Calculus

Problem 24301

$f$ est une fonction avec $u_{n} = f(n)$ . Si $f(x) \to -\infty$ quand $x \to +\infty$ , quelle est la limite de $u_{n}$ ?

See Solution

Problem 24302

Finde die Ableitung von $f(x) = -\sin(x)$ .

See Solution

Problem 24303

Find the rate of change of $y=\tan x$ at $x=0$ . Use $\frac{f(x+h)-f(x)}{h}$ to estimate it. What is the result?

See Solution

Problem 24304

Calculate the indefinite integral: $\int(4 \cos x + 7 \sin x) \, dx$

See Solution

Problem 24305

Find the values of $x$ where the rate of change for $f(x)=-x^{2}+x$ is negative. Options: 2, 1, $-3$ , $-10$ .

See Solution

Problem 24306

Find $f(6)-f(1)$ if $f^{\prime}(x)=6 x-8$ .

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Problem 24307

Evaluate the integral from 0 to 6 of the function $x - \frac{1}{2}$ .

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Problem 24308

Identify the false statement about inflection points based on the following conditions regarding $f^{\prime \prime}(x)$ .

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Problem 24309

Find the infimum, supremum, minimum, and maximum of the set $\{3,5,2\pi\} \cup \left\{\frac{1}{2n}, n \in \mathbb{N}^{*}\right\}$ . Justify your answer.
Show that the sequence $u_n = \frac{n-3}{n}$ is Cauchy. Also, prove for any $x \in \mathbb{R}$ , there exists a sequence of rationals $x_n$ such that $\lim_{n \to +\infty} x_n = x$ .

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Problem 24310

Find the integral of $\frac{1}{(2-5 x)^{2}}$ with respect to $x$ .

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Problem 24311

Find the slope of the tangent line to $y=\frac{1}{x^{2}}-2 x^{1/2}$ at $x=1$ . Choices: $-1$ , 3, 1, 0, $-3$ .

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Problem 24312

Identify the FALSE statement about the surge function $C(t)=a t e^{b t}$ from the options provided.

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Problem 24313

1. Prove that the sequence $u_{n}=\frac{n-3}{n}$ is Cauchy.
2. For any real $x$ , find a rational sequence $x_{n}$ such that $\lim_{n \to +\infty} x_{n}=x$ . Use $x_{n}=\frac{\mathrm{E}(n x)}{n}$ .

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Problem 24314

Find the tangent line equation for $f(x)=(e^{2 x}+7 x^{2})^{3}$ at $x=0$ . Choose from the options.

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Problem 24315

How long will it take for 770.33 grams of a radioactive isotope with a half-life of 3.4 years to decay to 120.69 grams? Round to the nearest tenth of a year.

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Problem 24316

Which function is not continuous at $x=2$ ? a. All are continuous b. $f(x)=\ln (x+2)$ c. $f(x)=2^{x-2}$ d. $f(x)=(x-2)^{2}$ e. $f(x)=\frac{2}{x}$

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Problem 24317

Find all $x$ where the function $f(x)=\frac{e^{5 x^{2}-3 x}}{3 x}$ has relative extrema.

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Problem 24318

Differentiate the equation $2 x^{2}-y^{2}=2$ implicitly to find $\frac{d y}{d x}$ .

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Problem 24319

Differentiate $2 x^{2}-y^{2}=2$ with respect to $x$ to find $\frac{d y}{d x}$ .

See Solution

Problem 24320

Find the volume using the shell method for the solid formed by revolving the region defined by $X=9-3y^2$ and $Y=\sqrt{3}$ around the $x$ -axis. The volume is $\square$ (exact answer in terms of $\pi$ ).

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Problem 24321

Evaluate the integral: $\int \frac{8}{\sqrt{x}(1+8 \sqrt{x})^{3}} d x$

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Problem 24322

Evaluate the integral: $\int \frac{1}{x^{2}} \sqrt{1+\frac{1}{x}} \, dx$

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Problem 24323

Find the limit as $x$ approaches 1 for the expression $\frac{2 \cos (x-1)-2 x}{2 \ln (3 x-2)}$ .

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Problem 24324

Find the volume of the solid formed by revolving the region bounded by $y=3$ , $x=\frac{1}{216}$ , and $y=1$ around the $x$ -axis using a. the washer method and b. the shell method.

