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A curve $y = m{e^{mx}}$ where m>0 intersects y-axis at a point P.
What is the slope of the curve at the point of intersection P?
A) $m$
B) \[{m^2}\]
C) $2m$
D) \[2{m^2}\]

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Last updated date: 13th Jul 2024
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Answer
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Hint: The slope of a curve at a point is equal to the slope of the straight line that is tangent to the curve at that point. Or we can say that slope is the ratio of vertical change to horizontal change.
As we know that
$ m = \dfrac{{\Delta y}}{{\Delta x}} = \dfrac{{vertical\,change}}{{Horizontal\,change}} = \dfrac{{rise}}{{run}}$.

Complete step by step solution:
Given,
$ \Rightarrow y = m{e^{mx}}$ ….(1)
To find the slope of curve
Differentiate the equation 1 w.r.t. ‘x’
$ \Rightarrow \dfrac{{dy}}{{dx}} = {m^2}{e^{mx}}$
Hence slope of line is
$ \Rightarrow \dfrac{{dy}}{{dx}} = {m^2}{e^{mx}}$ ….(2)
Line intersect at y axis,
$x = 0$
Put the value of $x = 0$in equation 2.
$ \Rightarrow \dfrac{{dy}}{{dx}} = {m^2}{e^{m0}}$
Simplify the equation
$\therefore {a^0} = 1$
\[ \Rightarrow {e^0} = 1\]
$ \Rightarrow \dfrac{{dy}}{{dx}} = {m^2}$
Therefore, the correct option is (B) \[{m^2}\].

Note:
Slope refers to the surface not level and allows gravity to move an object in the direction of gravity. In modern life slope application’s is used in the design of railways tracks, bridges, highways to construct the road. The slope is used in economics to show the growth. If the slope is zero is equivalent to level.