Calculus

Problem 13401

Find the derivative $f'(x)$ if $f(x)=\int_{-1}^{x^{2}} t^{2} d t$ .

See Solution

Problem 13402

Find the derivative of $P(t)=-\frac{1}{8240} t^{3}+\frac{1}{36} t^{2}$ .

See Solution

Problem 13403

Find the derivative $f'(x)$ if $f(x)=\int_{1}^{x^{3}} t^{5} dt$ .

See Solution

Problem 13404

Find where the function is concave up/down and identify points of inflection for $f(x)=6 x^{3}-5 x^{2}+5$ .

See Solution

Problem 13405

Calculate the integral $\int_{1}^{e^{8}} \frac{4}{x} dx$ .

See Solution

Problem 13406

Evaluate the integral from 0 to $\frac{\pi}{4}$ of $\sin(4t)$ .

See Solution

Problem 13407

Evaluate the integral from 1 to $\sqrt{5}$ of $\frac{7}{1+x^{2}}$ with respect to $x$ .

See Solution

Problem 13408

Calculate the integral from 0 to π of 10 sin(x) dx.

See Solution

Problem 13409

Find the value of $x$ where the local maximum of $f(x)=\int_{0}^{x} \frac{t^{2}-36}{1+\cos ^{2}(t)} dt$ occurs. $x=$

See Solution

Problem 13410

Find $f^{\prime}(3)$ for the function $f(x)=\frac{5 x^{3}-4 x^{2}}{\sqrt[3]{x^{4}}}$ .

See Solution

Problem 13411

Find where the function $f(x)=6 x^{3}-5 x^{2}+5$ is concave up/down and its points of inflection.

See Solution

Problem 13412

Find $f^{\prime \prime}(x)$ for $f(x)=\int_{0}^{x}(t^{3}+7 t^{2}+6) dt$ .

See Solution

Problem 13413

Evaluate the integral from 1 to 9 of $(2x^{2}+5)/\sqrt{x}$ dx.

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Problem 13414

Calculate the integral from -4 to 1 of the function $3 x^{7} + 2 x^{2}$ .

See Solution

Problem 13415

Find the derivative of $h(x)=\int_{-5}^{\sin (x)}\left(\cos \left(t^{5}\right)+t\right) d t$ . What is $h^{\prime}(x)$ ?

See Solution

Problem 13416

Graph the function $f(x)=x(x-2.4)^{3}$ and find its inflection point(s).

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Problem 13417

Calculate the integral from 3 to 2 of $\sin(t) \, dt$ .

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Problem 13418

Find the derivative of $f(x)=\int_{3}^{x}\left(\frac{1}{3} t^{2}-1\right)^{8} dt$ . What is $f'(x)$ ?

See Solution

Problem 13419

Find the derivative of $F(x)=\int_{x}^{3} \sin(t^{2}) dt$ . What is $F'(x)$ ?

See Solution

Problem 13420

Find the derivative of $f(x)=\int_{-3}^{x} \sqrt{t^{3}+3^{3}} dt$ . What is $f'(x)$ ?

See Solution

Problem 13421

Find the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for these functions: 1- $f(x, y)=e^{xy}+x$ , 2- $f(x, y)=\ln(x+y)$ .

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Problem 13422

Berechnen Sie die Produktionsmenge $x$ , bei der die Grenzkosten $25 \,€/$ Stück sind, gegeben die Kostenfunktion $K(x)=0,00002 x^{3}-0,02 x^{2}+21,4 x+1200$ .

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Problem 13423

Find the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for these functions: 1- $f(x, y)=e^{xy+x}$ 2- $f(x, y)=\ln(x+y)$ 3- $f(x, y)=(3y+xy)^{4}$ 4- $f(x, y)=y \ln(xy)$ 5- $f(x, y)=y \cos(xy)$ .

See Solution

Problem 13424

Find the value of $x$ where the local maximum of $f(x)=\int_{0}^{x} \frac{t^{2}-25}{1+\cos ^{2}(t)} d t$ occurs. $x=$

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Problem 13425

Find the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for these functions: 1- $f(x, y)=e^{xy+x}$ , 2- $f(x, y)=\ln(x+y)$ , 3- $f(x, y)=(3y+xy)^{4}$ , 4- $f(x, y)=y \ln(xy)$ , 5- $f(x, y)=y \cos(xy)$ .

See Solution

Problem 13426

Find the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for these functions: 1- $f(x, y)=e^{xy+x}$ 2- $f(x, y)=\ln(x+y)$ 3- $f(x, y)=(3y+xy)^{4}$ 4- $f(x, y)=y \ln(xy)$ 5- $f(x, y)=y \cos(xy)$ .

See Solution

Problem 13427

Find the slope of the tangent line to $9 x^{3}+x y+9 y^{4}=735$ at the point $(1,-3)$ .

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Problem 13428

Find the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for: 1- $f(x, y)=\mathrm{e}^{x y^{3}+x}$ , 2- $f(x, y)=\operatorname{Ln}(x+y)$ .

