Goal: Calculate partial derivatives without using limit The idea behind calculating partial derivatives is to keep all independent variables, other than the ones with respect to which you are differentiating, as constant. Then proceed as if you were using a function of single variable. To prove this let's fix y and pose g(x) = f(x, y) The same is true when differentiating with respect to y. In this case, we fix x and pose  h(y) = f(x,y)  as a function of y.                                  ..read more

 Goal: Calculate the partial derivatives of a function of two variables  Derivatives of a function of two variables We have seen that some methods of the function of one variable can be applied to the function of two variables. In the notation of Leibniz dy/dx, x is the independent variable and y is the dependent variable. In a function of two variables, how do we adapt this notation? What would be an acceptable definition for the derivative of a function of two variables. This leads to the notion of partial derivatives Definition Let f be a function of two variables. Then the partia ..read more

 Operations on continuity of a function of two variables: Theorem 1 The sum of continuous functions is continuous If a function f(x,y) is continuous at (x₀.y₀), then f(x,y) + g(x,y) is continuous at (x₀.y₀).  Theorem 2 The product of continuous functions is continuous If g(x) is continuous at x₀ and h(y) is continuous at y₀, then f(x, y) = g(x).h(y) is continuous at (x₀.y₀). Theorem 3 The composition of continuous functions is continuous Let g be a function of two variables from Let's suppose g continuous at  (x₀.y₀) and let z₀ = g(x₀.y₀). Let f be a function that relates eleme ..read more

 Goal: State the conditions for continuity of a function of two variables Definition The definition of the continuity of a function of two variables is similar to that of a function of one variable..  A function f(x, y) is continuous at a point (a,b) in its domain if the following conditions are verified: Example 1 Show that the function:                                                                  Solution In or ..read more

 Objective: Determine the limit of a function at a boundary point Before to define the limit of a function of two variables at a boundary point, let's define interior and boundary point. Interior point. Let's consider a subset S of R². A point P is an interior point of S if any δ disk centered at P is located completely inside of S. Example: the point (-1, 1) in the figure below is an interior point Boundary point A point P is a boundary point if any δ disk centered at P is not completely located inside of S. It means that  some points of the disk are located inside of S and others a ..read more

 Goal: define the limit of a function of two variables The limit of a function of two variables is based on the limit of a function of one variable. Let's recall the definition: Let f(x) be defined for all x≠a in an open interval containing a . Let L be a real number. Then  if  for every ϵ> 0 there exists ẟ> 0 such that if 0< ❙x - a❙ < δ for all x in the domain of f, then The idea of an open interval in a function of two variables is similar to the open interval in a single variable. Let's define an open interval in a function of two variables. Definition  A ..read more

 Civil engineers building roads in mountains use a topographical map that shows the different elevations of the mountain. Hikers walking through rugged trails use also a topographic map to show how steeply the trail changes.  A topographical map contains curved lines called contour lines. Each contour line corresponds to the points of the map that have the same elevation. A level curve of a function of two variables f(x,y) is similar to a contour line in a topographic map. The photo on the left is a topographical map of the Devil's Tower, Wyoming. USA. Lines that are close togethe ..read more

 In this post, we continue with examples. We are going to solve a practical problem concerning nuts and bolts. Problem A profit function for a hardware manufacturer is given by: where x is the number of nuts sold per month (measured in thousands) and y the number of bolts sold per month (measured in thousands). Profit is measured in thousands of dollars. Sketch a graph of this function Solution Let's determine the domain of the function. This function is a polynomial function with two variables. For a profit to occur, we need to have f(x,y) ≥ 0. In other terms:   ..read more

 Objective: Transform the function of a curve into parametric equations. Parameterization of a curve In the previous lesson we learn how to eliminate the parameter in the parametric equations of a curve. In this post we do the reverse meaning transforming a function of a curve into parametric equations. Example Find two different pairs of parametric equations to represent the graph of y = 2x² - 3 Solution One of the easiest way to do this is to write x(t) = t and substitute x in the function. The result is y = 2 t²-3. The first set of pair of parametric equations is x(t) = t y = 2t²-3 Sin ..read more

 Objectives: 1. Define parametric equations 2. Plot their curve Considerations Let's consider the orbit of the earth around the sun.  The earth revolves around the sun in 365.25 days. In this case, we consider 365 days. Each number in the figure represents the position of the earth in respect to the sun. The letter t represents the number of the Day. For example t = 1 means Day 1 that corresponds to January 1st, t = 274 means Day 274 and corresponds to October 1, and so on. According to Kepler's laws of planetary movement, the orbit of the earth around the sun is an ellipse wit ..read more