Question

In Problems 15and 16interpret each statement as a differential equation.

On the graph of$\mathit{y}\mathbf{=}\mathit{\varphi }\mathbf{\left(}\mathit{x}\mathbf{\right)}$the rate at which the slope changes with respect to x at a pointrole="math" localid="1663825517880" $\mathit{P}\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{)}$is the negative of the slope of the tangent line at$\mathit{P}\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{)}$.

Short answer

The differential equation is $y\text{'}\text{'}+y\text{'}=0$.

Step-by-step solution

Define a derivative of the function

The sensitivity of the function is derived by the derivative of a function of a real variable for modifying the argument.

Calculus uses derivatives as a fundamental tool. When a derivative of a single-variable function exists at a given input value, it is the slope of the tangent line to the function's graph at that point.

Determine the differential equation for the height

Let the slope of the tangent line of the graph of $y=\varphi \left(x\right)$at the point $P(x,y)$be $y\text{'}$.

And let the rate at which the slope changes be $y\text{'}\text{'}$.

Hence, the statement describes the differential equation,

$\begin{array}{rcl}y\text{'}\text{'}& =& -y\text{'}\\ y\text{'}\text{'}+y\text{'}& =& 0\end{array}$