Recent questions in Calculus and Analysis

Logarithmic Functions
Answered

Bierlehre59
2022-08-14

i.e. suppose f is a function and n is a real number, is $\mathrm{log}(f(x{)}^{n})=n\xb7\mathrm{log}(f(x))$?

Vectors
Answered

schnelltcr
2022-08-14

Trigonometry
Answered

podmitijuy0
2022-08-14

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

Implicit Differentiation
Answered

joyoshibb
2022-08-14

My workings:

$(1)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{d}{dx}[{x}^{2}]\phantom{\rule{thinmathspace}{0ex}}+\frac{d}{dx}[xy]\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\frac{d}{dx}[{y}^{3}]=\frac{d}{dx}[0]$

$(2)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x+y+\frac{d{y}^{3}}{dy}\frac{dy}{dx}=0$

$(3)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x+y+3{y}^{2}\phantom{\rule{thinmathspace}{0ex}}(\frac{dy}{dx})=0$

$(4)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{dy}{dx}=\overline{){\displaystyle -\frac{2x+y}{3{y}^{2}}}}$

But the answer says it should be:

$(3)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x+y+\frac{dxy}{dy}\frac{dy}{dx}+3{y}^{2}\phantom{\rule{thinmathspace}{0ex}}(\frac{dy}{dx})=0$

$(4)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x+y+\frac{dy}{dx}(x+3{y}^{2})=0$

$(5)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{dy}{dx}=-\frac{2x\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}y}{x\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}3{y}^{2}}$

Why?

joyoshibb
2022-08-13

if $f:\mathbb{R}\to \mathbb{R}$ is continuous and $\overline{f(\mathbb{N})}=\mathbb{R}$, then $f$ is onto (surjective)

by the intermediate value theorem?

garkochenvz
2022-08-13

$f(x)={x}^{3}{\mathrm{tan}}^{-1}(2x);\phantom{\rule{1em}{0ex}}|x|<\frac{1}{2}$

Cheyanne Jefferson
2022-08-13

Do you think it's possible?

Vectors
Answered

Mehlqv
2022-08-13

Evaluate the integral

${\iint}_{S}D\text{}dS,$

where S is the surface of a sphere with radius r=5 centered at the origin.

Vectors
Answered

Nica2t
2022-08-13

Polynomials
Answered

Jazmin Clark
2022-08-13

Show that $\frac{1}{2\pi i}\underset{|z|=R}{\int}\phantom{\rule{negativethinmathspace}{0ex}}{z}^{n-1}|f(z){|}^{2}dz={A}_{0}\overline{{A}_{n}}{R}^{2n}$\

Vectors
Answered

lollaupligey9
2022-08-13

(A′ being the transposed matrix)

Trigonometry
Answered

polynnxu
2022-08-13

Implicit Differentiation
Answered

Meossi91
2022-08-13

(i) Find the first partial derivatives ${G}_{x}$ and ${G}_{y}$.

(ii) Using (i) above, find $\frac{dy}{dx}$.

(iii) If $G(x,y)=0$, confirm your answer in part (ii) above, finding $\frac{dy}{dx}$ using implicit differentiation.

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