Calculus

Problem 30201

Soit $f(x)=\sqrt{|x^{2}-6x+5|}$ , étudiez sa continuité, dérivabilité, variations, asymptotes et symétrie sur $[1;5]$ .

See Solution

Problem 30202

Differentiate the function: $f(x) = \sqrt{x-2} (4x^2 + 1) + 2x^3$ .

See Solution

Problem 30203

Find the limit as $x$ approaches infinity for the expression $\frac{1+8 x}{3-2 x}$ .

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Problem 30204

Find the limits for $f(x)=\frac{x^{2}+1}{(2 x+3)(x-5)}$ as $x$ approaches 5 from left, right, and directly. Enter 'P', 'N', or 'D'.

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Problem 30205

Find $\frac{dy}{dx}$ for the function $y = x^{3}(x^{2} + 5)^{4}$ .

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Problem 30206

Evaluate these limits: (a) Find $\lim _{x \rightarrow \infty} \frac{1+8 x}{3-2 x}$ . (b) Find $\lim _{x \rightarrow-\infty} \frac{1+8 x}{3-2 x}$ .

See Solution

Problem 30207

Evaluate the integral: $\int \frac{3-x}{\sqrt{9-x^{2}}}$ .

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Problem 30208

Find the limit: $\lim _{x \rightarrow-\infty} \frac{\sqrt{7+4 x^{2}}}{3+8 x}$ .

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Problem 30209

Find the limit: $\lim _{x \rightarrow \infty} \frac{(11-x)(10+7 x)}{(3-4 x)(4+9 x)}$

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Problem 30210

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{7+4 x^{2}}}{3+8 x}$ .

See Solution

Problem 30211

Find the limit: $\lim _{x \rightarrow \infty} \frac{6 x^{3}-8 x^{2}-3 x}{8-11 x-3 x^{3}}$ .

See Solution

Problem 30212

Find the limit: $\lim _{x \rightarrow-\infty} \frac{(11-x)(10+7 x)}{(3-4 x)(4+9 x)}$ .

See Solution

Problem 30213

Find the limit: $\lim _{x \rightarrow-\infty} \frac{1+8 x}{3-2 x}$ .

See Solution

Problem 30214

Find the limit: $\lim _{x \rightarrow \infty} \frac{2 x+2}{10 x^{2}-2 x+8}$ .

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Problem 30215

Identify intervals where the function is increasing or decreasing, and points of relative maxima and minima.

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Problem 30216

Find the limit: $\lim _{x \rightarrow-\infty} \frac{6 x^{3}-8 x^{2}-3 x}{8-11 x-3 x^{3}}$ .

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Problem 30217

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(\sqrt{9 x^{2}+x}-3 x\right)$ .

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Problem 30218

Differentiate $y = \frac{7x}{2} \left( \frac{1}{x^2 + 1} \right)$ using the product rule and explain the steps.

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Problem 30219

Find the limits: (a) $\lim _{x \rightarrow \infty}(-26 x^{2}+28 x^{3})$ and (b) $\lim _{x \rightarrow -\infty}(-26 x^{2}+28 x^{3})$ .

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Problem 30220

Find the limit: $\lim _{x \rightarrow+\infty}\left(\sqrt{x^{2}+4}-\sqrt{x^{2}-6}\right)$ .

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Problem 30221

Sketch the graph of $y=x^{3}-6x^{2}-15x+10$ . Identify local max/min, end behavior, and intervals for $y$ , $y'$ , and $y''$ .

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Problem 30222

Calculate the limit: $\lim _{x \rightarrow \infty} \frac{10-\sqrt{x}}{10+\sqrt{x}}$ .

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Problem 30223

Find the derivative $f^{\prime}(-3)$ for the function $f(x)=x^{3}+4$ using the definition of a derivative.

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Problem 30224

E. coli bacteria grow at 1.9% per minute. If the current population is 172 million, what will it be in 7.2 minutes? a) 197.2 million b) 220.2 million c) 10 million d) 101.2 million

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Problem 30225

Find the limits: (a) $\lim _{x \rightarrow \infty} x^{2}(9 x)(-8 x)$ and (b) $\lim _{x \rightarrow -\infty} x^{2}(9 x)(-8 x)$ .

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Problem 30226

1. Find $\lim _{x \rightarrow 5} \frac{2}{(x-5)^{6}}$ .
2. Find $\lim _{x \rightarrow 3^{-}} \frac{2}{x-3}$ .
3. Find $\lim _{x \rightarrow 0} \frac{1}{x^{2}(x+7)}$ .
4. Find $\lim _{x \rightarrow-7^{-}} \frac{1}{x^{2}(x+7)}$ .

See Solution

Problem 30227

Find the average rate of change of $f(x)=x^{3}-5x^{2}$ from $x=5$ to $x=10$ .

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Problem 30228

Evaluate the line integral of $\mathbf{F}(x, y)=(x^{2}+y^{2}) \mathbf{i}+2xy \mathbf{j}$ along $\mathbf{r}(t)=2t \mathbf{i}+t^{3} \mathbf{j}$ for $0 \leq t \leq 1$ .

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Problem 30229

Find the integral of $e^{\sqrt[5]{x}}$ with respect to $x$ : $\int e^{\sqrt[5]{x}} d x$ .