See Solution

Problem 13429

Find the slope of the tangent line to $9 x^{3}+x y+9 y^{4}=735$ at $(1,-3)$ . The slope is $\square$ .

See Solution

Problem 13430

Bestimme die Wendepunkte und Sattelpunkte der Funktionen: $f(x)=x^{3}+3 x^{2}$ , $f(x)=-0,3 x^{3}+8,1$ , $f(x)=\frac{1}{3} x^{3}-4 x$ , $f(x)=\frac{1}{9} x^{3}-x^{2}$ , $f(x)=x^{3}-9 x^{2}+27 x-19$ , $f(x)=0,2 x^{3}+3 x^{2}-9,6 x$ , $f(x)=\frac{1}{8} x^{4}-3 x^{2}$ , $f(x)=0,25 x^{4}-2 x^{3}+4,5 x^{2}$ . Zeichne die Graphen und markiere die Wendepunkte.

See Solution

Problem 13431

Berechnen Sie die Ableitungen $f'(x)$ und $f''(x)$ für die Funktion $f(x) = x^3 + 3x^2$ .

See Solution

Problem 13432

Bestimme die erste Ableitung der Funktionen: $f(x)=x^{2}+3x^{4}$ und $f(x)=3x^{6}-14x^{2}$ .

See Solution

Problem 13433

Finde die erste Ableitung der folgenden Funktionen: a) $f(x)=x^{2}+3 x^{4}$ b) $f(x)=3 x^{6}-14 x^{2}$

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Problem 13434

Evaluate these integrals: (i) $\int \sec ^{5}\left(\frac{\theta}{2}\right) \tan \left(\frac{\theta}{2}\right) d \theta$ ; (ii) $\int \frac{\cos t}{1+\cos t} d t$ .

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Problem 13435

Find the moment about the $y$ -axis for the area under $y=\cos x$ , from $x=0$ to $x=\frac{\pi}{2}$ .

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Problem 13436

Untersuchen Sie lokale Extremalpunkte und skizzieren Sie die Graphen für: a) $f(x)=2 x^{2}+3 x-5$ , b) $f(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-3 x$ , c) $f(x)=\frac{1}{4} x^{3}-2$ .

See Solution

Problem 13437

Compute the differentials of the function $f(x, y) = x^{2} e^{5y} + x$ .

See Solution

Problem 13438

Calculate the integral from 1 to 2 of $\frac{2}{\sqrt{x}}$ with respect to $x$ .

See Solution

Problem 13439

Calculate the integral from 1 to 2 of $\frac{x^{2}+3}{x^{2}}$ with respect to $x$ .

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Problem 13440

Bilde die erste Ableitung ohne negative Exponenten für: a) $f_{a}(x)=4 x^{3}+3 x^{2}-2 x-1$ , b) $f_{b}(x)=x^{2}(x-5)$ , c) $f_{c}(x)=x^{3} \sqrt{\frac{1}{x}}$ , d) $f_{d}(x)=\sqrt{x^{2}-7}$ .

See Solution

Problem 13441

Finde die Seitenlänge der quadratischen Grundfläche eines pyramidenförmigen Zelts, das den Rauminhalt maximiert, bei $3 \mathrm{~m}$ Stäben.

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Problem 13442

Bestimme die Ableitungen der Funktionen: a) $f_{a}(x)=a x^{3}+b x^{2}+c x+d$ , b) $f_{b}(x)=\frac{x-6}{\sqrt{x}}$ , c) $f_{c}(x)=\left(x^{2}-4\right) \sqrt{x^{2}+4}$ .

See Solution

Problem 13443

Bestimme für die Funktion $f_{a}(t)=\frac{1}{4} t^{3}-a t^{2}+a^{2} t$ die Zeitpunkte ohne Wasserfluss, Maxima, Minima und starke Abfälle. Zeige, dass $F_{a}(t)=\frac{1}{16} t^{4}-\frac{a}{3} t^{3}+\frac{a^{2}}{2} t^{2}$ eine Stammfunktion ist. Finde den Zeitpunkt $t$ , an dem $F_{a1}=F_{a2}$ .

See Solution

Problem 13444

Gegeben ist die Funktion $f(x)=-\frac{5}{16} x^{4}+5 x^{3}$ .
a) Finde die Nullstellen von $f$ . b) Welche Symmetrie hat $f$ ? c) Bestimme Extrem- und Wendepunkte von $f$ . d) Nachweis und Berechnung des Sattelpunktes, falls vorhanden. e) Zeige, dass $f$ für $x>12$ streng monoton fallend ist. f) Finde die $x$ -Werte, wo die Steigung 1 beträgt und erkläre, warum die Gleichung nicht einfach gelöst werden kann.

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Problem 13445

Oblicz granicę dla ciągu $a_{n}=\frac{n(\sqrt{3 n}-7)^{2}}{(\sqrt{n+2}+2)^{4}}$ lub wyjaśnij, dlaczego nie istnieje.

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Problem 13446

Find the surface area generated by revolving $y=2\sqrt{x}$ from $x=6$ to $x=9$ about the $x$ -axis.

