Answer to Question #93640 in Calculus for Michael Michael

Question

1. Evaluate D(sin(x²+3x-1))
2. Given that f(x)=3x-1
g(x)=cosx
h(x)=e^2x+1
Find (fogoh)x and (hogof)x
3. Evaluate lim[⅔x] x->6-, [•] in the greatest integer formula.
4. Find lim (x(x+1)(2+x))/x+x³
5. Find the different coefficient of (i) xlog5^x (ii) x³log8^x

Answer 1

1.

$D(\sin(x^2+3x-1))=(x^2+3x-1)'\cos(x^2+3x-1)dx=(2x+3)\cos(x^2+3x-1)dx$

2.

$f\circ g\circ h(x)=f(g(h(x)))=f(\cos(e^{2x}+1))=3\cos(e^{2x}+1)-1$$h\circ g\circ f(x)=h(g(f(x)))=h(\cos(3x-1))=e^{2\cos(3x-1)}+1$

3.

$\lim\limits_{x\to6-}[\frac{2x}{3}]=3$ - For all 3<x<6 $\left[\frac{2x}{3}\right]=3$ . Here [x] is the greatest integer function.

4.

$\lim\frac{x(x+1)(x+2)}{x+x^3}$ no information on x here. If $x\to\infty$ then $\lim\limits_{x\to \infty}\frac{x(x+1)(x+2)}{x+x^3}=\lim\limits_{x\to \infty}\frac{x^3+3x^2+2x}{x+x^3}=1$

5.

$\frac{d}{dx}x\log5^x=\log5^x+x\frac{5^x\log5}{5^x}=\log5^x+x\log5$

$\frac{d}{dx}x^3\log8^x=3x^2\log8^x+x^3\frac{8^x\log8}{8^x}=3x^2\log8^x+x^3\log8$