See Solution

Problem 24325

Evaluate the integral: $\int \frac{5}{\sqrt{x}(9+5 \sqrt{x})^{6}} d x$

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Problem 24326

Differentiate the equation $2 x^{2}-5 y^{2}=3$ implicitly to find $\frac{d y}{d x}$ .

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Problem 24327

Given the sequence $f_{n}(x)=\frac{1}{(1+x^{2})^{n}}$ , show:
1. $f_{n} \rightarrow f$ pointwise on $\mathbb{R}$ , where $f(x)=1$ if $x=0$ , else $f(x)=0$ .
2. $f_{n}$ converges uniformly to $f$ on $[a,+\infty)$ for all $a>0$ .
3. Convergence is not uniform on $\mathbb{R}$ .

See Solution

Problem 24328

Evaluate the integral: $\int \frac{4}{\sqrt{x}(3+4 \sqrt{x})^{2}} d x$

See Solution

Problem 24329

Find the production level that minimizes the cost function $c(x)=\frac{x^{3}}{3}-x^{2}-8 x+400$ .

See Solution

Problem 24330

Estimate the cost of the 41st unit using marginal analysis from $C(q)=3q^2+q+500$ and calculate the actual cost.

See Solution

Problem 24331

Find the intercepts and asymptotes of $r(x)=(3x(x+2))/((x-1)(x-6))$ .

See Solution

Problem 24332

Differentiate $g(t)=t^{3} \cos t$ and find $g^{\prime}(t)$ .

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Problem 24333

Evaluate the limit: $\lim _{x \rightarrow \infty}\left(\frac{18 x}{18 x+4}\right)^{6 x}$ . Enter $-I$ , $I$ , or DNE for the answer.

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Problem 24334

Find the inflection point of the function $f(x)=\frac{x}{(x+1)^{2}}$ . Choose from the options: a, b, c, d.

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Problem 24335

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

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Problem 24336

Which function has an oblique asymptote? $f(x)=\frac{x^{2}+1}{x^{3}-x^{2}-1}$ , $f(x)=\frac{4x^{2}+x+1}{x^{2}}$ , $f(x)=\frac{x^{5}+1}{x^{4}+3x^{2}+2}$ , or $f(x)=\frac{x^{5}}{x^{2}-1}$ ?

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Problem 24337

Differentiate $y=\frac{x+4}{x^{3}+x-7}$ and verify if $y' = 1$ .

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Problem 24338

Differentiate the function $y=\frac{x+4}{x^{3}+x-7}$ and find $y'$ .

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Problem 24339

Differentiate the function $g(t)=\frac{t-\sqrt{t}}{t^{1/7}}$ .

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Problem 24340

Identify the type of discontinuity in the graph of $f(x)=\frac{x}{(x+1)(x-3)}$ : a) Two jump b) Three point c) Two asymptotic d) One asymptotic.

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Problem 24341

Differentiate $y=5 x^{2} \sin x \tan x$ .

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Problem 24342

Given $f(5)=2$ , $f^{\prime}(5)=6$ , $g(5)=-7$ , and $g^{\prime}(5)=1$ , find: (a) $(f g)^{\prime}(5)$ , (b) $(f / g)^{\prime}(5)$ , (c) $(g / f)^{\prime}(5)$ .

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Problem 24343

Find the equation of the tangent line to the curve $y=1/(1+x^{2})$ at the point $(-1, \frac{1}{2})$ . $y=$

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Problem 24344

Differentiate the function: $F(y)=\left(\frac{1}{y^{2}}-\frac{3}{y^{4}}\right)\left(y+5 y^{3}\right)$

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Problem 24345

Find the tangent line equation to the curve $y=\frac{x^{2}-1}{x^{2}+x+1}$ at the point (1,0).

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Problem 24346

Find the average rate of change of $f(x)=x^{3}-4$ from $x=-4$ to $x=3$ using $\frac{\Delta y}{\Delta x}$ .

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Problem 24347

Find the second derivative $f^{\prime \prime}(3)$ for the function $f(x)=\frac{x^{2}}{1+x}$ .

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Problem 24348

A hemispherical tank (radius $9 \mathrm{~m}$ ) fills at $4 \mathrm{~m}^{3}/\mathrm{min}$ . Find the water level rise rate when it's $7 \mathrm{~m}$ deep. Use $V = \frac{\pi h^{2}(27-h)}{3}$ and differentiate to find $\frac{d V}{d t}$ .