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Problem 30230

Find the limits as $x$ approaches $a$ : (a) $\lim [f(x)+2 g(x)]$ (b) $\lim [h(x)-3 g(x)+1]$ (c) $\lim [f(x) g(x)]$ (d) $\lim [g(x)]^{2}$ (e) $\lim \sqrt[3]{6+f(x)}$ (f) $\lim \frac{2}{g(x)}$

See Solution

Problem 30231

Find $\mathrm{F}^{\prime}(2)$ if $F(x)=5 f(x)+2 g(x)$ , given $f(2)=-3$ , $f^{\prime}(2)=4$ , $g(2)=1$ , $g^{\prime}(2)=-5$ .

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Problem 30232

A particle moves along the $x$ -axis with position $x(t)=t^{3}-15 t^{2}+72 t-9$ .
a. Find when it's farthest right for $3 \leq t \leq 9$ .
b. Determine its maximum speed on the same interval.

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Problem 30233

Calculate $\lim _{x \rightarrow-2^{+}}(x+3) \frac{|x+2|}{x+2}$ and $\lim _{x \rightarrow-2^{-}}(x+3) \frac{|x+2|}{x+2}$ .

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Problem 30234

Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ for $\mathrm{y}=\frac{2 \mathrm{u}}{1-4 \mathrm{u}}$ with $\mathrm{u}=(5 \mathrm{x}^{2}+1)^{4}$ .

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Problem 30235

Mr. Sullivan's apple juice function $A(t)$ is given. Estimate the rate of change at $t=10$ days and check if $A'(t)=\frac{2}{3}$ for $0 \leq t \leq 12$ .

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Problem 30236

Determine if the functions $x^{2}$ and $e^{-x^{2}}$ are acceptable in quantum mechanics. Plot and normalize if acceptable.

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Problem 30237

Find $F^{\prime}(2)$ for $F(x)=f(x)-3g(x)$ given $\mathrm{f}(2)=-3, \mathrm{f}^{\prime}(2)=4, \mathrm{g}(2)=1, \mathrm{g}^{\prime}(2)=-5$ .

See Solution

Problem 30238

Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ for $y=(u+3)^{2}$ , where $u=\sqrt{v-3}$ and $v=x^{2}$ .

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Problem 30239

Find the derivative of the function $f(x) = 2x^3 \ln(x)$ .

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Problem 30240

Find the derivative of the function $f(x) = \log_{2}|\csc(x)|$ .

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Problem 30241

Find $f(2)$ given that $f^{\prime}(x)=\sin \left(\frac{\pi e^{x}}{2}\right)$ and $f(0)=1$ .

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Problem 30242

Find the value of $a$ in the equation $\frac{d(\ln (1+y))}{d x}=\frac{a}{1+y} \frac{d y}{d x}$ .

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Problem 30243

Find $a$ and $b$ in the equation: $\frac{d(4(x^{2}+y^{2}))}{dx} = ax + by \frac{dy}{dx}$ .

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Problem 30244

Find $\frac{d x}{d t}$ when $y=2 \sin (x)$ and $\frac{d y}{d t}=1$ at $x=\frac{\pi}{3}$ .

See Solution

Problem 30245

Find $\frac{d y}{d t}$ when $y=12 \cos (x)$ and $\frac{d x}{d t}=1$ at $x=\frac{\pi}{6}$ .

See Solution

Problem 30246

Find the intervals where the function $f(x)=-x^{4}-12 x^{3}-36 x^{2}$ is increasing.

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Problem 30247

Find the intervals where the function $f(x)=-x^{4}+12 x^{3}-36 x^{2}$ is increasing.

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Problem 30248

Find the intervals where the function $f(x)=x^{4}-4 x^{3}$ is decreasing.

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Problem 30249

If $\frac{d A}{d t}=3 \frac{d C}{d t}$ , find $r$ given $\frac{d A}{d t}=2 \pi r \frac{d r}{d t}$ and $\frac{d C}{d t}=2 \pi \frac{d r}{d t}$ .

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Problem 30250

Find the intervals where the function $f(x)=-x^{5}+10 x^{4}-25 x^{3}$ is increasing.

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Problem 30251

Find $F(x)=\int_{\ln x}^{e^{2 x}} \sin t \arctan t \, dt$ using the Leibniz rule.

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Problem 30252

Find the average rate of change of the function $h(x)$ over the interval $4 \leq x \leq 6$ .

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Problem 30253

Find the largest area of a rectangle with its base on the $x$ -axis and top corners on $y = 6 - x^2$ .

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Problem 30254

Find the correct limit statements for the end behavior of $h(x)=-2 x(x-3)^{2}(x+4)^{3}$ . Options: (A), (B), (C), (D).

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Problem 30255

Differentiate $y^2$ with respect to $x$ : $\frac{d\left(y^{2}\right)}{d x}=f(y) \cdot \frac{d y}{d x}$ . What is $f(y)$ ?

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Problem 30256

Find the value of $\alpha$ in the equation: $\frac{d\left(x^{2}+y^{2}\right)}{d x}=2 x+\alpha y \cdot \frac{d y}{d x}$ .

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Problem 30257

Find $\frac{d y}{d t}$ when $y=-4 x^{2}$ , $\frac{d x}{d t}=-1$ unit/s, and $x=2$ units.

