Chapter 3.1, Problem 68E

To determine

To find:

The derivative of the curve and the value of the c

Answer to Problem 68E

The value of $c=6$

Explanation of Solution

Given:

The tangent to the curve

  $y=c\sqrt{x}$

Parallel to the line

  $y=\frac{3}{2}x+6$

Concept used:

The equation is in slope −intercept form, $y=mx+c$

An equation for the line through the point $\left(x_1,y_1\right)$ with slope m is

  $y-y_1=m\left(x-x_1\right)$

Calculation:

The function

  $y=c\sqrt{x}\mathrm{.......................}\left(1\right)$

The derivative of a function

  $\begin{array}{l}y=f\left(x\right)\\ y^{\prime }=f^{\prime }\left(x\right)=\frac{dy}{dx}=m\end{array}$

Differentiating the equation (1) with respect to $x$

  $\begin{array}{l}y=c\sqrt{x}\\ y^{\prime }=\frac{c}{2x^{\frac{1}{2}}}\mathrm{........................}\left(2\right)\end{array}$

The derivative is slope of the tangent line so in order to the slope of the tangent line

  $\begin{array}{l}y=\frac{3}{2}x+6\\ y^{\prime }=\frac{3}{2}\mathrm{.............................}\left(3\right)\end{array}$

The equation is in slope −intercept form

The slope of the line is $=\frac{3}{2}$

From equation (2) and equation (3)

  $\begin{array}{l}\frac{3}{2}=\frac{c}{2x^{\frac{1}{2}}}\\ c=3x^{\frac{1}{2}}\end{array}$

A system of three equations with the same number of unknowns

That system is

  $\left\{\begin{array}{l}c=3x^{\frac{1}{2}}\\ y=c\sqrt{x}\\ y=\frac{3}{2}x+6\end{array}\right\}$

Substitute these equations

  $\begin{array}{l}3x^{\frac{1}{2}}\sqrt{x}=\frac{3}{2}x+6\\ 3x=\frac{3}{2}x+6\\ \frac{3}{2}x=6\\ x=4\end{array}$

Therefore,

  $\begin{array}{l}c=3{\left(4\right)}^{\frac{1}{2}}\\ c=3\left(2\right)\\ c=6\end{array}$