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Problem 24349

How long does it take for a bacteria population to double at a growth rate of $5.3\%$ per hour? Round to the nearest hundredth.

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Problem 24350

Differentiate the function $f(x)=7 \sqrt{x} \sin x$ .

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Problem 24351

Differentiate the function $y=\frac{8 x}{7-\cot x}$ .

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Problem 24352

Find the derivative of the following functions: 1. $y=\sqrt{1-4 x}$ , 2. $f(x)=\ln (\cos 5 x)$ .

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Problem 24353

Find the rectangle with max area with one corner at (0,0) and the opposite at $(x, y)$ on the curve $y=\frac{3}{1+4 x^{2}}$ . Determine the $x$ and $y$ values for max area, where $0 \leq x \leq 3$ .

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Problem 24354

Differentiate $f(\theta)=\frac{\sec \theta}{9+\sec \theta}$ .

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Problem 24355

Find the derivative of $\left(\frac{\sqrt[3]{x}}{x^{n}}\right)^{3}$ with respect to $x$ .

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Problem 24356

Find the derivative of $F(x)=(x^{4}+9 x^{2}-9)^{8}$ .

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Problem 24357

Calculate the integral from -1 to 1 of $(3 x^{4}-9)$ . What is $\int_{-1}^{1}(3 x^{4}-9) dx=\square$ ?

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Problem 24358

Find the derivative of $f(x)=(2x-3)^{4}(x^{2}+x+1)^{5}$ .

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Problem 24359

Find the derivative of $y=\cos \left(a^{4}+x^{4}\right)$ .

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Problem 24360

Find the tangent line equation for $h(x)=x \sqrt{4+2 x}$ at $x=6$ .

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Problem 24361

Evaluate the integral: $\int_{-\pi / 3}^{\pi / 3} 4 \sec ^{2} x \, dx = \square$

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Problem 24362

Find the volume of the solid formed by rotating $f(x)=2e^{-\frac{x}{2}}$ over $I=[1 ; \infty)$ around the x-axis.

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Problem 24363

Evaluate the integral $\int 3(7-2 x)^{3 / 4} d x$ .

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Problem 24364

Calculate the integral $\int_{-\pi}^{\pi} 3 \sin x \, dx = \square$ .

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Problem 24365

Find the derivative of the function $y=\left(\frac{x^{2}+5}{x^{2}-5}\right)^{3}$ .

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Problem 24366

Find $g^{\prime}(2)$ for $g(x)=2+x^{3}+\tan^{-1}(f(x))$ with $f(x)=2$ and $f'(x)=5$ .

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Problem 24367

How long until half the ozone is gone if it decreases at a continuous rate of $1.02\%$ ?

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Problem 24368

Find the average distance from the parabola $y=10 x(16-x)$ to the $x$ -axis over the interval $[0,16]$ .

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Problem 24369

Find the derivative of the function $y = x \sin \frac{3}{x}$ .

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Problem 24370

Calculate the integral from -2 to 2 of (15 - |x|^3). What is the value of $\int_{-2}^{2}\left(15-|x|^{3}\right) d x$ ?

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Problem 24371

Find the integral of $(3x + 1)^{1/3}$ with respect to $x$ .

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Problem 24372

Calculate the volume $V=\frac{1}{2} \pi \int_{-\infty}^{0} e^{4 x} d x$ .

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Problem 24373

Find the max and min of $f(x)=4 \cos^{2} x$ on $[0, \pi]$ . Choose A, B, C, or D and fill in the blanks.

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Problem 24374

Find the slope of $g(\theta)=4 \cos ^{3}(2 \theta)$ at $\theta=\frac{\pi}{6}$ .

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Problem 24375

Find the derivative of $y=\cot^{2}(\cos \theta)$ .

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Problem 24376

A ladder 8 m long leans against a wall. If the top slips down at 6 m/s, how fast is the bottom moving when 4 m from the wall?

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Problem 24377

Find the first and second derivatives of $y=\sin(x^{2})$ .

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Problem 24378

Find the doubling time for the bacteria modeled by $n(t) = 500 e^{0.45 t}$ .

