# High school calculus questions and answers

Recent questions in Calculus and Analysis
Bierlehre59 2022-08-14

### Do the logarithmic rules work when taking logs of functions as opposed to numbers?i.e. suppose f is a function and n is a real number, is $\mathrm{log}\left(f\left(x{\right)}^{n}\right)=n·\mathrm{log}\left(f\left(x\right)\right)$?

schnelltcr 2022-08-14

### I'm wondering that why people doesn't care about vector's location. but, when we add the vectors, we move two or one vector to unite the vector's start point. and this mean location is important. but, when we learn very first of vector, we learn the vector doesn't matter where it is. in this point, I'm so confused because I can't find the reason that why vector's location doesn't matter. so i want to ask you - why the vector's location is not important?

podmitijuy0 2022-08-14

### Solve this:$1\right)\mathrm{sin}\left(\frac{7\pi }{6}\right)=?\phantom{\rule{0ex}{0ex}}2\right)\mathrm{cos}\left(-\frac{4\pi }{3}\right)=?\phantom{\rule{0ex}{0ex}}3\right)\mathrm{tan}\left(\frac{3\pi }{2}\right)=?\phantom{\rule{0ex}{0ex}}4\right)\mathrm{cot}\left(\frac{11\pi }{4}\right)=?$

joyoshibb 2022-08-14

### I am trying to solve the problem: ${x}^{2}+xy+{y}^{3}=0$ using implicit differentiation.My workings:$\left(1\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{d}{dx}\left[{x}^{2}\right]\phantom{\rule{thinmathspace}{0ex}}+\frac{d}{dx}\left[xy\right]\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\frac{d}{dx}\left[{y}^{3}\right]=\frac{d}{dx}\left[0\right]$$\left(2\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x+y+\frac{d{y}^{3}}{dy}\frac{dy}{dx}=0$$\left(3\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x+y+3{y}^{2}\phantom{\rule{thinmathspace}{0ex}}\left(\frac{dy}{dx}\right)=0$$\left(4\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{dy}{dx}=\overline{)-\frac{2x+y}{3{y}^{2}}}$But the answer says it should be:$\left(3\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x+y+\frac{dxy}{dy}\frac{dy}{dx}+3{y}^{2}\phantom{\rule{thinmathspace}{0ex}}\left(\frac{dy}{dx}\right)=0$$\left(4\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x+y+\frac{dy}{dx}\left(x+3{y}^{2}\right)=0$$\left(5\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{dy}{dx}=-\frac{2x\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}y}{x\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}3{y}^{2}}$Why?

rivasguss9 2022-08-13

### Use the Intermediate Value Theorem to prove $f:\left[0,1\right]\to \left[0,1\right]$ continuous and $C\in \left[0,1\right]$, there is some $c\in \left[0,1\right]$ such that $f\left(c\right)=C$.Using a similar technique to the proof of the intermediate value theorem, I can easily prove that there is an $f\left(x\right)=C$, but I am having trouble proving that a $f\left(c\right)=C$.This is what I have:Since $f$ is continuous on [0,1] there exists $f\left(a\right)=0$ and $f\left(b\right)=1$.Let $g\left(x\right)=F\left(x\right)-C$. We assume that $f\left(a\right) $\to$ $0when $x=a$, $g\left(a\right)$ is negative when $x=b$, $g\left(b\right)$ is positiveTherefore, $g\left(a\right)<0, and since $f$ is continuous on [0,1], so is $g$.Therefore there exists a $g\left(x\right)=0$, and $f\left(x\right)=C$.How can I prove there is a $f\left(c\right)=C$ ? Is this a rule for IVT?

Gauge Roach 2022-08-13

### Given $\stackrel{\to }{u}=\left(\genfrac{}{}{0}{}{{u}_{1}}{{u}_{2}}\right)$, if it is rotated sixty degrees anti-clockwise prove that the resulting vector will be $\stackrel{\to }{v}=\frac{1}{2}\left(\genfrac{}{}{0}{}{{u}_{1}-\sqrt{3}{u}_{2}}{\sqrt{3}{u}_{1}+{u}_{2}}\right)$I think there are better methods out there but one idea I had was to use$\mathrm{cos}60=\frac{1}{2}=\frac{\stackrel{\to }{u}\cdot \stackrel{\to }{v}}{|\stackrel{\to }{u}||\stackrel{\to }{v}|}\phantom{\rule{mediummathspace}{0ex}}$and solve for $\stackrel{\to }{v}$ but obviously this seems impossible to me with the $|\stackrel{\to }{v}|$ and I see no where to force a $\sqrt{3}$ to appear.Is this true? Or if this method works please share. I have other ideas which probably work but this one also came to mind but with no fruitition