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Problem 30258

Find the value of $\beta$ in the equation: $\frac{d\left(\cos \left(y^{2}\right)\right)}{d x}=\beta y \sin \left(y^{2}\right) \cdot \frac{d y}{d x}$ .

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Problem 30259

Solve the initial-value problem: $\frac{d y}{d x}=3 x^{2}-\sin x$ , with $y(0)=2$ .

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Problem 30260

Find these limits:
1. $\lim _{x \rightarrow \infty} \frac{2020 x^{2}-x}{x^{3}+x^{2}+1}$
2. $\lim _{x \rightarrow \infty} \frac{2020 x^{3}+10000}{x^{3}-999999}$
3. $\lim _{x \rightarrow 0} \frac{\sin 2 x}{5 x}$
4. $\lim _{x \rightarrow \infty} \frac{2 \sin x}{x}$
5. $\lim \frac{x^{2}-3 x+2}{x^{3}-2 x^{2}}$ as $x \rightarrow \infty$

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Problem 30261

Find the integral of the function $x e^{x^{2}-1}$ with respect to $x$ : $\int x e^{x^{2}-1} d x$ .

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Problem 30262

Find the correct horizontal asymptote for $h(x)=\frac{(x-2)^{5}-x^{5}}{3 x^{4}}$ : A) $\frac{1}{3}$ , B) $\frac{5}{3}$ , C) $-\frac{2}{3}$ , D) $-\frac{10}{3}$ , E) none.

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Problem 30263

Find the limit of $\frac{x^{2}-3x+2}{x^{3}-2x^{2}}$ as: a) $x \rightarrow 0^{+}$ , b) $x \rightarrow 2^{+}$ , c) $x \rightarrow 2^{-}$ , d) $x \rightarrow 2$ .

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Problem 30264

Find the correct horizontal asymptote for $h(x)=\frac{(x-2)^{5}-x^{5}}{3 x^{4}}$ : (A) $y=\frac{1}{3}$ , (B) $y=\frac{5}{3}$ , (C) $y=-\frac{2}{3}$ , (D) $y=-\frac{10}{3}$ , (E) no asymptote.

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Problem 30265

Find the limit: $\lim _{x \rightarrow \infty} \frac{2 \sin x}{x}=$

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Problem 30266

1. For bacteria growth, find the average rate of change from $t=0$ to $t=6$ using $N(t)=75000+64t^3$ .
2. For a particle's displacement $s(t)=4t^2-10t+13$ , calculate the average rate of change from $1s$ to $4s$ and estimate the instantaneous rate after $1s$ .
3. For a Ferris wheel, use $h=18\sin\left(\frac{\pi t}{100}\right)+20$ to find the average rate of change of height between given time intervals.

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Problem 30267

Find the derivative using the limit: $f^{\prime}(3)=\lim _{h \rightarrow 0} \frac{(9+6 h+h^{2}+15+5 h-24)}{h}$ .

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Problem 30268

Given $y=2 x-\sin (k x)$ and $x=2 k t$ , find: (a) $\frac{d y}{d x}=$ A (in terms of $k$ and $x$ ) (b) $\frac{d x}{d t}=$ A (in terms of $k$ and/or $t$ ) (c) For $k=1$ , at $t=\frac{\pi}{2}$ , find $\frac{d y}{d t}=$ A (to 1 decimal place)

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Problem 30269

Evaluate the integral $\int_{0}^{\frac{\pi}{2}} x \cos x \, dx$ .

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Problem 30270

Calculate the integral: $\int \frac{2 x^{3}+2 x^{2}+2 x+1}{x^{2}(x^{2}+1)} dx$ .

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Problem 30271

Find $f^{\prime}(3)$ for $f(x)=x^2+5x$ using the limit definition: $f^{\prime}(3)=\lim_{h \rightarrow 0} \frac{f(3+h)-f(3)}{h}$ .

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Problem 30272

Find the limit as $x$ approaches 0: $\lim _{x \rightarrow 0} \frac{\sqrt{x+36}-6}{x}$

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Problem 30273

1. Find a function $f(x)$ continuous everywhere except $x=1$ with a removable discontinuity. Explain why it's discontinuous there.
2. Find a function $g(x)$ continuous everywhere except $x=2$ with a nonremovable discontinuity. Explain why it's discontinuous.

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Problem 30274

Find the limit as $x$ approaches 6 from the left for the expression $\frac{2 x}{x^{2}-36}$ .

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Problem 30275

Show that $f(x)=\frac{x+2}{x^{2}+5x+6}$ has a continuous extension at $x=2$ , and determine the extension.

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Problem 30276

1. For $y^{\prime}=2 y-x$ , sketch slopes at (1,2), (1,1), (2,1).
2. Find a diff. eq. and initial condition for $y=\int_{1}^{x} \frac{1}{t} dt$ .
3. Apply Euler's method for $y^{\prime}=\frac{2 y}{x}$ , $y(1)=-1$ , $dx=0.5$ to get 3 points.

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Problem 30277

5) Determine convergence or divergence of $\sum_{n=1}^{\infty} \frac{8}{\sqrt{n+2}}$ using the integral test. 6) Determine convergence or divergence of $\sum_{n=1}^{\infty} \frac{1}{3^{n}+6}$ using the direct comparison test. 7) Determine convergence or divergence of $\sum_{n=1}^{\infty} \frac{\sqrt{n}}{n^{2}+1}$ using the limit comparison test.

