Question

What is the slope of y = log(x) when x=20 ?

The formula for the slope is _________ for h close to 0 (but not equal )

The best estimate for the slope is____

Answer 1

`{ \qquad\qquad\huge\underline{{\sf Answer}}}`

The formula for slope according to first principle is :

`\qquad \sf  \dashrightarrow \: \[ \displaystyle\lim_(x\to0) \: \: \sf (f(x + h) - f(x))/(h) \]`

`\qquad \sf  \dashrightarrow \: \[ \displaystyle\lim_(x\to0) \: \: \sf ( log(x + h) - log(x))/(h) \]`

`\qquad \sf  \dashrightarrow \: \[ \displaystyle\lim_(x\to0) \: \: \sf ( log( (x + h)/(x) ) )/(h) \]`

`\qquad \sf  \dashrightarrow \: \[ \displaystyle\lim_(x\to0) \: \: \sf ( log( ( h)/(x) + 1 ) )/(h) \]`

`\qquad \sf  \dashrightarrow \: \[ \displaystyle\lim_(x\to0) \: \: \sf ( log( ( h)/(x) + 1 ) )/( (h)/(x) * x) \]`

`\qquad \sf  \dashrightarrow \: \[ \displaystyle\lim_(x\to0) \: \: \sf (1)/(x) \bigg(( log_(e) ( ( h)/(x) + 1 ) )/( (h)/(x) * log_(e)(10) ) \] \bigg)`

`\qquad \sf  \dashrightarrow \: \[ (1)/(x \: log_(e)(10) ) \displaystyle\lim_(x\to0) \: \: \sf \bigg(( log_(e) ( ( h)/(x) + 1 ) )/( (h)/(x) ) \] \bigg)`

`\qquad \sf  \dashrightarrow \: \[ (1)/(x \: log_(e)(10) )`

Now, the best estimate of slope is at x = 20 :

`\qquad \sf  \dashrightarrow \: \[ (1)/(20* 2.303)`

`\qquad \sf  \dashrightarrow \: \[ (1)/(46.06)`

`\qquad \sf  \dashrightarrow \: \approx0.0217`