derivative Archives - Infinity is Really Big

October 29, 2019October 29, 2019

Differentiable Functions and Local Linearity

Calculus 1, Lectures 12 through 15B

An infinitely oscillating continuous and differentiable function. — *This function is both continuous and smooth at $x=0$ , in spite of oscillating infinitely often in any neighborhood of $x=0$ .*

*This function is both continuous and smooth at $x=0$ , in spite of oscillating infinitely often in any neighborhood of $x=0$ .*

In Steven Strogatz’s excellent book, Infinite Powers, there is a big emphasis on an idea he calls The Infinity Principle.

The concepts of a continuous function and of a differentiable function are … Read the rest

October 15, 2019October 15, 2019

Limit Definition, Continuity, and Derivatives

Calculus 1, Lectures 8A through 11

The limit definition of the derivative leads to the graph of a function with a hole (a removable discontinuity). The limit still exists, based on the precise epsilon delta definition of a limit. — The graph of the difference quotient $DQ(h)=\frac{(2+h)^{2}-8}{h}$ for computing the derivative $f'(2)$ when $f(x)=x^{3}$ has a removable discontinuity (a hole) at $h=0$ . The limit still exists so that $f'(2)=12$ .

The graph of the difference quotient $DQ(h)=\frac{(2+h)^{2}-8}{h}$ for computing the derivative $f'(2)$ when $f(x)=x^{3}$ has a removable discontinuity (a hole) at $h=0$ . The limit still exists so that $f'(2)=12$ .

Graphs with holes (missing points) might seem like anomalies — like they are mere curiosities hardly worth studying. However, they arise at the heart of differential calculus.

… Read the rest

September 20, 2019September 26, 2019

Calculus 1, Lectures 4A through 5B

The derivative of x^2 at an arbitrary value of x is 2x. This is the slope of the tangent line at that point. — The derivative of $y=f(x)=x^{2}$ at an arbitrary value of $x$ is equal by $2x$ . This is the slope of the tangent line at that point and is approximated by $2x+\Delta x$ , which are slopes of secant lines.

The derivative of $y=f(x)=x^{2}$ at an arbitrary value of $x$ is equal by $2x$ . This is the slope of the tangent line at that point and is approximated by $2x+\Delta x$ , which are slopes of secant lines.

Lectures 4A through 5B of my series of Calculus 1 lectures have a lot of content. I will summarize the

… Read the rest

September 9, 2019February 19, 2021

Calculus 1, Lecture 1

The area of a circle can be found with the Infinity Principle. This is described by Steven Strogatz in his book Infinite Powers.

Calculus is often described as the mathematics of change. In terms of a short description, this is apt.

You might even say this description gives us a view of the big … Read the rest

February 20, 2019February 21, 2019

New Video: Immunization, Part 5

Redington immunization is a concept that can be defined in terms of derivatives of present value functions.

Given asset cashflows $A_{0},A_{1},A_{2},\ldots,A_{n}$ and liability cashflows $L_{0},L_{1},L_{2},\ldots,L_{n},$ each set of flows occurring at times $t=0,1,2,\ldots,n,$ the expressions $P_{A}(i)=\displaystyle\sum_{t=0}^{n}A_{t}(1+i)^{-t}$ and $P_{L}(i)=\displaystyle\sum_{t=0}^{n}L_{t}(1+i)^{-t}$ represent the present values of these cashflows, as functions of an arbitrary periodic interest rate $i.$

If we let $h(i)=P_{A}(i)-P_{L}(i)=\displaystyle\sum_{t=0}^{n}C_{t}(1+i)^{-t},$ where $C_{t}=A_{t}-L_{t},$ then we say the liabilities are Redington immunized … Read the rest

February 4, 2019February 12, 2019

New Video: Immunization, Part 3

Immunization of a liability cashflow by an asset cashflow can help to cushion a company when interest rates change, but is it always possible?

My most recent video is “Actuarial Exam 2/FM: Liabilities Not Immunized by Assets in Spite of PV and Duration Matching” (Financial Math for Actuarial Exam 2 (FM), Video #171. October 2018 SOA Sample Exam, Problem #127). … Read the rest

February 1, 2019February 1, 2019

New Video: Immunization, Part 2

In the previous post, “New Video: Immunization, Part 1“, I introduced the idea of Redington immunization of liability cashflows by asset cashflows.

It was a post that delved into the details of my previous YouTube video “Actuarial Exam 2/FM Prep: Use Redington Immunization to Find a Ratio of Assets in a Portfolio” embedded … Read the rest

January 18, 2019January 28, 2019

New Video: Derivatives of Bond Prices w.r.t. Parallel Shift in Yield Curve

The series of payments that arises from a coupon bond is an example of a cashflow that can be assigned a “duration“.

A coupon bond can be purchased as an investment that pays “coupons”, which are fixed amounts of money, at certain equally-spaced times during the term of the investment, plus a redemption … Read the rest

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