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Problem 30278

1. Solve the initial value problem: $x y' + y = \sin x$ , $x > 0$ , $y\left(\frac{\pi}{2}\right) = 1$ .
2. For $a_n = \frac{2^n - 1}{2^n}$ , write the first four terms.
3. Find a formula for each sequence: a) $1, -4, 7, -10, 13, \ldots$ b) $\frac{5}{1}, \frac{8}{2}, \frac{11}{6}, \frac{14}{24}, \ldots$

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Problem 30279

Determine if the series converges or diverges. If converges, find its sum: a) $1-2+4-8+\cdots$ b) $\left(\frac{-2}{3}\right)^{2}+\left(\frac{-2}{3}\right)^{3}+\cdots$ c) $\sum_{n=0}^{\infty} \cos n \pi$ d) $\sum_{n=1}^{\infty} \frac{6}{(2 n-1)(2 n+1)}$

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Problem 30280

Find the limit: $\lim _{n \rightarrow \infty} \frac{\sqrt{2 n^{2}+n}}{\sqrt{n^{3}+2-2 n}}$

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Problem 30281

Find the minimum surface area $S=12 x^{2}+\frac{7}{8 x}$ for a container with capacity $0.75 \mathrm{~m}^{3}$ at $x=0.332 \mathrm{~m}$ . Justify the minimum.

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Problem 30282

Determine if the series $\sum_{n=1}^{\infty}\left(\frac{n+a}{n+b}\right)^{n^{2}}$ converges or diverges for $b<a$ . Justify.

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Problem 30283

Aufgabe: Analysiere die Funktion $g(x)=-0,25 x^{3}+2,7 x^{2}-6 x$ für Damm und Graben. Finde Breiten, maximale Höhe, Steigung und Monotonie im Intervall $[2; 5]$ .

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Problem 30284

Skizzieren Sie $f$ und bestimmen Sie die lokale Änderungsrate an $x_{0}$ : a) $f(x)=0,5 x^{2}, x_{0}=2$ b) $f(x)=1-x^{2}, x_{0}=2$ c) $f(x)=\frac{1}{x}, x_{0}=1$ .

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Problem 30285

Gegeben ist die Weg-Zeit-Funktion $s(t)=-0,00006 \cdot t^{3}+0,01983 \cdot t^{2}$ . Bestimme die Geschwindigkeitsfunktion $v(t)$ , finde den Zeitpunkt der maximalen Geschwindigkeit und berechne die mittlere Geschwindigkeit im Intervall [50 s; 150 s].

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Problem 30286

Find the derivative of $f(x)=\frac{2}{\sqrt{x}}$ and calculate $f^{\prime}(4)$ .

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Problem 30287

Berechnen Sie die lokale Änderungsrate von $f$ an $x_{0}$ mit der h-Methode oder der $(x-x_{0})$ -Methode: a) $f(x)=0,5 x^{2}, x_{0}=2$ b) $f(x)=1-x^{2}, x_{0}=2$ c) $f(x)=2 x+1, x_{0}=3$

See Solution

Problem 30288

Find the integral: $\int \frac{x^{3}}{2 x^{4}+3} d x$

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Problem 30289

Zad 1. Sprawdź zbieżność szeregów: (a) $\sum_{n=3}^{+\infty}\left(\sqrt[3]{n^{3}+2}-\sqrt{n^{2}+7}\right)$ .

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Problem 30290

Find the limit as $x$ approaches 0 from the right: $\lim _{x \rightarrow 0^{+}} \ln x$ .

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Problem 30291

Prove that for all $x, y \in \mathbb{R}$ , the inequality $|\arctan(x^3) - \arctan(y^3)| \leq M|x-y|$ holds, where $M = \sup_{t \in \mathbb{R}} \frac{3t^2}{1+t^6}$ .

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Problem 30292

Find the limit: $\lim_{x \rightarrow \infty} \ln x$ .

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Problem 30293

Find the infimum and supremum of $f(t)=\frac{3 t^{2}}{1+t^{6}}$ , and determine if it has a max or min. Sketch the graph.

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Problem 30294

Bestimmen Sie die Ableitung von $f(x)=e^{x}+x^{3}$ . Was ist $f^{\prime}(x)=?$

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Problem 30295

Find the limit as $x$ approaches infinity for $\sin x$ .

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Problem 30296

Find the derivative of the function $f(x) = 4x^{2} - e^{x}$ . What is $f^{\prime}(x)$ ?

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Problem 30297

If $p$ is an even positive integer, find $\lim _{x \rightarrow-\infty} x^{p}$ .

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Problem 30298

Find the limit: $\lim _{x \rightarrow \infty}\left(1+\frac{a-1}{4 n^{2}+1}\right)^{2 n^{2}-1}$ .

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Problem 30299

Find the limit as $x$ approaches $-\infty$ for $x^{p}$ . What is $\lim _{x \rightarrow-\infty} x^{p}$ ?

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Problem 30300

8 Wassermelonen wachsen täglich um $12\%$ . Berechne Zeiträume für $1\%, 100\%, 500\%$ Gewichtszunahme und beantworte weitere Fragen.