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Problem 13447

Find the surface area from revolving $y=2 \sqrt{x}$ around the $x$ -axis for $6 \leq x \leq 9$ .

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Problem 13448

Find the volume of the solid formed by revolving the area between $y=x$ , $y=11$ , and $x=0$ around $y=12$ .

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Problem 13449

Find the local maxima and minima of $f(x)=x-2 \sin x$ for $0 \leq x \leq 2\pi$ .

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Problem 13450

Approximate the arc length of $y=\sin 2x$ from $0$ to $2\pi$ . Round to three decimal places. Choices: (A) 10.340 (B) 5.270 (C) 7.640 (D) 10.540 (E) 11.740

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Problem 13451

Find the volume of the solid formed by revolving the area between $y=\frac{1}{x}$ , $y=0$ , $x=8$ , and $x=15$ around the $x$ -axis.

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Problem 13452

Find the volume of the solid formed by revolving the region bounded by $y=x^{3}$ , $y=0$ , and $x=9$ around $x=18$ . Options: (A) $\frac{118,098}{5} \pi$ , (B) $\frac{177,147}{5} \pi$ , (C) $\frac{59,049}{5} \pi$ , (D) $\frac{354,294}{5} \pi$ , (E) $\frac{236,196}{5} \pi$ .

See Solution

Problem 13453

Find the volume of the solid formed by revolving the region bounded by $y=2-x$ , $y=0$ , and $x=0$ around the $x$ -axis.

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Problem 13454

Find the average value of $f(x, y) = y$ over $y$ from -2 to 2 and $x$ to 4. Options: (A) 0 (B) 1.3 (C) 10 (D) 2

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Problem 13455

Find the acceleration of the particle at time $t=\pi$ given $x(t)=\sin (2 t)-\cos (3 t)$ .

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Problem 13456

Find the volume of the solid formed by revolving the area between $y=\sin x$ , $y=0$ , from $0$ to $\frac{\pi}{2}$ around $y=11$ .

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Problem 13457

Calculate the area between $f(x)=\frac{14 x}{x^{2}+1}$ and $y=0$ for $0 \leq x \leq 5$ . Round to three decimal places.

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Problem 13458

Find the volume of the solid formed by revolving the area between $y=19 x^{2}$ , $y=0$ , and $x=2$ around $x=2$ .

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Problem 13459

Find the area of the surface formed by revolving $y=\frac{1}{8} x^{3}$ from $0$ to $8$ around the $x$ -axis.

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Problem 13460

Find the volume of the solid formed by revolving the area between $y=x^{5}$ and $y=32$ around the $y$ -axis.

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Problem 13461

Calculate the integral $\int_{0}^{3} \int_{0}^{5}\left(y e^{x}-x-y\right) d x d y$ . Choose the correct answer from the options.

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Problem 13462

Find the volume of the solid formed by revolving the region bounded by $y=\sqrt{x+156}, y=x, y=0$ around the $x$ -axis using the shell method. Choose the correct answer from the options provided.

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Problem 13463

Find the volume of the solid formed by revolving the area bounded by $y=x^{\frac{4}{5}}$ , $y=1$ , and $x=0$ around the $y$ -axis.

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Problem 13464

Find the volume of the solid formed by revolving the region bounded by $y=x^{6}$ , $x=0$ , and $y=64$ around the $x$ -axis using the shell method.

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Problem 13465

Find the volume of the solid formed by revolving the area between $y=\frac{15}{x^{2}}$ , $y=0$ , $x=1$ , and $x=8$ around the $x$ -axis. Round to two decimal places.

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Problem 13466

Find the volume of the solid formed by revolving the area between $y=8$ and $y=16-\frac{x^{2}}{16}$ around the $x$ -axis.

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Problem 13467

Find the tangent to $f(x)=x^{3}$ parallel to the line $g: 3x-y=1$ .

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Problem 13468

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

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Problem 13469

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

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Problem 13470

Find the limit: $\lim _{x \rightarrow-4}\left(x^{2}-2 x-8\right)$ . What is the answer?

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Problem 13471

Find the derivative of the constant function $f(x)=5$ using first principles.

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Problem 13472

Calculate the integral from 0 to 1 and 0 to 2-y: $\int_{0}^{1} \int_{0}^{2-y} x \, dx \, dy$ .

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Problem 13473

Find the volume of the solid formed by revolving the area bounded by $y=3 x^{3}, y=0, x=3$ around the $x$ -axis.

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Problem 13474

Find the area between $f(x)=\frac{8 x}{x^{2}+1}$ and $y=0$ from $0 \leq x \leq 4$ . Options: (A) $A=15 \ln 4$ , (B) $A=4 \ln 17$ , (C) $A=17 \ln 4$ , (D) $A=8 \ln 17$ , (E) $A=4 \ln 15$ .

See Solution

Problem 13475

Find the volume of the solid formed by revolving the region in the first quadrant bounded by $x^{\frac{2}{3}}+y^{\frac{2}{3}}=3^{\frac{2}{3}}$ about (i) the $x$ -axis and (ii) the $y$ -axis.

