Calculus Answers

For the infinite geometric sequence (x to base n) whose first four terms are 1.5, 3.6, 8.64 & 20.736

1) find the values of the first term a and the common ratio r and write down a recurrence system for this sequence.
2) Write a closed form for this sequence
3) Calculate the 7th term of sequence
4) how many terms of sequence are less than 30000
Thank you for your help

the position of an object at time t is given by s(t)= -9-3t. Find the instantaneous velocity a t=8 by finding the deriative

(a) Let H(t) be the total weight of all the lobsters, in kg, after t months.
Find a formula for H(t), expressing it in simplest form, Since the number of lobsters in the tank in months is expresses as N(t)= 6000/(t^2+25) and the mean weight of lobsters in months is expressed as W(t)=1.5ln⁡(t/4+1)

(b) draw this function and find its first derivative, H'(t),


(c) Consider the graph of H'(t). By choosing a suitable option, determine the rate at which the total weight of all the lobsters is changing after 2 months and after 10 months. Explain the meaning of your answers with reference to their magnitude and sign.

The Taylor series 0 for the function f(x)=1/(1-x) is 1+x+x^2+x^3+x^4+x^5+.......Use this to find the Taylor series for 1(1-x^3),giving fhe first 3 non-zero terms.

sketch the graph of y=sinxcosx for -2pi<x<2pi

use the formula cosQ=1/2(e^iQ+e^-iQ) to obtain the identity cos^5 Q=1/16(cos5Q+5cos3Q+10cosQ).

Find the Derivative of y=x^2e^(-3x)

a)use the binomial series to find the Taylor series about 0 for the function f(x)=(1+x)^-1/5, giving all terms up to the one in x^4, and calculating each coefficient as an integer or a fraction.
b)state an interval of validity for this series.
c)use this series and the Taylor series for cosx to find the cubic Taylor polynomial about 0 for the function f(x)= cosx/(1+x)^1/5.