Calculus

Problem 22601

Find the average rate of change of $g(x)=-x^{2}-5x+11$ from $x=-8$ to $x=4$ .

See Solution

Problem 22602

Find the function $f(x)$ given that $f^{\prime}(x)=0.8 e^{-0.7 x}$ and $f(0)=1$ .

See Solution

Problem 22603

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

See Solution

Problem 22604

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

See Solution

Problem 22605

Find the function $f(x)$ given $f'(x) = 0.2 e^{-0.3 x}$ and $f(0) = 3.$

See Solution

Problem 22606

Find the function $f(x)$ given its derivative $f^{\prime}(x) = 0.2 e^{-0.3 x}$ and $f(0) = 3$ .

See Solution

Problem 22607

Evaluate the integral $\int(\ln (z))^{6} \frac{1}{z} d z$ using the substitution $u=\ln (z)$ and $d u=\frac{1}{z} d z$ .

See Solution

Problem 22608

Find the antiderivative of $f^{\prime}(x)=\sqrt[4]{x}+1$ with the condition $f(1)=0$ .

See Solution

Problem 22609

Calculate the area between the curve $f(x)=36-x^{2}$ and the x-axis from $-8$ to $8$ .

See Solution

Problem 22610

Find the area between the curve $f(x)=e^{2 x}-11$ and the $x$ -axis from $0$ to $\frac{\ln 11}{2}$ .

See Solution

Problem 22611

Calculate the area between the curve $f(x)=e^{3 x}-6$ and the $x$ -axis from $0$ to $\frac{\ln 6}{3}$ .

See Solution

Problem 22612

Find the derivatives using the Fundamental Theorem of Calculus:
1. For $f(x)=\int_{-2}^{x} \sqrt{t^{3}+8} d t$ , $f^{\prime}(x)=\sqrt{x^{3}+8}$ .
2. For $g(x)=\int_{5}^{x} \frac{1}{1+t^{4}} d t$ , $g^{\prime}(x)=\frac{1}{1+x^{4}}-\sigma^{8}$ .

See Solution

Problem 22613

Calculate the area between the curves $y=4 x^{2}-4$ and $y=-3 x^{2}+3$ for $x$ in $[-1, 1]$ .

See Solution

Problem 22614

Find the derivative of $f(x)=\int_{-2}^{x} \sqrt{t^{3}+8} dt$ and $g(x)=\int_{5}^{x} \frac{1}{1+t^{4}} dt$ .

See Solution

Problem 22615

Find the area between the curves $y=x^{6}$ and $y=8x^{3}$ from their intersection points.

See Solution

Problem 22616

Find the area between the curves $y=-9 x^{2}$ and $y=-\sqrt{x}$ at their intersection points.

See Solution

Problem 22617

Evaluate the integral from -7 to -2 of $6u^{2} - 6u + 8$ and round the answer to three decimal places.

See Solution

Problem 22618

Calculate the average value of $f(x) = x^{2} - 15$ over the interval from $x = 0$ to $x = 9$ .

See Solution

Problem 22619

Calculate the volume of the solid formed by rotating $y=\sqrt{36-x^{2}}$ from $x=-6$ to $x=6$ around the $x$ -axis.

See Solution

Problem 22620

Find the average value of $f(x) = \frac{1}{x}$ from $x = \frac{1}{12}$ to $x = 12$ .

See Solution

Problem 22621

Find the volume of the solid formed by revolving the area between $y=3 x^{2}$ and the $x$ -axis from $x=0$ to $x=3$ around the $x$ -axis.

See Solution

Problem 22622

Prove that if $f(x)=x^{n}$ , then $f^{\prime}(x)=n x^{n-1}$ using implicit or logarithmic differentiation.

See Solution

Problem 22623

Find the volume of the solid formed by revolving $y=\sqrt{x}$ from $x=0$ to $x=14$ around the $x$ -axis.

See Solution

Problem 22624

Prove the Quotient Rule: If $f(x)=\frac{u(x)}{v(x)}$ , show that $f^{\prime}(x)=\frac{u^{\prime} v - u v^{\prime}}{v^{2}}$ .

See Solution

Problem 22625

Find the area between the curves $x=y^{2}+1$ and $x=5$ by integrating with respect to $y$ . Provide the exact answer.

See Solution

Problem 22626

Calculate the integral: $\int\left(x^{4 / 3}-3 x^{5 / 2}\right) d x$

See Solution

Problem 22627

Find the sum: $\sum_{k=1}^{\infty} \frac{1}{\sqrt{k^{2}+1}}$ .

See Solution

Problem 22628

Calculate the integral: $\int 7 x^{-3} d x$

See Solution

Problem 22629

Find the bounds $a$ and $b$ for the integral $\int_{a}^{b} f(x) dx$ given that $a=2$ and $b=11$ .