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Calculus

Problem 30201

Soit f(x)=∣x2−6x+5∣f(x)=\sqrt{|x^{2}-6x+5|}f(x)=∣x2−6x+5∣​, étudiez sa continuité, dérivabilité, variations, asymptotes et symétrie sur [1;5][1;5][1;5].

Problem 30202

Differentiate the function: f(x)=x−2(4x2+1)+2x3f(x) = \sqrt{x-2} (4x^2 + 1) + 2x^3f(x)=x−2​(4x2+1)+2x3.

Problem 30203

Find the limit as xxx approaches infinity for the expression 1+8x3−2x\frac{1+8 x}{3-2 x}3−2x1+8x​.

Problem 30204

Find the limits for f(x)=x2+1(2x+3)(x−5)f(x)=\frac{x^{2}+1}{(2 x+3)(x-5)}f(x)=(2x+3)(x−5)x2+1​ as xxx approaches 5 from left, right, and directly. Enter 'P', 'N', or 'D'.

Problem 30205

Find dydx\frac{dy}{dx}dxdy​ for the function y=x3(x2+5)4y = x^{3}(x^{2} + 5)^{4}y=x3(x2+5)4.

Problem 30206

Evaluate these limits: (a) Find lim⁡x→∞1+8x3−2x\lim _{x \rightarrow \infty} \frac{1+8 x}{3-2 x}limx→∞​3−2x1+8x​. (b) Find lim⁡x→−∞1+8x3−2x\lim _{x \rightarrow-\infty} \frac{1+8 x}{3-2 x}limx→−∞​3−2x1+8x​.

Problem 30207

Evaluate the integral: ∫3−x9−x2\int \frac{3-x}{\sqrt{9-x^{2}}}∫9−x2​3−x​.

Problem 30208

Find the limit: lim⁡x→−∞7+4x23+8x\lim _{x \rightarrow-\infty} \frac{\sqrt{7+4 x^{2}}}{3+8 x}limx→−∞​3+8x7+4x2​​.

Problem 30209

Find the limit: lim⁡x→∞(11−x)(10+7x)(3−4x)(4+9x)\lim _{x \rightarrow \infty} \frac{(11-x)(10+7 x)}{(3-4 x)(4+9 x)}limx→∞​(3−4x)(4+9x)(11−x)(10+7x)​

Problem 30210

Find the limit: lim⁡x→∞7+4x23+8x\lim _{x \rightarrow \infty} \frac{\sqrt{7+4 x^{2}}}{3+8 x}limx→∞​3+8x7+4x2​​.

Problem 30211

Find the limit: lim⁡x→∞6x3−8x2−3x8−11x−3x3\lim _{x \rightarrow \infty} \frac{6 x^{3}-8 x^{2}-3 x}{8-11 x-3 x^{3}}limx→∞​8−11x−3x36x3−8x2−3x​.

Problem 30212

Find the limit: lim⁡x→−∞(11−x)(10+7x)(3−4x)(4+9x)\lim _{x \rightarrow-\infty} \frac{(11-x)(10+7 x)}{(3-4 x)(4+9 x)}limx→−∞​(3−4x)(4+9x)(11−x)(10+7x)​.

Problem 30213

Find the limit: lim⁡x→−∞1+8x3−2x\lim _{x \rightarrow-\infty} \frac{1+8 x}{3-2 x}limx→−∞​3−2x1+8x​.

Problem 30214

Find the limit: lim⁡x→∞2x+210x2−2x+8\lim _{x \rightarrow \infty} \frac{2 x+2}{10 x^{2}-2 x+8}limx→∞​10x2−2x+82x+2​.

Problem 30215

Identify intervals where the function is increasing or decreasing, and points of relative maxima and minima.

Problem 30216

Find the limit: lim⁡x→−∞6x3−8x2−3x8−11x−3x3\lim _{x \rightarrow-\infty} \frac{6 x^{3}-8 x^{2}-3 x}{8-11 x-3 x^{3}}limx→−∞​8−11x−3x36x3−8x2−3x​.

Problem 30217

Find the limit as xxx approaches infinity: lim⁡x→∞(9x2+x−3x)\lim _{x \rightarrow \infty}\left(\sqrt{9 x^{2}+x}-3 x\right)limx→∞​(9x2+x​−3x).

Problem 30218

Differentiate y=7x2(1x2+1)y = \frac{7x}{2} \left( \frac{1}{x^2 + 1} \right)y=27x​(x2+11​) using the product rule and explain the steps.

Problem 30219

Find the limits: (a) lim⁡x→∞(−26x2+28x3)\lim _{x \rightarrow \infty}(-26 x^{2}+28 x^{3})limx→∞​(−26x2+28x3) and (b) lim⁡x→−∞(−26x2+28x3)\lim _{x \rightarrow -\infty}(-26 x^{2}+28 x^{3})limx→−∞​(−26x2+28x3).

Problem 30220

Find the limit: lim⁡x→+∞(x2+4−x2−6)\lim _{x \rightarrow+\infty}\left(\sqrt{x^{2}+4}-\sqrt{x^{2}-6}\right)limx→+∞​(x2+4​−x2−6​).

Problem 30221

Sketch the graph of y=x3−6x2−15x+10y=x^{3}-6x^{2}-15x+10y=x3−6x2−15x+10. Identify local max/min, end behavior, and intervals for yyy, y′y'y′, and y′′y''y′′.

