[GRE 1268, #7]
The figure below shows the graph of the derivative \(f'\) of a function \(f\), where \(f\)
is continuous on the interval \([0, 4]\) and differentiable on the interval \((0, 4)\).
List the numbers \(f(0), f(2)\), and \(f(4)\) in increasing order.
\(\img{1268_7.png}{0em}{}{10em}\)
[GRE 1268, #9]
Let \(g\) be a continuous real-valued function defined on \(\mathbb R\) with the following properties.
\begin{eqnarray*}
g'(0)&=& 0\\
g''(-1)& \gt & 0\\
g''(x)& \lt & 0 \hspace{2ex}\mbox{ if } 0\lt x \lt 2.
\end{eqnarray*}
Which of the following could be part of the graph of \(g\) ?
\(\img{1268_9f1.png}{0em}{}{8em}\)
\(\img{1268_9f2.png}{0em}{}{8em}\)
\(\img{1268_9f3.png}{0em}{}{8em}\)
\(\img{1268_9f4.png}{0em}{}{8em}\)
\(\img{1268_9f5.png}{0em}{}{8em}\)
[GRE 1268, #13]
If \(f\) is a continuously differentiable real-valued function defined on the open interval
\((-1, 4)\) such that \(f(3) = 5\) and \(f'(x) \geq -1\) for all \(x\),
what is the greatest possible value of \(f(0)\)?
[GRE 1268, #14]
Suppose \(g\) is a continuous real-valued function such that \(3x^5 + 96 =\int_c^x g(t)dt\) for each
\(x\) where \(c\) is a constant. What is the value of \(c\)?
[GRE 1268, #20]
Let \(g\) be the function defined by \(g(x)=e^{2x+1}\) for all real \(x\). Find
\(\displaystyle \lim_{x\rightarrow 0}\frac{g(g(x)) -g(e)}{x}\)
[GRE 1268, #21]
What is the value of
\(\displaystyle \int_{-\pi/4}^{\pi/4} \left(\cos t + \sqrt{1+t^2}\sin^3 t \cos^3 t\right) dt\)?
[GRE 1268, #26]
\begin{eqnarray*}
3x\equiv 5 \pmod{11}\\
2y\equiv 7 \pmod{11}
\end{eqnarray*}
If \(x\) and \(y\) are integers that satisfy the congruences above, then what is \(x+y \pmod{11}\)?
[GRE 1268, #27]
Find \(\left(1+i\right)^{10}\).
[GRE 1268, #28]
Let \(f\) be a one-to-one (injective), positive-valued function defined on \(\mathbb R\).
Assume that \(f\) is differentiable at \(x = 1\) and that in the \(xy\)-plane the line
\(y - 4 = 3(x - 1)\) is tangent to the graph of \(f\) at \(x = 1\). Let \(g\) be the
function defined by \(g(x)=\sqrt{x}\) for \(x \geq 0\). Find the following:
\(f'(1)\)
\(\left(f^{-1}\right)' (4)\)
\((fg)'(1)\)
\( (g\circ f)'(1)\)
\((g\circ f)(1)\)
[GRE 1268, #30]
For what positive value of \(c\) does the equation \(\log x = cx^4\) have exactly one real solution for \(x\) ?
[GRE 1268, #55]
If \(a\) and \(b\) are positive numbers, what is the value of
\(\displaystyle \int_0^\infty \frac{e^{ax}-e^{bx}}{(1+e^{ax})(1+e^{bx})}dx\)?
[GRE 8767, #24]
Let \(f\) and \(g\) be functions defined on the positive integers and related in the following way:
\[f(n)=\begin{cases} 1 & \mbox{if } n=1\\
2f(n-1) & \mbox{if } n\neq 1
\end{cases}\]
and
\[g(n)=\begin{cases}3g(n+1) & \mbox{if } n\neq 3 \\
f(n) & \mbox{if } n=3
\end{cases}.\]
Find \(g(1)\).
[GRE 9768, #4]
If \(b \gt 0\) and \(\displaystyle \int_0^b xdx = \int_0^bx^2dx\), find the area of the
shaded region in the figure.
\(\img{9768_4.png}{0em}{}{12em}\)
[GRE 8767, #21]
For all \(x \gt 0\), if \(f(\log x)=\sqrt{x}\), find \(f(x)\).
[GRE 8767, #15]
If \(f\) is a linear transformation from the plane to the real numbers and if \(f(1,1)=1\) and
\(f(-1,0)=2\), find \(f(3,5)\).
[GRE 9367, #6]
For what value of \(b\) is the value of \(\int _b^{b+1}(x^2+x)dx\) a minimum?
[GRE 9367, #8]
Suppose that the function \(f\) is defined by the formula \(f(x)=\sqrt{\tan^2 x -1}\), and its domain is contained in
\([0,2\pi]\). Find the domain of \(f\).
[GRE 9367, #15]
Let \(f(x)=\int_1^x\frac{1}{1+t^2}dt\) for all real \(x\). Find the equation of the tangent line to the graph of \(f(x)\)
at the point \((2,f(2))\). Write your answer in the form \(ax+by=c\), where \(a, b,c\) are real numbers, and do not use
decimals or fractions.
[GRE 9367, #23]
Let \(f\) be a real-valued function continuous on the closed interval \([0,1]\) and differentiable on the open interval
\((0,1)\) with \(f(0)=1\) and \(f(1)=0\). For each of the following statements, decide if the statement is true or false.
If it is true, explain the reasoning for your decision. If it is false, show by example why it is false.
There exists \(x\in (0,1)\) such that \(f(x)=x\).
There exists \(x\in (0,1)\) such that \(f'(x)=-1\)
\(f(x)\gt 0 \) for all \(x\in [0,1)\)
[GRE 9768, #2]
If \(F(1)=2\) and \(\displaystyle F(n)=F(n-1)+\frac{1}{2}\) for all integers \(n\gt 1\), find \(F(101)\).
[GRE 9768, #7]
Draw the graph of \(\displaystyle \left\{ (\sin t,\cos t): -\frac{\pi}{2}\leq t \leq 0 \right\}\) in the \(xy\)-plane.
[GRE 9367, #18]
If \(\displaystyle f(x)= \sum_{n=0}^\infty (-1)^n x^{2n}\) for all \(x\in (0,1)\), find \(f'(x)\).
[GRE 9367, #26]
Let \({\mathbf i}=(1,0,0)\), \({\mathbf j}=(0,1,0)\), and \({\mathbf k}=(0,0,1)\). The vectors \({\mathbf v_1}\) and
\({\mathbf v_2}\) are orthogonal if \({\mathbf v_1}={\mathbf i}+ {\mathbf j} - {\mathbf k}\) and \({\mathbf v_2}=\)
(A) \({\mathbf i}+ {\mathbf j} - {\mathbf k}\)
(B) \({\mathbf i}- {\mathbf j} + {\mathbf k}\)
(C) \({\mathbf i}+ {\mathbf k}\)
(D) \( {\mathbf j} - {\mathbf k}\)
(E) \({\mathbf i}+ {\mathbf j}\)
[GRE 9367, #41]
If \( \displaystyle A= \begin{pmatrix} 1 & 2 \\ 0 & -1\end{pmatrix}\), find the set of all vectors \(\mathbf v\) such that
\(A{\mathbf v} = {\mathbf v}\).