Recent questions tagged first-order

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Determine the domain of each of the following.\(f(x)=2 x^3+4 x^2+6\)
172
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Compute the definite integral of the following function over the interval \( [0, \pi ] \)\[f(x)=\sin ^{2}(x)+\cos ^{2}(x)\]
194
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a) Which function, \(f\) or \(g\), has the equation \(y=a^x\) ?b) What is the range of \(g(x)\) ?c) Give the equation of the asymptote of \(g(x)\).d) Give the equation of...
807
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What is the difference between a first-order optimization algorithm and a second-order optimization algorithm?
489
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The domain of the function \(f(x)=\sin ^{-1}\left(\frac{|x|+5}{x^2+1}\right)\) is \((-\infty,-a] \cup[a, \infty)\), Then a is equal to:(1) \(\frac{1+\sqrt{17}}{2}\)(2) \(...
257
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If \(|x|<1,|y|<1\) and \(x \neq y\), then the sum to infinity of the following series \((x+y)+\left(x^2+x y+y^2\right)+\left(x^3+x^2 y+\right.\) \(\left.x y^2+y^3\right)+...
317
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If the tangent to the curve \(y=x+\) siny at a point \((a, b)\) is parallel to the line joining \(\left(0, \frac{3}{2}\right)\) and \(\left(\frac{1}{2}, 2\right)\), then(...
242
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A line parallel to the straight line \(2 x-y=0\) is tangent to the hyperbola \(\frac{x^2}{4}-\frac{y^2}{2}=1\) at the point \(\left(x_1, y_1\right)\). Then \(x_1^2+5 y_1^...
674
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Give an example of a function where \(f^{\prime}(x) \neq 0\) and \(f^{\prime \prime}(x)=\) \(0 .\)
418
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Give an example of a function \(f(x)\) where \(f^{\prime}(x)=0\).
407
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Give an example of a function \(f(x)\) where \(f^{\prime}(x)=f(x)\)
203
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Let \(f(x)=\sin x+2 x+1\). Approximate \(f(3)\) using an appropriate tangent line.
200
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Experimental Measurements determine that a function \(f(x)\) satisfies \(f(0)=0, f^{\prime}(0)=1\) and \(f(1)=2\).(a) Estimate \(f(1 / 3)\) using tangent line (linear) ap...
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If \(h(x)=f^{2}(x)-g^{2}(x), f^{\prime}(x)=-g(x)\) and \(g^{\prime}(x)=f(x)\), then \(h^{\prime}(x)\) is:A) 0B 1C) \(-4 f(x) g(x)\)D) \((-g(x))^{2}-(f(x))^{2}\)
295
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How many square units are in the region satisfying the inequalities \(y \geq|x|\) and \(y \leq-|x|+3\) ? Express your answer as a decimal.
720
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What is the domain of the function\[f(x)=\frac{(2 x-3)(2 x+5)}{(3 x-9)(3 x+6)} ?\]Express your answer as an interval or as a union of intervals.
168
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For every \(a, b, b \neq a\) prove that\[\frac{a^{2}+b^{2}}{2}>\left(\frac{a+b}{2}\right)^{2} .\]
233
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Put \(f(0,0)=0\), and\[f(x, y)=\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}\]if \((x, y) \neq(0,0)\). Prove that(a) \(f, D_{1} f\), and \(D_{2} f\) are continuous in \...
481
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If \(f(0,0)=0\) and\[f(x, y)=\frac{x y}{x^{2}+y^{2}} \quad \text { if }(x, y) \neq(0,0)\]prove that \(\left(D_{1} f\right)(x, y)\) and \(\left(D_{2} f\right)(x, y)\) exis...
326
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Calculate\[\exp \left(t x^{3} \frac{d}{d x}\right) x\]All terms must be summed up. Describe the connection with the solution of the initial value problem of the different...
250
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Show that the homogeneous equation\[\frac{d u}{d x}+f\left(\frac{u}{x}\right)=0\]is transformed by\[y=x, \quad v(y(x))=\frac{u(x)}{x}\]into the separable equation\[v+y \f...
