# Recent questions tagged first-order

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Determine the domain of each of the following.$$f(x)=2 x^3+4 x^2+6$$
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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)$
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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...
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What is the difference between a first-order optimization algorithm and a second-order optimization algorithm?
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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 views 0 answers 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 views 0 answers 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)$$, then(...
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A line parallel to the straight line $$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 views 0 answers Give an example of a function where \(f^{\prime}(x) \neq 0$$ and $$f^{\prime \prime}(x)=$$ $$0 .$$
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Give an example of a function $$f(x)$$ where $$f^{\prime}(x)=0$$.
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Give an example of a function $$f(x)$$ where $$f^{\prime}(x)=f(x)$$
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Let $$f(x)=\sin x+2 x+1$$. Approximate $$f(3)$$ using an appropriate tangent line.
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Experimental Measurements determine that a function $$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}$$
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How many square units are in the region satisfying the inequalities $$y \geq|x|$$ and $$y \leq-|x|+3$$ ? Express your answer as a decimal.
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What is the domain of the function$f(x)=\frac{(2 x-3)(2 x+5)}{(3 x-9)(3 x+6)} ?$Express your answer as an interval or as a union of intervals.
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For every $$a, b, b \neq a$$ prove that$\frac{a^{2}+b^{2}}{2}>\left(\frac{a+b}{2}\right)^{2} .$
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Put $$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$$ are continuous in \...
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If $$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...
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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...
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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...
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Solve for $$x$$ in $$\dfrac{x+1}{2}+4=7$$
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Solve for $$x$$ in $$\frac{1}{x-4}+\frac{2}{x^{2}-16}=\frac{3}{x+4}$$
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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...
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A function $$f(x, y)$$ is continuous at a point $$(a, b)$$ if
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With $$a<b$$, the total variation of $$f(x)$$ on a finite or infinite interval $$(a, b)$$ is
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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 $$(0,1)$$.
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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 $$f^{\prime}$$.
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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 ...
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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 ...
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If $f(x)=x+\sqrt{x^{2}+1}+\frac{1}{x-\sqrt{x^{2}+1}}$, what is the value of $f\left(2016^{2017}\right) ?$
Solve \$2,500(1+0.06 t)+\$1,000(1+0.04 t)= \$3,553.62 388 views 1 answers Solve for$x$in$\dfrac{5}{x}+\dfrac{1}{x}=\dfrac{2}{x}+2$1.8k views 0 answers If$h(x)=f^{2}(x)-g^{2}(x), f^{\prime}(x)=-g(x)$and$g^{\prime}(x)=f(x)$, then$h^{\prime}(x)$is: 1.1k views 0 answers Which of the following answers choices is a solution to the differential equation$y^{\prime \prime \prime}-6 y^{\prime \prime}+11 y^{\prime}-6 y=0 ?$2.8k views 1 answers If$h(x)=2 x \cdot \cos x$, find$h^{\prime \prime}(\pi)$298 views 1 answers Solve for$x$:$\quad 2^{x}=\dfrac{1}{16}$433 views 1 answers Let$f(x)=-2 x^{2}+1$and$g(x)=4 x-3$. Find$(f \circ g)(x)$. 331 views 1 answers Let$f(x)=3 x^{2}-1$and$g(x)=-2 x+7$. Find$(f+g)(-3)$. 594 views 1 answers 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 views 1 answers 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 views 1 answers 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 views 1 answers Solve for$r$in the equation:$V=\pi r^{2} h(r>0)$323 views 1 answers Solve for$x$in$\frac{3 x-2}{2}=x+1$746 views 0 answers Let$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)$. 535 views 0 answers Let$f(x)=x^{2} \sin \left(\frac{1}{x}\right)$if$x \neq 0$, and$f(0)=0 .$Find$f^{\prime}(0)$(or say why it doesn't exist.) 611 views 0 answers Convert$\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}(...
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 \$...