The tutor explains concavity and point of inflection with an example. Concavity refers to an aspect of graph shape. My first-year calculus professor explained it this way: concave upward will collect rain, while concave downward will shed rain. Numerically, when …
The tutor shows how to remind yourself that the derivative of tanx is sec2x. Let’s imagine you don’t recall that (tanx)’=sec2x. Here’s how to reconstruct it: Recall that tanx=sinx/cosx Take the derivative of sinx/cosx using the quotient rule:(u/v)’ = (vu’-uv’)/v2 …
Calculus: proving derivative of tan: quotient rule Read more →
The tutor explores how to detect and solve exact differential equations with a very simple example. When a differential equation of the form P(x,y) +Q(x,y)y’ = 0 results from the implicit differentiation of an original equation F(x,y)=c, the equation P(x,y) …
Differential equations: exact differential equation Read more →
The tutor explains the idea of odd function vs even function. An odd function is one that gives the opposite output for -x as for x: f(x) odd, then f(-x) = -f(x) x3 and sinx are both odd functions: (-2)3=-8, …
Math: what is an odd function? What is an even function? Read more →
The tutor shows the integral function on the TI-83 Plus. To evaluate an integral on the TI-83 Plus, go to the Math menu, then arrow down to fnInt(, which is choice 9. It accepts the parameters as follows: fnInt(function, variable …
Calculator usage: definite integrals on the TI-83 Plus Read more →
The tutor discusses the limit, as n goes to infinity, of the nth root of n factorial. It’s probably easy to agree that, for large n, n! > (n/6)^n. limn→∞ ((n/6)^n)^(1/n) = limn→∞n/6 = ∞ Therefore, limn→∞ (n!)^(1/n) = ∞, …
The tutor uses the ratio test to show the infinite series Σ4n/n! converges. Example: Check Σ0∞4n/n! for convergence or divergence. Solution: The ratio test says that, if limn→∞|an+1/an| < 1, then the series converges. In this case, the terms are …
Calculus: ratio test: checking Σ4n/n! for convergence Read more →
The tutor shows how to integrate sec³x. My post here might be a helpful warm-up towards the technique below. For ∫sec³xdx we use integration by parts: ∫uv’ = uv – ∫vu’ With sec³x we imagine v’=sec²xdx, u=secx. Note that if …
The tutor shows that limx→0 sinx/x = 1 using l’Hôpital’s rule. Put very simply, l’Hôpital’s rule states that for a situation where Limit f(x)/g(x) = 0/0, Limit f(x)/g(x) = Limit f'(x)/g'(x), provided f(x), g(x) are both differentiable and so on. …
Calculus: l’Hôpital’s rule: another proof that limx→0 sinx/x = 1 Read more →
The tutor shows a reason for the famous limit x→0 sinx/x = 1. The area of a sector of a circle about angle Ɵ is Ɵr2/2. (A sector is a pie slice.) The area of a triangle is base(height)/2. In …