Exam 5 Flashcards
Applications of Definite Integrals, Improper Integrals, Partial Derivatives (18 cards)
How do you get the bounds of a definite integral given two functions?
Set them equal to each other and solve for x
How do you solve P1, P2 equations?
1) Set the equations equal to each other to get the years or upper bound
2) Evaluate the integral
How do you solve a maximum profit function?
P(x) = D(x) - C(x)
1) Multiply entire price function by x to get demand
2) Subtract C(x) from it and solve for x to get bounds
3) Evaluate the integral using the longer CS equation
Willingness to Spend/Consumer Surplus Equation
q0
l D(q)dq
0
Consumer Surplus Equation
q0
l D(q)q - p0q0
0
Average Value formula
1 b
(b-a) l f(x)dx
a
Future Value formula
T
e to the rT l f(t)e to the -rt dt
0
Present Value formula
T
l f(t)e to the -rt dt
0
Lim e to the x =
x – ∞
+∞
Lim e to the -x =
x – ∞
0
Lim (x to the P)/e to the x
x – ∞
P = any power
0
Lim ln(x)
x – ∞
+∞
How do you solve equations with infinity as the upper bound?
1) replace ∞ with N, and as N approached ∞
2) Evaluate integral
3) Try to solve it so that N = 0, or the useful limits
What phrase will tell you to use improper integrals in applications? How do you check them?
“In perpetuity”
Can check by increasing upper bound in calculator until the result plateaus
How do you solve functions of 2 variables when C is involved?
Set each equation equal to C, with should equal 0, and solve for x. These will give you intercepts to draw the graph with
How do you solve partial derivatives?
1) Find fx and fy
2) For fx, take the derivative of the x pieces and let y pieces equal 0, but if there are y pieces attached to an x, leave them as is
3) Do the opposite for y
How do you solve demand problems that involve partials?
For D1, treat p1 as a constant, and see what p2 equals
For D2, treat p2 as constant, and see what p1 equals
If they are both positive - substitutes
If they are both negative - complements
Positive and negative - neither
How do you take second partial derivatives
1) Take fx and fy as usual
2) Take fxx and fxy of fx,which means taking the derivative wrt x on xx and wrt y for xy
3) Take yy and fyzx for fx, which means taking the derivative wrt y for yy
4) Xy and yx should match
