Calculus

Problem 8401

True or False: For $f(x)=(3 x-1)(5 x+3)$ , does $f^{\prime}(x)$ yield the same result using product or power rule?

See Solution

Problem 8402

Find the max and min of $f(x, y) = 6xy$ subject to $\frac{x^{2}}{16}+\frac{y^{2}}{36}=1$ using Lagrange multipliers.

See Solution

Problem 8403

Find the absolute max and min of $f(x)=-6 x^{3}-72 x^{2}-270 x$ on $[-7,7]$ as an ordered pair.

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Problem 8404

Analyze the curve $y+\cos y=x+1$ for $0 \leq y \leq 2 \pi$ : (a) Find $\frac{d y}{d x}$ in terms of $y$ . (b) Determine vertical tangent equations. (c) Find $\frac{d^{2} y}{d x^{2}}$ in terms of $y$ .

See Solution

Problem 8405

Find the derivative of $g(t)=2 t e^{3 t}$ . Options: $g^{\prime}(t)=(3 t+6) e^{3 t}$ , $g^{\prime}(t)=(6 t+2) e^{3 t}$ , $g^{\prime}(t)=2 e^{3 t}$ , $g^{\prime}(t)=6 t e^{3 t}$ , $g^{\prime}(t)=(2 t+6) e^{3 t}$ .

See Solution

Problem 8406

Find the limit: $\lim _{x \rightarrow \infty} \arctan (x)$ .

See Solution

Problem 8407

Find the derivative $f^{\prime}(2)$ of the drug quantity function $Q=f(t)=312 t(0.5488)^{t}$ . Round to 2 decimal places.

See Solution

Problem 8408

Find the derivative of $f(x)=\sqrt{x} \cdot 2^{-x}$ .

See Solution

Problem 8409

What is the speed of water hitting the bottom of Victoria Falls, given a height of $108 \mathrm{~m}$ and horizontal flow of $3.60 \mathrm{~m/s}$ ?

See Solution

Problem 8410

Bestimme den Flächeninhalt unter $f(x)=x^{2}$ von 0 bis $x$ mit $n$ Streifen (Obersummenfläche).

See Solution

Problem 8411

Berechnen Sie die Ableitung von $f(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-2 x$ .

See Solution

Problem 8412

Calculate the limit of $\frac{f(x+h)-f(x)}{h}$ for $f(x)=x^{2}-5x-1$ .

See Solution

Problem 8413

Berechne die Durchschnittsgeschwindigkeit des Schlittens mit $s(t)=\frac{1}{2} t^{2}$ in den ersten 5 Sekunden. Bestimme die Momentangeschwindigkeit bei $x_{0}=5$ und erkläre den Unterschied zwischen mittlerer und lokaler Änderungsrate.

See Solution

Problem 8414

Find $\frac{f(x+h)-f(x)}{h}$ for $f(x)=3 x^{2}-x+5$ , where $h \neq 0$ , and simplify.

See Solution

Problem 8415

Find the horizontal asymptote of $f(x)=5 \frac{(x+2)(8 x-1)}{(6-x)(8 x+2)}$ . What is $y$ ?

See Solution

Problem 8416

Given the curve $x y^{2}-x^{3} y=6$ , find: (a) $\frac{d y}{d x}$ , (b) points with $x=1$ and tangent line equations, (c) $x$ -coordinates of vertical tangents.

See Solution

Problem 8417

Find the rate of decrease of the surface area of a sphere when the radius is $6 \mathrm{~m}$ and decreases at $5 \mathrm{~m/s}$ .

See Solution

Problem 8418

Find the radius $r$ of a cylinder with volume 500 cm³ that minimizes cost $C$ given material prices. $r=\square$ cm.

See Solution

Problem 8419

Find $\frac{d z}{d t}$ at $(2,1)$ for $z^{2}=x^{2}+y^{2}$ with $\frac{d x}{d t}=2$ and $\frac{d y}{d t}=2$ .

See Solution

Problem 8420

Find the limit as $h$ approaches 0 for the expression $\frac{\frac{1}{2+h}-\frac{1}{2}}{h}$ .

See Solution

Problem 8421

A triangle has sides of $4 \mathrm{ft}$ and $5 \mathrm{ft}$ with the angle between them increasing at $0.2 \mathrm{rad/s}$ . Find the area change rate when the angle is $\frac{\pi}{3}$ . Answer in $\mathrm{ft}^{2}/\mathrm{s}$ .

See Solution

Problem 8422

Find the rate of water depth decrease in a cone (height 17 ft, radius 3 ft) when water is 12 ft high, leaking at 11 ft³/min. Round to 3 decimal places.

See Solution

Problem 8423

Find the rate of change of the area $A$ of a circle when $r=4$ and $\frac{d r}{d t}=5$ .