See Solution

Problem 22630

Evaluate the integral: $\int\left(t^{t}+e^{3 t}\right) d t$

See Solution

Problem 22631

Calculate the integral: $\int\left(t^{4}+e^{3 t}\right) d t$

See Solution

Problem 22632

Find the tangent line to $y=f(x)$ where $f(x)=2 \csc ^{2}(x)$ at $x=\frac{2 \pi}{3}$ .

See Solution

Problem 22633

Evaluate the integral: $\int \frac{x d x}{\left(7 x^{2}+3\right)^{5}}$

See Solution

Problem 22634

Find the tangent line equation for $y=-3 \sec ^{2}(x)$ at $x=0$ .

See Solution

Problem 22635

Find $C(x)$ given $C'(x)=5 x^{2}-7 x+4$ and $C(6)=260$ .

See Solution

Problem 22636

Solve the initial-boundary value problem:
$\frac{\partial u}{\partial t}=3 \frac{\partial^{2} u}{\partial x^{2}}, \, u(0, t)=u(\pi, t)=0, \, u(x, 0)=-4 x^{2}.$
Find $u(x, t)$ as a series in terms of $n$ .

See Solution

Problem 22637

Ein Bauer will mit einem 200 m langen Zaun die größte rechteckige Weide am Kanal abstecken. Bestimme die Zielfunktion und das Maximum.

See Solution

Problem 22638

Bestimmen Sie die Flächeninhalte der Funktion $f(x)=4$ im Intervall $[0 ; 3]$ .

See Solution

Problem 22639

Bestimme die Flächeninhalte der Funktionen im Intervall: b) $f(x)=-2,5$ für $[0 ; 3,5]$ c) $f(x)=2x$ für $[0 ; 2]$ d) $f(x)=-2x$ für $[0 ; 3]$ e) $f(x)=3x^{2}$ für $[0 ; 1]$ f) $f(x)=x^{2}$ für $[0 ; 2]$

See Solution

Problem 22640

Berechne die Flächeninhalte der Funktionen im Intervall: b) $f(x)=-2,5$ für $[0 ; 3,5]$ c) $f(x)=2 x$ für $[0 ; 2]$ d) $f(x)=-2 x$ für $[0 ; 3]$ e) $f(x)=3 x^{2}$ für $[0 ; 1]$ f) $f(x)=x^{2}$ für $[0 ; 2]$

See Solution

Problem 22641

Ein Bauer möchte mit einem 200 m langen Zaun eine rechteckige Weide am Kanal abstecken. Bestimmen Sie die maximalen Maße der Weide.

See Solution

Problem 22642

Find the position function $s(t)$ if $a(t)=4t+4$ , $v(0)=4$ , and $s(0)=w$ .

See Solution

Problem 22643

Find the position function $s(t)$ for an object with acceleration $a(t)=4t+4$ , initial velocity $v(0)=4$ , and $s(0)=10$ .

See Solution

Problem 22644

Calculate the average rate of change of $f(x)=\sqrt{x}$ from $x_{1}=16$ to $x_{2}=36$ . The answer is $\square$ .

See Solution

Problem 22645

Find the marginal cost of producing the 9,001st cell phone using $M(n)=0.0000015 n^{2}+0.0001 n+4$ . Answer in dollars.

See Solution

Problem 22646

Bestimme die Ableitung von $f(t)=1,5 \cdot t \cdot e^{-0,5 t+1}+37$ .

See Solution

Problem 22647

Bestimmen Sie die lokale Änderungsrate von $f$ an den Stellen a mit der h-Methode für die Funktionen a) bis f) und vergleichen Sie die Grenzwerte mit dem Differenzenquotienten für $h=0,000001$ .

See Solution

Problem 22648

Milch bei $6^{\circ} \mathrm{C}$ erwärmt sich im Raum mit $25^{\circ} \mathrm{C}$ . Bestimme die Erwärmungsgeschwindigkeit und die Temperatur in Abhängigkeit von der Zeit.

See Solution

Problem 22649

Berechne den Flächeninhalt unter $f(x)=3 \cdot x^{2}$ im Intervall $[0; 2]$ mit $n=16$ gleich großen Teilintervallen.

See Solution

Problem 22650

Berechne den Flächeninhalt unter $f(x)=3 \cdot x^{2}$ im Intervall $[0 ; 2]$ mit $n=8$ Untersummen. $U_{8} =$

See Solution

Problem 22651

Berechne den Flächeninhalt zwischen dem Graphen von $f(x)=3 \cdot x^{2}$ und der $x$ -Achse im Intervall $[0;2]$ für $n=16$ .

See Solution

Problem 22652

Izračunajte limitu: $\lim _{n \rightarrow \infty} \frac{2^{n}-3^{n}}{1+3^{n}}$ .

See Solution

Problem 22653

Find the derivative $\frac{d y}{d x}$ for the polar function $r=2 \sin (3 \theta)$ .

