GRE - Variant Problems

[GRE 1268, #1a] Find \(\displaystyle \lim_{x\rightarrow 0}\frac{e^{2x}-1-2x}{x^2}\).

[GRE 1268, #3a] Find \(\displaystyle \int_{e^{-2}}^{e^{-1}}\frac{\sqrt{|\log x|}}{x}dx\).

[GRE 1268, #7a] The graph of the derivative \(f'(x)\) of a function \(f(x)\) is shown in the picture. The area labeled \(A\) is 17, the area labeled \(B\) is 67, and the area labeled \(C\) is 31. \(\img{1268_7a.png}{0em}{}{10em}\) Suppose \(f(-2)=7\). Find \(f(-4), f(1)\) and \(f(3)\).

[GRE 1268, #9a] The graph of a function \(f\) is shown in the picture. \(\img{Q9a.png}{0em}{}{18em}\) Fill the table below with \(0\), \(+\) or \(-\):

\(f'(-2)\)	\(\hspace{3ex}\)	\(f''(0)\)	\(\hspace{3ex}\)	\(f'(1)\)	\(\hspace{3ex}\)	\(f''(2)\)	\(\hspace{3ex}\)
\(f'(-1)\)	\(\hspace{3ex}\)	\(f''(-1)\)	\(\hspace{3ex}\)	\(f(3)\)	\(\hspace{3ex}\)	\(f'(3)\)	\(\hspace{3ex}\)

[GRE 1268, #13a] Suppose \(f\) is a differentiable function such that \(f(1)=3\) and \(x\leq f'(x)\leq x^2\) for all \(x\) in the interval \(0\leq x \leq 3\). Find the smallest closed interval that must contain \(f(2)\).

[GRE 1268, #14a] Suppose that \(f\) is a continuous function, and \(a,b\) are real numbers such that \[\int_{a}^x f(t)dt = \int_a ^b f(t)dt+24+3x^3 \hspace{2ex} \mbox{ for all } x.\] Find \(b\).

[GRE 1268, #20a] Let \(f(x)=\ln (1+e^x)\). Find \(\displaystyle \lim_{x\rightarrow 0} \frac{f(f(x))-f(\ln 2)}{x} \).

[GRE 1268, #21a] Find the value of \(\displaystyle \int_{-1}^1 |x| \left(x^2+\sin\left(x^3\right)\right) dx\).

[GRE 1268, #26a] Suppose that \[ \begin{aligned} 5x &\equiv 4 \pmod{7} \\ 6y &\equiv 5 \pmod{7} \end{aligned} \] Find an integer \(a\) such that \(x+2y+a\) is divisible by 7.

[GRE 1268, #27a] Find \(\left(i\sqrt{3}-1\right)^{13}\).

[GRE 1268, #28a] Suppose that \(f\) is an invertible function, and the line \(y=3x+1\) is tangent to the graph of \(f(x)\) at \(x=0\). Let \(g(x)=\ln (1+x)\). Find the following:

1. \(f'(0)\)
2. \(\left(f^{-1}\right)' (1)\)
3. \((fg)'(0)\)
4. \( (f\circ g)'(0)\)
5. \((g\circ f)'(0)\)

[GRE 1268, #30a] Find all values of \(c\) for which the equation \(x^5+c=x\) has exactly two solutions.

[GRE 1268, #30b] Let \(k\) be a non-negative integer, and define the set \(S_k\) by \[S_k=\{c\in {\mathbb R}: \mbox{ the equation } 3x^5+45x=20x^3+c \mbox{ has exactly $k$ solutions} \}.\] Find the set \(S_k\) for all \(k\geq 0\).

[GRE 1268, #30c] Let \(k\) be a non-negative integer, and define the set \(S_k\) by \[S_k=\{c\in {\mathbb R}: \mbox{ the equation } 3x^4+c=4x^3+36x^2 \mbox{ has exactly $k$ solutions} \}.\] Find the set \(S_k\) for all \(k\geq 0\).

[GRE 1268, #32a] Let \(f(x)= \displaystyle \int_{2x}^{x^2}t^tdt\). Find \(f'(2)\).

[GRE 1268, #33a] Let \(f(x) = x\sin x\). Find the \(77\)-th derivative of \(f(x)\).

[GRE 1268, #45a] How many solutions does the equation \(\ln x = \sin(89x)\) have?

[GRE 1268, #55a] Find the value of \(\displaystyle \int_1^e \frac{1-\log x}{x(2+3\log x)(3+2\log x)}dx\).