See Solution

Problem 8424

Find the height increase rate when water is $10 \mathrm{~m}$ deep in a pyramid tank with base $4 \mathrm{~m}$ and height $15 \mathrm{~m}$ , pumped at $\frac{2}{3} \mathrm{~m}^{3}/\mathrm{s}$ . Round to 3 decimal places. Use $V=\frac{s^{2} h}{3}$ .

See Solution

Problem 8425

A company sells widgets with sales $x$ given by $x=\frac{30000}{\sqrt{5 p+1}}$ . Find revenue change rate at $p=\$ 160$ .

See Solution

Problem 8426

Gravel is dumped at 40 cubic ft/min forming a cone with equal base diameter and height. Find height increase rate at 12 ft.

See Solution

Problem 8427

Bestimme die Tangentengleichung für die Funktionen und Punkte: a) $f(x)=1,5^{x}$ bei $A(1 \mid 1,5)$ b) $f(x)=\left(\frac{1}{4}\right)^{x}$ bei $A(-1 \mid 4)$ c) $f(x)=\left(\frac{4}{3}\right)^{x}$ bei $A(2 \mid f(2))$

See Solution

Problem 8428

Bestimmen Sie die Tangentengleichung an $f$ im Punkt $A$ : a) $f(x)=1,5^{x}$ , $A(1 \mid 1,5)$ ; b) $f(x)=\left(\frac{1}{4}\right)^{x}$ , $A(-1 \mid 4)$ ; c) $f(x)=\left(\frac{4}{3}\right)^{x}$ , $A(2 \mid f(2))$ ; d) $f(x)=x^{2}-15^{x}$ , $A(2 \mid f(2))$ .

See Solution

Problem 8429

Bestimme die Stammfunktion $F$ für die gegebenen $f$ mit den Bedingungen: a) $f: x \mapsto \frac{1}{x}+4 x^{3}$ , $F(1)=-4$ b) $f: x \mapsto 1+\sin x$ , $F(0)=3$ c) $f: x \mapsto \ln x$ , $F(e)=e$

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Problem 8430

Evaluate the limit: $\lim _{x \rightarrow 2} \frac{x^{3}-x^{2}-4}{x-2}$ .

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Problem 8431

Differentiate $y=\sec(\theta)\tan(\theta)$ .

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Problem 8432

Evaluate these limits: (a) $\lim _{x \rightarrow 2} \frac{x^{3}-x^{2}-4}{x-2}$ , (b) $\lim _{x \rightarrow 1} \frac{\sqrt{3+x}-2}{x^{2}-1}$ .

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Problem 8433

Find the slope of the tangent line for these curves at specified points: (a) $x^{3}+y^{3}=2$ at $(1,1)$ (b) $2 x^{2}+3 \sqrt{y}=-4 x+y$ at $(-2,9)$ (c) $x^{2}-x y+y^{2}=3$ at $x=1$

See Solution

Problem 8434

Find the slope of the tangent line to $x^{3}+y^{3}=2$ at the point $(1,1)$ .

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Problem 8435

Given $f(x)=x^{4}-\frac{x^{2}+1}{\sqrt{x+1}}+3$ for $x>-1$ , find: (a) $f(0)$ , (b) $f^{\prime}(x)$ , (c) tangent line at $x=0$ , (d) normal line at $x=0$ .

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Problem 8436

Find the derivative of $F(t)=e^{2 t \sin (2 t)}$ .

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Problem 8437

Emma invested \$19,000 at a continuous interest rate of 4.2%. How long to reach \$40,000? Round to the nearest year.

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Problem 8438

Jayden invested \$ 22,000 at 2.5\% interest, compounded continuously. How long to reach \$ 30,400? Round to 0.1 years.

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Problem 8439

Find the slope of the tangent line for $f(x)=9 x^{2}-3 x+1$ at $x=10$ .

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Problem 8440

Find the slope of the tangent line for $f(x)=x^{2}-5$ at $x=-12$ .

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Problem 8441

Find the first and second derivatives of $y=\sqrt{x} \ln(x)$ . What are $y^{\prime}$ and $y^{\prime \prime}$ ?

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Problem 8442

Find the slope of the tangent line for $f(x)=x^{2}+10$ at $x=10$ .

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Problem 8443

Differentiate the following functions without simplifying:
(a) $f(x)=\left(x^{5}-\sqrt{x^{2}+1}\right)\left(2 x^{3}+\sqrt{2 x+1}\right)$
(b) $f(x)=\frac{x^{3}+3}{x^{2}+x+1}$
(c) $f(x)=\sqrt{3 x^{1/3}+1}$ ; find $\frac{\mathrm{d} f}{\mathrm{~d} x}$ and $\frac{\mathrm{d}^{2} f}{\mathrm{~d} x^{2}}$ .

See Solution

Problem 8444

Find the tangent line equation for the function $f(x)=-4 x^{2}+5 x-11$ at the point where $x=4$ .