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Problem 13476

Find the limit as $x$ approaches $a$ of $\frac{f(x)-f(a)}{x-a}$ for $f(x)=3x-5$ .

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Problem 13477

Identify the integrand, limits, and variable in the integral: $\int_{1}^{3.6} \sqrt{7 y-3} d y$ .
Answer: The integrand is $\sqrt{7 y-3}$ , the lower bound is $1$ , the upper bound is $3.6$ , and the variable of integration is $y$ .

See Solution

Problem 13478

Express the limit of $L_{n}=\frac{1}{n} \sum_{i=1}^{n}\left(3+\frac{i-1}{n}\right)$ as a definite integral.
$\int_{0}^{1} \left(3+x\right) dx$

See Solution

Problem 13479

Express the limit as an integral: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(10\left(x_{i}^{}\right)^{5}-7 x_{i}^{}-9\right) \Delta x$ over $[3,4]$ .

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Problem 13480

Which option shows the limit definition of the integral $\int_{1}^{7} 9 x d x$ using a left-endpoint Riemann sum?

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Problem 13481

Identify the integrand, limits, and variable in the integral: $\int_{1}^{4 \pi}-\frac{8 \sin (7 t)}{t} d t$ .
Answer: Integrand is $\square$ , lower bound is $\square$ , upper bound is $\square$ , variable is $\square$ .

See Solution

Problem 13482

Find the derivatives: 1) $f^{\prime}(2)$ for $f(x)=\sin \sqrt{x}$ ; 2) $f^{\prime}(1.3)$ for $f(x)=\ln \left(\frac{1}{5-x}\right)$ .

See Solution

Problem 13483

Zeige, dass $F$ eine Stammfunktion von $f$ ist und nenne drei weitere Stammfunktionen für die folgenden Paare: a) $F(x)=2 x^{3}$ , $f(x)=6 x^{2}$ ; b) $F(x)=5 x^{4}+2$ , $f(x)=20 x^{3}$ ; c) $F(x)=\frac{2}{15} x^{5}-3$ , $f(x)=\frac{2}{3} x^{4}$ ; d) $F(x)=2 x^{4}-3 x^{3}+5$ , $f(x)=8 x^{3}-9 x^{2}$ .

See Solution

Problem 13484

1. Find $f^{\prime}(2)$ for $f(x)=\sin \sqrt{x}$ .
2. Find $f^{\prime}(1.3)$ for $f(x)=\ln \left(\frac{1}{5-x}\right)$ .
3. Write the tangent line equation to $y=\sqrt{\frac{x}{x^{3}+1}}$ at $x=1$ .
4. Estimate $f^{\prime}(3)$ from the table of $f(x)$ values.
5. Estimate $w^{\prime}(100)$ from the table of $w(x)$ values.

See Solution

Problem 13485

Évaluez la limite suivante : $\lim _{t \rightarrow-3} \frac{t^{2}-9}{2 t^{2}+7 t+3}$ .

See Solution

Problem 13486

Find the limit: $\lim _{h \rightarrow 0} \frac{(-5+h)^{2}-25}{h}$ .

See Solution

Problem 13487

Calculate the average rate of change of $f(x)=3$ over the interval $[-2,0]$ .

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Problem 13488

Find the volume of the solid formed by rotating a right triangle (sides 3 and 5) about the side of length 3.

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Problem 13489

Sketch the graph of $y=27x-x^{3}$ using its derivatives. Verify with a graphing calculator.

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Problem 13490

Analyze case D: with $f' < 0$ and $f'' < 0$ , determine if the function is increasing/decreasing and find points of inflection.

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Problem 13491

Find the limit: $\lim _{t \rightarrow 0}\left(\frac{1}{t}-\frac{1}{t^{2}+t}\right)$ .

See Solution

Problem 13492

Calculate the integral from 2 to 4 of $(x^{2}+1)^{2}$ dx.

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Problem 13493

Graph $f(x)=\frac{e^{2 x}}{2 x-1}$ . Find the domain: A. $\left(-\infty, \frac{1}{2}\right) \cup\left(\frac{1}{2}, \infty\right)$ , B. $\left(0, \frac{1}{2}\right) \cup\left(\frac{1}{2}, \infty\right)$ , C. $(-\infty, 0) \cup(0, \infty)$ , D. $(-\infty, \infty)$ .

See Solution

Problem 13494

Find the limit: $\lim _{x \rightarrow-4} \frac{\sqrt{x^{2}+9}-5}{x+4}$ .

See Solution

Problem 13495

Given the curve $2 y^{2}-6=y \sin x$ for $y>0$ :
(a) Prove that $\frac{d y}{d x}=\frac{y \cos x}{4 y-\sin x}$ .
(b) Find the tangent line equation at $(0, \sqrt{3})$ .
(c) Find where the tangent line is horizontal for $0 \leq x \leq \pi$ .
(d) Determine if $f$ has a relative min, max, or neither at the point from (c). Justify.