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Problem 24379

Find points on the curve $y=x^{3}+2x+7$ where the tangent is parallel to $3x-y=-3$ . Are there multiple points?

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Problem 24380

Evaluate the integral $\int_{0}^{2} \frac{3 x d x}{\sqrt[3]{1+2 x^{2}}}$ .

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Problem 24381

Find the tangent line equation for $h(x)=x \sqrt{4+2 x}$ at $x=6$ .

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Problem 24382

Use the function $f(x)=x^{2}+7$ to solve: (a) Find $f'(2)$ using the limit definition. (b) Calculate $f(2)$ and the tangent line. (c) Graph $f(x)$ and the tangent line at $(2, f(2))$ .

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Problem 24383

Find the derivative of the function $e^{-2x}$ .

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Problem 24384

Solve the IVP: $y^{(5)} - 16y' = 0$ , with $y(0)=y'(0)=y''(0)=0$ , $y'''(0)=y^{(4)}(0)=1$ . Analyze behavior as $x \to \infty$ .

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Problem 24385

Find the derivatives: 6. $y=\ln(4x)$ , 7. $Y=e^{-2x}$ , 8. $Y=\operatorname{Arctan}(x^2)$ .

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Problem 24386

Find $F^{\prime}(2)$ for $F(x)=f(g(x))$ given $f(-1)=2$ , $f'(-1)=6$ , $g(2)=-1$ , and $g'(2)=6$ .

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Problem 24387

Determine where the function $f(x)=4(x+1)^{\frac{5}{2}}(4x-7)$ is concave up or down and find inflection points on $[-1, \infty)$ .

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Problem 24388

Evaluate the integral $\int_{1}^{6} \frac{2 d x}{(3 x-2)^{3 / 4}}$ .

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Problem 24389

Find the average rate of change of $y=-x^{3}+6 x^{2}+1$ from $x=2$ to $x=8$ .

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Problem 24390

A train's position is $s(t)=\frac{140}{t}$ for $2 \leq t \leq 5$ .
(a) Graph $s(t)$ . (b) Find the average velocity from $t=2$ to $t=5$ : $\square \mathrm{km/hr}$ . Round to the nearest tenth.

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Problem 24391

Find the derivative of the function: $\frac{d}{dx} \left( \arctan(x^2) \right)$ .

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Problem 24392

Evaluate the integral: $\int x^{2}(1-2 x^{3})^{4} \, dx$

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Problem 24393

Find the derivative of $y=\frac{10}{x}+3 \sec x$ .

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Problem 24394

Find the area between the graph of $f(x)=e^{3x-5}+1$ and the $x$ -axis for $1 \leq x \leq 2$ .

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Problem 24395

Solve the equation $y^{\prime \prime \prime}-7 y^{\prime \prime}-5 y^{\prime}+75 y=0$ with $y(0)=-4$ , $y^{\prime}(0)=-5$ , $y^{\prime \prime}(0)=-78$ . Find $y(t)=\square$ .

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Problem 24396

Given $N(t)$ satisfies $\frac{d N}{d t}=r N$ , find the per capita growth rate and analyze $N(1)$ if $r<0$ and $N(0)=20$ .

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Problem 24397

Find points on the curve $y=x^{3}+4x+5$ where the tangent is parallel to $5x-y=3$ . Are there multiple points?

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Problem 24398

Solve the equation $y''' - 6y'' - y' + 30y = 0$ with conditions $y(0)=-2$ , $y'(0)=-13$ , $y''(0)=-39$ . Find $y(t)$ .

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Problem 24399

A scientist models the density $D(x) = \frac{x+7}{4x^2 - 6x + 5}$ . Find $x$ that maximizes $D(x)$ and the max density.

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Problem 24400

Evaluate the integral: $\int \frac{(2 - 3x^{2}) \, dx}{(2x - x^{3})^{2}}$

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Calculus

Problem 24301

fff est une fonction avec un=f(n)u_{n} = f(n)un​=f(n). Si f(x)→−∞f(x) \to -\inftyf(x)→−∞ quand x→+∞x \to +\inftyx→+∞, quelle est la limite de unu_{n}un​ ?

Problem 24302

Finde die Ableitung von f(x)=−sin⁡(x)f(x) = -\sin(x)f(x)=−sin(x).

