Calculus

Problem 5301

Find the derivative of $f(x)=2 x^{6}-8 x^{4}+x^{3}-20$ and divide it by $x+2$ .

See Solution

Problem 5302

Find local extrema of the function $f(x)=-2 x^{\frac{3}{2}}+12 x+10$ using the First Derivative Test.

See Solution

Problem 5303

Find the intervals where $W(t)=41+\frac{54 t^{2}}{t^{2}+17}$ is increasing or decreasing. Express as open intervals.

See Solution

Problem 5304

Find the intervals where revenue $R(x)=54x-0.04x^{2}$ is increasing and decreasing for $0 \leq x \leq 1100$ .

See Solution

Problem 5305

Find the intervals where $W(t)=41+\frac{55 t^{2}}{t^{2}+16}$ is increasing or decreasing. Use open intervals for your answers.

See Solution

Problem 5306

Find the intervals where the revenue $R(x)=54x-0.04x^{2}$ is increasing and decreasing for $0 \leq x \leq 1100$ .

See Solution

Problem 5307

Find local extrema for the function $f(x)=\frac{64 x^{2}-1}{x}$ using the First Derivative Test.

See Solution

Problem 5308

What is the limit of the recording speed $W(t)=41+\frac{55 t^{2}}{t^{2}+16}$ as $t$ approaches infinity?

See Solution

Problem 5309

Find local extrema of the function $f(x)=\frac{36 x^{2}-25}{x}$ using the First Derivative Test. Enter as ordered pairs.

See Solution

Problem 5310

Find the derivative $q^{\prime}(p)$ of $q(p)=\ln(p^{4})$ . Which option is correct?

See Solution

Problem 5311

Given the curve $y^{2}=2+x y$ , find $\frac{d y}{d x}$ , points with slope $\frac{1}{2}$ , horizontal tangents, and $\frac{d x}{d t}$ at $t=5$ when $y=3$ and $\frac{d y}{d t}=6$ .

See Solution

Problem 5312

Berechne die Flächeninhalte für die Funktionen: a) $f(x)=x^{2}-x+1$ über $[0 ; 2]$ b) $f(x)=\frac{1}{x^{2}}$ über $[1 ; 3]$ c) $f(x)=x^{3}-x$ im 4. Quadranten. d) $f(x)=x^{3}-x$ über $[0 ; 2]$ (zwischen Kurve und $x$ -Achse).

See Solution

Problem 5313

Given the curve $x y^{2}-x^{3} y=6$ :
(a) Prove that $\frac{d y}{d x}=\frac{3 x^{2} y-y^{2}}{2 x y-x^{3}}$ .
(b) Find points with $x=1$ and write tangent line equations.
(c) Determine $x$ -coordinates where tangent lines are vertical.

See Solution

Problem 5314

Prove a value exists for $f(x)=8 x^{4}-3 x^{2}+4 x-1$ in $[-2,0]$ by finding $f(-2)$ .

See Solution

Problem 5315

Determine if the function $f(x)$ has a jump discontinuity at $x=2$ by finding $\lim _{x \rightarrow 2^{-}} f(x)$ and $\lim _{x \rightarrow 2^{+}} f(x)$ . Graph $f(x)$ .

See Solution

Problem 5316

Analyze $f(x)=13 x^{4}-5 x^{2}+7 x-1$ on [0,3] using the Intermediate Value Theorem. Find local/global extrema or specific $x$ .

See Solution

Problem 5317

Find $f(5)$ and $f^{\prime}(5)$ given the tangent line to $y=f(x)$ at $(5,1)$ passes through $(7,-4)$ .

See Solution

Problem 5318

If $S=x^{8}$ and $x=\ln (t)$ , what is $\frac{d S}{d t}$ ? Choose the correct option from the list.

See Solution

Problem 5319

Determine the curvilinear asymptote of the function $f(x)=\frac{x^{4}-x^{2}-8}{x^{2}-7}$ . Find $y=$ .

See Solution

Problem 5320

Find the 6th derivative of $f(x)=\cos(\ln(1+x^{2}))$ at $x=0$ using the Taylor series.

See Solution

Problem 5321

Verify Green's Theorem for the function $\phi(xy+y^2)$ over the curve $C$ bounded by $y=x$ and $y=x^2$ .

See Solution

Problem 5322

Evaluate the limit: $\lim _{n \rightarrow \infty} \frac{360}{n}$ . What is the result?

See Solution

Problem 5323

In the space provided, type the two-word phrase that completes the statement: If $f(a)>f(x)$ nearby, then $f(a)$ is a/an ___ of $f$ .

See Solution

Problem 5324

Find the function $f(x)$ and the value $a$ from the limit representing the derivative: $\lim _{h \rightarrow 0} \frac{(3+h)^{2}-9}{h}$ .

