Section 2.1¶
Problem 2¶
\(\(\lim_{h\rightarrow0}\frac{(10+h)^3-1000}{h}\)\)¶
Problem 7¶
\(\(\lim_{h\rightarrow0}\frac{(3+h)^2-9}{h}\)\)¶
\(\(m=\frac{\textrm{rise}}{\textrm{run}}=\frac{y_2-y_1}{x_2-x_1}=\frac{\Delta x}{\Delta y}\)\)¶
\(6\%\textrm{ grade}=100' \textrm{ run},6'\textrm{ rise}\)
Whiteboard Question¶
\(\(m=\frac{f(x+h)-f(x)}{x+h-x}=\frac{f(x+h)-f(x)}{h}\)\)¶
\(\(f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}\)\)¶
Example¶
\(\(f(x)=x^2\)\)¶
\(\(f(x+h)=(x+h)^2\)\)¶
\(\(f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}\)\)¶
\[f'(x)=\lim_{h\rightarrow0}\frac{(x+h)^2-x^2}{h}=\lim_{h\rightarrow0}\frac{x^2+2xh+h^2-x^2}{h}=\lim_{h\rightarrow0}\frac{2xh+h^2}{h}=\lim_{h\rightarrow0}\frac{h(2x+h)}{h}=\lim_{h\rightarrow0}2x+h$$
### $$\lim_{h\rightarrow0}2x+h=2x+0=2x$$
### $$f'(x)=aNx^{N-1}$$
###### Example
### $$f(x)=x^3$$
### $$f(x+H)=(x+h)^3$$
### $$f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}$$
$$f'(x)=\lim_{h\rightarrow0}\frac{(x+h)^3-x^3}{h}=\lim_{h\rightarrow0}\frac{(2xh+h^2)(x+h)-x^3}{h}=\lim_{h\rightarrow0}\frac{3x^2h+3h^2x+h^3}{h}=\lim_{h\rightarrow0}\frac{3x^2h+3h^2x+h^3}{h}$$
### $$\lim_{h\rightarrow0}\frac{h(3x^2+3hx+h^2)}{h}=\lim_{h\rightarrow0}3x^2+3hx+h^2=3x^2\]