See Solution

Problem 13496

Find the integral of $(3 x+2)^{5}$ with respect to $x$ .

See Solution

Problem 13497

Calculate the integral from 3 to 9 of $\sqrt[3]{x}(x-4)$ .

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Problem 13498

Find the position function $s(t)$ and the velocity function $v(t)$ for a particle with acceleration $a(t)=6t$ and $s(0)=0$ .

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Problem 13499

Ordne den Funktionen $f(x)$ die passenden Stammfunktionen $F(x)$ zu: $f(x)=2 x$ , $x^{4}$ , $4 x^{3}+3 x^{2}$ , $10 x^{4}-2$ , $\frac{14}{9} x^{6}-4 x^{3}$ .

See Solution

Problem 13500

Graph $f(x)=\frac{e^{2 x}}{2 x-1}$ . Find critical points using $f^{\prime}(x)=\frac{4(x-1)e^{2 x}}{(2 x-1)^{2}}$ . Choices: A. No points. B. At $x=\square$ .

See Solution

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Calculus

Problem 13401

Find the derivative f′(x)f'(x)f′(x) if f(x)=∫−1x2t2dtf(x)=\int_{-1}^{x^{2}} t^{2} d tf(x)=∫−1x2​t2dt.

Problem 13402

Find the derivative of P(t)=−18240t3+136t2P(t)=-\frac{1}{8240} t^{3}+\frac{1}{36} t^{2}P(t)=−82401​t3+361​t2.

Problem 13403

Find the derivative f′(x)f'(x)f′(x) if f(x)=∫1x3t5dtf(x)=\int_{1}^{x^{3}} t^{5} dtf(x)=∫1x3​t5dt.

Problem 13404

Find where the function is concave up/down and identify points of inflection for f(x)=6x3−5x2+5f(x)=6 x^{3}-5 x^{2}+5f(x)=6x3−5x2+5.

Problem 13405

Calculate the integral ∫1e84xdx\int_{1}^{e^{8}} \frac{4}{x} dx∫1e8​x4​dx.

Problem 13406

Evaluate the integral from 0 to π4\frac{\pi}{4}4π​ of sin⁡(4t)\sin(4t)sin(4t).

Problem 13407

Evaluate the integral from 1 to 5\sqrt{5}5​ of 71+x2\frac{7}{1+x^{2}}1+x27​ with respect to xxx.

Problem 13408

Calculate the integral from 0 to π of 10 sin(x) dx.

Problem 13409

Find the value of xxx where the local maximum of f(x)=∫0xt2−361+cos⁡2(t)dtf(x)=\int_{0}^{x} \frac{t^{2}-36}{1+\cos ^{2}(t)} dtf(x)=∫0x​1+cos2(t)t2−36​dt occurs. x=x=x=

Problem 13410

Find f′(3)f^{\prime}(3)f′(3) for the function f(x)=5x3−4x2x43f(x)=\frac{5 x^{3}-4 x^{2}}{\sqrt[3]{x^{4}}}f(x)=3x4​5x3−4x2​.

Problem 13411

Find where the function f(x)=6x3−5x2+5f(x)=6 x^{3}-5 x^{2}+5f(x)=6x3−5x2+5 is concave up/down and its points of inflection.

Problem 13412

Find f′′(x)f^{\prime \prime}(x)f′′(x) for f(x)=∫0x(t3+7t2+6)dtf(x)=\int_{0}^{x}(t^{3}+7 t^{2}+6) dtf(x)=∫0x​(t3+7t2+6)dt.

Problem 13413

Evaluate the integral from 1 to 9 of (2x2+5)/x(2x^{2}+5)/\sqrt{x}(2x2+5)/x​ dx.

Problem 13414

Calculate the integral from -4 to 1 of the function 3x7+2x23 x^{7} + 2 x^{2}3x7+2x2.

Problem 13415

Find the derivative of h(x)=∫−5sin⁡(x)(cos⁡(t5)+t)dth(x)=\int_{-5}^{\sin (x)}\left(\cos \left(t^{5}\right)+t\right) d th(x)=∫−5sin(x)​(cos(t5)+t)dt. What is h′(x)h^{\prime}(x)h′(x)?

Problem 13416

Graph the function f(x)=x(x−2.4)3f(x)=x(x-2.4)^{3}f(x)=x(x−2.4)3 and find its inflection point(s).

Problem 13417

Calculate the integral from 3 to 2 of sin⁡(t) dt\sin(t) \, dtsin(t)dt.

Problem 13418

Find the derivative of f(x)=∫3x(13t2−1)8dtf(x)=\int_{3}^{x}\left(\frac{1}{3} t^{2}-1\right)^{8} dtf(x)=∫3x​(31​t2−1)8dt. What is f′(x)f'(x)f′(x)?

Problem 13419

Find the derivative of F(x)=∫x3sin⁡(t2)dtF(x)=\int_{x}^{3} \sin(t^{2}) dtF(x)=∫x3​sin(t2)dt. What is F′(x)F'(x)F′(x)?