See Solution

Problem 5325

Berechne die Grenzwerte der Folgen: a) $a_{n}=\frac{1+2 n}{1+n}$ , b) $a_{n}=\frac{7 n^{3}+1}{n^{3}-10}$ , c) $a_{n}=\frac{n^{2}+2 n+1}{1+n+n^{2}}$ , d) $a_{n}=\frac{\sqrt{n}+n+n^{2}}{\sqrt{2 n}+n^{2}}$ , e) $a_{n}=\frac{n^{5}-n^{4}}{6 n^{5}-1}$ , f) $a_{n}=\frac{\sqrt{n+1}}{\sqrt{n+1}+2}$ , g) $a_{n}=\frac{(5-n)^{4}}{(5+n)^{4}}$ , h) $a_{n}=\frac{(2+n)^{10}}{(1+n)^{10}}$ , i) $a_{n}=\frac{(1+2 n)^{10}}{(1+n)^{10}}$ , j) $a_{n}=\frac{(1+2 n)^{k}}{(1+3 n)^{k}}$ . Bestimme die Grenzwerte: a) $\lim _{n \rightarrow \infty} \frac{2^{n}-1}{2^{n}}$ , b) $\lim _{n \rightarrow \infty} \frac{2^{n}-1}{2^{n-1}}$ , c) $\lim _{n \rightarrow \infty} \frac{2^{n}}{1+4^{n}}$ , d) $\lim _{n \rightarrow \infty} \frac{2^{n}-3^{n}}{2^{n}+3^{n}}$ , e) $\lim _{n \rightarrow \infty} \frac{2^{n}+3^{n+1}}{2 \cdot 3^{n}}$ . Berechne erneut die Grenzwerte: a) $a_{n}=\frac{n-3}{4 n}$ , b) $a_{n}=\frac{n \cdot(n+1)}{4 n^{2}}$ , c) $a_{n}=\frac{4+3^{n}}{3^{n}}$ , d) $a_{n}=\frac{3^{n}}{4+3^{n}}$ .

See Solution

Problem 5326

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

See Solution

Problem 5327

Differentiate both sides of $\sin(x+y) = 6x + 2\cos(y)$ .

See Solution

Problem 5328

Solve $\frac{d}{d x}(e^{2 x y}) = \frac{d}{d x}(\sin(y^7))$ for $x$ , with $y$ as a constant or function of $x$ .

See Solution

Problem 5329

Solve for the function $y = y(x)$ in the equation $\frac{d y}{d x}\left(e^{2 x y}\right)=\frac{d y}{d x}\left(\sin \left(y^{7}\right)\right)$ .

See Solution

Problem 5330

Find $\frac{dy}{dx}$ for the equation $e^{2xy} = \sin(y^7)$ .

See Solution

Problem 5331

Find the derivative of Y with respect to $X$ for the equation $6 x^{2}+2 x^{3} y=3 x+y$ .

See Solution

Problem 5332

Find the tangent line equation to the curve $xy + x^2y^2 = 20$ at the point $(4, 1)$ .

See Solution

Problem 5333

Find the tangent line equation to the curve $x^{2}+\sin (y)=x y^{2}+9$ at the point $(3,0)$ .

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Problem 5334

Consider the function $g(x)=\frac{1}{2 x-3}$ .
a) What is the domain restriction? b) Sketch the graph of $g(x)$ . c) What happens as $x$ approaches the restriction from the left and right? d) What happens as $x \rightarrow \infty$ and $x \rightarrow -\infty$ ? e) Estimate the slope at $x=1.6, 2, 10$ and $x=1.4, 0, -10$ . f) Describe the graph in terms of asymptotes and slope.

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Problem 5335

Calculate the following using the provided values for $f(x)$ and $g(x)$ at $x=3$ :
1. $\frac{f(3)}{g(3)+5}$
2. $\left(\frac{f}{g}\right)^{\prime}(3)$
3. $(f-g)^{\prime}(3)$
4. $(f g)^{\prime}(3)$

See Solution

Problem 5336

Find points on the graph of $3 x^{2}+4 y^{2}+3 x y=16$ where the tangent line is horizontal.

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Problem 5337

Find $y^{\prime \prime}$ in terms of $x$ and $y$ for the equation $y^{\prime}=\frac{x}{y^{4}}$ .

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Problem 5338

Find the tangent line equation at the point (3, 9) for the curve $e^{x^{2}-y}=\frac{x^{2}}{y}$ .

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Problem 5339

Calculate the following using the given functions:
1. $(f g)^{\prime}(-3)$
2. $(f-g)^{\prime}(-3)$
3. $(f g)(4)$
4. $\frac{f(-3)}{g(-3)+5}$
Given results:
1. 4290, 2. -63, 3. 2400, 4. 7.14.

See Solution

Problem 5340

Find the derivative of $f(x)=(x+x^{5})(-x-2)$ using the product rule. What is $f^{\prime}(x)$ ?

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Problem 5341

Calculate the following using the given functions and their derivatives: $(f g)(2)$ , $(f g)^{\prime}(3)$ , $\frac{f(3)}{g(3)+5}$ , and $(f+g)^{\prime}(3)$ .

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Problem 5342

Find the derivative using the Product Rule for $f(x)=(2 x-6)(2 e^{x}+1)$ . What is $f^{\prime}(x)=?$

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Problem 5343

Show that the polynomial $f(x)=8x^{4}-3x^{2}+4x-1$ has a zero in $[-2,0]$ using the Intermediate Value Theorem. Find $f(-2)$ . Simplify: $f(-2)=\square$ .