260
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Solve for \(x\) in \(\dfrac{x+1}{2}+4=7\)
210
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Solve for \(x\) in \(\frac{1}{x-4}+\frac{2}{x^{2}-16}=\frac{3}{x+4}\)
939
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Four technicians regularly make repairs when breakdowns occur on an automated production line. Janet, who services \(20 \%\) of the breakdowns, makes an incomplete repair...
367
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1 answers
A function \(f(x, y)\) is continuous at a point \((a, b)\) if
471
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With \(a<b\), the total variation of \(f(x)\) on a finite or infinite interval \((a, b)\) is
250
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Give an example of a quadratic function of the form \(f(x)=x^{2}+b x+c\) whose tangent line is \(y=3 x+1\) at the point \((0,1)\).
323
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Give an example of a function that satisfies \(f(-1)=0, f(10=0\), and \(f^{\prime}(x)>0\) for all \(x\) in the domain of \(f^{\prime}\).
430
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An amount of \(A_{0}\) CAD is invested against yearly interest of \(p \%\). Give the expression for \(A(t)\), the value of the investment in CAD after \(t\) years if the ...
696
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Interval of definition. By looking at an initial value problem $d y / d x=f(x, y)$ with $y\left(x_{0}\right)=y_{0}$, it is not always possible to determine the domain of ...
413
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If $f(x)=x+\sqrt{x^{2}+1}+\frac{1}{x-\sqrt{x^{2}+1}}$, what is the value of $f\left(2016^{2017}\right) ?$
386
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Solve \$ 2,500(1+0.06 t)+\$1,000(1+0.04 t)= \$3,553.62
388
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Solve for $x$ in $\dfrac{5}{x}+\dfrac{1}{x}=\dfrac{2}{x}+2$
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If $h(x)=f^{2}(x)-g^{2}(x), f^{\prime}(x)=-g(x)$ and $g^{\prime}(x)=f(x)$, then $h^{\prime}(x)$ is:
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Which of the following answers choices is a solution to the differential equation $y^{\prime \prime \prime}-6 y^{\prime \prime}+11 y^{\prime}-6 y=0 ?$
2.8k
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If $h(x)=2 x \cdot \cos x$, find $h^{\prime \prime}(\pi)$
298
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Solve for $x$ : $\quad 2^{x}=\dfrac{1}{16}$
433
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1 answers
Let $f(x)=-2 x^{2}+1$ and $g(x)=4 x-3$. Find $(f \circ g)(x)$.
331
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Let $f(x)=3 x^{2}-1$ and $g(x)=-2 x+7$. Find $(f+g)(-3)$.
594
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Simplify:$$\frac{2}{x^{2}-x}+\frac{x^{2}+x+1}{x^{3}-1}-\frac{x}{x^{2}-1}, \quad(x \neq 0 ; x \neq \pm 1)$$
380
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Simplify:$$\frac{x-2}{x^{2}-4}+\frac{x^{2}}{x-2}-\frac{x^{3}+x-4}{x^{2}-4}, \quad(x \neq \pm 2)$$
338
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Simplify:$$\frac{x^{2}-x-2}{x^{2}-4} \div \frac{x^{2}+x}{x^{2}+2 x}, \quad(x \neq 0 ; x \neq \pm 2)$$
367
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Solve for $r$ in the equation: $V=\pi r^{2} h(r>0)$
323
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Solve for $x$ in $\frac{3 x-2}{2}=x+1$
746
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Let $f(x)=2 x+\cos x$. Say why $f(x)$ is an increasing function for all $x$. Let $g(x)=f^{-1}(x)$, and calculate $g^{\prime}(0)$.
535
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Let $f(x)=x^{2} \sin \left(\frac{1}{x}\right)$ if $x \neq 0$, and $f(0)=0 .$ Find $f^{\prime}(0)$ (or say why it doesn't exist.)
611
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Convert $\frac{d^{3} x}{d t^{3}}+x=0$ to a first-order differential equation. Solve this equation over the interval $[0,1]$ for the initial conditions $x^{\prime \prime}(...
561
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1 answers
Convert $\frac{d^{2} x}{d t^{2}}+x=0$ to a first-order differential equation. Solve over the interval $[0, \pi]$ with $h=\frac{\pi}{10}$ assuming the initial conditions $...
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