Problem 30222

Calculate the limit: lim⁡x→∞10−x10+x\lim _{x \rightarrow \infty} \frac{10-\sqrt{x}}{10+\sqrt{x}}limx→∞​10+x​10−x​​.

Problem 30223

Find the derivative f′(−3)f^{\prime}(-3)f′(−3) for the function f(x)=x3+4f(x)=x^{3}+4f(x)=x3+4 using the definition of a derivative.

Problem 30224

E. coli bacteria grow at 1.9% per minute. If the current population is 172 million, what will it be in 7.2 minutes? a) 197.2 million b) 220.2 million c) 10 million d) 101.2 million

Problem 30225

Find the limits: (a) lim⁡x→∞x2(9x)(−8x)\lim _{x \rightarrow \infty} x^{2}(9 x)(-8 x)limx→∞​x2(9x)(−8x) and (b) lim⁡x→−∞x2(9x)(−8x)\lim _{x \rightarrow -\infty} x^{2}(9 x)(-8 x)limx→−∞​x2(9x)(−8x).

Problem 30226

Problem 30227

Find the average rate of change of f(x)=x3−5x2f(x)=x^{3}-5x^{2}f(x)=x3−5x2 from x=5x=5x=5 to x=10x=10x=10.

Problem 30228

Evaluate the line integral of F(x,y)=(x2+y2)i+2xyj\mathbf{F}(x, y)=(x^{2}+y^{2}) \mathbf{i}+2xy \mathbf{j}F(x,y)=(x2+y2)i+2xyj along r(t)=2ti+t3j\mathbf{r}(t)=2t \mathbf{i}+t^{3} \mathbf{j}r(t)=2ti+t3j for 0≤t≤10 \leq t \leq 10≤t≤1.

Problem 30229

Find the integral of ex5e^{\sqrt[5]{x}}e5x​ with respect to xxx: ∫ex5dx\int e^{\sqrt[5]{x}} d x∫e5x​dx.

Problem 30230

Problem 30231

Find F′(2)\mathrm{F}^{\prime}(2)F′(2) if F(x)=5f(x)+2g(x)F(x)=5 f(x)+2 g(x)F(x)=5f(x)+2g(x), given f(2)=−3f(2)=-3f(2)=−3, f′(2)=4f^{\prime}(2)=4f′(2)=4, g(2)=1g(2)=1g(2)=1, g′(2)=−5g^{\prime}(2)=-5g′(2)=−5.

Problem 30232

A particle moves along the xxx-axis with position x(t)=t3−15t2+72t−9x(t)=t^{3}-15 t^{2}+72 t-9x(t)=t3−15t2+72t−9. a. Find when it's farthest right for 3≤t≤93 \leq t \leq 93≤t≤9. b. Determine its maximum speed on the same interval.

Problem 30233

Calculate lim⁡x→−2+(x+3)∣x+2∣x+2\lim _{x \rightarrow-2^{+}}(x+3) \frac{|x+2|}{x+2}limx→−2+​(x+3)x+2∣x+2∣​ and lim⁡x→−2−(x+3)∣x+2∣x+2\lim _{x \rightarrow-2^{-}}(x+3) \frac{|x+2|}{x+2}limx→−2−​(x+3)x+2∣x+2∣​.

Problem 30234

Find dydx\frac{\mathrm{dy}}{\mathrm{dx}}dxdy​ for y=2u1−4u\mathrm{y}=\frac{2 \mathrm{u}}{1-4 \mathrm{u}}y=1−4u2u​ with u=(5x2+1)4\mathrm{u}=(5 \mathrm{x}^{2}+1)^{4}u=(5x2+1)4.

Problem 30235

Mr. Sullivan's apple juice function A(t)A(t)A(t) is given. Estimate the rate of change at t=10t=10t=10 days and check if A′(t)=23A'(t)=\frac{2}{3}A′(t)=32​ for 0≤t≤120 \leq t \leq 120≤t≤12.

Problem 30236

Determine if the functions x2x^{2}x2 and e−x2e^{-x^{2}}e−x2 are acceptable in quantum mechanics. Plot and normalize if acceptable.

Problem 30237

Find F′(2)F^{\prime}(2)F′(2) for F(x)=f(x)−3g(x)F(x)=f(x)-3g(x)F(x)=f(x)−3g(x) given f(2)=−3,f′(2)=4,g(2)=1,g′(2)=−5\mathrm{f}(2)=-3, \mathrm{f}^{\prime}(2)=4, \mathrm{g}(2)=1, \mathrm{g}^{\prime}(2)=-5f(2)=−3,f′(2)=4,g(2)=1,g′(2)=−5.

Problem 30238

Find dydx\frac{\mathrm{dy}}{\mathrm{dx}}dxdy​ for y=(u+3)2y=(u+3)^{2}y=(u+3)2, where u=v−3u=\sqrt{v-3}u=v−3​ and v=x2v=x^{2}v=x2.

Problem 30239

Find the derivative of the function f(x)=2x3ln⁡(x)f(x) = 2x^3 \ln(x)f(x)=2x3ln(x).

Problem 30240

Find the derivative of the function f(x)=log⁡2∣csc⁡(x)∣f(x) = \log_{2}|\csc(x)|f(x)=log2​∣csc(x)∣.