See Solution

Problem 5344

Find the world population growth rate in millions/year for the years 1920, 1951, and 2000 using the model $P(t)=(1436.53)(1.01395)^{t}$ .

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Problem 5345

Find the derivative of $h(s)=\left(s^{-1 / 2}+4 s\right)\left(5-s^{-1}\right)$ at $s=16$ . $\left.\frac{d h}{d s}\right|_{s=16}=$

See Solution

Problem 5346

Find the limit: $\lim _{x \rightarrow 6} \frac{x^{5}-7776}{x-6}$ using the factorization formula for $x^{n}-a^{n}$ .

See Solution

Problem 5347

Prove that $\lim _{x \rightarrow 0}|x|=0$ by evaluating $\lim _{x \rightarrow 0^{-}}|x|$ and $\lim _{x \rightarrow 0^{+}}|x|$ .

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Problem 5348

Find the local max and min points and their values for the function $f$ with peak at (-1.5, 1.7) and lowest at (1, 1.8).

See Solution

Problem 5349

Find the limit: $\lim _{x \rightarrow 6} \frac{x^{5}-7776}{x-6}$ using the factorization formula for $x^{n}-a^{n}$ .

See Solution

Problem 5350

Find the limit: $\lim _{x \rightarrow 6,561} \frac{\sqrt[4]{x}-9}{x-6,561}$ .

See Solution

Problem 5351

Find the limit as $x$ approaches 10,000 for $\frac{\sqrt[4]{x}-10}{x-10,000}$ .

See Solution

Problem 5352

Find the derivative of $f(x)=(x+x^{4})(-x-1)$ using the product rule. What is $f^{\prime}(x)=?$

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Problem 5353

Find the limit: $\lim _{x \rightarrow 6} \frac{x^{5}-7776}{x-6}$ .

See Solution

Problem 5354

Find the derivative of $y=\frac{x^{3}}{(3 x^{2}+2)^{2}}$ using the log shortcut method. What is $\frac{d y}{d x}$ ?

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Problem 5355

Find the derivative of $y=e^{(5x^{3}-2x+1)^{4}}$ . What is $\frac{dy}{dx}$ ?

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Problem 5356

Find the derivative of $y=e^{3x}$ . What is $\frac{dy}{dx}$ ? Options: $3 e^{3 x}$ , $3 e^{2 x}$ , $3 x e^{3 x}$ , $e^{2 x}$ , $3 e^{x}$ .

See Solution

Problem 5357

Find the limit: $\lim _{x \rightarrow 6} \frac{x^{5}-7776}{x-6}$ using the factorization formula for $x^{n}-a^{n}$ .

See Solution

Problem 5358

Find the derivative of $y=\ln \frac{3}{x^{2}}$ . What is $\frac{d y}{d x}$ ?

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Problem 5359

Find the local minimum of the function $f$ given its peak at $(-1.5, 1.7)$ and lowest point at $(1, -1.8)$ .

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Problem 5360

Find $\lim_{x \rightarrow 0^{+}} g(x)$ and $\lim_{x \rightarrow 0^{-}} g(x)$ given $g(x)=f(9-x)$ and limits of $f(x)$ at 9.

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Problem 5361

Calculate the average rate of change of $h(x)=5-2 x^{2}$ from $x=-2$ to $x=4$ .

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Problem 5362

Given functions $p(x)=\frac{9 x}{5 x+3}$ and $q(x)=4 x-1$ , find $r^{\prime}$ for $r(x)=\frac{p(x)}{q(x)}$ .

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Problem 5363

Find $\lim _{x \rightarrow 3} \frac{f(x)}{g(x)}$ if $f(x) \rightarrow -250$ and $g(x) \rightarrow 0^+$ as $x \rightarrow 3$ .

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Problem 5364

Calculate the average rate of change of $g(x)=3 x^{3}-1$ over the interval $[-3,3]$ .

See Solution

Problem 5365

Given $p(x)=\frac{9 x}{5 x+3}$ and $q(x)=4 x-1$ , find $r^{\prime}$ for $r(x)=\frac{p(x)}{q(x)}$ .

See Solution

Problem 5366

Find $\lim _{x \rightarrow 5} \frac{f(x)}{g(x)}$ given that $f(x) \rightarrow -200$ and $g(x) \rightarrow 0^+$ as $x \rightarrow 5$ .

See Solution

Problem 5367

Find the derivative of $h(s)=\left(s^{-1 / 2}+2 s\right)\left(7-s^{-1}\right)$ at $s=4$ . Compute $\left.\frac{d h}{d s}\right|_{s=4}=$

See Solution

Problem 5368

Find the limit as $x$ approaches 6 from the left for the expression $\frac{1}{x-6}$ .

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Problem 5369

Find the linearization of $f(x, y)=\sqrt{76-4 x^{2}-4 y^{2}}$ at $(3,3)$ . What is $L(x, y)=?$

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Problem 5370

Find the limit: $\lim _{x \rightarrow 2^{+}} \frac{x^{2}-8 x+15}{(x-2)^{2}}$

See Solution

Problem 5371

Integrate $\int \frac{\ln(15 x^{5})}{x} dx$ . What is the result?

