Question 1

Let $y=e^x sinx$.
Find an expression for $\frac{dy}{dx}$.

Accepted Answer

$e^x sinx+e^x cosx$

Answer

$xe^x sinx+e^x cosx$

Answer

$e^x cosx$

Answer

$xe^x cosx$

Question 2

Let $y=e^x sinx$.
Find $\frac{d^2y}{dx^2}$.

Accepted Answer

$2e^x cosx$

Answer

$2e^x sinx+2e^x cosx$

Answer

$2e^x sinx$

Answer

$0$

Question 3

Consider the function $f$  defined by $f(x)=e^x sinx$, $0≤x≤π$.
Find the $x$-coordinate of the point of inflexion of the graph of $f$.

Accepted Answer

$\frac{π}{2}$

Answer

$\frac{3π}{2}$

Answer

$0$

Answer

$\frac{3π}{4}$

Question 4

Let $l$ be the tangent to the curve $y=xe^{2x}$ at the point $(1, e^2)$.
Find the coordinates of the point where $l$ meets the $x$-axis.

Accepted Answer

$(\frac{2}{3},  0)$

Answer

$(-\frac{5}{3},  0)$

Answer

$(0,  0)$

Answer

$(\frac{1}{2},  0)$

Question 5

The function $f$ is defined as $f(x)=e^{(3x+1)}$, $x∈R$.  Find $f^{-1} (x)$

Accepted Answer

$f^{-1} (x)=\frac{1}{3}(lnx-1)$

Answer

$f^{-1} (x)=3e^ 
{(3x+1)}$

Answer

$f^{-1} (x)=\frac{1}{3}(lnx+1)$

Answer

$f^{-1} (x)=3(lnx-1)$

Question 6

The function $f$ is defined as $f(x)=e^{(3x+1)}$, $x∈R$. The function $g$ is defined as $g(x)=lnx, x∈R^+$. The graph of $y=g(x)$ and the graph of $y=f^{-1} (x)$ intersect at the point $P$. Find the coordinates of $P$.

Accepted Answer

$(e^{-\frac{1}{2}}, -\frac{1}{2})$

Answer

$(e, 1)$

Answer

$(e^{\frac{1}{2}}, \frac{1}{2})$

Answer

$(e^{-\frac{1}{3}}, -\frac{1}{3})$

Question 7

Let $f(x)=\frac{(e^{2x}+1)}{(e^x-2)}$.   Find $\frac{dy}{dx}$.

Accepted Answer

$\frac{dy}{dx}=\frac{(2(e^x-2) e^{2x}-(e^{2x}+1) e^x)}{(e^x-2)^2}$

Answer

$\frac{dy}{dx}=\frac{(2(e^x-2) e^{x}-(e^{2x}+1) e^x)}{(e^x-2)^2}$

Answer

$\frac{dy}{dx}=\frac{((e^{2x}+1) e^x-2(e^x-2) e^{2x})}{(e^x-2)^2}$

Answer

$\frac{dy}{dx}=\frac{((e^x-2) e^{2x}-(e^{2x}+1) e^x)}{(e^x-2)^2}$

Question 8

Let $f(x)=\frac{(e^{2x}+1)}{(e^x-2)}$.   The curve has exactly one point where its tangent is horizontal. Find the coordinates of this point.

Accepted Answer

$(1.44, 8.47)$

Answer

$(1.34, 8.57)$

Answer

$(1.54, 8.54)$

Answer

$(1.48, 8.48)$

Question 9

Let $f(x)=\sqrt{\frac{x}{1-x}}$, $0<x<1$.  Find$f^{'} (x)$

Accepted Answer

$f^{'} (x)=\frac{1}{2} x^{-\frac{1}{2}} (1-x)^{-\frac{3}{2}}$

Answer

$f^{'} (x)=\frac{1}{2} x^{-\frac{1}{2}} (1-x)^2$

Answer

$f^{'} (x)=x^{-\frac{1}{2}} (1-x)^{-\frac{3}{2}}$

Answer

$f^{'} (x)=\frac{1}{2} x^{-\frac{1}{2}} (1-x)^{-\frac{1}{2}}$

Question 10

Let $f(x)=\sqrt{\frac{x}{1-x}}$, $0<x<1$.  The curve $y=f(x)$ has one point of inflexion,  find its coordinates.

Accepted Answer

$(\frac{1}{4},\frac{1}{\sqrt{3}})$

Answer

$(\frac{1}{2},1)$

Answer

$(\frac{1}{5},\frac{1}{2})$

Answer

$(\frac{1}{3},\frac{1}{\sqrt{2}})$

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