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Problem 8445

A man walks north at $4 \mathrm{ft/s}$ ; a woman walks south at $5 \mathrm{ft/s}$ . Find their separation rate after $15$ min.

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Problem 8446

Find the tangent line equation for $f(x)=10 x^{2}+2 x+7$ at $x=6$ .

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Problem 8447

Find the tangent line equation for the function $f(x)=-10 x^{2}+6 x+6$ at the point where $x=-10$ .

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Problem 8448

Find the derivative of $f(x)=\ln (x-3)$ using the limit definition as $h$ approaches $0$ . No need to simplify.

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Problem 8449

Verify that $(1,1)$ satisfies the equation $3 y^{3} x^{1/3} - 2 x^{2} \sqrt{y} + x y = 2$ and find $\frac{dy}{dx}$ at $(1,1)$ .

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Problem 8450

Find the derivative of $f(x)=2 \sec x$ using the limit definition as $h$ approaches $0$ . No need to simplify.

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Problem 8451

Find the derivative of the function $f(x)=5 x^{4}+3 x^{5}$ using the limit definition. No need to simplify.

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Problem 8452

Find the derivative of the function $f(x)=\sqrt{x-4}$ using the limit definition. Simplification is not required.

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Problem 8453

Find the derivative of $f(x)=3 x^{2}-x^{3}$ at $x=5$ using the limit definition. No need to simplify.

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Problem 8454

Find the slope of the tangent line for $f(x)=x^{4}+x^{3}$ at $x=-8$ using the limit method. No simplification needed.

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Problem 8455

Find $\frac{d h}{d t}$ for the equation $3 h^{3}-2 h t+4 t^{3}-2 t=0$ .

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Problem 8456

Find the tangent line equation for $f(x)=x^{3}-\cos(3x)$ at $x=-0.5$ . Round decimals to 3 places.

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Problem 8457

Find the derivative of the function $y=2 e^{-7 x}$ . What is $\frac{d y}{d x}=$ ?

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Problem 8458

Find the rate of decrease of surface area when the radius is $6 \mathrm{~m}$ and decreasing at $5 \mathrm{~m/s}$ .

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Problem 8459

Find the height increase rate when water is $10 \mathrm{~m}$ deep in a pyramid tank being filled at $\frac{2}{3} \mathrm{~m}^{3}/\mathrm{s}$ .

See Solution

Problem 8460

Find the rate at which the water depth decreases when the water is $12 \mathrm{ft}$ high in a cone leaking at $11 \mathrm{ft}^{3}/\mathrm{min}$ .

See Solution

Problem 8461

Differentiate $f(x)=\mathrm{e}^{x^{3}} \sqrt{x^{5}}$ . Find $f^{\prime}(x)$ .

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Problem 8462

A 6-ft person walks at $3 \mathrm{ft/s}$ toward a wall. How fast does their shadow's height change when $10 \mathrm{ft}$ away? Answer in $\mathrm{ft/s}$ .

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Problem 8463

Differentiate $y=7^{x^{3}+5}$ . Find $\frac{d y}{d x}$ .

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Problem 8464

Find the derivative of the function: $7 x \ln (x) - 5 x$ .

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Problem 8465

Find the critical numbers of the function $f(x)=2 x^{3}+x^{2}+2 x$ . Enter as a comma-separated list or DNE if none.

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Problem 8466

Calculate the half-life of an element that decays at a rate of $3.421\%$ per day.

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Problem 8467

Find $f^{\prime}(x)$ for the function $f(x)=\log _{7}(3 x^{2}-5 x+1)$ .

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Problem 8468

Check if $f(x)=6-16 x+4 x^{2}$ meets Rolle's Theorem on $[1,3]$ and find $c$ values. $c=$

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Problem 8469

Find $y^{\prime}$ if $y=\ln(\mathrm{e}^{-5 x}+5 x \mathrm{e}^{-5 x})$ . Options: $-\frac{x}{1+5 x}$ , $\frac{25 x}{1+5 x}$ , $-\frac{1}{(1+5 x)^{2}}$ , $-\frac{25 x}{1+5 x}$ .

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Problem 8470

Sketch the graph of $f(x)=\frac{5}{x}$ for $x \geq 5$ and find its max/min values. List answers as: absolute max, absolute min, local max, local min.

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Problem 8471

Differentiate the function $y=7^{x+5}$ . What is $\frac{d y}{d x}$ ?

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Problem 8472

Determine vertical and horizontal asymptotes for the function $f(x)=\frac{7 x^{3}+1}{x+5}$ .

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Problem 8473

Determine the vertical and horizontal asymptotes of the function $f(x)=\frac{6 x^{5}-4 x^{2}+5}{6 x^{2}+5 x-4}$ .

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Problem 8474

Bestimmen Sie die Steigung von $f(x)=2 x^{3}$ bei $x_{0}=-1$ .

