Q. 80 - Calculus

Question

$$\begin{gathered} \mathrm{Prove} \mathrm{that} \mathrm{if} f \mathrm{is} \mathrm{continuous} \mathrm{on} [a, b] \mathrm{and} C \mathrm{is} \mathrm{any} \mathrm{real} \mathrm{number}, \mathrm{then} f\left(x\right) \\ \mathrm{and} f\left(x\right)+C \mathrm{have} \mathrm{the} \mathrm{same} \mathrm{arc} \mathrm{length} \mathrm{on} [a, b]. \mathrm{Then} \mathrm{explain} \mathrm{why} \mathrm{this} \\ \mathrm{makes} \mathrm{sense} \mathrm{graphically}. \end{gathered}$$

Step-by-Step Solution

Answer

$$\begin{gathered} \mathrm{The} \mathrm{arc} \mathrm{length} \mathrm{of} f\left(x\right) \mathrm{from} x=a \mathrm{to} x=b \mathrm{can} \mathrm{be} \mathrm{represented} \mathrm{by} \mathrm{the} \\ \mathrm{definite} \mathrm{integral}:{\int }_{\mathrm{a}}^{\mathrm{b}}\sqrt{1 + ( \mathrm{f}\text{'}{\left(\mathrm{x}\right))}^2}d\mathrm{x} \\ \mathrm{Here}, \mathrm{when} f\left(x\right) \mathrm{and} f\left(x\right)+C \mathrm{are} \mathrm{differentiated}, \mathrm{we} \mathrm{get} \mathrm{the} \mathrm{same} f\text{'}\left(x\right) \mathrm{as} \\ \mathrm{differentiation} \mathrm{of} \mathrm{a} \mathrm{constant} \mathrm{is} \mathrm{zero}. \mathrm{when} \mathrm{this} \mathrm{is} \mathrm{substituted} \mathrm{in} \mathrm{the} \mathrm{definite} \\ \mathrm{integral} \mathrm{we} \mathrm{get} \mathrm{the} \mathrm{same} \mathrm{arc} \mathrm{length} \mathrm{in} \mathrm{the} \mathrm{same} \mathrm{interval}. \\ \mathrm{Graphically} \mathrm{this} \mathrm{makes} \mathrm{sense} \mathrm{as} \mathrm{the} f\left(x\right)+C \mathrm{is} \mathrm{vertically} \mathrm{shifted} \mathrm{version} \\ \mathrm{of} f\left(x\right) \mathrm{and} \mathrm{arc} \mathrm{lengths} \mathrm{of} \mathrm{both} \mathrm{the} \mathrm{curves} \mathrm{are} \mathrm{same} \mathrm{in} \mathrm{the} \mathrm{same} \mathrm{interval}. \end{gathered}$$

Step 1. Given information

$f\left(x\right) \mathrm{is} \mathrm{continuous} \mathrm{on} [a, b] \mathrm{and} C \mathrm{is} \mathrm{any} \mathrm{real} \mathrm{number}$.

Step 2. Finding arc lengths of f ( x )   and   f ( x ) + C and proving why both are same.

$$\begin{gathered} \mathrm{The} \mathrm{arc} \mathrm{length} \mathrm{of} f\left(x\right) \mathrm{from} x=a \mathrm{to} x=b \mathrm{can} \mathrm{be} \mathrm{represented} \mathrm{by} \mathrm{the} \\ \mathrm{definite} \mathrm{integral}:{\int }_{\mathrm{a}}^{\mathrm{b}}\sqrt{1 + ( \mathrm{f}\text{'}{\left(\mathrm{x}\right))}^2}d\mathrm{x} \\ \mathrm{Here}, \mathrm{when} f\left(x\right) \mathrm{and} f\left(x\right)+C \mathrm{are} \mathrm{differentiated}, \mathrm{we} \mathrm{get} \mathrm{the} \mathrm{same} f\text{'}\left(x\right) \mathrm{as} \\ \mathrm{differentiation} \mathrm{of} \mathrm{a} \mathrm{constant} \mathrm{is} \mathrm{zero}. \mathrm{when} \mathrm{this} \mathrm{is} \mathrm{substituted} \mathrm{in} \mathrm{the} \mathrm{definite} \\ \mathrm{integral} \mathrm{we} \mathrm{get} \mathrm{the} \mathrm{same} \mathrm{arc} \mathrm{length} \mathrm{in} \mathrm{the} \mathrm{same} \mathrm{interval}. \\ \mathrm{Graphically} \mathrm{this} \mathrm{makes} \mathrm{sense} \mathrm{as} \mathrm{the} f\left(x\right)+C \mathrm{is} \mathrm{vertically} \mathrm{shifted} \mathrm{version} \\ \mathrm{of} f\left(x\right) \mathrm{and} \mathrm{arc} \mathrm{lengths} \mathrm{of} \mathrm{both} \mathrm{the} \mathrm{curves} \mathrm{are} \mathrm{same} \mathrm{in} \mathrm{the} \mathrm{same} \mathrm{interval}. \end{gathered}$$