Problem 13420

Find the derivative of f(x)=∫−3xt3+33dtf(x)=\int_{-3}^{x} \sqrt{t^{3}+3^{3}} dtf(x)=∫−3x​t3+33​dt. What is f′(x)f'(x)f′(x)?

Problem 13421

Find the partial derivatives ∂f∂x\frac{\partial f}{\partial x}∂x∂f​ and ∂f∂y\frac{\partial f}{\partial y}∂y∂f​ for these functions: 1- f(x,y)=exy+xf(x, y)=e^{xy}+xf(x,y)=exy+x, 2- f(x,y)=ln⁡(x+y)f(x, y)=\ln(x+y)f(x,y)=ln(x+y).

Problem 13422

Berechnen Sie die Produktionsmenge xxx, bei der die Grenzkosten 25 €/25 \,€/25€/ Stück sind, gegeben die Kostenfunktion K(x)=0,00002x3−0,02x2+21,4x+1200K(x)=0,00002 x^{3}-0,02 x^{2}+21,4 x+1200K(x)=0,00002x3−0,02x2+21,4x+1200.

Problem 13423

Problem 13424

Find the value of xxx where the local maximum of f(x)=∫0xt2−251+cos⁡2(t)dtf(x)=\int_{0}^{x} \frac{t^{2}-25}{1+\cos ^{2}(t)} d tf(x)=∫0x​1+cos2(t)t2−25​dt occurs. x= x= x=

Problem 13425

Problem 13426

Problem 13427

Find the slope of the tangent line to 9x3+xy+9y4=7359 x^{3}+x y+9 y^{4}=7359x3+xy+9y4=735 at the point (1,−3)(1,-3)(1,−3).

Problem 13428

Find the partial derivatives ∂f∂x\frac{\partial f}{\partial x}∂x∂f​ and ∂f∂y\frac{\partial f}{\partial y}∂y∂f​ for: 1- f(x,y)=exy3+xf(x, y)=\mathrm{e}^{x y^{3}+x}f(x,y)=exy3+x, 2- f(x,y)=Ln⁡(x+y)f(x, y)=\operatorname{Ln}(x+y)f(x,y)=Ln(x+y).

Problem 13429

Find the slope of the tangent line to 9x3+xy+9y4=7359 x^{3}+x y+9 y^{4}=7359x3+xy+9y4=735 at (1,−3)(1,-3)(1,−3). The slope is □\square□.

Problem 13430

Problem 13431

Berechnen Sie die Ableitungen f′(x)f'(x)f′(x) und f′′(x)f''(x)f′′(x) für die Funktion f(x)=x3+3x2f(x) = x^3 + 3x^2f(x)=x3+3x2.

Problem 13432

Bestimme die erste Ableitung der Funktionen: f(x)=x2+3x4f(x)=x^{2}+3x^{4}f(x)=x2+3x4 und f(x)=3x6−14x2f(x)=3x^{6}-14x^{2}f(x)=3x6−14x2.

Problem 13433

Finde die erste Ableitung der folgenden Funktionen: a) f(x)=x2+3x4f(x)=x^{2}+3 x^{4}f(x)=x2+3x4 b) f(x)=3x6−14x2f(x)=3 x^{6}-14 x^{2}f(x)=3x6−14x2

Problem 13434

Evaluate these integrals: (i) ∫sec⁡5(θ2)tan⁡(θ2)dθ\int \sec ^{5}\left(\frac{\theta}{2}\right) \tan \left(\frac{\theta}{2}\right) d \theta∫sec5(2θ​)tan(2θ​)dθ; (ii) ∫cos⁡t1+cos⁡tdt\int \frac{\cos t}{1+\cos t} d t∫1+costcost​dt.

Problem 13435

Find the moment about the yyy-axis for the area under y=cos⁡xy=\cos xy=cosx, from x=0x=0x=0 to x=π2x=\frac{\pi}{2}x=2π​.

Problem 13436

Untersuchen Sie lokale Extremalpunkte und skizzieren Sie die Graphen für: a) f(x)=2x2+3x−5f(x)=2 x^{2}+3 x-5f(x)=2x2+3x−5, b) f(x)=13x3+12x2−3xf(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-3 xf(x)=31​x3+21​x2−3x, c) f(x)=14x3−2f(x)=\frac{1}{4} x^{3}-2f(x)=41​x3−2.

Problem 13437

Compute the differentials of the function f(x,y)=x2e5y+xf(x, y) = x^{2} e^{5y} + xf(x,y)=x2e5y+x.

Problem 13438

Calculate the integral from 1 to 2 of 2x\frac{2}{\sqrt{x}}x​2​ with respect to xxx.

Problem 13439

Calculate the integral from 1 to 2 of x2+3x2\frac{x^{2}+3}{x^{2}}x2x2+3​ with respect to xxx.

Problem 13440

Problem 13441

Finde die Seitenlänge der quadratischen Grundfläche eines pyramidenförmigen Zelts, das den Rauminhalt maximiert, bei 3 m3 \mathrm{~m}3 m Stäben.