Problem 24303

Find the rate of change of y=tan⁡xy=\tan xy=tanx at x=0x=0x=0. Use f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ to estimate it. What is the result?

Problem 24304

Calculate the indefinite integral: ∫(4cos⁡x+7sin⁡x) dx\int(4 \cos x + 7 \sin x) \, dx∫(4cosx+7sinx)dx

Problem 24305

Find the values of xxx where the rate of change for f(x)=−x2+xf(x)=-x^{2}+xf(x)=−x2+x is negative. Options: 2, 1, −3-3−3, −10-10−10.

Problem 24306

Find f(6)−f(1)f(6)-f(1)f(6)−f(1) if f′(x)=6x−8f^{\prime}(x)=6 x-8f′(x)=6x−8.

Problem 24307

Evaluate the integral from 0 to 6 of the function x−12x - \frac{1}{2}x−21​.

Problem 24308

Identify the false statement about inflection points based on the following conditions regarding f′′(x)f^{\prime \prime}(x)f′′(x).

Problem 24309

Problem 24310

Find the integral of 1(2−5x)2\frac{1}{(2-5 x)^{2}}(2−5x)21​ with respect to xxx.

Problem 24311

Find the slope of the tangent line to y=1x2−2x1/2y=\frac{1}{x^{2}}-2 x^{1/2}y=x21​−2x1/2 at x=1x=1x=1. Choices: −1-1−1, 3, 1, 0, −3-3−3.

Problem 24312

Identify the FALSE statement about the surge function C(t)=atebtC(t)=a t e^{b t}C(t)=atebt from the options provided.

Problem 24313

1. Prove that the sequence un=n−3nu_{n}=\frac{n-3}{n}un​=nn−3​ is Cauchy. 2. For any real xxx, find a rational sequence xnx_{n}xn​ such that lim⁡n→+∞xn=x\lim_{n \to +\infty} x_{n}=xlimn→+∞​xn​=x. Use xn=E(nx)nx_{n}=\frac{\mathrm{E}(n x)}{n}xn​=nE(nx)​.

Problem 24314

Find the tangent line equation for f(x)=(e2x+7x2)3f(x)=(e^{2 x}+7 x^{2})^{3}f(x)=(e2x+7x2)3 at x=0x=0x=0. Choose from the options.

Problem 24315

How long will it take for 770.33 grams of a radioactive isotope with a half-life of 3.4 years to decay to 120.69 grams? Round to the nearest tenth of a year.

Problem 24316

Which function is not continuous at x=2x=2x=2? a. All are continuous b. f(x)=ln⁡(x+2)f(x)=\ln (x+2)f(x)=ln(x+2) c. f(x)=2x−2f(x)=2^{x-2}f(x)=2x−2 d. f(x)=(x−2)2f(x)=(x-2)^{2}f(x)=(x−2)2 e. f(x)=2xf(x)=\frac{2}{x}f(x)=x2​

Problem 24317

Find all xxx where the function f(x)=e5x2−3x3xf(x)=\frac{e^{5 x^{2}-3 x}}{3 x}f(x)=3xe5x2−3x​ has relative extrema.

Problem 24318

Differentiate the equation 2x2−y2=22 x^{2}-y^{2}=22x2−y2=2 implicitly to find dydx\frac{d y}{d x}dxdy​.

Problem 24319

Differentiate 2x2−y2=22 x^{2}-y^{2}=22x2−y2=2 with respect to xxx to find dydx\frac{d y}{d x}dxdy​.

Problem 24320

Find the volume using the shell method for the solid formed by revolving the region defined by X=9−3y2X=9-3y^2X=9−3y2 and Y=3Y=\sqrt{3}Y=3​ around the xxx-axis. The volume is □\square□ (exact answer in terms of π\piπ).

Problem 24321

Evaluate the integral: ∫8x(1+8x)3dx\int \frac{8}{\sqrt{x}(1+8 \sqrt{x})^{3}} d x∫x​(1+8x​)38​dx

Problem 24322

Evaluate the integral: ∫1x21+1x dx\int \frac{1}{x^{2}} \sqrt{1+\frac{1}{x}} \, dx∫x21​1+x1​​dx

Problem 24323

Find the limit as xxx approaches 1 for the expression 2cos⁡(x−1)−2x2ln⁡(3x−2)\frac{2 \cos (x-1)-2 x}{2 \ln (3 x-2)}2ln(3x−2)2cos(x−1)−2x​.

