Calculus Answers

Prove that for any 0 < x < 1, the series (1−x)+(x²−x³)+... converges and show that it converges to 1/(1+x)

Let ∞n=1an be a series of real numbers and N0 ∈ N. Show that ∞n=1an converges if and only if ∞n=N0 an converges

Prove that if ∞n=0an is a convergent series, then limn→∞an = 0 (i.e., starting with n = 0 in the series yield the same result)

Let {an}∞n=1 be a convergent sequence with limit L. Then prove that any subsequence of {an}∞n=1 is also convergent and has limit L

Find the mass of a triangular lamina whose vertices are (0,0), (1, 1), (0.1). if the density of the lamina is p(x, y) = sin (y^2).

Let R be bounded by the curves y = x2 + 2 and y = 2x + 5. What integrals will give the volume of the solid of revolution formed by revolving R around the line x = −3?

Find the differential in y=3√6x

The velocity of a moving vehicle is given by the equation "\ud835\udc63 = (2\ud835\udc61 + 3)^4". Use the Chain Rule to determine an equation for the acceleration when 𝑎 = 𝑑𝑣 𝑑𝑡 .

1 The equation for a distance, s(m), travelled in time t(s) by an object starting with an initial velocity u(ms-1 ) and uniform acceleration a(ms-2 ) is: 𝑠 = 𝑢𝑡 + 1 2 𝑎𝑡 2 The tasks are to: a) Plot a graph of distance (s) vs time (t) for the first 10s of motion if 𝑢 = 10𝑚𝑠 −1 and 𝑎 = 5𝑚𝑠 −2 . b) Determine the gradient of the graph at 𝑡 = 2𝑠 and 𝑡 = 6𝑠. c) Differentiate the equation to find the functions for i) Velocity (𝑣 = 𝑑𝑠 𝑑𝑡) ii) Acceleration (𝑎 = 𝑑𝑣 𝑑𝑡 = 𝑑 2 𝑠 𝑑𝑡2) d) Use your result from part c to calculate the velocity at 𝑡 = 2𝑠 and 𝑡 = 6𝑠. e) Compare your results for part b and part d.

The position of a particle is given by the function s (t) = (2t - 3)e^2-t for t > 0. What is the average velocity of the particle from t = 1 to t = 3?

Leave your answer in terms of e, not a decimal.