MEAN value theorem
Optimization
1st Derivs are #1
100
Which is not necessarily a requirement for the MVT to be applied:
a) f(x) is continuous
b) f(x) is differentiable
c) f(a)=f(b)
c), f(a)=f(b) is a requirement for ROLLE'S THEOREM
100
2 positive numbers that have a sum of 300 and a maximum product
What are the numbers, 150 and 150?
100
The relative maximum of the equation f(x) = x + 1/x
What is (-1,-2)?
200
What is a guaranteed derivative of f(x)=28x2-12x+2 on the interval (4,8) by the mean value theorem?
a derivative of 324 is guaranteed by the MVT
200
You are building a rectangular backyard in minecraft with a stack of fences (64). Your backyard connects to the northeast and southeast corners of your rectangular house. The dimensions of the backyard that would maximize the amount of chickens (area) that would fit are...
What are the dimensions, 16 blocks by 32 blocks?
200
The intervals when f(x) =x2 - 2x - 8 is increasing.
What is (1,infinity)?
300
f(0)=9, f(3)=4, f(9)=-2, f(10)=2, f(12)=1
If f(x) is a differentiable function, on what interval is there a guaranteed value c where f ’(c)= 1?
[9, 12]
300
Garfield is sitting near a round table, whose graph is represented by x^2 + y^2 = 4. He fits a lasagna plate inside the table such that the corners of his plate are on the edges of the table. The dimensions of the lasagna plate such that the amount of lasagna is at a maximum are...
What are the dimensions, 2 by 2sqrt3?
300
The velocity of a marble is denoted with the equation v(t) = -3x5+5x3. At what point in time (in seconds) does the marble change from positive to negative velocity?
What is 1 second?
400
What is the average acceleration of a snowball represented by the position function a(t)=-8t^2-4t+6 on the time interval (0, 5)?
-16
400
Chloe Ting is trying to maximize her booty band revenue. Let p be the price of her If she charges less than $5, she will go broke. If she charges more than $50, she will get cancelled for scamming people. Because she has limited stock, the amount of bands she can sell a day is represented by the function b(p)=300-6p. Chloe should charge this much per band to maximize her revenue.
What are $25 per band?
400
Students are entering and leaving the call at the end of Mr.Harari’s zoom class. The number of students present in the zoom room is represented by the equation f(x) = ½ x - sinx What is the max number of students in Mr.Harari’s room in the time between when he ends the class t = 0 and when he ends the call to hang out with his daughter t=6?
What is
5pi/6 + sqrt3 / 2
3.484 students
3 students?
500
The amount of boba a local boba shop sells each day is modeled by the equation f(t)=10t2-4t where t is the number of hours they are open. Between what time interval where the times are one hour apart is there a rate at which the shop is selling boba equal to 46 drinks per hour?
Between the time interval (2, 3)
500
Dwight kidnapped Ryan to his beet farm, which is 6 mi west of a pier. Ryan’s friend lives 2 miles north of the pier in the bay. Ryan can run at 8mph and swim at 3mph. Ryan should run this distance before swimming to his friend’s house to escape in the least amount of time possible.
Hint: distance=ratextime
What are 5.19 miles?
500
An anime protagonist and villain are fighting using their powerful laser beams. As they bust out their final moves, they each send a huge blast towards each other. Now, whoever’s blast manages to reach the opponent is declared the winner. The anime protagonist’s blast distance travelled is x(t) =sin(x)x2. At what time t does the anime protagonist start to move forward and begin to overcome the villain’s beam between time 1<= t <= 8.
What is 4.325 seconds?