MATH 121 Lecture Notes - Lecture 6: Maxima And Minima, Gruel, Max B

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Unit #6 - families of functions, taylor polynomials, l"hopital"s rule. Some problems and solutions selected or adapted from hughes-hallett calculus. The x2 + 12 factor is always positive. At x = 0, the sign of the numerator can change, and there are asymptotes at x = 2 and x = 2. Therefore, we look at the intervals ( 19, 2), ( 2, 0), (0, 2), and (2, 19). (the 19"s due to the explicit domain given. ) Ection because they aren"t points on the graph, but vertical asymptotes. The function is increasing and decreasing where f(cid:48)(x) > 0 and f(cid:48)(x) < 0; critical points are where f(cid:48)(x) = 0. For this function, f(cid:48)(x) = 6x2 54 = There is no cancellation with the numerator when the denominator is zero, so x = 2 and 2 are ver- tical asymptotes. Expand: (x2 4)3 (x2 4)3 (4x5 24x3 16x3 96x) (4x5 48x3) (a) true. (b) true. (c) true. (d) true. (e) true.

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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. One of the tables below contains (x,y) values that weregenerated by a linear function. Determine which table, and thenwrite the equation of the linear function represented by the

Table #1 x 2 5 8 11 14 17 20 y 1 3 7 13 21 31 43

Table #2 x 1 2 3 4 5 6 7 y 10 13 18 21 26 29 34

Table #3 x 2 4 6 8 10 12 14 y 1 6 11 16 21 26 31

2. Give an example of each of the following:

A function whosedomain is [2, infinity)

An arithmeticsequence

A system of equationswith no solutions

3. For f(x) =1/x^2-3, find:

f(3)
f(2+h)

4. For f(x) = 1/x-5and g(x) = x^2+2, find:

1. (fog)(x)

2. (gof)(6)

5. For the sequencegiven by a (subscript n) = 4n+5 an=4n+5, answer the following:

Find the first fiveterms

Find the sum of thefirst 25 terms.

Is this an arithmeticsequence? If so, how can you tell? If not, why not?

6. Graph the areabound by y<1/2x+6, x+3y>=12, x>=0, and x<=12

7. For the functiondefined by:

f(x)= {x^2,x<=1}

{2x+1,x>1}

a. evaluate f(0)

b. graph f(x)

8. Solve thefollowing system of equations algebraically. Verify your solutioneither graphically or by using matrices.

3x-y=0

5x+2y=22

9. Solve thefollowing system of equations algebraically. Verify your solutioneither graphically or by using matrices

8x-2y=5

-12x+3y=7

10.Given matricesA, B, and C below, perform the indicatedoperations if possible. If the operation is not possible, explainwhy.

A (3x3) [1st 2, -1,0] [2nd 0, 5, 0.3] [3rd 1, 4, 10]

B (3x3) [1st 5, 0, 2][2nd 1, -3, 9] [3rd2, 0, 4]

C (1x1) [1, 3, 5]

1.3A+B

2. B+C

3. CA

11. Given the tablebelow, evaluate the following:

x	-3	-2	-1	0	1	2	3	4	5
f(x)	10	20	30	40	50	60	70	80	90
g(x)	-1	-2	-3	-4	-5	-6	-7	-8	-9

a.(3f+2g(1) b. (fog)(-1)

12.Express the following function, F(x) as a composition of twofunctions f and g

f(x)= x^2/(x^2+4)

13.You have a coupon for your favorite clothing store for $25 off anypurchase of more than $50. The store is also running a 20%-off saleon its entire inventory. Let x be the original price, f(x)be the price with the $25 coupon applied, and g(x) be the pricewith the 20% discount applied.

a.Write an expression for f(x)

b.Write an expression for g(x)

c.What would the expression (fog)(x) represent?

d.What would the expression (gof)(x) represent?

e.If the store allows you to apply both the 20% discount and the$25-off coupon, does it matter which you apply first? How do youknow?