Midterm 2 Study Guide
Tuesday, March 27, 2018 1:23 PM
Limits of Multivariable Functions
the function is continuous:ConstantsSine CosinePolynomials+ ‐ * of these functions
the domain of f.Partial Derivatives
It is denoted, for a function f(x,y), fClairaut's Theorem says that if fcontinuous, then they are equal.
Approach a limit from different paths
f the result for the limit varies, then there exists no limit You can just use the value of a function as the limit if you know
Division and square root (of these functions) are
continuous on their domains
Composition of functions: f(g(x)), assuming range(g(x)) is in
A partial derivative is a derivative of an expression or function of multiple variables. We treat everything but our desired partial variable as constants and take the derivative as normal. x or fy.
xy (taking (fx)y) and fyx are
- f(x,y) must have continuous partials fx and fy near (x0, y0). - The linear approximation is: We also discuss several other topics like What is the name of telemachus’ mother?
Cheat Sheets Page 1
○ L = f(x0,y0) + fx(x0,y0)(x‐x0) + fy(x0,y0)(y‐y0)
The Chain Rule
- The basic chain rule learned in Calculus 1:
- We extend this to the multivariable case with partials:
- We can represent this using a tree diagram: Cheat Sheets Page 2
Cheat Sheets Page 3
To verify a solution to a given differential equation, we attempt to find the quantities in the differential equation and see if those are indeed equal, as defined by the equation.
Directional Derivatives: Gradient
- Definition of the directional derivative:
Cheat Sheets Page 4
- Gradients are tangent to level curves of a function. Tangent lines and planes
- Don't forget about the age old question of What are the assumptions, in monopolistic competition?
Tangent line to a curve can be thought of as a tangent line to an equivalent level curve of a surface:
Cheat Sheets Page 5
Find the gradient, and the point you want to be tangent at.
- Since gradient is tangent to the level curve:
The general equation for a tangent line/plane at point P is:
Local min/max of a function of two variables. - Combine the first and second derivative tests. - Second derivative of a function:
○ D > 0, fxx > 0: minimum
○ D > 0, fxx < 0: maximum We also discuss several other topics like What do we mean by self-esteem or addiction?
○ D < 0: saddle point
○ D = 0: unsure. Try a different test.
Cheat Sheets Page 6
Finding Global Extrema
Very similar to global extrema in single variable.
Check for critical points and then check the endpoints. However, with more dimensions, our endpoints may be actual equations or even surfaces. -
A bounded region has a boundary, or it can be contained within a shape of finite size. Don't forget about the age old question of What is the difference between essential and non-essential of amino acids?
If you want to learn more check out Who uses water?
- A close region is a region that contains its boundary. - Finding the global extrema
Find the critical points of the function, where the gradient of f = <0,0> or <0,0,0> If you want to learn more check out What does the law of superposition?
Calculate equations for the boundaries of f, or use Lagrange multipliers.
With equations, find their min/max with critical points, and find their endpoints as well. ▪
With Lagrange multipliers, solve this system of equations:
Cheat Sheets Page 7
○ Then, compare all your points
Note. Lagrange multipliers doesn't always solve for all the min/max on a region. You have to checks spots where the gradients or the derivatives are discontinuous.
Cheat Sheets Page 8
Approximations are the same with these, but we simply test with more points because there are extra dimensions.
Cheat Sheets Page 9
Polar coordinate integrals
- You can do the same integral, substituting properly: Cheat Sheets Page 10
But, you have to multiply your f(r,theta) by r after
substituting. This is the Jacobian, a factor based on how you actually changed the area when you changed to polar coordinates.
Spherical coordinates integrals
- Spherical coordinates:
Cheat Sheets Page 11
Change of Variables, change of basis
When changing from x, y to u, v to calculate some integral, we need to find the Jacobian for that new space. This is what we've done for polar and for spherical coordinates.
Find x = f(u,v)
y = g(u,v)
Cheat Sheets Page 12
The Jacobian for more variables extends to the determinant of a larger matrix (3 by 3 for three, 4 by 4 for four, etc.)
Cheat Sheets Page 13