01:640:151 Midterm: 01:640:151 Exam 1 2016 Fall Solution

How large should we take n in order to guarantee that the Trapezoidal and Midpoint Rule approximations for 2 1 x dx 1 are accurate to within 0.00002? SOLUTION If f(x) = 1 x , then f '(x) = , and f ''(x) = . Since 1 ≤ x ≤ 2, we have 1 x ≤ 1, so |f ''(x)| = ≤ 2 13 = We take K = 2, a = 1, and b = 2. Accuracy to within 0.00002 means that the size of the error should be less than 0.00002. Therefore we chose n so that 2 3 12n2 ≤ 0.00002 Solving the inequality for n, we get n2 > 2 3 12(0.00002) or n > 1 0.00012 ≈ 91.29 Thus n = (rounded up to the nearest integer) will ensure the desired accuracy. For the same accuracy with the Midpoint Rule we choose n so that 2 3 24n2 < 0.00002 which gives the following. (Round your answer up to the nearest integer.) n > 1 0.00024 ≈

Consider the concentration, C, in mg per liter, of a drug in theblood as a function of x, the amount, in mg, of the drug given andt, the time in hours since the injection. For 0 < (or equal to)x < (or equal to) 4 and t > (or equal to) we have C = f (x,t)= te^(-t(5-x)

Make graphs of these (you dont need to)

A) f(4,t)

B f(x,3)

Using your graph in A, where is f(4,t)

1) What is the maximum of T?

2) What is the Minimum of T?

Using your graph in B, where is f(x,3)

3) Where is the maximum of X?

4) Where is the minimum of X?

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:

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

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:

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?