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Problem 8475

Find $f^{\prime}(x)$ for the function $f(x)=\log_{7}(3x^{2}-5x+1)$ .

See Solution

Problem 8476

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x y - y^{2} = 3$ .

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Problem 8477

Find the derivative $\frac{d y}{d x}$ for $y=\ln \left(x^{2 / 3}\right)[0.5 p t]$ .

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Problem 8478

Evaluate the integral $\int_{2}^{6} \frac{1}{5} x^{2} dx$ using a Riemann sum approach.

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Problem 8479

Find $f^{\prime}(x)$ for $f(x)=\frac{(1+x^{2})^{5}(1+x^{3})^{8}}{(5x+3)^{2/3}}$ using logarithmic differentiation.

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Problem 8480

Find the derivative, integral, roots, and behavior of $f(x)=\frac{\sqrt{7+x^{2}}}{8+x}$ . Check for continuity and horizontal asymptote at $y=0$ .

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Problem 8481

Find the limits: 53. $\lim _{x \rightarrow 0} \frac{\sin 3 x}{5 x^{8}-4 x}$ and 55. $\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta+\tan \theta}$ .

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Problem 8482

Find the limits: 55. $\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta+\tan \theta}$ , 57. $\lim _{\theta \rightarrow 0} \frac{\cos \theta-1}{2 \theta^{2}}$ .

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Problem 8483

Find the speed of an object dropped from a 650 m cliff after 2 seconds, given its height function: $650 - 4.9 t^{2}$ .

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Problem 8484

Find the critical points of $f^{\prime}(x)=(x-2)^{2/3}(2)+(2x+1)\left[\frac{2}{3}(x-2)^{-1/3}\right]$ .

See Solution

Problem 8485

Simplify: $\frac{27}{n^{3}} \sum \frac{n(n+1)(2 n+1)}{6} + \frac{36}{n^{2}} \sum \frac{n(n+1)}{2} - \frac{6}{n} n$ and apply limits.

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Problem 8486

Find the limit as $x$ approaches $2^+$ of $\frac{f(2)-f(2-\frac{1}{x})}{2-(2-\frac{1}{x})}$ , where $f(x) = \frac{1}{x}$ .

See Solution

Problem 8487

Find the critical points of the function $f(x) = x^{3}(x-2)^{2}$ .

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Problem 8488

A 1.0 kg mass on a spring (90 N/m) is released from 0.40 m. What is the maximum distance it will fall?

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Problem 8489

Find critical points of $f(x)=x-\ln(2x)$ .

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Problem 8490

Find the derivative of $y=(\sin(x))^{4x^{3}-8x+17}$ . Provide the answer in exact form: $y'=$

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Problem 8491

Differentiate the function $y=\frac{\ln x}{x^{4}}$ .

See Solution

Problem 8492

Differentiate the function: $f(x)=\ln (\ln (3 x))$ . Find $f^{\prime}(x)$ .

See Solution

Problem 8493

Find the derivative of $y=\frac{x(x+8)^{3}}{(4 x-1)^{2}}$ using logarithmic differentiation. What is $y^{\prime}=$ ?

See Solution

Problem 8494

Find the derivative of $y=\frac{x(x+8)^{3}}{(4x-1)^{2}}$ using logarithmic differentiation.

See Solution

Problem 8495

Find the derivative $f^{\prime}(x)$ for the function $f(x)=2 x^{4} \ln ^{2} x$ .

See Solution

Problem 8496

Find the $z$ -coordinate(s) for $x=1, y=2$ in $f(x, y, z)=3yz+xz^2-x^3z=-4$ . Then, find a normal vector and the tangent plane equation.

See Solution

Problem 8497

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

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Problem 8498

Find the day $t$ (1 ≤ $t$ ≤ 46) when the maximum growth rate of car sales $N(t) = 3000 + 36t^2 - t^3$ occurs.

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Problem 8499

Differentiate $f(x)=(\sinh (x))^{7}$ and provide the exact form of $f^{\prime}(x)=$ .

See Solution

Problem 8500

Find the limit as $x$ approaches -1 of $\frac{f(x)-f(-1)}{x+1}$ for $f(x)=\frac{x}{x+2}$ .

See Solution

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Calculus

Problem 8401

True or False: For f(x)=(3x−1)(5x+3)f(x)=(3 x-1)(5 x+3)f(x)=(3x−1)(5x+3), does f′(x)f^{\prime}(x)f′(x) yield the same result using product or power rule?

Problem 8402

Find the max and min of f(x,y)=6xyf(x, y) = 6xyf(x,y)=6xy subject to x216+y236=1\frac{x^{2}}{16}+\frac{y^{2}}{36}=116x2​+36y2​=1 using Lagrange multipliers.

