Math Approaches

Satisfying the conditions (Lagrange)

1. Check to see if any of the first order conditions can be factorized, and if not, proceed to step 2 2. Isolate 𝛌

1st and 2nd order partial derivatives

1. Derivative with respect to "x" = F'(x) 2. Derivative with respect to "y" = F'(y) 3. Second order derivative of F'(x) with respect to "x" = F''xx 4. Second order derivative of F'(y) with respect to "y" = F''yy 5. Second order derivative, either F''xy or F''yx

Finding the Tangent

1. Determine the derivative of the function 2. Determine the value of the derivative at the point given; this is the slope at that point hence the slope of the tangent 3. Plug into tangent line formula 4. State your solution with your target line

Continuity

1. Fill in the values of "x" where the formulas spilt for both functions, and if it is not included in the interval, write it as a limit 2. Set the two formulas equal to each other 3. State that this proves what was stated in the instructions, or that the functions are only continuous at those points.

Partial Differentiation Max/Min/Saddle

1. Find F''xx, F''yy, and F''xy 2. Find the stationary points using partial differentiation 3. Plug the stationary points into this formula: (F''xx) x (F''yy) - ((F''xy)^2) 4. Plug in the stationary points into the equations in the formula 5. If it is negative its a maximum, if its positive its a minimum

Lagrange Max/Min

1. Find Lagrange function 2. Find L'x and L'y. Solve for 𝛌 or a variable, plug into other equation, then use the variable you got to plug into g(x) 3. Once you have the "x"s and "y's plug them back into L'x and L'y to get 𝛌 as a number. This way you will have x1, x2, y1, y2, 𝛌1, and 𝛌2. 4. Find F'x, F''xx, F''xy, F'y, F''yy, g'x, g''xx, g''xy, g'y, g''yy. 5. Plug them into the D = formula 6. Use the first set of variables for the number, and if the ending number is >0 it is a minimum, and if it is <o it is a maximum. 7. If the first set creates a minimum, the second set must be a maximum, and vice versa

Regular Minimum/ Maximum Points

1. Find stationary points 2. Test "x" values surrounding stationary points 3. Either minimum (-,+), maximum (+,-), or saddle (+,+).

Determining Intervals at which a function is Concave/Convex

1. Find the first derivative 2. Find the second derivative 3. Set second derivative to zero. Convexity: bigger than zero, Concavity: smaller than zero. Solve the inequality 4. Use the intervals you got to make a sign diagram

Change in formula (Lagrange)

1. Formula : New Max = old max value + ∆change x 𝛌 2. ∆change = the difference between what number g(x) equals 𝛌 = the rate of change, which is what we found 𝛌 was equal to in the first order derivative for Lagrange you find the only max value by using the max coordinates and plugging them into f(x).

Non-vertical Asymptotes / Oblique asymptotes

1. Perform Polynomial division 2. Fill on the limit as x goes to infinity 3. Write what ever is not your remainder as your answer

Limits

1. Plug in the limit (just to double check) 2. If/When you get either 0/0 or infinity/infinity, perform l'hopital 3. If you get 0/0 again or infinity/infinity again, keep on doing l'hopital until you don't

Homogeneity (proving and at what degree)

1. Put a "t" in front of every "x" and "y" and put the "tx" "ty" between brackets 2. Work out the brackets so that the "t"s are seperate 3. Try to group all the "t"s together in front of the original function, and then substitute the original function for f(x,y). 4. The power is to what degree, state the degree (it can be zero)

Logarithmic Differentiation

1. Put ln() on both sides of the function 2. Differentiate with respect to "x" 3. Multiply it by the original equation

Solving inequalities

1. Set all denominators equal 2. Set everything to one side put in on fraction 3. See when it is positive or negative using the sign diagram

Partial Derivative Stationary Points

1. Stationary points are when F'(x) = 0 and when F'(y) = 0. If that is not possible because, for example you have more than one variable, then go to step 2. 2. Write F'(x) as x=... or y=... and then substitute that into the other equation such as F'(y) 3. Solve for the values, and then plug them into the x=... or y=... equation to get the pull points.

Implicit Differentiation

1. Take derivative with respect to x (everything on one side) 2. Take derivative with respect to y (everything on one side) 3. Divide F'(x) by F'(y) multiplied my -1 4. Then plug in points given to you in the instructions

Regular Stationary Points

1. Take the derivative 2. Set it equal to zero

Regular Inflection point

1. Take the first and second derivative 2. Set it equal to zero

Partial Elasticity

1. Take the partial derivative of f(x) and if it is complex remember logarithmic rules 2. Multiply the derivative by x/f(x) 3. Fill in the value of "x" given in the instructions into the new equation

Lagrange: 1st order

1. Take the two functions and plug them into f(x) - 𝛌g(x) 2. Take partial derivative with respect to "x" and set equal to zero 3. Take partial derivative with respect to "y" and set equal to zero 4. When setting to zero, find 𝛌 or a variable 5. Plug that into the other equation 6. Then when isolating as variable here, plug that into the g(x) equation 7. Solve g(x) and plug the units found into the original g(x) function (proving the points are correct) 8. Then state the intervals you got

Proving differentiability

1. The function can only be differentiable if it is continuous, so prove continuity first 2. For the function to be differentiable, this formula has to hold. Limit as "h" approaches zero (negative) ( (f(x+h) - f(x))/ h ) = the same function but "h" zero (positive). 3. Solve and get the intervals 4. Compare these numbers to the intervals you got for continuity 5. If they are the same, then it proves differentiability, if only one then only differentiable at that point, and if none then it is not differentiable.

Domain/Range of of an inverse

1. Write down: the domain of the inverse is the range of the original function 2. Determine whether the function is increasing or decreasing 3. Find the value of Y at the start of the interval 4. See what the limit of f(x) is as it approaches infinity 5. Note those as your interval