Problem 24324

Find the volume of the solid formed by revolving the region bounded by y=3y=3y=3, x=1216x=\frac{1}{216}x=2161​, and y=1y=1y=1 around the xxx-axis using a. the washer method and b. the shell method.

Problem 24325

Evaluate the integral: ∫5x(9+5x)6dx\int \frac{5}{\sqrt{x}(9+5 \sqrt{x})^{6}} d x∫x​(9+5x​)65​dx

Problem 24326

Differentiate the equation 2x2−5y2=32 x^{2}-5 y^{2}=32x2−5y2=3 implicitly to find dydx\frac{d y}{d x}dxdy​.

Problem 24327

Problem 24328

Evaluate the integral: ∫4x(3+4x)2dx\int \frac{4}{\sqrt{x}(3+4 \sqrt{x})^{2}} d x∫x​(3+4x​)24​dx

Problem 24329

Find the production level that minimizes the cost function c(x)=x33−x2−8x+400c(x)=\frac{x^{3}}{3}-x^{2}-8 x+400c(x)=3x3​−x2−8x+400.

Problem 24330

Estimate the cost of the 41st unit using marginal analysis from C(q)=3q2+q+500C(q)=3q^2+q+500C(q)=3q2+q+500 and calculate the actual cost.

Problem 24331

Find the intercepts and asymptotes of r(x)=(3x(x+2))/((x−1)(x−6))r(x)=(3x(x+2))/((x-1)(x-6))r(x)=(3x(x+2))/((x−1)(x−6)).

Problem 24332

Differentiate g(t)=t3cos⁡tg(t)=t^{3} \cos tg(t)=t3cost and find g′(t)g^{\prime}(t)g′(t).

Problem 24333

Evaluate the limit: lim⁡x→∞(18x18x+4)6x\lim _{x \rightarrow \infty}\left(\frac{18 x}{18 x+4}\right)^{6 x}limx→∞​(18x+418x​)6x. Enter −I-I−I, III, or DNE for the answer.

Problem 24334

Find the inflection point of the function f(x)=x(x+1)2f(x)=\frac{x}{(x+1)^{2}}f(x)=(x+1)2x​. Choose from the options: a, b, c, d.

Problem 24335

Find the limit: lim⁡x→−12x2−x−3x+1\lim _{x \rightarrow-1} \frac{2 x^{2}-x-3}{x+1}limx→−1​x+12x2−x−3​.

Problem 24336

Problem 24337

Differentiate y=x+4x3+x−7y=\frac{x+4}{x^{3}+x-7}y=x3+x−7x+4​ and verify if y′=1y' = 1y′=1.

Problem 24338

Differentiate the function y=x+4x3+x−7y=\frac{x+4}{x^{3}+x-7}y=x3+x−7x+4​ and find y′y'y′.

Problem 24339

Differentiate the function g(t)=t−tt1/7g(t)=\frac{t-\sqrt{t}}{t^{1/7}}g(t)=t1/7t−t​​.

Problem 24340

Identify the type of discontinuity in the graph of f(x)=x(x+1)(x−3)f(x)=\frac{x}{(x+1)(x-3)}f(x)=(x+1)(x−3)x​: a) Two jump b) Three point c) Two asymptotic d) One asymptotic.

Problem 24341

Differentiate y=5x2sin⁡xtan⁡xy=5 x^{2} \sin x \tan xy=5x2sinxtanx.

$f$ est une fonction avec $u_{n} = f(n)$ . Si $f(x) \to -\infty$ quand $x \to +\infty$ , quelle est la limite de $u_{n}$ ?

Finde die Ableitung von $f(x) = -\sin(x)$ .

Find the rate of change of $y=\tan x$ at $x=0$ . Use $\frac{f(x+h)-f(x)}{h}$ to estimate it. What is the result?

Calculate the indefinite integral: $\int(4 \cos x + 7 \sin x) \, dx$

Find the values of $x$ where the rate of change for $f(x)=-x^{2}+x$ is negative. Options: 2, 1, $-3$ , $-10$ .

Find $f(6)-f(1)$ if $f^{\prime}(x)=6 x-8$ .

Evaluate the integral from 0 to 6 of the function $x - \frac{1}{2}$ .