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Problem 5372

Find the limit as $x$ approaches 0 for the expression $\frac{x^{6}+2 x^{5}}{x^{5}}$ .

See Solution

Problem 5373

Find the limit as $x$ approaches 6 from the right: $\lim _{x \rightarrow 6^{+}} \frac{1}{x-6}$ .

See Solution

Problem 5374

Compute the integral: $\int \frac{4 x}{(x^{2}-1)^{3}} dx$ . What is the result?

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Problem 5375

Compute the integral of $x^{2} e^{x/2}$ dx using integration by parts multiple times.

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Problem 5376

Find the derivative of $h(s)=(s^{-1 / 2}+2 s)(7-s^{-1})$ at $s=4$ using the Product Rule. $\left.\frac{d h}{d s}\right|_{s=4}=$

See Solution

Problem 5377

Evaluate the limit: $\lim _{x \rightarrow 5^{+}} \frac{x+6}{\sqrt{x-5}}$ .

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Problem 5378

Find the limit as $\theta$ approaches $\frac{7 \pi^{+}}{2}$ for $\frac{1}{4} \tan \theta$ .

See Solution

Problem 5379

Find the antiderivative of the logarithm: $I(x)=\int \ln x \, dx$ . Use integration by parts.

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Problem 5380

Find the limit as $x$ approaches 2 from the left: $\lim _{x \rightarrow 2^{-}} \frac{x^{2}-8 x+15}{(x-2)^{2}}$ .

See Solution

Problem 5381

Integrate the function: $\int \frac{\ln(15 x^{5})}{x} \, dx$

See Solution

Problem 5382

Find the limit: $\lim _{x \rightarrow 1^{+}} \frac{x^{2}-7 x+10}{x-1}$ .

See Solution

Problem 5383

Find the rate of change of surface area $S=4 \pi r^{2}$ with respect to radius for $r=9$ and $r=14$ .

See Solution

Problem 5384

Find the derivative of $f(x)=\frac{\tan x-5}{\sec x}$ and compute $f^{\prime}\left(\frac{\pi}{3}\right)$ exactly.

See Solution

Problem 5385

Find $r^{\prime}$ where $r(x)=\frac{p(x)}{q(x)}$ , with $p(x)=\frac{3x}{7x+3}$ and $q(x)=2x-1$ .

See Solution

Problem 5386

Find the derivative of $f(x)=\frac{9}{x}$ at $x=2$ using the limit definition: $f^{\prime}(2)=\lim _{h \rightarrow 0}\left(\frac{f(2+h)-f(2)}{h}\right)=\lim _{h \rightarrow 0}(\square)=\square$ .

See Solution

Problem 5387

Evaluate the integral using integration by parts: $\int x^{2} e^{x / 2} d x$ .

See Solution

Problem 5388

Find the derivative of $f(x)=\frac{9}{x}$ using the definition: $f^{\prime}(x)=\lim _{h \rightarrow 0}\left(\frac{f(x+h)-f(x)}{h}\right)$ . Simplify and evaluate the limit.

See Solution

Problem 5389

Find the derivative of the function $h(x)=\frac{4 x+1}{3 x-4}$ . Calculate $h^{\prime}(x)$ .

See Solution

Problem 5390

Find the derivative of $f(x)=\frac{x+6}{x^{2}+5 x+1}$ using the Quotient Rule.

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Problem 5391

Find the derivative of $g(t)=\frac{t^{4}+4}{t^{4}-4}$ at $t=1$ using the Quotient Rule. $\left.\frac{d g}{d t}\right|_{t=1}=$

See Solution

Problem 5392

Find the domain of the vector function $\vec{r}(t)=\langle 5 \sin (t), 3 \cos (t), \ln (t+5)\rangle$ . Domain: $\{t \mid t > -5\}$ .

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Problem 5393

Find the half-life of a radioactive substance with a decay rate of $4.1\%$ per day. Round to the nearest hundredth.

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Problem 5394

Find the hourly growth rate of a bacteria population that grows from 2100 to 2517 in 4 hours. Express as a percentage.

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Problem 5395

Find the time in hours for a bacteria population to double with a growth rate of $9.2\%$ per hour. Round to the nearest hundredth.

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Problem 5396

Find $\lim _{x \rightarrow -4} f(x)$ given that $1 x+1 \leq f(x) \leq x^{2}+9 x+17$ . What theorem did you use?

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Problem 5397

Find the derivative of $f(x)=\frac{5}{1+e^{x}}$ using the Quotient Rule. $f^{\prime}(x)=$

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Problem 5398

Evaluate the limit $\lim _{x \rightarrow 6} f(x)$ using the Squeeze Theorem, given $12 x-36 \leq f(x) \leq x^{2}$ on $[4,8]$ .

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Problem 5399

Find the derivative of $w(x)=\frac{x^{7}}{\sqrt{x}+x}$ at $x=1$ using the Quotient Rule. What is $\left.\frac{d w}{d x}\right|_{x=1}=$ ?

