Calculus Answers

What is the coefficient of (x+1)^3 in the cubic Taylor polynomial about -1 for the function f(x)=e^x. The coefficient of (x+1)^3 is?

The Taylor series about 0 for the function f(x)=sin(x) is x^2-(x^3)/6+(x^5)/120-(x^7)/5040+(x^9)/362880.... and the Taylor series about 0 for the function g(x)=e^x is 1+x+(x^2)/2+(x^3)/6+(x^4)/24+(x^5)/120+....What is the coefficient of x^3 in the series for e^x sin(x)?

The movement of the crest of a wave is modelled with the equation h(t)=0.3cos3t+0.4sin3t. Find the maximum height of the wave and the time at which it occurs.

I found the max which is 0.5 but I do not know how to find the time?

Please help solve explaining step by step

44. Capital value: find the capital value of an asset that generates $7200 yearly income if the interest rate is as follows.

a.) 5% compounded continuously
b.) 10% compounded continuously

Solve explaining step by step:

Find the cost function for each marginal cost function.

46. c'(x)=0.2x^2+5x; Fixed cost is $10

solve dy/dx=y3+3x2y/x3+3xy2

I need help solving step by step the following question below.

Velocity: The position of a particle at time t is given by s, Find the velocity ds/dt.

2s^2+√st -4=3t

Evaluate ∫_0^2 ∫_0^(π/2) sin⁡2θ/√(cos^2⁡θ⁡+⁡16 ) dθdr

Evaluate ∫_(1/2)^1 ∫_0^2x cos⁡(πx^2 ) dy dx

A ball is launched into the air from below a cliff, such that after t seconds its height above the cliff top is h metres, and is given by the equation

h =  − 4.9t^2 + 19.6t − 9.6.

Calculate, to the nearest metre, the maximum height the ball achieves above the cliff top.