Problem 8403

Find the absolute max and min of f(x)=−6x3−72x2−270xf(x)=-6 x^{3}-72 x^{2}-270 xf(x)=−6x3−72x2−270x on [−7,7][-7,7][−7,7] as an ordered pair.

Problem 8404

Analyze the curve y+cos⁡y=x+1y+\cos y=x+1y+cosy=x+1 for 0≤y≤2π0 \leq y \leq 2 \pi0≤y≤2π: (a) Find dydx\frac{d y}{d x}dxdy​ in terms of yyy. (b) Determine vertical tangent equations. (c) Find d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ in terms of yyy.

Problem 8405

Problem 8406

Find the limit: lim⁡x→∞arctan⁡(x)\lim _{x \rightarrow \infty} \arctan (x)limx→∞​arctan(x).

Problem 8407

Find the derivative f′(2)f^{\prime}(2)f′(2) of the drug quantity function Q=f(t)=312t(0.5488)tQ=f(t)=312 t(0.5488)^{t}Q=f(t)=312t(0.5488)t. Round to 2 decimal places.

Problem 8408

Find the derivative of f(x)=x⋅2−xf(x)=\sqrt{x} \cdot 2^{-x}f(x)=x​⋅2−x.

Problem 8409

What is the speed of water hitting the bottom of Victoria Falls, given a height of 108 m108 \mathrm{~m}108 m and horizontal flow of 3.60 m/s3.60 \mathrm{~m/s}3.60 m/s?

Problem 8410

Bestimme den Flächeninhalt unter f(x)=x2f(x)=x^{2}f(x)=x2 von 0 bis xxx mit nnn Streifen (Obersummenfläche).

Problem 8411

Berechnen Sie die Ableitung von f(x)=13x3+12x2−2xf(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-2 xf(x)=31​x3+21​x2−2x.

Problem 8412

Calculate the limit of f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=x2−5x−1f(x)=x^{2}-5x-1f(x)=x2−5x−1.

Problem 8413

Berechne die Durchschnittsgeschwindigkeit des Schlittens mit s(t)=12t2s(t)=\frac{1}{2} t^{2}s(t)=21​t2 in den ersten 5 Sekunden. Bestimme die Momentangeschwindigkeit bei x0=5x_{0}=5x0​=5 und erkläre den Unterschied zwischen mittlerer und lokaler Änderungsrate.

Problem 8414

Find f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=3x2−x+5f(x)=3 x^{2}-x+5f(x)=3x2−x+5, where h≠0h \neq 0h=0, and simplify.

Problem 8415

Find the horizontal asymptote of f(x)=5(x+2)(8x−1)(6−x)(8x+2)f(x)=5 \frac{(x+2)(8 x-1)}{(6-x)(8 x+2)}f(x)=5(6−x)(8x+2)(x+2)(8x−1)​. What is yyy?

Problem 8416

Given the curve xy2−x3y=6x y^{2}-x^{3} y=6xy2−x3y=6, find: (a) dydx\frac{d y}{d x}dxdy​, (b) points with x=1x=1x=1 and tangent line equations, (c) xxx-coordinates of vertical tangents.

Problem 8417

Find the rate of decrease of the surface area of a sphere when the radius is 6 m6 \mathrm{~m}6 m and decreases at 5 m/s5 \mathrm{~m/s}5 m/s.

Problem 8418

Find the radius rrr of a cylinder with volume 500 cm³ that minimizes cost CCC given material prices. r=□r=\squarer=□ cm.

Problem 8419

Find dzdt\frac{d z}{d t}dtdz​ at (2,1)(2,1)(2,1) for z2=x2+y2z^{2}=x^{2}+y^{2}z2=x2+y2 with dxdt=2\frac{d x}{d t}=2dtdx​=2 and dydt=2\frac{d y}{d t}=2dtdy​=2.

Problem 8420

Find the limit as hhh approaches 0 for the expression 12+h−12h\frac{\frac{1}{2+h}-\frac{1}{2}}{h}h2+h1​−21​​.

Problem 8421

A triangle has sides of 4ft4 \mathrm{ft}4ft and 5ft5 \mathrm{ft}5ft with the angle between them increasing at 0.2rad/s0.2 \mathrm{rad/s}0.2rad/s. Find the area change rate when the angle is π3\frac{\pi}{3}3π​. Answer in ft2/s\mathrm{ft}^{2}/\mathrm{s}ft2/s.

Problem 8422

Find the rate of water depth decrease in a cone (height 17 ft, radius 3 ft) when water is 12 ft high, leaking at 11 ft³/min. Round to 3 decimal places.

Problem 8423

Find the rate of change of the area AAA of a circle when r=4r=4r=4 and drdt=5\frac{d r}{d t}=5dtdr​=5.

