Quiz 3 Calculus 1 Math 2210-12 James Wibowo¶

\[f(x)=11\textrm{ and }f(x+h)=11$$ #### $$f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}\Rightarrow f'(x)=\lim_{h\rightarrow0}\frac{(11)-(11)}{h}\Rightarrow f'(x)=\lim_{h\rightarrow0}\frac{0}{h}\Rightarrow \boxed{f'(x)=0}$$ ###### Q2 Using Difference Quotients find $f'(x)$: $$f(x)=\sqrt{x}$$ ###### A2 $$f(x)=x^{1/2}\textrm{ and }f(x+h)=(x+h)^{1/2}$$ $$f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}\Rightarrow f'(x)=\lim_{h\rightarrow0}\frac{\sqrt{x+h}-\sqrt{x}}{h}*\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}\Rightarrow f'(x)=\lim_{h\rightarrow0}\frac{x+h-x}{h(\sqrt{x+h}+\sqrt{x})}$$ $$f'(x)=\lim_{h\rightarrow0}\frac{h}{h(\sqrt{x+h}+\sqrt{x})}\Rightarrow f'(x)=\lim_{h\rightarrow0}\frac{1}{\sqrt{x+h}+\sqrt{x}}\Rightarrow f'(x)=\frac{1}{\sqrt{x}+\sqrt{x}}\Rightarrow \boxed{f'(x)=\frac{1}{2\sqrt{x}}}$$ $$(\sqrt{a}-\sqrt{b})(\sqrt{a}+\sqrt{b})=\frac{a-b}{\sqrt{a}+\sqrt{b}}$$ ###### Q3 Using Difference Quotients find $f'(x)$: $$f(x)=x^3+x$$ ###### A3 $$f(x)=x^3+x\textrm{ and }f(x+h)=(x+h)^3+x+h$$ $$f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}\Rightarrow f'(x)=\lim_{h\rightarrow0}\frac{(x+h)^3+x+h-x^3+x}{h}\Rightarrow f'(x)=\lim_{h\rightarrow0}\frac{(x+h)^3+2x+h-x^3}{h}$$ $$f'(x)=\lim_{h\rightarrow0}\frac{(x+h)(x+h)(x+h)+2x+h-x^3}{h}\Rightarrow f'(x)=\lim_{h\rightarrow0}\frac{(x^2+h^2+2xh)(x+h)+2x+h-x^3}{h}$$ $$f'(x)=\lim_{h\rightarrow0}\frac{2x^2h+2h^2x+h^2x+h^3+2x+h-x^3}{h}\Rightarrow f'(x)=\lim_{h\rightarrow0}\frac{h(2x^2+2hx+hx+h^2)+2x-x^3}{h}$$ $$\lim_{h\rightarrow0}(2x^2+2hx+hx+h^2)+2x-x^3\Rightarrow f'(x)=(2x^2+2(0)x+(0)x+(0)^2)+2x-x^3\Rightarrow \boxed{f'(x)=3x^2+1}\]