[GRE 8767, #24a] The functions \(f\) and \(g\) are defined as follows: \[f(x)=\begin{cases} 2 & \mbox{if } x=3\\ 3f(x+1) & \mbox{if } x\neq 3\end{cases},\] \[g(x)=\begin{cases} 2g(x-1) & \mbox{if } x\neq 0 \\ f(x) & \mbox{if } x=0\end{cases}.\] Find \(g(3)\).

[GRE 9768, #4a] Suppose that \(\displaystyle \int_0^b \frac{3x}{1+x^2}dx = \int_0^bxdx\). Find the area of the shaded region in the figure. \(\img{9768_4a.png}{0em}{}{12em}\)

[GRE 8767, #21a] Suppose \(\displaystyle{f\left(\frac{e^x}{1-e^x}\right)=e^{2x}}\). Find \(f(x)\).

[GRE 8767, #15a] Suppose \(\mathbf{v}\) is a vector in \({\mathbb R}^2\) and \(\mathbf{u_1}=(1,-1)\), \(\mathbf{u_2}=(2,1)\), \(\mathbf{u_3}=(8,5)\). If \(\mathbf{v}\cdot \mathbf{u_1}=-1\) and \(\mathbf{v}\cdot \mathbf{u_2}=11\), find \(\mathbf{v}\cdot \mathbf{u_3}\).

[GRE 9367, #6a] For what positive value of \(x\) does the integral \(\int_x^{2x}t^3 e^{-t}dt\) have a maximum?

[GRE 9367, #8a] Let \(f(x)=\tan\left(\frac{1}{x}\right)\).

1. Find the largest interval in \([0,1]\) on which \(f(x)\) is continuous.
2. Repeat part a. with \([0,1]\) replaced by \([0,\pi/2]\).

[GRE 9367, #15a] Let \(f(x) = \int_{1/2}^x \frac{1}{\sqrt{1-t^2}}dt \). Find the equation of the tangent line to the graph of \(f(x)\) at the point \((1/\sqrt{2},f(1/\sqrt{2}))\). Write the answer in the form \(ax+by=c\), where \(a,b,c\) are real numbers, without decimals or fractions.

[GRE 9367, #23a] Suppose \(f(x)\) and \(g(x)\) are continuous and differentiable functions on \([0,1]\), and \(f(0)=g(1)=1\), \(g(0)=f(1)=0\). For each of the following statement, 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.

1. There is some \(x\in (0,1)\) such that \(f(x)=g(x)\).
2. There is some \(x\in (0,1)\) such that \(f'(x)=g'(x)\)
3. \(f(x)\leq 1\) and \(g(x)\geq 0\) for all \(x\in (0,1)\)
4. There is some \(x\in (0,1)\) such that \(g'(x)=f'(x)+2\)

If \(n\) is a positive integer, denote by \(\overline{n}\) the positive integer obtained by reading the digits of \(n\) (in base 10) backwards. So for example if \(n=349802\), then \(\overline{n}=208943\).

1. Explain why \(n-\overline{n}\) is always divisible by 9.
2. Explain why \(n-\overline{n}\) is always divisible by 99 when the number of digits of \(n\) is odd.

[GRE 9768, #7a] Draw the curve defined by the paramteric equations \[ \begin{cases} x=1+2\sin(t)\\ y=3+2\cos(t) \end{cases}, \hspace{2ex} \frac{\pi}{4}\leq t \leq \frac{5\pi}{4} \]

[GRE 9768, #2a] Let \(a_n\) be defined by \[a_n=\frac{1}{2}a_{n-1} +1, \hspace{2ex} a_1=3.\] Find \(a_n\) as a function of \(n\).

[GRE 9768, #2b] Let \(r\), \(b\) and \(c\) be real numbers and suppose \(a_n\) is defined by \[a_1=c, \hspace{2ex} a_n=ra_{n-1} +b \hspace{2ex} \mbox{ for all } n\geq 2.\] Find \(a_n\) as a function of \(n,r,b\) and \(c\).

[GRE 9367, #18a] Let \(f(x)\) be the function defined for \(|x|\lt 1\) by the formula \[f(x)=\sum_{n=1}^\infty (-1)^n\frac{x^{2n}}{n}.\] Find a formula for \(f'(x)\) that does not involve any sum.

[GRE 9367, #41a] Suppose \(A=\begin{pmatrix} 1 & 2 \\ x & 3 \end{pmatrix}\) and \(\mathbf v\) is non-zero vector such that \(A {\mathbf v} =5{\mathbf v}\). Find \(x\).