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: …
The tutor investigates a problem involving the remainder of a power. On page 48 of his Elementary Number Theory, second edition, Underwood Dudley requests the remainder when 20012001 is divided by 26. Solution: 2001 mod 26 = 25 ⇒ 2001 …
Number theory: congruence: another problem from Underwood Dudley Read more »
The tutor investigates a problem involving composite numbers. For problem 4b, page 19, of his Elementary Number Theory (second edition), Dudley invites the reader to prove there are infinite n such that both 6n-1 and 6n+1 are composite. (Composite means …
Number theory: another problem from Dudley’s Elementary Number Theory Read more »
The tutor shows an essential example of integration by parts. Back in my July 23 post, I show the formula for integration by parts as ∫uv’=uv-∫vu’ Let’s consider the example ∫xsinxdx Integration by parts is used to integrate a product …
Calculus: integration by parts: how to integrate xsinx Read more »
The tutor solves a system of linear congruences. Back in my post from March 25, 2014, I explain that “mod” means remainder: for instance, 7 mod 3 = 1. Two numbers that, divided by a number n, give the same …