the definition of continuity to determine whether the function g is continuous at 2. Show your work. (b) (3 pts) Evaluate the limit. Justify your answer. Show your work. Write "does not exist" only if the limit does not exist and is not + or -0. lim g(x) = lim -2+ = +00 X 2 + X since the form of the limit is NUM: POS, DENOM: POS (c) (2 pts) Write the equation(s) of all vertical asymptotes of g. Write "none" if appropriate. The result in part (b) implies that the line x= -2 is a vertical asymptote. This is the only vertical asymptote, since the function g is continuous on its domain (d) (2 pts) Determine the largest) intervals of continuity of g. Use interval notation to write your answer. Let g be the function given by 12+ if x < 2, x# -2 2 + x g(x)= x2-x-1 x-1 if x>2. (e) (6 pts) Evaluate each limit. Do not use L'Hôpital's Rule. Write "does not exist" only if the limit does not exist and is not too or -00. Show your work. 6-1 i lim g(x) = lim -00 2 + X = lim -002 + x 0-1 + 1 x -00 x x x -+- 2 +1 0 0 + 11 x2 - x-1 1 ii. lim g(x) = lim x++ V x2 – x-1 X-1 x2 - x-1 X - 1 = lim x +oc x +0 - lim Vam M93001 (f) (2 pts) Write the equation(s) of all horizontal asymptotes of g. Write "none" if appropriate.

1. (18 pts) The graph of a function f, defined on (-0, +o), is given in the figure below. - - y = f(x) - - - - - - N - - 4 A -2-1 144 -1 - - - - - - - -- - - -- Use the graph of f to answer questions (a)-(f) below. (a) Determine the value or write "does not exist". i. f(-3) = -2 ii. f(0) = 1 (b) Determine the limit or write "does not exist" (DNE). Write "does not exist" only if a limit does not exist and is not too or -0. i. lim f(x) = 0 iv. lim f(x) = +00 X- 3 x2 ii. lim f(x) = -4 v. lim f(x) = -0 X- 0- x -00 iii. lim f(x) = DNE vi. lim f(x) = -2 x0 x++00 (C) MULTIPLE CHOICE! Circle ONLY one answer! Find a number a for which the following statement is true (the function f is shown above): STATEMENT: The function f is continuous at a, and lim f(x) = 0. i. a=-3 iii. a=0 V. a=3 ii. a = -2 iv. a = 2 vi. Such a number does not exist. (d) Determine the largest intervals of continuity of f. Use interval notation to write your answer. (-0, -3), (-3,0), (0, 2), (2, +00) (e) Write the equation(s) of all vertical asymptotes of f. Write "none" if appropriate. x= 2 (f) Write the equation(s) of all horizontal asymptotes of f. Write "none" if appropriate. y = -2

Question1.

Let L be the line containing thepoint P(2,-5,-1) and perpendicular to theplane 2x1+3x2+2x3=-3.

Find a vector equationfor L.

My answer: <2,-5,-1>+ t<2,3,2> (correct)

At what pointdoes

Lintersect the yz-plane?

Question2:

x1 = -6 t - 4

x2 = - t+ 2

x3 = 5t - 6

and the second set ofparametric equations

x1 = 6t

x2 = ?

x3 = ?

describe the same line inR3. Find x2 and x3.

Question3:

It is one of 3 options foreach one: Parallel, perpindicular, or neither. Please explainwhy.

http://imgur.com/UbDuH8Z

Question4:

Find the point ofintersection of the two lines

L1: x=1+t, y=-10-2t, z=-5-2t,

L2: x=7-4s, y=-10+2s, z=-7+3s.

Note that in general, twolines in 3D do not intersect. These two lines are carefully chosenso that they do

Find the point where theline

x=1+2t,y=-4+t,z=-2-2t

intersects theplane

4x-y+3z=3.

Thanks. I have a lot ofpoints for this and will give it to someone that can helpme.

ochreimpala126

MATH 151 Midterm: MATH 151 TAMU Y2009 2009c Exam 1a Solutions

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