Fundamental Theorem
& Beyond
& Beyond
If A(x) = 2∫x cos(t)dt, what is A'(x)?
What is cos(x)?
Use the fact that ex = 1 + x + x2/2! + x3/3! + x4/4! + ... to find the series for g(x) = e-x^2
What is 1 + x2 + x4/2! + x6/3! + x8/4! ?
A tank has a solution of salt water ("brine") flowing into it at a rate of 10 liters per minute, with each liter containing 5 grams of salt. The contents of the tank are kept thoroughly mixed, and the contents flow out at 10 liters per minute. The tank is initially full with 100 liters of brine. Let S(t) be the total amount of salt in the tank in grams at time t minutes. Note that the amount of salt in each liter in the tank at time t minutes is S(t)/100 grams per liter.
Give an expression for the rate (in g/min) that salt is leaving the tank.
What is 0.10 S(t)?
The function F is defined by F(x) =1∫x h(t)dt, where h is the function graphed. The table gives the values of the shaded areas.
What is the value of F(-2)?
What is F(-2) = 1∫-2 h(t)dt = - -2∫1 h(t)dt = -A1
x∫2 d/dt[ln(t2)]dt
What is ln(4) - ln(x2)
Determine whether the sum is geometric; if it is, identify r, a and n.
22/3 + 23/32 + 24/33 + 25/34 + 26/35 + ... + (-2)m/3m-1
What is yes. a = 4/3, r = -2/3, n = m-1
A tank has a solution of salt water ("brine") flowing into it at a rate of 10 liters per minute, with each liter containing 5 grams of salt. The contents of the tank are kept thoroughly mixed, and the contents flow out at 10 liters per minute. The tank is initially full with 100 liters of brine. Let S(t) be the total amount of salt in the tank in grams at time t minutes. Note that the amount of salt in each liter in the tank at time t minutes is S(t)/100 grams per liter.
If the tank initially contains 300 g of salt, will the amount of salt initially increase or decrease? Explain how you know.
What is increase because ds/dt is positive while s is less than 500?
Determine whether the function is a solution to the differential equation.
y' + y = e-x
y = e-2x + e-x
What is y = e-2x + e-x is not a solution?
d/dx[-1∫x^4 cos(t2)dt]
What is cos(x8) * 4x3?
Write out the first 3 terms of the sum and simplify as much as possible.
k = 0∑∞ (x2k+1)/(3k + 2)What is x/2 + x3/5 + x5/8 + ... ?
A tank has a solution of salt water ("brine") flowing into it at a rate of 10 liters per minute, with each liter containing 5 grams of salt. The contents of the tank are kept thoroughly mixed, and the contents flow out at 10 liters per minute. The tank is initially full with 100 liters of brine. Let S(t) be the total amount of salt in the tank in grams at time t minutes. Note that the amount of salt in each liter in the tank at time t minutes is S(t)/100 grams per liter.
If the tank initially contains 10,000 grams of salt, how much salt will be in the tank after a long time? Explain how you know.
What is the amount of salt will decrease until it reaches equilibrium at 500g. This is because for s > 500, ds/dt < 0?
Suppose that the population of a particular species is described by the function P(t) where P is expressed in millions. Suppose further that the population's rate of change is governed by the differential equation dP/dt = ƒ(P) where ƒ(P) is graphed.
Identify all the equilibrium solutions of the DE and say whether they are stable or unstable.
What is P = 0 is unstable, P = 2 is stable and P = 3 is unstable ?
The function F is defined by F(x) = 1∫x h(t)dt where h is plotted on the graph provided and has the following feature:
h'(-5.74) = 0 h'(-.911) = 0 h'(4.397) = 0
h(-7) = 0 h(-4) = 0 h(2) = 0 h(6) = 0
Is F(-4) positive, negative or zero? Explain how you know.
What is F(-4) is negative because the area under h between -4 and 1 is positive and -4∫1h(t)dt equals that area?
Express the sum in sigma notation with k starting at 1 and 3.
22/3 + 23/32 + 24/33 + 25/34 + 26/35
What is k = 1∑∞ (-1)k+1 . 2(2/3)k and k = 3∑∞ (-1)k+1 . 2(2/3)k-2 ?
A can of cold soda is sitting in a 70 degree room. The temperature of the soda was 40 degrees when it was set down in the room. The soda warms at a rate of 10% of the difference between the soda's temperature and the room's temperature every minute. Let T(t) be the temperature of the soda t minutes after it is set down in the room.
Give the differential equation that models this situation.
What is dT/dt = 0.10(70 - T)?
Sketch a slope field for the DE y' = xy on a 4x4 graph
What is *the graph on slides* ?
Based on the graph, list all the intervals where F is increasing (be careful to indicate whether endpoints of the intervals are included in your answer).
What is (-∞, -2), (0,3),(8,∞)?
Find T5(x), the Taylor Polynomial of degree 5 of g(x) = six at a = π.
What is T5(x) = -(x - π) + (x - π)3/3! - (x - π)5/5! ?
A can of cold soda is sitting in a 70 degree room. The temperature of the soda was 40 degrees when it was set down in the room. The soda warms at a rate of 10% of the difference between the soda's temperature and the room's temperature every minute. Let T(t) be the temperature of the soda t minutes after it is set down in the room.
Find the solution to the initial value problem.
What is T(t) = 70 - 30e-0.10t?
A tank holding 4000 liters contains 40 kilograms of salt. A salt solution with a concentration of 10 kilograms per liter enters the tank at a rate of 20 liters per minute. It is mixed with the existing solution. Meanwhile, the mixed solution drains from the tank at the rate of 20 liters per minute. Let S(t) be the amount of salt in kilograms in the tank after t minutes and set up a differential equation that models this situation.
What is S’(t) = (10 kg/L)(20 L/min)- (S(t)/4000 kg/L) (20 L/min) ?