Calculus Answers

 You plan to make a simple, open topped box from a piece of sheet metal by cutting a square – of equal size – from each corner and folding up the sides as shown in the diagram: If 𝑙 = 200𝑚𝑚 and 𝑤 = 150𝑚𝑚 calculate: a) The value of x which will give the maximum volume b) The maximum volume of the box c) Comment of the value obtained in part b.

the equation I = 2.4e-6t defines how current varies with time

Evaluate \int \:\int _D\left(x+y\right)^{-\frac{1}{2}}dA\: over the region 𝑥 − 2𝑦 ≤ 1 and 𝑥 ≥ 𝑦2 + 1

Which of the following function has a removable discontinuity at the given point?

a) f(x)=x/|x| at a=0

b) f(x)=(x²+x-6)/(x³-3x²+2) at a=0

c) f(x)=(x²+x-6)/(x³-3x²+2) at a=2

d) f(x)=x/(x-2) at a=2

e) f(x)=x/(x-2) at a=0

A particle moves along the x-axis so that at time t >= 0 (t is greater than or equal to "t") its position is given by x(t)=-t^3+11t^2-24t. Determine the velocity of the particle at t=7.

Prove that a sequence that diverges to +∞ (resp. −∞) is divergent

Suppose {an} ∞ n=1 be a sequence of positive real numbers and 0 < x < 1. If an+1 < x · an for every n ∈ N, prove that limn→∞ an = 0.

Suppose limn→∞ (Sn − 1)/ (Sn + 1) = 0. Prove that limn→∞ Sn = 1.

Let {an} ∞ n=1 be a non-decreasing (resp. non-increasing) sequence which converges to a. Then prove that an ≤ a (resp. a ≤ an) for every n ∈ N.

Prove that a sequence is bounded if and only if it is both bounded above and bounded below