Problem 13442

Bestimme die Ableitungen der Funktionen: a) fa(x)=ax3+bx2+cx+df_{a}(x)=a x^{3}+b x^{2}+c x+dfa​(x)=ax3+bx2+cx+d, b) fb(x)=x−6xf_{b}(x)=\frac{x-6}{\sqrt{x}}fb​(x)=x​x−6​, c) fc(x)=(x2−4)x2+4f_{c}(x)=\left(x^{2}-4\right) \sqrt{x^{2}+4}fc​(x)=(x2−4)x2+4​.

Find the derivative $f'(x)$ if $f(x)=\int_{-1}^{x^{2}} t^{2} d t$ .

Find the derivative of $P(t)=-\frac{1}{8240} t^{3}+\frac{1}{36} t^{2}$ .

Find the derivative $f'(x)$ if $f(x)=\int_{1}^{x^{3}} t^{5} dt$ .

Find where the function is concave up/down and identify points of inflection for $f(x)=6 x^{3}-5 x^{2}+5$ .

Calculate the integral $\int_{1}^{e^{8}} \frac{4}{x} dx$ .

Evaluate the integral from 0 to $\frac{\pi}{4}$ of $\sin(4t)$ .

Evaluate the integral from 1 to $\sqrt{5}$ of $\frac{7}{1+x^{2}}$ with respect to $x$ .

Find the value of $x$ where the local maximum of $f(x)=\int_{0}^{x} \frac{t^{2}-36}{1+\cos ^{2}(t)} dt$ occurs. $x=$

Find $f^{\prime}(3)$ for the function $f(x)=\frac{5 x^{3}-4 x^{2}}{\sqrt[3]{x^{4}}}$ .

Find where the function $f(x)=6 x^{3}-5 x^{2}+5$ is concave up/down and its points of inflection.

Find $f^{\prime \prime}(x)$ for $f(x)=\int_{0}^{x}(t^{3}+7 t^{2}+6) dt$ .

Evaluate the integral from 1 to 9 of $(2x^{2}+5)/\sqrt{x}$ dx.

Calculate the integral from -4 to 1 of the function $3 x^{7} + 2 x^{2}$ .

Find the derivative of $h(x)=\int_{-5}^{\sin (x)}\left(\cos \left(t^{5}\right)+t\right) d t$ . What is $h^{\prime}(x)$ ?

Graph the function $f(x)=x(x-2.4)^{3}$ and find its inflection point(s).

Calculate the integral from 3 to 2 of $\sin(t) \, dt$ .

Find the derivative of $f(x)=\int_{3}^{x}\left(\frac{1}{3} t^{2}-1\right)^{8} dt$ . What is $f'(x)$ ?

Find the derivative of $F(x)=\int_{x}^{3} \sin(t^{2}) dt$ . What is $F'(x)$ ?

Find the derivative of $f(x)=\int_{-3}^{x} \sqrt{t^{3}+3^{3}} dt$ . What is $f'(x)$ ?

Find the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for these functions: 1- $f(x, y)=e^{xy}+x$ , 2- $f(x, y)=\ln(x+y)$ .

Berechnen Sie die Produktionsmenge $x$ , bei der die Grenzkosten $25 \,€/$ Stück sind, gegeben die Kostenfunktion $K(x)=0,00002 x^{3}-0,02 x^{2}+21,4 x+1200$ .

Find the value of $x$ where the local maximum of $f(x)=\int_{0}^{x} \frac{t^{2}-25}{1+\cos ^{2}(t)} d t$ occurs. $x=$

Find the slope of the tangent line to $9 x^{3}+x y+9 y^{4}=735$ at the point $(1,-3)$ .

Find the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for: 1- $f(x, y)=\mathrm{e}^{x y^{3}+x}$ , 2- $f(x, y)=\operatorname{Ln}(x+y)$ .

Find the slope of the tangent line to $9 x^{3}+x y+9 y^{4}=735$ at $(1,-3)$ . The slope is $\square$ .

Berechnen Sie die Ableitungen $f'(x)$ und $f''(x)$ für die Funktion $f(x) = x^3 + 3x^2$ .

Bestimme die erste Ableitung der Funktionen: $f(x)=x^{2}+3x^{4}$ und $f(x)=3x^{6}-14x^{2}$ .

Finde die erste Ableitung der folgenden Funktionen: a) $f(x)=x^{2}+3 x^{4}$ b) $f(x)=3 x^{6}-14 x^{2}$

Evaluate these integrals: (i) $\int \sec ^{5}\left(\frac{\theta}{2}\right) \tan \left(\frac{\theta}{2}\right) d \theta$ ; (ii) $\int \frac{\cos t}{1+\cos t} d t$ .

Find the moment about the $y$ -axis for the area under $y=\cos x$ , from $x=0$ to $x=\frac{\pi}{2}$ .

Untersuchen Sie lokale Extremalpunkte und skizzieren Sie die Graphen für: a) $f(x)=2 x^{2}+3 x-5$ , b) $f(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-3 x$ , c) $f(x)=\frac{1}{4} x^{3}-2$ .