See Solution

Problem 5400

Evaluate the limit as $t$ approaches infinity: $\lim _{t \rightarrow \infty}\left\langle 2 e^{-t}, e^{-2 t}, \ln (t-2)\right\rangle$ .

See Solution

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Calculus

Problem 5301

Find the derivative of f(x)=2x6−8x4+x3−20f(x)=2 x^{6}-8 x^{4}+x^{3}-20f(x)=2x6−8x4+x3−20 and divide it by x+2x+2x+2.

Problem 5302

Find local extrema of the function f(x)=−2x32+12x+10f(x)=-2 x^{\frac{3}{2}}+12 x+10f(x)=−2x23​+12x+10 using the First Derivative Test.

Problem 5303

Find the intervals where W(t)=41+54t2t2+17W(t)=41+\frac{54 t^{2}}{t^{2}+17}W(t)=41+t2+1754t2​ is increasing or decreasing. Express as open intervals.

Problem 5304

Find the intervals where revenue R(x)=54x−0.04x2R(x)=54x-0.04x^{2}R(x)=54x−0.04x2 is increasing and decreasing for 0≤x≤11000 \leq x \leq 11000≤x≤1100.

Problem 5305

Find the intervals where W(t)=41+55t2t2+16W(t)=41+\frac{55 t^{2}}{t^{2}+16}W(t)=41+t2+1655t2​ is increasing or decreasing. Use open intervals for your answers.

Problem 5306

Find the intervals where the revenue R(x)=54x−0.04x2R(x)=54x-0.04x^{2}R(x)=54x−0.04x2 is increasing and decreasing for 0≤x≤11000 \leq x \leq 11000≤x≤1100.

Problem 5307

Find local extrema for the function f(x)=64x2−1xf(x)=\frac{64 x^{2}-1}{x}f(x)=x64x2−1​ using the First Derivative Test.

Problem 5308

What is the limit of the recording speed W(t)=41+55t2t2+16W(t)=41+\frac{55 t^{2}}{t^{2}+16}W(t)=41+t2+1655t2​ as ttt approaches infinity?

Problem 5309

Find local extrema of the function f(x)=36x2−25xf(x)=\frac{36 x^{2}-25}{x}f(x)=x36x2−25​ using the First Derivative Test. Enter as ordered pairs.

Problem 5310

Find the derivative q′(p)q^{\prime}(p)q′(p) of q(p)=ln⁡(p4)q(p)=\ln(p^{4})q(p)=ln(p4). Which option is correct?

Problem 5311

Given the curve y2=2+xyy^{2}=2+x yy2=2+xy, find dydx\frac{d y}{d x}dxdy​, points with slope 12\frac{1}{2}21​, horizontal tangents, and dxdt\frac{d x}{d t}dtdx​ at t=5t=5t=5 when y=3y=3y=3 and dydt=6\frac{d y}{d t}=6dtdy​=6.

Problem 5312

Problem 5313

Problem 5314

Prove a value exists for f(x)=8x4−3x2+4x−1f(x)=8 x^{4}-3 x^{2}+4 x-1f(x)=8x4−3x2+4x−1 in [−2,0][-2,0][−2,0] by finding f(−2)f(-2)f(−2).

Problem 5315

Determine if the function f(x)f(x)f(x) has a jump discontinuity at x=2x=2x=2 by finding lim⁡x→2−f(x)\lim _{x \rightarrow 2^{-}} f(x)limx→2−​f(x) and lim⁡x→2+f(x)\lim _{x \rightarrow 2^{+}} f(x)limx→2+​f(x). Graph f(x)f(x)f(x).

Problem 5316

Analyze f(x)=13x4−5x2+7x−1f(x)=13 x^{4}-5 x^{2}+7 x-1f(x)=13x4−5x2+7x−1 on [0,3] using the Intermediate Value Theorem. Find local/global extrema or specific xxx.

Problem 5317

Find f(5)f(5)f(5) and f′(5)f^{\prime}(5)f′(5) given the tangent line to y=f(x)y=f(x)y=f(x) at (5,1)(5,1)(5,1) passes through (7,−4)(7,-4)(7,−4).

Problem 5318

If S=x8S=x^{8}S=x8 and x=ln⁡(t)x=\ln (t)x=ln(t), what is dSdt\frac{d S}{d t}dtdS​? Choose the correct option from the list.

Problem 5319

Determine the curvilinear asymptote of the function f(x)=x4−x2−8x2−7f(x)=\frac{x^{4}-x^{2}-8}{x^{2}-7}f(x)=x2−7x4−x2−8​. Find y=y=y=.

Problem 5320

Find the 6th derivative of f(x)=cos⁡(ln⁡(1+x2))f(x)=\cos(\ln(1+x^{2}))f(x)=cos(ln(1+x2)) at x=0x=0x=0 using the Taylor series.

Problem 5321

Verify Green's Theorem for the function ϕ(xy+y2)\phi(xy+y^2)ϕ(xy+y2) over the curve CCC bounded by y=xy=xy=x and y=x2y=x^2y=x2.

Problem 5322

Evaluate the limit: lim⁡n→∞360n\lim _{n \rightarrow \infty} \frac{360}{n}limn→∞​n360​. What is the result?

