+89 1. (a) Sketch the graph of the function f(t) = 45(1 + e −.01 t ) for t ≥ 0 indicating clearly the values of f(t) at t = 0, 1, 3, 5, 10. What is the limiting value of f(t) as t becomes infinitely large?
(b) Find the annual percentage rate (APR) for the following interest rates:
(i) 3% compounded monthly,
(ii) 2.9% compounded continuously.
Which of the above rates of interest gives the better choice for a savings account?
(c) Find the total value of a savings account after six years, where e 400 is paid in at the start of each month for six years into an account paying 6% compounded monthly.
(d) Differentiate the functions y = ln(x 4 ) + e x and z = e 2x 2 + 6.
Hence find the derivative of the function p = ln(x 4 ) + e x e 2x2 + 6 , when x = 1.
(e) Let f(x) = 1 3 x 3 + x − 2x 2 . Find and classify all critical points of f(x).
+9675 I don't know all those but I do know how to do (d).
dydx=ddx(ln(x4)+ex)=ddx(4lnx)+ddx(ex)
=4ddx(lnx)+ddxex
=4x+ex
z0=euu=2x2dz0dx=dz0du×dudx← Chain rule=eu×4x=4xe2x2
You can see that I am differentiating z0 =e^2x^2 using chain rule.
dzdx=ddx(z0)+ddx(6)=4xe2x2
∴dpdx=dydx+dzdx=4x+ex+4xe2x2
