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 …
The tutor comments on the divergence of a series. On page 545, number 17, Larson and Hostetler1 ask about the convergence or divergence of ln2/2 +ln3/3 +ln4/4 +…. The series must diverge: for q≥3, lnq >1. Therefore, ln3/3 +ln4/4 +ln5/5+….> …
The tutor checks an infinite series for convergence. Number 22, page 545, of Larson and Hostetler1 asks about the convergence of the series Σ2∞(n-1(n2-1)-0.5) Solution: First, we realize that, for n>1, n2-1 > (n-1)2 Therefore, Σ2∞(n-1(n2-1)-0.5) < Σ2∞(n-1((n-1)2)-0.5)=Σ2∞(n(n-1))-1 In turn, …
Calculus: convergence (or divergence?): Σ2∞(n-1(n2-1)-0.5) Read more »
The tutor gives an example of the disc method for finding volume of revolution. Usually, the disc method is preferred when the graph is revolved about the x axis. Example: Find the volume generated, x=7 to x=10, when y=-(x-9)2+5 is …
Calculus: finding volume of revolution: the disc method Read more »
The tutor shows how to integrate exsinxdx by parts. I’ve written a couple of posts on integration by parts: here and here. The method is used on products, and depends on choosing one function to integrate, then differentiating the other: …