Problem 8424

Find the height increase rate when water is 10 m10 \mathrm{~m}10 m deep in a pyramid tank with base 4 m4 \mathrm{~m}4 m and height 15 m15 \mathrm{~m}15 m, pumped at 23 m3/s\frac{2}{3} \mathrm{~m}^{3}/\mathrm{s}32​ m3/s. Round to 3 decimal places. Use V=s2h3V=\frac{s^{2} h}{3}V=3s2h​.

Problem 8425

A company sells widgets with sales xxx given by x=300005p+1x=\frac{30000}{\sqrt{5 p+1}}x=5p+1​30000​. Find revenue change rate at p=$160p=\$ 160p=$160.

Problem 8426

Gravel is dumped at 40 cubic ft/min forming a cone with equal base diameter and height. Find height increase rate at 12 ft.

Problem 8427

Problem 8428

Problem 8429

Bestimme die Stammfunktion FFF für die gegebenen fff mit den Bedingungen: a) f:x↦1x+4x3f: x \mapsto \frac{1}{x}+4 x^{3}f:x↦x1​+4x3, F(1)=−4F(1)=-4F(1)=−4 b) f:x↦1+sin⁡xf: x \mapsto 1+\sin xf:x↦1+sinx, F(0)=3F(0)=3F(0)=3 c) f:x↦ln⁡xf: x \mapsto \ln xf:x↦lnx, F(e)=eF(e)=eF(e)=e

Problem 8430

Evaluate the limit: lim⁡x→2x3−x2−4x−2\lim _{x \rightarrow 2} \frac{x^{3}-x^{2}-4}{x-2}limx→2​x−2x3−x2−4​.

Problem 8431

Differentiate y=sec⁡(θ)tan⁡(θ)y=\sec(\theta)\tan(\theta)y=sec(θ)tan(θ).

Problem 8432

Evaluate these limits: (a) lim⁡x→2x3−x2−4x−2\lim _{x \rightarrow 2} \frac{x^{3}-x^{2}-4}{x-2}limx→2​x−2x3−x2−4​, (b) lim⁡x→13+x−2x2−1\lim _{x \rightarrow 1} \frac{\sqrt{3+x}-2}{x^{2}-1}limx→1​x2−13+x​−2​.

Problem 8433

Find the slope of the tangent line for these curves at specified points: (a) x3+y3=2x^{3}+y^{3}=2x3+y3=2 at (1,1)(1,1)(1,1) (b) 2x2+3y=−4x+y2 x^{2}+3 \sqrt{y}=-4 x+y2x2+3y​=−4x+y at (−2,9)(-2,9)(−2,9) (c) x2−xy+y2=3x^{2}-x y+y^{2}=3x2−xy+y2=3 at x=1x=1x=1

Problem 8434

Find the slope of the tangent line to x3+y3=2x^{3}+y^{3}=2x3+y3=2 at the point (1,1)(1,1)(1,1).

Problem 8435

Given f(x)=x4−x2+1x+1+3f(x)=x^{4}-\frac{x^{2}+1}{\sqrt{x+1}}+3f(x)=x4−x+1​x2+1​+3 for x>−1x>-1x>−1, find: (a) f(0)f(0)f(0), (b) f′(x)f^{\prime}(x)f′(x), (c) tangent line at x=0x=0x=0, (d) normal line at x=0x=0x=0.

Problem 8436

Find the derivative of F(t)=e2tsin⁡(2t)F(t)=e^{2 t \sin (2 t)}F(t)=e2tsin(2t).

Problem 8437

Emma invested \$19,000 at a continuous interest rate of 4.2%. How long to reach \$40,000? Round to the nearest year.

Problem 8438

Jayden invested \$ 22,000 at 2.5\% interest, compounded continuously. How long to reach \$ 30,400? Round to 0.1 years.

Problem 8439

Find the slope of the tangent line for f(x)=9x2−3x+1f(x)=9 x^{2}-3 x+1f(x)=9x2−3x+1 at x=10x=10x=10.

Problem 8440

Find the slope of the tangent line for f(x)=x2−5f(x)=x^{2}-5f(x)=x2−5 at x=−12x=-12x=−12.

Problem 8441

Find the first and second derivatives of y=xln⁡(x)y=\sqrt{x} \ln(x)y=x​ln(x). What are y′y^{\prime}y′ and y′′y^{\prime \prime}y′′?

True or False: For $f(x)=(3 x-1)(5 x+3)$ , does $f^{\prime}(x)$ yield the same result using product or power rule?

Find the max and min of $f(x, y) = 6xy$ subject to $\frac{x^{2}}{16}+\frac{y^{2}}{36}=1$ using Lagrange multipliers.

Find the absolute max and min of $f(x)=-6 x^{3}-72 x^{2}-270 x$ on $[-7,7]$ as an ordered pair.

Analyze the curve $y+\cos y=x+1$ for $0 \leq y \leq 2 \pi$ : (a) Find $\frac{d y}{d x}$ in terms of $y$ . (b) Determine vertical tangent equations. (c) Find $\frac{d^{2} y}{d x^{2}}$ in terms of $y$ .