Problem 30241

Soit $f(x)=\sqrt{|x^{2}-6x+5|}$ , étudiez sa continuité, dérivabilité, variations, asymptotes et symétrie sur $[1;5]$ .

Differentiate the function: $f(x) = \sqrt{x-2} (4x^2 + 1) + 2x^3$ .

Find the limit as $x$ approaches infinity for the expression $\frac{1+8 x}{3-2 x}$ .

Find the limits for $f(x)=\frac{x^{2}+1}{(2 x+3)(x-5)}$ as $x$ approaches 5 from left, right, and directly. Enter 'P', 'N', or 'D'.

Find $\frac{dy}{dx}$ for the function $y = x^{3}(x^{2} + 5)^{4}$ .

Evaluate these limits: (a) Find $\lim _{x \rightarrow \infty} \frac{1+8 x}{3-2 x}$ . (b) Find $\lim _{x \rightarrow-\infty} \frac{1+8 x}{3-2 x}$ .

Evaluate the integral: $\int \frac{3-x}{\sqrt{9-x^{2}}}$ .

Find the limit: $\lim _{x \rightarrow-\infty} \frac{\sqrt{7+4 x^{2}}}{3+8 x}$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{(11-x)(10+7 x)}{(3-4 x)(4+9 x)}$

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{7+4 x^{2}}}{3+8 x}$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{6 x^{3}-8 x^{2}-3 x}{8-11 x-3 x^{3}}$ .

Find the limit: $\lim _{x \rightarrow-\infty} \frac{(11-x)(10+7 x)}{(3-4 x)(4+9 x)}$ .

Find the limit: $\lim _{x \rightarrow-\infty} \frac{1+8 x}{3-2 x}$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{2 x+2}{10 x^{2}-2 x+8}$ .

Find the limit: $\lim _{x \rightarrow-\infty} \frac{6 x^{3}-8 x^{2}-3 x}{8-11 x-3 x^{3}}$ .

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(\sqrt{9 x^{2}+x}-3 x\right)$ .

Differentiate $y = \frac{7x}{2} \left( \frac{1}{x^2 + 1} \right)$ using the product rule and explain the steps.

Find the limits: (a) $\lim _{x \rightarrow \infty}(-26 x^{2}+28 x^{3})$ and (b) $\lim _{x \rightarrow -\infty}(-26 x^{2}+28 x^{3})$ .

Find the limit: $\lim _{x \rightarrow+\infty}\left(\sqrt{x^{2}+4}-\sqrt{x^{2}-6}\right)$ .

Sketch the graph of $y=x^{3}-6x^{2}-15x+10$ . Identify local max/min, end behavior, and intervals for $y$ , $y'$ , and $y''$ .

Calculate the limit: $\lim _{x \rightarrow \infty} \frac{10-\sqrt{x}}{10+\sqrt{x}}$ .

Find the derivative $f^{\prime}(-3)$ for the function $f(x)=x^{3}+4$ using the definition of a derivative.

Find the limits: (a) $\lim _{x \rightarrow \infty} x^{2}(9 x)(-8 x)$ and (b) $\lim _{x \rightarrow -\infty} x^{2}(9 x)(-8 x)$ .

Find the average rate of change of $f(x)=x^{3}-5x^{2}$ from $x=5$ to $x=10$ .

Evaluate the line integral of $\mathbf{F}(x, y)=(x^{2}+y^{2}) \mathbf{i}+2xy \mathbf{j}$ along $\mathbf{r}(t)=2t \mathbf{i}+t^{3} \mathbf{j}$ for $0 \leq t \leq 1$ .

Find the integral of $e^{\sqrt[5]{x}}$ with respect to $x$ : $\int e^{\sqrt[5]{x}} d x$ .

Find $\mathrm{F}^{\prime}(2)$ if $F(x)=5 f(x)+2 g(x)$ , given $f(2)=-3$ , $f^{\prime}(2)=4$ , $g(2)=1$ , $g^{\prime}(2)=-5$ .

A particle moves along the $x$ -axis with position $x(t)=t^{3}-15 t^{2}+72 t-9$ .
a. Find when it's farthest right for $3 \leq t \leq 9$ .
b. Determine its maximum speed on the same interval.

Calculate $\lim _{x \rightarrow-2^{+}}(x+3) \frac{|x+2|}{x+2}$ and $\lim _{x \rightarrow-2^{-}}(x+3) \frac{|x+2|}{x+2}$ .

Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ for $\mathrm{y}=\frac{2 \mathrm{u}}{1-4 \mathrm{u}}$ with $\mathrm{u}=(5 \mathrm{x}^{2}+1)^{4}$ .

Mr. Sullivan's apple juice function $A(t)$ is given. Estimate the rate of change at $t=10$ days and check if $A'(t)=\frac{2}{3}$ for $0 \leq t \leq 12$ .

Determine if the functions $x^{2}$ and $e^{-x^{2}}$ are acceptable in quantum mechanics. Plot and normalize if acceptable.

Find $F^{\prime}(2)$ for $F(x)=f(x)-3g(x)$ given $\mathrm{f}(2)=-3, \mathrm{f}^{\prime}(2)=4, \mathrm{g}(2)=1, \mathrm{g}^{\prime}(2)=-5$ .

Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ for $y=(u+3)^{2}$ , where $u=\sqrt{v-3}$ and $v=x^{2}$ .