Matonya 2022-08-13

### Consider the space${Ł}^{p}\left(\mathrm{\Omega }\right):=\left({L}^{p}\left(\mathrm{\Omega }\right){\right)}^{N}={L}^{p}\left(\mathrm{\Omega }\right)×{L}^{p}\left(\mathrm{\Omega }\right)×...×{L}^{p}\left(\mathrm{\Omega }\right),\phantom{\rule{thinmathspace}{0ex}}N\ge 1,\phantom{\rule{thinmathspace}{0ex}}\mathrm{\Omega }\subset {\mathbb{R}}^{\mathbb{n}},$as the vectorial ${L}^{p}$ space associated to the scalar one.My questions are:What is the norm $||\cdot |{|}_{{Ł}^{p}}$ and can we relate it to $||\cdot |{|}_{{L}^{p}}$?How to apply the Hölder Inequality on${\int }_{\mathrm{\Omega }}u\left(x\right)\cdot v\left(x\right)dx,$where$u\in {Ł}^{q}$ , $v\in {Ł}^{p}$ , $\frac{1}{p}+\frac{1}{q}=1$, $u=\left({u}_{1},...,{u}_{N}{\right)}^{t}$, $v=\left({v}_{1},....,{v}_{N}{\right)}^{t}$ , ${u}_{i}\in {L}^{q}\left(\mathrm{\Omega }\right)$, ${v}_{i}\in {L}^{p}\left(\mathrm{\Omega }\right)$, and $u\left(x\right)\cdot v\left(x\right)$ is the scalar product between?

Dorsheele0p 2022-08-13

### Find the gradient of the curve $x\mathrm{ln}y-\frac{x}{y}=2$ at (−1,1)I've done something wrong as I got the gradient to be 0 when the answer in the back says −0.5Can someone help me with this question?edit:I got: $1+\frac{1}{y}\frac{dy}{dx}-\frac{y-x\frac{dy}{dx}}{{y}^{2}}$${y}^{2}+y\frac{dy}{dx}-y-x\frac{dy}{dx}=0$$\frac{dy}{dx}=\frac{y-{y}^{2}}{y-x}$Substituting the values in did not get me the answer

joyoshibb 2022-08-13

### What do these terms mean? $v\left(a\right)+R\left(h\right)$In regards to the differentiability of a function at a.I know that is it differentiable if the limit of $\frac{f\left(a+h\right)-f\left(a\right)}{h}$ exists.Then I am given the following equation, supposing f is differentiable:$f\left(a+h\right)-f\left(a\right)=h\cdot v\left(a\right)+R\left(h\right)$ with $li{m}_{h\to 0}\frac{R\left(h\right)}{||h||}$What is v(a) and R(h). Where did they come from?

garkochenvz 2022-08-13

### Find Maclaurin series for this$f\left(x\right)={x}^{3}{\mathrm{tan}}^{-1}\left(2x\right);\phantom{\rule{1em}{0ex}}|x|<\frac{1}{2}$

Cheyanne Jefferson 2022-08-13

### I want to give an informal proof of the intermediate value theorem to my calculus students. They have just learned about continuity and they don't know yet about derivatives among more other advanced topics.Do you think it's possible?

Mehlqv 2022-08-13

### Consider the vector $D=3\mathrm{sin}\left(\theta \right)$ ar (unit vector).Evaluate the integralwhere S is the surface of a sphere with radius r=5 centered at the origin.

Nica2t 2022-08-13

### Determine $\beta$ and $\alpha$ by using vectors such that A, B and C lie in the same plane, given that vector $\stackrel{\to }{AB}=-4\stackrel{\to }{ı}-\stackrel{\to }{ȷ}-2\stackrel{\to }{k}$ and vector $\stackrel{\to }{BC}=4\stackrel{\to }{ı}+\left(\beta +3\right)\stackrel{\to }{ȷ}+\left(\alpha -6\right)\stackrel{\to }{k}$

yongenelowk 2022-08-13

### How do you factor $\left(10x+24{\right)}^{2}-{x}^{4}$

Jazmin Clark 2022-08-13

### Let $f\left(z\right)={A}_{0}+{A}_{1}z+{A}_{2}{z}^{2}+\dots +{A}_{n}{z}^{n}$ be a complex polynomial of degree $n>0$Show that $\frac{1}{2\pi i}\underset{|z|=R}{\int }\phantom{\rule{negativethinmathspace}{0ex}}{z}^{n-1}|f\left(z\right){|}^{2}dz={A}_{0}\overline{{A}_{n}}{R}^{2n}$\

lollaupligey9 2022-08-13

### If A is a $3×3$-matrix, while x and y are a vector point with x,y,z. Why is the dot product of $⟨{A}^{\ast }x,y⟩$ the same as the dot product of $⟨{A}^{\prime \ast }y,x⟩$(A′ being the transposed matrix)

Lacey Rojas 2022-08-13