Question

If a tangent to the curve \[y=2+\sqrt{4x+1}\] has slope \[\dfrac{2}{5}\] at appoint, then the point is
A) \[(0,2)\]
B) \[\left( \dfrac{3}{4},4 \right)\]
C) \[(2,5)\]
D) \[(7,6)\]
E) \[(6,7)\]

Answer 1

Hint: In this particular problem, slope \[\dfrac{2}{5}\] is given that is\[\dfrac{dy}{dx}=\dfrac{2}{5}\] and tangent to the curve equation is given that is \[y=2+\sqrt{4x+1}\] and we have to find the value x and y to find the we have to differentiate with respect to x and by further simplification and substitution the equation of y we will get the points.

Complete step by step solution:
In this type of problem, the equation of y is given as \[y=2+\sqrt{4x+1}\]
So we have to differentiate with respect to x.
\[\dfrac{dy}{dx}=\dfrac{d\left( 2+\sqrt{4x+1} \right)}{dx}\]
By applying the basic derivative rule in this above equation and as we know that derivative of any constant is zero.
\[\dfrac{dy}{dx}=\dfrac{d\left( \sqrt{4x+1} \right)}{dx}--(2)\]
But by using the rule of derivative and also using the composite function we get:
\[\dfrac{d\left( \sqrt{4x+1} \right)}{dx}=\dfrac{1}{2\sqrt{4x+1}}\times \dfrac{d\left( 4x+1 \right)}{dx}\]
Further simplification we get:
\[\dfrac{d\left( \sqrt{4x+1} \right)}{dx}=\dfrac{4}{2\sqrt{4x+1}}\]
Further solving we get:
\[\dfrac{d\left( \sqrt{4x+1} \right)}{dx}=\dfrac{2}{\sqrt{4x+1}}\] Substitute this values on equation (2) we get:
\[\dfrac{dy}{dx}=\dfrac{2}{\sqrt{4x+1}}\]
As we know that, slope is \[\dfrac{2}{5}\] given that means \[\dfrac{dy}{dx}=\dfrac{2}{5}\]substitute in above equation we get:
\[\dfrac{2}{5}=\dfrac{2}{\sqrt{4x+1}}\]
After simplification we get:
\[\sqrt{4x+1}=5\]
By squaring on both sides we get:
\[4x+1=25\]
After simplification we get:
\[x=\dfrac{24}{4}\]
After solving we get:
\[x=6\]
Substitute the above values on equation (1) we get:
\[y=2+\sqrt{4(6)+1}\]
After simplifying further we get:
\[y=2+25\]
Further solving we get:
\[y=27\]
Therefore, the required point is \[(6,7)\]. So, the correct option is option (E).

Note:
In this type of problem, remember all the derivation rules which we used in this problem. Note one more thing is that in the question it has given the value of slope that means the slope of a line is a measure of its steepness. Mathematically, slope is calculated as "rise over run" (change in y divided by change in x). It can be represented as slope\[=\dfrac{dy}{dx}\].