Find the derivative of the function $f(x) = 2x^3 \ln(x)$ .

Find the derivative of the function $f(x) = \log_{2}|\csc(x)|$ .

Find $f(2)$ given that $f^{\prime}(x)=\sin \left(\frac{\pi e^{x}}{2}\right)$ and $f(0)=1$ .

Find the value of $a$ in the equation $\frac{d(\ln (1+y))}{d x}=\frac{a}{1+y} \frac{d y}{d x}$ .

Find $a$ and $b$ in the equation: $\frac{d(4(x^{2}+y^{2}))}{dx} = ax + by \frac{dy}{dx}$ .

Find $\frac{d x}{d t}$ when $y=2 \sin (x)$ and $\frac{d y}{d t}=1$ at $x=\frac{\pi}{3}$ .

Find $\frac{d y}{d t}$ when $y=12 \cos (x)$ and $\frac{d x}{d t}=1$ at $x=\frac{\pi}{6}$ .

Find the intervals where the function $f(x)=-x^{4}-12 x^{3}-36 x^{2}$ is increasing.

Find the intervals where the function $f(x)=-x^{4}+12 x^{3}-36 x^{2}$ is increasing.

Find the intervals where the function $f(x)=x^{4}-4 x^{3}$ is decreasing.

If $\frac{d A}{d t}=3 \frac{d C}{d t}$ , find $r$ given $\frac{d A}{d t}=2 \pi r \frac{d r}{d t}$ and $\frac{d C}{d t}=2 \pi \frac{d r}{d t}$ .

Find the intervals where the function $f(x)=-x^{5}+10 x^{4}-25 x^{3}$ is increasing.

Find $F(x)=\int_{\ln x}^{e^{2 x}} \sin t \arctan t \, dt$ using the Leibniz rule.

Find the average rate of change of the function $h(x)$ over the interval $4 \leq x \leq 6$ .

Find the largest area of a rectangle with its base on the $x$ -axis and top corners on $y = 6 - x^2$ .

Find the correct limit statements for the end behavior of $h(x)=-2 x(x-3)^{2}(x+4)^{3}$ . Options: (A), (B), (C), (D).

Differentiate $y^2$ with respect to $x$ : $\frac{d\left(y^{2}\right)}{d x}=f(y) \cdot \frac{d y}{d x}$ . What is $f(y)$ ?

Find the value of $\alpha$ in the equation: $\frac{d\left(x^{2}+y^{2}\right)}{d x}=2 x+\alpha y \cdot \frac{d y}{d x}$ .

Find $\frac{d y}{d t}$ when $y=-4 x^{2}$ , $\frac{d x}{d t}=-1$ unit/s, and $x=2$ units.

Find the value of $\beta$ in the equation: $\frac{d\left(\cos \left(y^{2}\right)\right)}{d x}=\beta y \sin \left(y^{2}\right) \cdot \frac{d y}{d x}$ .

Solve the initial-value problem: $\frac{d y}{d x}=3 x^{2}-\sin x$ , with $y(0)=2$ .

Find the integral of the function $x e^{x^{2}-1}$ with respect to $x$ : $\int x e^{x^{2}-1} d x$ .

Find the correct horizontal asymptote for $h(x)=\frac{(x-2)^{5}-x^{5}}{3 x^{4}}$ : A) $\frac{1}{3}$ , B) $\frac{5}{3}$ , C) $-\frac{2}{3}$ , D) $-\frac{10}{3}$ , E) none.

Find the limit of $\frac{x^{2}-3x+2}{x^{3}-2x^{2}}$ as: a) $x \rightarrow 0^{+}$ , b) $x \rightarrow 2^{+}$ , c) $x \rightarrow 2^{-}$ , d) $x \rightarrow 2$ .

Find the correct horizontal asymptote for $h(x)=\frac{(x-2)^{5}-x^{5}}{3 x^{4}}$ : (A) $y=\frac{1}{3}$ , (B) $y=\frac{5}{3}$ , (C) $y=-\frac{2}{3}$ , (D) $y=-\frac{10}{3}$ , (E) no asymptote.

Find the limit: $\lim _{x \rightarrow \infty} \frac{2 \sin x}{x}=$

Find the derivative using the limit: $f^{\prime}(3)=\lim _{h \rightarrow 0} \frac{(9+6 h+h^{2}+15+5 h-24)}{h}$ .

Evaluate the integral $\int_{0}^{\frac{\pi}{2}} x \cos x \, dx$ .

Calculate the integral: $\int \frac{2 x^{3}+2 x^{2}+2 x+1}{x^{2}(x^{2}+1)} dx$ .

Find $f^{\prime}(3)$ for $f(x)=x^2+5x$ using the limit definition: $f^{\prime}(3)=\lim_{h \rightarrow 0} \frac{f(3+h)-f(3)}{h}$ .

Find the limit as $x$ approaches 0: $\lim _{x \rightarrow 0} \frac{\sqrt{x+36}-6}{x}$

1. Find a function $f(x)$ continuous everywhere except $x=1$ with a removable discontinuity. Explain why it's discontinuous there.
2. Find a function $g(x)$ continuous everywhere except $x=2$ with a nonremovable discontinuity. Explain why it's discontinuous.

Find the limit as $x$ approaches 6 from the left for the expression $\frac{2 x}{x^{2}-36}$ .