Compute the differentials of the function $f(x, y) = x^{2} e^{5y} + x$ .

Calculate the integral from 1 to 2 of $\frac{2}{\sqrt{x}}$ with respect to $x$ .

Calculate the integral from 1 to 2 of $\frac{x^{2}+3}{x^{2}}$ with respect to $x$ .

Finde die Seitenlänge der quadratischen Grundfläche eines pyramidenförmigen Zelts, das den Rauminhalt maximiert, bei $3 \mathrm{~m}$ Stäben.

Bestimme die Ableitungen der Funktionen: a) $f_{a}(x)=a x^{3}+b x^{2}+c x+d$ , b) $f_{b}(x)=\frac{x-6}{\sqrt{x}}$ , c) $f_{c}(x)=\left(x^{2}-4\right) \sqrt{x^{2}+4}$ .

Oblicz granicę dla ciągu $a_{n}=\frac{n(\sqrt{3 n}-7)^{2}}{(\sqrt{n+2}+2)^{4}}$ lub wyjaśnij, dlaczego nie istnieje.

Find the surface area generated by revolving $y=2\sqrt{x}$ from $x=6$ to $x=9$ about the $x$ -axis.

Find the surface area from revolving $y=2 \sqrt{x}$ around the $x$ -axis for $6 \leq x \leq 9$ .

Find the volume of the solid formed by revolving the area between $y=x$ , $y=11$ , and $x=0$ around $y=12$ .

Find the local maxima and minima of $f(x)=x-2 \sin x$ for $0 \leq x \leq 2\pi$ .

Approximate the arc length of $y=\sin 2x$ from $0$ to $2\pi$ . Round to three decimal places. Choices: (A) 10.340 (B) 5.270 (C) 7.640 (D) 10.540 (E) 11.740

Find the volume of the solid formed by revolving the area between $y=\frac{1}{x}$ , $y=0$ , $x=8$ , and $x=15$ around the $x$ -axis.

Find the volume of the solid formed by revolving the region bounded by $y=2-x$ , $y=0$ , and $x=0$ around the $x$ -axis.

Find the average value of $f(x, y) = y$ over $y$ from -2 to 2 and $x$ to 4. Options: (A) 0 (B) 1.3 (C) 10 (D) 2

Find the acceleration of the particle at time $t=\pi$ given $x(t)=\sin (2 t)-\cos (3 t)$ .

Find the volume of the solid formed by revolving the area between $y=\sin x$ , $y=0$ , from $0$ to $\frac{\pi}{2}$ around $y=11$ .

Calculate the area between $f(x)=\frac{14 x}{x^{2}+1}$ and $y=0$ for $0 \leq x \leq 5$ . Round to three decimal places.

Find the volume of the solid formed by revolving the area between $y=19 x^{2}$ , $y=0$ , and $x=2$ around $x=2$ .

Find the area of the surface formed by revolving $y=\frac{1}{8} x^{3}$ from $0$ to $8$ around the $x$ -axis.

Find the volume of the solid formed by revolving the area between $y=x^{5}$ and $y=32$ around the $y$ -axis.

Calculate the integral $\int_{0}^{3} \int_{0}^{5}\left(y e^{x}-x-y\right) d x d y$ . Choose the correct answer from the options.

Find the volume of the solid formed by revolving the region bounded by $y=\sqrt{x+156}, y=x, y=0$ around the $x$ -axis using the shell method. Choose the correct answer from the options provided.

Find the volume of the solid formed by revolving the area bounded by $y=x^{\frac{4}{5}}$ , $y=1$ , and $x=0$ around the $y$ -axis.

Find the volume of the solid formed by revolving the region bounded by $y=x^{6}$ , $x=0$ , and $y=64$ around the $x$ -axis using the shell method.

Find the volume of the solid formed by revolving the area between $y=\frac{15}{x^{2}}$ , $y=0$ , $x=1$ , and $x=8$ around the $x$ -axis. Round to two decimal places.

Find the volume of the solid formed by revolving the area between $y=8$ and $y=16-\frac{x^{2}}{16}$ around the $x$ -axis.

Find the tangent to $f(x)=x^{3}$ parallel to the line $g: 3x-y=1$ .

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

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

Find the limit: $\lim _{x \rightarrow-4}\left(x^{2}-2 x-8\right)$ . What is the answer?

Find the derivative of the constant function $f(x)=5$ using first principles.

Calculate the integral from 0 to 1 and 0 to 2-y: $\int_{0}^{1} \int_{0}^{2-y} x \, dx \, dy$ .

Find the volume of the solid formed by revolving the area bounded by $y=3 x^{3}, y=0, x=3$ around the $x$ -axis.

Find the area between $f(x)=\frac{8 x}{x^{2}+1}$ and $y=0$ from $0 \leq x \leq 4$ . Options: (A) $A=15 \ln 4$ , (B) $A=4 \ln 17$ , (C) $A=17 \ln 4$ , (D) $A=8 \ln 17$ , (E) $A=4 \ln 15$ .