Problem 5323

In the space provided, type the two-word phrase that completes the statement: If f(a)>f(x)f(a)>f(x)f(a)>f(x) nearby, then f(a)f(a)f(a) is a/an ___ of fff.

Problem 5324

Find the function f(x)f(x)f(x) and the value aaa from the limit representing the derivative: lim⁡h→0(3+h)2−9h\lim _{h \rightarrow 0} \frac{(3+h)^{2}-9}{h}h→0lim​h(3+h)2−9​.

Problem 5325

Problem 5326

Find the limit as xxx approaches 8 for 1x+1−3\frac{1}{\sqrt{x+1}-3}x+1​−31​.

Problem 5327

Differentiate both sides of sin⁡(x+y)=6x+2cos⁡(y)\sin(x+y) = 6x + 2\cos(y)sin(x+y)=6x+2cos(y).

Problem 5328

Solve ddx(e2xy)=ddx(sin⁡(y7))\frac{d}{d x}(e^{2 x y}) = \frac{d}{d x}(\sin(y^7))dxd​(e2xy)=dxd​(sin(y7)) for xxx, with yyy as a constant or function of xxx.

Problem 5329

Solve for the function y=y(x)y = y(x)y=y(x) in the equation dydx(e2xy)=dydx(sin⁡(y7))\frac{d y}{d x}\left(e^{2 x y}\right)=\frac{d y}{d x}\left(\sin \left(y^{7}\right)\right)dxdy​(e2xy)=dxdy​(sin(y7)).

Problem 5330

Find dydx\frac{dy}{dx}dxdy​ for the equation e2xy=sin⁡(y7)e^{2xy} = \sin(y^7)e2xy=sin(y7).

Problem 5331

Find the derivative of Y with respect to XXX for the equation 6x2+2x3y=3x+y6 x^{2}+2 x^{3} y=3 x+y6x2+2x3y=3x+y.

Problem 5332

Find the tangent line equation to the curve xy+x2y2=20xy + x^2y^2 = 20xy+x2y2=20 at the point (4,1)(4, 1)(4,1).

Problem 5333

Find the tangent line equation to the curve x2+sin⁡(y)=xy2+9x^{2}+\sin (y)=x y^{2}+9x2+sin(y)=xy2+9 at the point (3,0)(3,0)(3,0).

Problem 5334

Problem 5335

Calculate the following using the provided values for f(x)f(x)f(x) and g(x)g(x)g(x) at x=3x=3x=3: 1. f(3)g(3)+5\frac{f(3)}{g(3)+5}g(3)+5f(3)​2. (fg)′(3)\left(\frac{f}{g}\right)^{\prime}(3)(gf​)′(3)3. (f−g)′(3)(f-g)^{\prime}(3)(f−g)′(3)4. (fg)′(3)(f g)^{\prime}(3)(fg)′(3)

Problem 5336

Find points on the graph of 3x2+4y2+3xy=163 x^{2}+4 y^{2}+3 x y=163x2+4y2+3xy=16 where the tangent line is horizontal.

Problem 5337

Find y′′y^{\prime \prime}y′′ in terms of xxx and yyy for the equation y′=xy4y^{\prime}=\frac{x}{y^{4}}y′=y4x​.

Problem 5338

Find the tangent line equation at the point (3, 9) for the curve ex2−y=x2ye^{x^{2}-y}=\frac{x^{2}}{y}ex2−y=yx2​.

Problem 5339

Calculate the following using the given functions: 1. (fg)′(−3)(f g)^{\prime}(-3)(fg)′(−3)2. (f−g)′(−3)(f-g)^{\prime}(-3)(f−g)′(−3)3. (fg)(4)(f g)(4)(fg)(4)4. f(−3)g(−3)+5\frac{f(-3)}{g(-3)+5}g(−3)+5f(−3)​Given results: 1. 4290, 2. -63, 3. 2400, 4. 7.14.

Problem 5340

Find the derivative of f(x)=(x+x5)(−x−2)f(x)=(x+x^{5})(-x-2)f(x)=(x+x5)(−x−2) using the product rule. What is f′(x)f^{\prime}(x)f′(x)?

Problem 5341

Calculate the following using the given functions and their derivatives: (fg)(2)(f g)(2)(fg)(2), (fg)′(3)(f g)^{\prime}(3)(fg)′(3), f(3)g(3)+5\frac{f(3)}{g(3)+5}g(3)+5f(3)​, and (f+g)′(3)(f+g)^{\prime}(3)(f+g)′(3).

Problem 5342

Find the derivative of $f(x)=2 x^{6}-8 x^{4}+x^{3}-20$ and divide it by $x+2$ .

Find local extrema of the function $f(x)=-2 x^{\frac{3}{2}}+12 x+10$ using the First Derivative Test.

Find the intervals where $W(t)=41+\frac{54 t^{2}}{t^{2}+17}$ is increasing or decreasing. Express as open intervals.

Find the intervals where revenue $R(x)=54x-0.04x^{2}$ is increasing and decreasing for $0 \leq x \leq 1100$ .