Find the limit: $\lim _{x \rightarrow \infty} \arctan (x)$ .

Find the derivative $f^{\prime}(2)$ of the drug quantity function $Q=f(t)=312 t(0.5488)^{t}$ . Round to 2 decimal places.

Find the derivative of $f(x)=\sqrt{x} \cdot 2^{-x}$ .

What is the speed of water hitting the bottom of Victoria Falls, given a height of $108 \mathrm{~m}$ and horizontal flow of $3.60 \mathrm{~m/s}$ ?

Bestimme den Flächeninhalt unter $f(x)=x^{2}$ von 0 bis $x$ mit $n$ Streifen (Obersummenfläche).

Berechnen Sie die Ableitung von $f(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-2 x$ .

Calculate the limit of $\frac{f(x+h)-f(x)}{h}$ for $f(x)=x^{2}-5x-1$ .

Berechne die Durchschnittsgeschwindigkeit des Schlittens mit $s(t)=\frac{1}{2} t^{2}$ in den ersten 5 Sekunden. Bestimme die Momentangeschwindigkeit bei $x_{0}=5$ und erkläre den Unterschied zwischen mittlerer und lokaler Änderungsrate.

Find $\frac{f(x+h)-f(x)}{h}$ for $f(x)=3 x^{2}-x+5$ , where $h \neq 0$ , and simplify.

Find the horizontal asymptote of $f(x)=5 \frac{(x+2)(8 x-1)}{(6-x)(8 x+2)}$ . What is $y$ ?

Given the curve $x y^{2}-x^{3} y=6$ , find: (a) $\frac{d y}{d x}$ , (b) points with $x=1$ and tangent line equations, (c) $x$ -coordinates of vertical tangents.

Find the rate of decrease of the surface area of a sphere when the radius is $6 \mathrm{~m}$ and decreases at $5 \mathrm{~m/s}$ .

Find the radius $r$ of a cylinder with volume 500 cm³ that minimizes cost $C$ given material prices. $r=\square$ cm.

Find $\frac{d z}{d t}$ at $(2,1)$ for $z^{2}=x^{2}+y^{2}$ with $\frac{d x}{d t}=2$ and $\frac{d y}{d t}=2$ .

Find the limit as $h$ approaches 0 for the expression $\frac{\frac{1}{2+h}-\frac{1}{2}}{h}$ .

A triangle has sides of $4 \mathrm{ft}$ and $5 \mathrm{ft}$ with the angle between them increasing at $0.2 \mathrm{rad/s}$ . Find the area change rate when the angle is $\frac{\pi}{3}$ . Answer in $\mathrm{ft}^{2}/\mathrm{s}$ .

Find the rate of change of the area $A$ of a circle when $r=4$ and $\frac{d r}{d t}=5$ .

Find the height increase rate when water is $10 \mathrm{~m}$ deep in a pyramid tank with base $4 \mathrm{~m}$ and height $15 \mathrm{~m}$ , pumped at $\frac{2}{3} \mathrm{~m}^{3}/\mathrm{s}$ . Round to 3 decimal places. Use $V=\frac{s^{2} h}{3}$ .

A company sells widgets with sales $x$ given by $x=\frac{30000}{\sqrt{5 p+1}}$ . Find revenue change rate at $p=\$ 160$ .

Bestimme die Stammfunktion $F$ für die gegebenen $f$ mit den Bedingungen: a) $f: x \mapsto \frac{1}{x}+4 x^{3}$ , $F(1)=-4$ b) $f: x \mapsto 1+\sin x$ , $F(0)=3$ c) $f: x \mapsto \ln x$ , $F(e)=e$

Evaluate the limit: $\lim _{x \rightarrow 2} \frac{x^{3}-x^{2}-4}{x-2}$ .

Differentiate $y=\sec(\theta)\tan(\theta)$ .

Evaluate these limits: (a) $\lim _{x \rightarrow 2} \frac{x^{3}-x^{2}-4}{x-2}$ , (b) $\lim _{x \rightarrow 1} \frac{\sqrt{3+x}-2}{x^{2}-1}$ .

Find the slope of the tangent line for these curves at specified points: (a) $x^{3}+y^{3}=2$ at $(1,1)$ (b) $2 x^{2}+3 \sqrt{y}=-4 x+y$ at $(-2,9)$ (c) $x^{2}-x y+y^{2}=3$ at $x=1$

Find the slope of the tangent line to $x^{3}+y^{3}=2$ at the point $(1,1)$ .

Given $f(x)=x^{4}-\frac{x^{2}+1}{\sqrt{x+1}}+3$ for $x>-1$ , find: (a) $f(0)$ , (b) $f^{\prime}(x)$ , (c) tangent line at $x=0$ , (d) normal line at $x=0$ .

