Exponential Questions (1 Viewer)

boredofstudiesuser1 · Mar 1, 2017

1. The line y=mx is a tangent to the curve y=e^3x. Find m.

2. Show that y=e^x-1 and y=e^-x intersect when e^2x-e^x-1=0

InteGrand · Mar 1, 2017

boredofstudiesuser1 said:
1. The line y=mx is a tangent to the curve y=e^3x. Find m.

2. Show that y=e^x-1 and y=e^-x intersect when e^2x-e^x-1=0

$\noindent 1) Assume the point of tangency is at $\left(t, e^{3t}\right)$. On the line, the $y$-value is $mt$, so we must have $mt = e^{3t}$. The slope here on the curve $y = e^{3x}$ is $3e^{3t}$ and the line's slope is $m$, so we must have $m = 3e^{3t}$. We have established two equations in two unknowns ($m$ and $t$), which we can solve to get the answer.$

$\noindent 2) The two curves intersect when $e^{x} -1 = e^{-x} \iff e^{x} -1 -e^{-x} = 0$. Multiplying through by $e^{x}$, this is equivalent to $e^{2x} -e^{x} -1 = 0$.$

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Exponential Questions (1 Viewer)

boredofstudiesuser1

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InteGrand

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