Identify the false statement about inflection points based on the following conditions regarding $f^{\prime \prime}(x)$ .

Find the integral of $\frac{1}{(2-5 x)^{2}}$ with respect to $x$ .

Find the slope of the tangent line to $y=\frac{1}{x^{2}}-2 x^{1/2}$ at $x=1$ . Choices: $-1$ , 3, 1, 0, $-3$ .

Identify the FALSE statement about the surge function $C(t)=a t e^{b t}$ from the options provided.

1. Prove that the sequence $u_{n}=\frac{n-3}{n}$ is Cauchy.
2. For any real $x$ , find a rational sequence $x_{n}$ such that $\lim_{n \to +\infty} x_{n}=x$ . Use $x_{n}=\frac{\mathrm{E}(n x)}{n}$ .

Find the tangent line equation for $f(x)=(e^{2 x}+7 x^{2})^{3}$ at $x=0$ . Choose from the options.

Which function is not continuous at $x=2$ ? a. All are continuous b. $f(x)=\ln (x+2)$ c. $f(x)=2^{x-2}$ d. $f(x)=(x-2)^{2}$ e. $f(x)=\frac{2}{x}$

Find all $x$ where the function $f(x)=\frac{e^{5 x^{2}-3 x}}{3 x}$ has relative extrema.

Differentiate the equation $2 x^{2}-y^{2}=2$ implicitly to find $\frac{d y}{d x}$ .

Differentiate $2 x^{2}-y^{2}=2$ with respect to $x$ to find $\frac{d y}{d x}$ .

Find the volume using the shell method for the solid formed by revolving the region defined by $X=9-3y^2$ and $Y=\sqrt{3}$ around the $x$ -axis. The volume is $\square$ (exact answer in terms of $\pi$ ).

Evaluate the integral: $\int \frac{8}{\sqrt{x}(1+8 \sqrt{x})^{3}} d x$

Evaluate the integral: $\int \frac{1}{x^{2}} \sqrt{1+\frac{1}{x}} \, dx$

Find the limit as $x$ approaches 1 for the expression $\frac{2 \cos (x-1)-2 x}{2 \ln (3 x-2)}$ .

Find the volume of the solid formed by revolving the region bounded by $y=3$ , $x=\frac{1}{216}$ , and $y=1$ around the $x$ -axis using a. the washer method and b. the shell method.

Evaluate the integral: $\int \frac{5}{\sqrt{x}(9+5 \sqrt{x})^{6}} d x$

Differentiate the equation $2 x^{2}-5 y^{2}=3$ implicitly to find $\frac{d y}{d x}$ .

Evaluate the integral: $\int \frac{4}{\sqrt{x}(3+4 \sqrt{x})^{2}} d x$

Find the production level that minimizes the cost function $c(x)=\frac{x^{3}}{3}-x^{2}-8 x+400$ .

Estimate the cost of the 41st unit using marginal analysis from $C(q)=3q^2+q+500$ and calculate the actual cost.

Find the intercepts and asymptotes of $r(x)=(3x(x+2))/((x-1)(x-6))$ .

Differentiate $g(t)=t^{3} \cos t$ and find $g^{\prime}(t)$ .

Evaluate the limit: $\lim _{x \rightarrow \infty}\left(\frac{18 x}{18 x+4}\right)^{6 x}$ . Enter $-I$ , $I$ , or DNE for the answer.

Find the inflection point of the function $f(x)=\frac{x}{(x+1)^{2}}$ . Choose from the options: a, b, c, d.

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

Differentiate $y=\frac{x+4}{x^{3}+x-7}$ and verify if $y' = 1$ .

Differentiate the function $y=\frac{x+4}{x^{3}+x-7}$ and find $y'$ .

Differentiate the function $g(t)=\frac{t-\sqrt{t}}{t^{1/7}}$ .

Identify the type of discontinuity in the graph of $f(x)=\frac{x}{(x+1)(x-3)}$ : a) Two jump b) Three point c) Two asymptotic d) One asymptotic.

Differentiate $y=5 x^{2} \sin x \tan x$ .

Given $f(5)=2$ , $f^{\prime}(5)=6$ , $g(5)=-7$ , and $g^{\prime}(5)=1$ , find: (a) $(f g)^{\prime}(5)$ , (b) $(f / g)^{\prime}(5)$ , (c) $(g / f)^{\prime}(5)$ .

