Category: Mathematics for Marine Engineering.

Double and Triple Integrals

Methods of Integration

Integration is not as easy as differentiation. This is first due to its nature. Finding a derivative of a given…

Cartesian Coordinates

Using Cartesian Coordinates we mark a point on a graph by how far along and how far up it is: They are also called Rectangular Coordinates because it…

Integral Calculus

Gottfried Wilhelm Leibnitz (1646-1716) and Sir Isaac Newton (1643-1727) independently discovered calculus in the mid-17th century. Leibnitz, a German philosopher, mathematician, and…

Lagrange Multipliers

In the previous section we optimized (i.e. found the absolute extrema) a function on a region that contained its boundary. Finding…

Finding Maxima and Minima using Derivatives

Where is a function at a high or low point? Calculus can help! A maximum is a high point and…

Examples on Chain Rule (Differentiation Rules)

Examples on Chain Rule Example 10.8 Find F′(x) if F (x) = √[x2 +1] Solution Take u = g(x) = x2 + 1 and f (u) =√u ∴ F (x) = ( fog)(x) = f (g(x)) Example 10.9 Differentiate…

Euler’s Homogeneous Function Theorem

Let  be a homogeneous function of order  so that (1) Then define  and . Then (2) (3) (4) Let , then (5) This can be generalized to…

Homogeneous Functions

To be Homogeneous a function must pass this test: f(zx,zy) = znf(x,y) In other words Homogeneous is when we can take a function:f(x,y)…

Higher-Order Partial Derivatives

Fred has mastered differentiating when a function has only one variable. Somehow, he missed the lessons on partial differentiation. He’s…

Geometry of partial derivatives

Partial Derivatives

Now that we have the brief discussion on limits out of the way we can proceed into taking derivatives of…