Find the derivative of $F(t)=e^{2 t \sin (2 t)}$ .

Find the slope of the tangent line for $f(x)=9 x^{2}-3 x+1$ at $x=10$ .

Find the slope of the tangent line for $f(x)=x^{2}-5$ at $x=-12$ .

Find the first and second derivatives of $y=\sqrt{x} \ln(x)$ . What are $y^{\prime}$ and $y^{\prime \prime}$ ?

Find the slope of the tangent line for $f(x)=x^{2}+10$ at $x=10$ .

Find the tangent line equation for the function $f(x)=-4 x^{2}+5 x-11$ at the point where $x=4$ .

A man walks north at $4 \mathrm{ft/s}$ ; a woman walks south at $5 \mathrm{ft/s}$ . Find their separation rate after $15$ min.

Find the tangent line equation for $f(x)=10 x^{2}+2 x+7$ at $x=6$ .

Find the tangent line equation for the function $f(x)=-10 x^{2}+6 x+6$ at the point where $x=-10$ .

Find the derivative of $f(x)=\ln (x-3)$ using the limit definition as $h$ approaches $0$ . No need to simplify.

Verify that $(1,1)$ satisfies the equation $3 y^{3} x^{1/3} - 2 x^{2} \sqrt{y} + x y = 2$ and find $\frac{dy}{dx}$ at $(1,1)$ .

Find the derivative of $f(x)=2 \sec x$ using the limit definition as $h$ approaches $0$ . No need to simplify.

Find the derivative of the function $f(x)=5 x^{4}+3 x^{5}$ using the limit definition. No need to simplify.

Find the derivative of the function $f(x)=\sqrt{x-4}$ using the limit definition. Simplification is not required.

Find the derivative of $f(x)=3 x^{2}-x^{3}$ at $x=5$ using the limit definition. No need to simplify.

Find the slope of the tangent line for $f(x)=x^{4}+x^{3}$ at $x=-8$ using the limit method. No simplification needed.

Find $\frac{d h}{d t}$ for the equation $3 h^{3}-2 h t+4 t^{3}-2 t=0$ .

Find the tangent line equation for $f(x)=x^{3}-\cos(3x)$ at $x=-0.5$ . Round decimals to 3 places.

Find the derivative of the function $y=2 e^{-7 x}$ . What is $\frac{d y}{d x}=$ ?

Find the rate of decrease of surface area when the radius is $6 \mathrm{~m}$ and decreasing at $5 \mathrm{~m/s}$ .

Find the height increase rate when water is $10 \mathrm{~m}$ deep in a pyramid tank being filled at $\frac{2}{3} \mathrm{~m}^{3}/\mathrm{s}$ .

Find the rate at which the water depth decreases when the water is $12 \mathrm{ft}$ high in a cone leaking at $11 \mathrm{ft}^{3}/\mathrm{min}$ .

Differentiate $f(x)=\mathrm{e}^{x^{3}} \sqrt{x^{5}}$ . Find $f^{\prime}(x)$ .

A 6-ft person walks at $3 \mathrm{ft/s}$ toward a wall. How fast does their shadow's height change when $10 \mathrm{ft}$ away? Answer in $\mathrm{ft/s}$ .

Differentiate $y=7^{x^{3}+5}$ . Find $\frac{d y}{d x}$ .

Find the derivative of the function: $7 x \ln (x) - 5 x$ .

Find the critical numbers of the function $f(x)=2 x^{3}+x^{2}+2 x$ . Enter as a comma-separated list or DNE if none.

Calculate the half-life of an element that decays at a rate of $3.421\%$ per day.

Find $f^{\prime}(x)$ for the function $f(x)=\log _{7}(3 x^{2}-5 x+1)$ .

Check if $f(x)=6-16 x+4 x^{2}$ meets Rolle's Theorem on $[1,3]$ and find $c$ values. $c=$

Find $y^{\prime}$ if $y=\ln(\mathrm{e}^{-5 x}+5 x \mathrm{e}^{-5 x})$ . Options: $-\frac{x}{1+5 x}$ , $\frac{25 x}{1+5 x}$ , $-\frac{1}{(1+5 x)^{2}}$ , $-\frac{25 x}{1+5 x}$ .

Sketch the graph of $f(x)=\frac{5}{x}$ for $x \geq 5$ and find its max/min values. List answers as: absolute max, absolute min, local max, local min.

Differentiate the function $y=7^{x+5}$ . What is $\frac{d y}{d x}$ ?

Determine vertical and horizontal asymptotes for the function $f(x)=\frac{7 x^{3}+1}{x+5}$ .

Determine the vertical and horizontal asymptotes of the function $f(x)=\frac{6 x^{5}-4 x^{2}+5}{6 x^{2}+5 x-4}$ .

Bestimmen Sie die Steigung von $f(x)=2 x^{3}$ bei $x_{0}=-1$ .