Find the equation of the tangent line to the curve $y=1/(1+x^{2})$ at the point $(-1, \frac{1}{2})$ . $y=$

Differentiate the function: $F(y)=\left(\frac{1}{y^{2}}-\frac{3}{y^{4}}\right)\left(y+5 y^{3}\right)$

Find the tangent line equation to the curve $y=\frac{x^{2}-1}{x^{2}+x+1}$ at the point (1,0).

Find the average rate of change of $f(x)=x^{3}-4$ from $x=-4$ to $x=3$ using $\frac{\Delta y}{\Delta x}$ .

Find the second derivative $f^{\prime \prime}(3)$ for the function $f(x)=\frac{x^{2}}{1+x}$ .

A hemispherical tank (radius $9 \mathrm{~m}$ ) fills at $4 \mathrm{~m}^{3}/\mathrm{min}$ . Find the water level rise rate when it's $7 \mathrm{~m}$ deep. Use $V = \frac{\pi h^{2}(27-h)}{3}$ and differentiate to find $\frac{d V}{d t}$ .

How long does it take for a bacteria population to double at a growth rate of $5.3\%$ per hour? Round to the nearest hundredth.

Differentiate the function $f(x)=7 \sqrt{x} \sin x$ .

Differentiate the function $y=\frac{8 x}{7-\cot x}$ .

Find the derivative of the following functions: 1. $y=\sqrt{1-4 x}$ , 2. $f(x)=\ln (\cos 5 x)$ .

Find the rectangle with max area with one corner at (0,0) and the opposite at $(x, y)$ on the curve $y=\frac{3}{1+4 x^{2}}$ . Determine the $x$ and $y$ values for max area, where $0 \leq x \leq 3$ .

Differentiate $f(\theta)=\frac{\sec \theta}{9+\sec \theta}$ .

Find the derivative of $\left(\frac{\sqrt[3]{x}}{x^{n}}\right)^{3}$ with respect to $x$ .

Find the derivative of $F(x)=(x^{4}+9 x^{2}-9)^{8}$ .

Calculate the integral from -1 to 1 of $(3 x^{4}-9)$ . What is $\int_{-1}^{1}(3 x^{4}-9) dx=\square$ ?

Find the derivative of $f(x)=(2x-3)^{4}(x^{2}+x+1)^{5}$ .

Find the derivative of $y=\cos \left(a^{4}+x^{4}\right)$ .

Find the tangent line equation for $h(x)=x \sqrt{4+2 x}$ at $x=6$ .

Evaluate the integral: $\int_{-\pi / 3}^{\pi / 3} 4 \sec ^{2} x \, dx = \square$

Find the volume of the solid formed by rotating $f(x)=2e^{-\frac{x}{2}}$ over $I=[1 ; \infty)$ around the x-axis.

Evaluate the integral $\int 3(7-2 x)^{3 / 4} d x$ .

Calculate the integral $\int_{-\pi}^{\pi} 3 \sin x \, dx = \square$ .

Find the derivative of the function $y=\left(\frac{x^{2}+5}{x^{2}-5}\right)^{3}$ .

Find $g^{\prime}(2)$ for $g(x)=2+x^{3}+\tan^{-1}(f(x))$ with $f(x)=2$ and $f'(x)=5$ .

How long until half the ozone is gone if it decreases at a continuous rate of $1.02\%$ ?

Find the average distance from the parabola $y=10 x(16-x)$ to the $x$ -axis over the interval $[0,16]$ .

Find the derivative of the function $y = x \sin \frac{3}{x}$ .

Calculate the integral from -2 to 2 of (15 - |x|^3). What is the value of $\int_{-2}^{2}\left(15-|x|^{3}\right) d x$ ?

Find the integral of $(3x + 1)^{1/3}$ with respect to $x$ .

Calculate the volume $V=\frac{1}{2} \pi \int_{-\infty}^{0} e^{4 x} d x$ .

Find the max and min of $f(x)=4 \cos^{2} x$ on $[0, \pi]$ . Choose A, B, C, or D and fill in the blanks.

Find the slope of $g(\theta)=4 \cos ^{3}(2 \theta)$ at $\theta=\frac{\pi}{6}$ .