+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).
+9664 I don't know all those but I do know how to do (d).
\(\dfrac{dy}{dx}\\=\dfrac{d}{dx} \left(\ln \left(x^4\right)+e^x\right)\\=\dfrac{d}{dx}(4\ln x)+\dfrac{d}{dx}(e^x)\)
\(=4\dfrac{d}{dx}(\ln x)+\dfrac{d}{dx} e^x\)
\(=\dfrac{4}x+e^x\)
\(\begin{array}{rl}z_0=e^u&u=2x^2\\\end{array}\\ \dfrac{dz_0}{dx}=\dfrac{dz_0}{du}\times \dfrac{du}{dx}\leftarrow\text{ Chain rule}\\ \;\;\;\;\;\;\!=e^u\times 4x\\ \;\;\;\;\;\;\!=4xe^{2x^2}\)
You can see that I am differentiating z0 =e^2x^2 using chain rule.
\(\dfrac{dz}{dx}\\ =\dfrac{d}{dx}(z_0)+\dfrac{d}{dx}(6)\\ =4xe^{2x^2}\)
\(\therefore\dfrac{dp}{dx}=\dfrac{dy}{dx}+\dfrac{dz}{dx}=\dfrac{4}{x}+e^x+4xe^{2x^2}\)
