Final Flashcards
What kind of function is a one to one function?
A function where there is only one y value for every x value; when the function is either increasing or decreasing
Is a parabola or a function with relative min or max a one to one?
No because the curve makes it not one to one (ie fails horizonal line test). Saddle points would mean its a one to one
How do you find the inverse function?
By interchanging the values ie switch x and y
How do you find out if two functions are inverse functions using the composition property?
Plug one function into the other function for x, and then vice versa. If you get the same answer both times, they are inverse functions.
How do you do growth and decay rates
Use the formula : P= the number of the population times (1-the rate)^number of years
Its 1- the rate if its decreasing
How do you find linear regression?
Store some values and then get that expreg from F5
What do you do when you have to solve for consumer surplus
Find the integral of the top functions over the interval and multiply it by the x and y values of the market equilibrium point (the poiint where the two functions equal each other)
For the producer surplus, we use the bottom function for the integral and We subtract it from the xy value (known as xe and pe) because its believed that the consumer surplus is derived by paying a lower than equilibrium price. Thus, the producer surplus is derived by taking xepe - the integral (the vice versa) because economists define producer surplus as the value derived by the producer chsrging a lower price
How to find present value
Present value = the sum of money * ((1 - e^(-rt))/r)
How to find final value
The presenr value * e^(intrrest rate * years)
How to find critical points
Find the derivative. Then find what the denominator equals at zero AND what the numerator equals at zero. or if its factoraboe, what the factors (of the derivative!) are are at zero. To get the y, plug the x back into the original
Does this function have any critical points? 1/x?
No. So doesnt 1/(x+1) or things that look like that because apparently it can easily become a zero as a denominator.
How do you find the largest possible product? Of x and y?
Fiind the constraint so you can represent y or x in terms of the other term. Then find the vertex -b/2a , which gives you your value and then plug that back into the thing they need you to find (eg sum or product)
What do they do when they ask you to maximize revenue?
Revenue is usually x*p, and p is the price or the demand function while x is the #of items.
So multiply p by x, find the derivative, then find what x is at the max (meaning set it to zero)
How to tell if something is a relative max or min?
Take the second derivative and check for positive or ngative
How do you find max or min average cost?
Do c(x)/x
Then differentiate and find critical poonts for max or min. Fnd out max and min using 2nd deriviative if you need it
True or false, marginal cost = average cost
True
How do you find out maximum profit
You have to do R(x) - c(x) then differentiate and critical points
Marginal revenue?
Derviative of revenue
Mrginal cost?
Derivative of cost
Elasticity of demand
Is p/x *dx/dp
Dx/dp is just ther derivative of x
If bacteria is decaying do you add a negative to the exponent in a=pert
Yah
If if not compounded what do you use
Eg 30 dollars for 7 percent a year for 10 years is 30*(1.07)^10
When something is added to the exponent does it shift left or right or up or down
It shifts left
When do you use u substituitoon
When theres something complicated inside something simple for itnegrals
What is the formula for present value
Principle *((1-e^(-rt))/r)
What is the formula for elasticity of demand
P/x *dx/dp
True or false: The derivative of e^(x^2-3x) is (e^(x^2-3x))*(2x-3)
True
How to find average value
1/(b-a) * definite integrla of whatever they give you
Horizontal asympototes, three conditions:
Top equation bigger-> none
Bottom bigger -> zero
Same size-> divide by the constant in front of the leading coefficients
Vertical asympototes
Factor the denominator and set equal to zero
Its concave up when
2nd derivative is positive in the interval
It is concave down when
Second derivative is negative in the interval
What to do with an exponential function with 2 points
Y=ab^x
Plug in one set of points for y and x. Try to find a in terms of b
Then plug in the second set using only one variable b
Solve for a, now you have a function
True or false : d/dx of ((e^x)^10) is 10e^10x
True
What is the log rule for loga(x^b)
B*loga(x)
What can you do with ln(1/e^5)
Turn it into -5
What is e^0?
1
Formula for producer surplus
Xepe * integral of supply
(Xepe found by setting both equations equal to each other, solving for x, then plugging x into one of the equations for p)
Frmula for consumer surplus
Intrgral of demand - xepe
Xepe found by equaling both euations to each other, solving for x and then plugging x into either for p
Let y = 5 x^2. Find the change in y, \Delta y when x= 1 and \Delta x = 0.3 Find the differential dy when x= 1?
Let y = 5x^2.
Find the change in y, ∆y when x= 1 and ∆x = 0.3
Find the differential dy when x= 1 and dx = 0.3
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Best Answer: ∆y = 5(x + ∆x)^2 - 5x^2
= 5(x^2 + 2x∆x + ∆x^2) - 5x^2
= 5(2x∆x + ∆x^2)
= 10x∆x + 5∆x^2
So when x = 1 and ∆x = 0.3,
∆y = 1010.3 + 5*(0.3^2)
= 3 + 0.45
= 3.45
dy = 10x dx
So when x = 1 and dx = 0.3,
dy = 1010.3
= 3
Note: I have given you the answer you evidently wanted, but strictly speaking, dx and dy are differentials (infinitely small quantities) and as such cannot have real number values.
chauncy · 6 years ago
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What do you do with that delta y stuff?
Every time you see an x, replace wth x+delta x, and at the end of that, subtract the original equation
What is the chain rule for derivatives
Dy/du * du/dx
Basically if you have e^2x,you first find u which is 2x then find du which is 2. Then you take your y which is now e^u and find rhe derivstive of tht which is just e^u. Then you multiply your du with your dx. So yu get 2*e^u and then you plug in 2x for u.
What is the derivstive of 4e^u?
4e^u
Nothing happens to the constant
How to calculate net present value
A= p*((1-e^-n)/r
Ie all those ‘flows uniformly’ questions
How to calculate actual change?
F(b) -f(a) on the interval a,b
How calculare average rateof change
(F(b) - f(a)) /(b-a)
What does closed interval mean?
Does include endpoints
What does open interval mean?
Does not include endpoints
When you see those interval problems what do you do?
Make a chart
How do you do epsilon
((F(x+delta x) - f(x))/ delta x) -f’(x)
What do you do with e to the exponent?
E to the exponent times the derivstive of the exponent
Wat do you do when you see solve for dy/dx = somethig?
Multiply both sides by dx
Find the integral
How do you tell if a function is increasing or decreasing?
Use the first derivative test. If the derivative is positive on that interval, it is increasing and if the derivative is negative on that interval, it is decreasing.
