Category Archives: Differential

Finding Extrema of Multivariable Functions

1. Remember for a curve , we have maxima and minima occur whenever (1) What’s the multivariable analog to this notion?

Posted in Calculus, Gradient, Optimization, Vector Calculus | Tagged | 1 Comment

Differentiation Technique #1: Logarithmic Differentiation

1. We should recall the chain rule applied to gives us (1) where . So what? Well, if is some nightmarish function (e.g. ), then we have (2). Quite a quick way to compute nightmarish derivatives!

Posted in Calculus, Differential, Natural Logarithm | Tagged | 2 Comments

Taylor Series Uses #1: Complicated Integrals

1. Can you compute ? Well, first we consider the antiderivative for …which doesn’t exist. What do we do? Cry. No, what I mean is, use Taylor series!

Posted in Analysis, Calculus, Infinite Series, Power Series, Taylor Series | Leave a comment

Thinking “Infinitesimally”

1. SO I’d like to reiterate the intuitive picture one should have when working with calculus. We should think of a differential as a “really small” change in …well, it’s the “smallest” possible change! The reason I bring this up, … Continue reading

Posted in Calculus, Differential | Tagged | 2 Comments

Taylor Polynomials

1. So last time we concluded discussing Taylor series with constructing a polynomial using the first terms in the Taylor series. This “Taylor polynomial” approximated our function, and we want to know how well does it approximate? We will derive … Continue reading

Posted in Analysis, Calculus, Differential, Infinite Series, Power Series, Taylor Series | Leave a comment

Taylor Series

0. Can we approximate a function using a power series? With the calculus’ help, we can!

Posted in Analysis, Approximation, Calculus, Differential, Infinite Series, Power Series, Taylor Series | Tagged , | 3 Comments

Differentiation Under the Integral Sign

1. Sometimes we work with tricky integrals. For example, consider the integral (1). What is ??? We can evaluate (1) directly as (2). Thus its derivative is (3). But what about the general case?

Posted in Calculus, Differential, Integral, Integration Techniques | 2 Comments