Question

) Find the slope m of the tangent to the curve
y = 7/radical x at the point where x = a > 0.

Answer 1

y= \[7\div \sqrt{x}\]

Answer 2

\[\Large y = \frac{7}{\sqrt{x}}\]


\[\Large y^{\prime} = \frac{d}{dx}\left(\frac{7}{\sqrt{x}}\right)\]


\[\Large y^{\prime} = \frac{d}{dx}\left(7x^{-\frac{1}{2}}\right)\]


\[\Large y^{\prime} = 7\frac{d}{dx}\left(x^{-\frac{1}{2}}\right)\]


\[\Large y^{\prime} = 7(-\frac{1}{2})x^{-\frac{3}{2}}\]


\[\Large y^{\prime} = -\frac{7}{2x^{\frac{3}{2}}}\]


\[\Large y^{\prime} = -\frac{7}{2x\sqrt{x}}\]

Answer 3

That answer was not taken by the website?

Answer 4

that's just the derivative function

Answer 5

you use that to find the slope at the given point on the function

Answer 6

oh, it says at the point where x=a>0

Answer 7

in your case, x = a, so replace each x with a to get

\[\Large y^{\prime} = -\frac{7}{2x\sqrt{x}}\]


\[\Large y^{\prime} = -\frac{7}{2a\sqrt{a}}\]


So

\[\Large m = -\frac{7}{2a\sqrt{a}}\]

Answer 8

at x = a

Answer 9

ok, how do i find equations of the tangent lines at the points (1, 7) and (4, 7/2)?

Answer 10

two separate equations for each point..

Answer 11

same function?

Answer 12

yes

Answer 13

plug in a = 1 (for the first point) to get the slope


\[\Large m = -\frac{7}{2a\sqrt{a}}\]


\[\Large m = -\frac{7}{2(1)\sqrt{1}}\]


\[\Large m = -\frac{7}{2}\]


Now use the general line equation y = mx+b along with x = 1 and x = 7 to get


y = mx+b


7 = (-7/2)(1)+b


7 = -7/2 + b


7+7/2 = b


21/2 = b

So the equation of the tangent line at the point (1,7) is \[\Large y = -\frac{7}{2}x+\frac{21}{2}\]


So the same with the second point

Answer 14

okay, thank you!