Find the intervals where $W(t)=41+\frac{55 t^{2}}{t^{2}+16}$ is increasing or decreasing. Use open intervals for your answers.

Find the intervals where the revenue $R(x)=54x-0.04x^{2}$ is increasing and decreasing for $0 \leq x \leq 1100$ .

Find local extrema for the function $f(x)=\frac{64 x^{2}-1}{x}$ using the First Derivative Test.

What is the limit of the recording speed $W(t)=41+\frac{55 t^{2}}{t^{2}+16}$ as $t$ approaches infinity?

Find local extrema of the function $f(x)=\frac{36 x^{2}-25}{x}$ using the First Derivative Test. Enter as ordered pairs.

Find the derivative $q^{\prime}(p)$ of $q(p)=\ln(p^{4})$ . Which option is correct?

Given the curve $y^{2}=2+x y$ , find $\frac{d y}{d x}$ , points with slope $\frac{1}{2}$ , horizontal tangents, and $\frac{d x}{d t}$ at $t=5$ when $y=3$ and $\frac{d y}{d t}=6$ .

Prove a value exists for $f(x)=8 x^{4}-3 x^{2}+4 x-1$ in $[-2,0]$ by finding $f(-2)$ .

Determine if the function $f(x)$ has a jump discontinuity at $x=2$ by finding $\lim _{x \rightarrow 2^{-}} f(x)$ and $\lim _{x \rightarrow 2^{+}} f(x)$ . Graph $f(x)$ .

Analyze $f(x)=13 x^{4}-5 x^{2}+7 x-1$ on [0,3] using the Intermediate Value Theorem. Find local/global extrema or specific $x$ .

Find $f(5)$ and $f^{\prime}(5)$ given the tangent line to $y=f(x)$ at $(5,1)$ passes through $(7,-4)$ .

If $S=x^{8}$ and $x=\ln (t)$ , what is $\frac{d S}{d t}$ ? Choose the correct option from the list.

Determine the curvilinear asymptote of the function $f(x)=\frac{x^{4}-x^{2}-8}{x^{2}-7}$ . Find $y=$ .

Find the 6th derivative of $f(x)=\cos(\ln(1+x^{2}))$ at $x=0$ using the Taylor series.

Verify Green's Theorem for the function $\phi(xy+y^2)$ over the curve $C$ bounded by $y=x$ and $y=x^2$ .

Evaluate the limit: $\lim _{n \rightarrow \infty} \frac{360}{n}$ . What is the result?

In the space provided, type the two-word phrase that completes the statement: If $f(a)>f(x)$ nearby, then $f(a)$ is a/an ___ of $f$ .

Find the function $f(x)$ and the value $a$ from the limit representing the derivative: $\lim _{h \rightarrow 0} \frac{(3+h)^{2}-9}{h}$ .

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

Differentiate both sides of $\sin(x+y) = 6x + 2\cos(y)$ .

Solve $\frac{d}{d x}(e^{2 x y}) = \frac{d}{d x}(\sin(y^7))$ for $x$ , with $y$ as a constant or function of $x$ .

Solve for the function $y = y(x)$ in the equation $\frac{d y}{d x}\left(e^{2 x y}\right)=\frac{d y}{d x}\left(\sin \left(y^{7}\right)\right)$ .

Find $\frac{dy}{dx}$ for the equation $e^{2xy} = \sin(y^7)$ .

Find the derivative of Y with respect to $X$ for the equation $6 x^{2}+2 x^{3} y=3 x+y$ .

Find the tangent line equation to the curve $xy + x^2y^2 = 20$ at the point $(4, 1)$ .

Find the tangent line equation to the curve $x^{2}+\sin (y)=x y^{2}+9$ at the point $(3,0)$ .

Calculate the following using the provided values for $f(x)$ and $g(x)$ at $x=3$ :
1. $\frac{f(3)}{g(3)+5}$
2. $\left(\frac{f}{g}\right)^{\prime}(3)$
3. $(f-g)^{\prime}(3)$
4. $(f g)^{\prime}(3)$

Find points on the graph of $3 x^{2}+4 y^{2}+3 x y=16$ where the tangent line is horizontal.

Find $y^{\prime \prime}$ in terms of $x$ and $y$ for the equation $y^{\prime}=\frac{x}{y^{4}}$ .

Find the tangent line equation at the point (3, 9) for the curve $e^{x^{2}-y}=\frac{x^{2}}{y}$ .

Calculate the following using the given functions:
1. $(f g)^{\prime}(-3)$
2. $(f-g)^{\prime}(-3)$
3. $(f g)(4)$
4. $\frac{f(-3)}{g(-3)+5}$
Given results:
1. 4290, 2. -63, 3. 2400, 4. 7.14.

Find the derivative of $f(x)=(x+x^{5})(-x-2)$ using the product rule. What is $f^{\prime}(x)$ ?

Calculate the following using the given functions and their derivatives: $(f g)(2)$ , $(f g)^{\prime}(3)$ , $\frac{f(3)}{g(3)+5}$ , and $(f+g)^{\prime}(3)$ .

