unit 8

find the average value of each function on the given interval 1. f(x) = x^2 on [2,4] 2. f(x) = x^2 - 2x on [0,3] 3. f(x) = sinx on [0,π] 4. f(x) = √x on [0,16] 5. f(x) = 1/x^2 on [-4,-2] on the given interval, find the XS-value where the function is equivalent to the average value on that interval 7. f(x) = 2x - 2 on [1,4] 8. f(x) = -x^2/2 on [0,3] 9. f(x) = 2x^2 + 16x + 28 on [-5,-2] find the average rate of change on the given interval 10. f(x) = -(2x - 6)^(2/3) on [1,3] 11. f(x) = y = x^3 - 2x^2 + 2 on [-1,1] 12. y = ln√x on [1,e]

1. 28/3 2. 0 3. 2/π 4. 8/3 5. 1/8 7. 3; x=5/2 8. -1.5; x=√3 9. -2; x = -5 and x = -3 10. 1.26 11. 1 12. 0.291

1. the average value of f(x) = x^3 over the interval a≤x≤b is 2. the average value of the function f(x) = ln^(2)x on the interval [2,4] is 3. the function f is continuous on the closed interval [1,3] and has the values given in the table. the equation g(x) = 5/4 must have at least two intersections with f(x) in the interval [1,3] if k = x: 1,2,3 f(x): 2,k,4 4. a particle moves along the x-axis so that its position at time t is given by x(t) = t^2 - 7t + 12. for what value of t is the velocity of the particle zero? 5. the function g is given by g(x) = 3x^2/e^(3x). on which of the following intervals is g increasing?

1. b^4-a^4 / 4b-4a 2. 1.204 3. 1/4 4. 3.5 5. (0,2/3)

average value of a function

1/(b-a) S [a,b] f(x)dx

MVT for integrals:

1/(b-a) S [a,b] f(x)dx * (b-a)

look at 21 on 9.3 ap problems for graph

YUHHH!!!!

mr. brust is driving across town to mr. Sullivan's house to play with a new set of Star Wars figures. mr. brust's speed would obviously vary throughout the drive, but because he is so cool, he came up with a function that represents his velocity (miles per minute) at any given time t (minutes) since he left his house during the 30 minute drive. v(t) = sin(0.3t) + ln(t+1) - 2 set up the expressions for the following scenarios. use a calculator to solve. a. how far is mr. brust from his house after 10 minutes? b. how far is he after 15 minutes? c. if he arrives at the other house after 30 minutes, how far away does he live? d. how many miles did he drive?

a. S [0,10] v(t)dt = 3.01 miles b. S [0,15] v(t)dt = 3.397 miles c. S [0,30] v(t)dt = 22.824 miles d. S [0,30] Iv(t)Idt = 28.497 miles

so basically I have given up on quizlet, so units 8, 9, and 10 are not on quizlet...

you will prolly never see this bc you never actually use these quizlets

find where the instantaneous rate of change is equivalent to the average rate of change (where y' = avg ROC) 13. y = x^2 - 4x + 3 on [0,4] 14. y = √(9-3x) on [-2,3] 15. the temperature (in degrees F) t hours after 9am is approximated by the function T(t) = 50 + 14sin(πt/12). find the average temperature during the time period 9am to 9pm 16. the depth of water in mr. bursts hot tub can be represented by the formula h(t) = -cost + 2, where t is the time in minutes since he begins pouring in water and h(t) is measured in feet. what is the average depth of the water during the first three minutes? set up the expression and use a calculator to help solve. 17. find the number(s) b such that the average value of y = 2 + 7x - x^3 on the interval [0,b] is equal to 2. 18. find the number(s) b such that the average value of y = 2 + 6x - 3x^2 on the interval [0,b] is equal to 3.

13. x = 2 14. x = 1.75 15. 58.9126 degrees F 16. 1.9529 ft 17. b = +/-√14 18. b = (3 +/- √5)/2

comparing average rate of change (secant slope) and average value of a function: set up the equation for each question, but do not solve it. what units will the answer be? 2. h(t) = -16t^2 + 41t + 10. h is height (feet) and t is time (seconds) a. what is the average height during the first 3 seconds? b. what is the average velocity during the first 3 seconds? 3. r(x) = 2sinx - 1, where r is the rate at which Mr. Brust's waistline is changing (inches per month) and x is time (months) a. what is the average rate that Mr. Brust's waistline changes from the 10th to the 12th month? b. what is the average change of this rate during the first 5 months?

2. a. 1/3 S[0,3] h(t)dt; feet b. h(3)-h(0) / 3-0; feet/sec 3. a. 1/2 S[10,12] r(x)dx; inches/month b. r(5)-r(0) / 5-0; inches/month^2

1. find the average value of f(x) = 6-x^2 on [-1,3] when does the function assume this value?

3.67 1.578

20. a metal wire of length 9 cm is heated at one end. the table above gives selected values of the temperature T(x), in degrees Celsius, of the wire x cm from the heated end. the function T is decreasing and twice differentiable. distance (x (cm)): 0,1,5,6,8 temperature (T(x) (degrees C)): 100, 93, 70, 62, 55 b. write an integral expression in terms of T(x) for the average temperature of the wire. estimate the average temperature of the wire using a trapezoidal sum with the four subintervals indicated by the data in the table. indicate units of measure. c. find S[0,8] T'(x)dx, and indicate units of measure. explain the meaning of S[0,8] T'(x)dx in terms of the temperature of the wire. d. are the data in the table consistent with the assertion that T"(x)>0 for every x in the interval 0<x<8? explain your answer.

b. 1/8 S[0,8] T(x)dx; average temp = 75.6875 degrees C c. -45 degrees C; the temperature drops 45 degrees C from the heated end of the wire to the other end of the wire d. no, by the MVT, T'(c1) = -5.75 for some c1 in the interval (1,5) and T'(c2) = -8 for some c2 in the interval (5,6). it follows that T' must decrease somewhere in the integral (c1,c2). therefore T" is not positive for every x in the [0,8]

19. traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. the traffic flow at a particular intersection is modeled by the function F defined by F(t) = 82 + 4sin(t/2) for 0≤t≤30, where F(t) is measured in cars per minute and t is measured in minutes c. what is the average value of the traffic flow over the time interval 10≤t≤15? indicate units of measure d. what is the average rate of change of the traffic flow over the time interval 10≤t≤15? indicate units of measure

c. 81.899 cars/min d. 1.518 cars/min^2

mean value theorem (MVT) for derivatives

f'(c) = f(b)-f(a)/b-a

average rate of change

f(b)-f(a)/b-a

S rate of change =

net change

when you integRATE a RATE, you get...

net change