When adding or subtracting two functions and trying to find the derivative, take each derivative separately then add/subtract. Ex: d/dx ( f(x) ± g(x) ) =d/dx f(x) ± d/dx g(x) What is the Sum and Difference Rule? What is the Constant Multiple Rule? What is the Quotient Rule? What is the Constant Rule?

What is the Sum and Difference Rule?

If you have a constant in front of a variable while trying to get a derivative, multiply the exponent by constant to get a coefficient Ex: d/dx (C)= C(d/dx) f(x) What is the Power Rule? What is the Sum and Difference Rule? What is the Constant Multiple Rule? What is the Product Rule?

What is the Constant Multiple Rule?

ROC between 2 points is a secant line, not derivative oscilating discontinuity How to draw a graph that meets limits: What to remember about rate of change removable discontinuity

What to remember about rate of change

1) draw asymptotes 2) notice what limit is as x approaches left and right side of asymptotes How to draw a graph that meets limits: oscilating discontinuity removable discontinuity jump discontinuity

How to draw a graph that meets limits:

k*∫ f(x) dx ∫ xⁿ dx Power Rule Integral ∫ k*f(x) dx Multiplication by constant Summation formula: Sum(from i=1 to n) of i³ =

∫ k*f(x) dx Multiplication by constant

∫ f(x) dx - ∫ g(x) dx ∫ [f(x)+g(x)] dx Sum Rule ∫ [f(x)-g(x)] dx Difference Rule Integration by parts ∫ xⁿ dx Power Rule

∫ [f(x)-g(x)] dx Difference Rule

ln[A] = - kt + ln[A]0 first order integrated rate law osmotic pressure formula zero order half life zero order integrated rate law

first order integrated rate law

1/[A]t = kt + 1/[A]0 first order integrated rate law second order integrated rate law Arrenius equation zero order half life

second order integrated rate law

ΔTf = (Kf)(m)(i) second order integrated rate law osmotic pressure formula freezing point depression formula Arrenius equation

freezing point depression formula

Take the partial derivative to x of f(x,y) and the partial derivative to y of f(x,y) and plug in the numbers Compute the marginal productivity of x and y when x equals... and y equals... dz If D<0 Critical point

Compute the marginal productivity of x and y when x equals... and y equals...

approximates the change in output due to changes in multiple inputs Second derivative test Restricted growth (carrying capacity) variables The total differential of a multi-variate function If D>0 and fxx(a,b)>0

The total differential of a multi-variate function

f(x1,y1)-f(x0,y0) Euler's Method table columns Method of second derivative test delta z (the actual change in output) Critical point

delta z (the actual change in output)

1. Find the first-order partial derivatives 2. Evaluate the derivatives at the anchor point (x0,y0) 3. Find the one-dimensional differentials 4. assemble the total differential dz Unrestricted growth (exponential) differential equation Legrange multipliers method Method to find total differential

Method to find total differential

1. Find the first partial derivatives 2. Set each partial derivative equal to zero and solve for critical points 3. Find the second partial derivatives and evaluate at each critical point 4. Calculate the discriminant for each critical point and use the second derivative test to classify The total differential of a multi-variate function Unrestricted growth (exponential) variables Method of second derivative test Method to find total differential

Method of second derivative test

calculate D = fxx*fyy-[fxy]^2 Euler's Method table columns For each critical point in the second derivative test Unrestricted growth (exponential) variables Restricted growth (carrying capacity) solution

For each critical point in the second derivative test

1st order, seperable if k is positive general form for 1st order separable diff eq equation for A logistic You can only solve a differential equation algebraically when it is a

You can only solve a differential equation algebraically when it is a

highest derivative that is in equation always remember when solving diff eqs algebraically... logistic model is much more realistic because if k is negative What order depends on for differential equations

What order depends on for differential equations

substitute y=f(x) and corresponding derivatives into diff eq Euler's method produces a estimate that can be... What order depends on for differential equations to find if y=f(x) is a solution to a diff eq... if k is positive

to find if y=f(x) is a solution to a diff eq...

dy/dx = f(x) * g(x) = diff functions have to be multiplied or divided by eachother You can only solve a differential equation algebraically when it is a to find if y=f(x) is a solution to a diff eq... Euler's method produces a estimate that can be... general form for 1st order separable diff eq

general form for 1st order separable diff eq

C! always remember when solving diff eqs algebraically... exponential regular equation and what variables represent general form for 1st order separable diff eq logistic model is much more realistic because

always remember when solving diff eqs algebraically...

Euler's method, slope fields diff eqs can be used to model exponential growth and decay as long as the particular rate of change of some event is ... to find if y=f(x) is a solution to a diff eq... how to estimate solution to 1st order non-seperable diff eqs- 2 methods exponential diff eq and what variables represent

how to estimate solution to 1st order non-seperable diff eqs- 2 methods

an over or under est general form for 1st order separable diff eq What order depends on for differential equations Euler's method produces a estimate that can be... exponential regular equation and what variables represent

Euler's method produces a estimate that can be...

dp/dt = k*P k is proportionality constant or relative growth/decay rate logistic model function exponential diff eq and what variables represent diff eq for logistic model You can only solve a differential equation algebraically when it is a

exponential diff eq and what variables represent

P(t) = Ae^kt A is initial population- must be positive k is proportionality constant or relative growth/decay rate to find if y=f(x) is a solution to a diff eq... if k is negative exponential regular equation and what variables represent diff eqs can be used to model exponential growth and decay as long as the particular rate of change of some event is ...

exponential regular equation and what variables represent

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