Find the derivative using the Product Rule for $f(x)=(2 x-6)(2 e^{x}+1)$ . What is $f^{\prime}(x)=?$

Show that the polynomial $f(x)=8x^{4}-3x^{2}+4x-1$ has a zero in $[-2,0]$ using the Intermediate Value Theorem. Find $f(-2)$ . Simplify: $f(-2)=\square$ .

Find the world population growth rate in millions/year for the years 1920, 1951, and 2000 using the model $P(t)=(1436.53)(1.01395)^{t}$ .

Find the derivative of $h(s)=\left(s^{-1 / 2}+4 s\right)\left(5-s^{-1}\right)$ at $s=16$ . $\left.\frac{d h}{d s}\right|_{s=16}=$

Find the limit: $\lim _{x \rightarrow 6} \frac{x^{5}-7776}{x-6}$ using the factorization formula for $x^{n}-a^{n}$ .

Prove that $\lim _{x \rightarrow 0}|x|=0$ by evaluating $\lim _{x \rightarrow 0^{-}}|x|$ and $\lim _{x \rightarrow 0^{+}}|x|$ .

Find the local max and min points and their values for the function $f$ with peak at (-1.5, 1.7) and lowest at (1, 1.8).

Find the limit: $\lim _{x \rightarrow 6} \frac{x^{5}-7776}{x-6}$ using the factorization formula for $x^{n}-a^{n}$ .

Find the limit: $\lim _{x \rightarrow 6,561} \frac{\sqrt[4]{x}-9}{x-6,561}$ .

Find the limit as $x$ approaches 10,000 for $\frac{\sqrt[4]{x}-10}{x-10,000}$ .

Find the derivative of $f(x)=(x+x^{4})(-x-1)$ using the product rule. What is $f^{\prime}(x)=?$

Find the limit: $\lim _{x \rightarrow 6} \frac{x^{5}-7776}{x-6}$ .

Find the derivative of $y=\frac{x^{3}}{(3 x^{2}+2)^{2}}$ using the log shortcut method. What is $\frac{d y}{d x}$ ?

Find the derivative of $y=e^{(5x^{3}-2x+1)^{4}}$ . What is $\frac{dy}{dx}$ ?

Find the derivative of $y=e^{3x}$ . What is $\frac{dy}{dx}$ ? Options: $3 e^{3 x}$ , $3 e^{2 x}$ , $3 x e^{3 x}$ , $e^{2 x}$ , $3 e^{x}$ .

Find the limit: $\lim _{x \rightarrow 6} \frac{x^{5}-7776}{x-6}$ using the factorization formula for $x^{n}-a^{n}$ .

Find the derivative of $y=\ln \frac{3}{x^{2}}$ . What is $\frac{d y}{d x}$ ?

Find the local minimum of the function $f$ given its peak at $(-1.5, 1.7)$ and lowest point at $(1, -1.8)$ .

Find $\lim_{x \rightarrow 0^{+}} g(x)$ and $\lim_{x \rightarrow 0^{-}} g(x)$ given $g(x)=f(9-x)$ and limits of $f(x)$ at 9.

Calculate the average rate of change of $h(x)=5-2 x^{2}$ from $x=-2$ to $x=4$ .

Given functions $p(x)=\frac{9 x}{5 x+3}$ and $q(x)=4 x-1$ , find $r^{\prime}$ for $r(x)=\frac{p(x)}{q(x)}$ .

Find $\lim _{x \rightarrow 3} \frac{f(x)}{g(x)}$ if $f(x) \rightarrow -250$ and $g(x) \rightarrow 0^+$ as $x \rightarrow 3$ .

Calculate the average rate of change of $g(x)=3 x^{3}-1$ over the interval $[-3,3]$ .

Given $p(x)=\frac{9 x}{5 x+3}$ and $q(x)=4 x-1$ , find $r^{\prime}$ for $r(x)=\frac{p(x)}{q(x)}$ .

Find $\lim _{x \rightarrow 5} \frac{f(x)}{g(x)}$ given that $f(x) \rightarrow -200$ and $g(x) \rightarrow 0^+$ as $x \rightarrow 5$ .

Find the derivative of $h(s)=\left(s^{-1 / 2}+2 s\right)\left(7-s^{-1}\right)$ at $s=4$ . Compute $\left.\frac{d h}{d s}\right|_{s=4}=$

Find the limit as $x$ approaches 6 from the left for the expression $\frac{1}{x-6}$ .

Find the linearization of $f(x, y)=\sqrt{76-4 x^{2}-4 y^{2}}$ at $(3,3)$ . What is $L(x, y)=?$

Find the limit: $\lim _{x \rightarrow 2^{+}} \frac{x^{2}-8 x+15}{(x-2)^{2}}$

Integrate $\int \frac{\ln(15 x^{5})}{x} dx$ . What is the result?

Find the limit as $x$ approaches 0 for the expression $\frac{x^{6}+2 x^{5}}{x^{5}}$ .

Find the limit as $x$ approaches 6 from the right: $\lim _{x \rightarrow 6^{+}} \frac{1}{x-6}$ .

Compute the integral: $\int \frac{4 x}{(x^{2}-1)^{3}} dx$ . What is the result?