See Solution

Problem 22654

Find the slope of the tangent line to the polar curve $r=2 \theta$ at $\theta=\frac{\pi}{2}$ .

See Solution

Problem 22655

Bestimme die Ableitung von $f(x)=2 \sin(x) + x^{-3}$ .

See Solution

Problem 22656

Estimate the area under $g(x) = 5x^2 + 5$ on $[1,3]$ using 8 rectangles. Find the area bounds.

See Solution

Problem 22657

Graph the function $f(x)=3x+2$ and estimate the area under it from $x=0$ to $x=3$ using rectangles.

See Solution

Problem 22658

Evaluate the definite integral using the limit definition: $\int_{1}^{6} 4 \, dx$ .

See Solution

Problem 22659

Find the absolute minimum of $f(x)=x^{4}-8 x^{2}+10$ on the interval $[-3,1]$ .

See Solution

Problem 22660

Find the absolute minimum value of the function $f(x)=x^{4}-8 x^{2}+10$ on the interval $[-3,1]$ .

See Solution

Problem 22661

Find intervals where the function $f$ is decreasing given that $f^{\prime}(x)=-x^{3}+18 x^{2}-72 x$ .

See Solution

Problem 22662

Find the intervals where the function $f(x)=x^{4}-16 x^{3}$ is decreasing.

See Solution

Problem 22663

Find the intervals where the function $f(x)=2 x^{3}-3 x^{2}-12 x+20$ is increasing.

See Solution

Problem 22664

Berechnen Sie die Ableitungen von $f$ für die folgenden Funktionen: a) $f(x)=x^{3}+x^{2}$ b) $f(x)=1-x^{4}$ c) $f(x)=x^{3}+x^{5}+x+2$

See Solution

Problem 22665

Find the absolute maximum of the function $f(x)=x^{3}-9 x^{2}+27 x-5$ on the interval $[2,4]$ .

See Solution

Problem 22666

Finde die zwei falschen Aussagen über Ableitungen: (1) $f(x)=x^{3} \Rightarrow f^{\prime}(x)=3 \cdot x^{2}$ , (2) $f(x)=x^{x} \Rightarrow f^{\prime}(x)=x \cdot x^{x-1}$ , (3) $f(x)=x^{2 a} \Rightarrow f^{\prime}(x)=2 \cdot x^{a}$ , (4) $f(x)=x^{a+1} \Rightarrow f^{\prime}(x)=(a+1) \cdot x^{a}$ .

See Solution

Problem 22667

Find the second derivative of $f$ where $f'(x)=-2+x+3 e^{-\cos(4x)}$ and analyze $f$ on the interval $0 < x < ?$ .

See Solution

Problem 22668

Find the first derivative of $f(x) = \frac{5}{x^3} - \frac{x^3}{5}$ .

See Solution

Problem 22669

Find the horizontal asymptote of the function $f(x)=\frac{9 x^{3}+3 x^{2}-2 x}{9 x^{2}+3 x-2}$ .

See Solution

Problem 22670

Bestimme die erste Ableitung von $f(x) = (x + 4)^2$ .

See Solution

Problem 22671

Find the second derivative of the function $f$ given that $f^{\prime}(x)=-2+x+3 e^{-\cos (4 x)}$ .

See Solution

Problem 22672

Find the value of $x$ that minimizes the average cost function for $C(x)=489888+1.7x+42x^{2}$ , where $1 \leq x \leq 285$ .

See Solution

Problem 22673

Berechnen Sie die erste Ableitung von $f(x) = \sqrt[4]{x} + \frac{1}{x^{2}}$ .

See Solution

Problem 22674

The drug concentration $C(t)=\frac{1}{2t^{2}+1}$ :
a. Find the horizontal asymptote and describe concentration behavior as $t$ increases. b. Graph $C(t)$ ; state the domain and range with reasoning. c. Find when the concentration is highest.

See Solution

Problem 22675

Find the horizontal asymptote of $C(t)=\frac{1}{2t^{2}+1}$ and explain the concentration behavior as $t$ increases. Graph it with appropriate domain and range, then find when the concentration is highest.

See Solution

Problem 22676

Find the increase in total costs when production rises from 62 to 82 units, given marginal cost $262+\frac{212}{\sqrt{x}}$ .

See Solution

Problem 22677

Find the solution to the differential equation: $y^{\prime}=2 x\left(\log x^{2}+\frac{1}{\ln 10}\right)$ .

See Solution

Problem 22678

Find the differential of the function $A=20 \pi x^{2}$ .

See Solution

Problem 22679

Estimate the cost change using differentials for $C(x)=278+5 x+0.16 x^{2}$ from 71 to 74 units.

See Solution

Problem 22680

Find the first derivative of $f(t) = 2 \cdot (3t - 1)^2$ .

See Solution

Problem 22681

Find the derivative of the function $f(x) = (5x + 2)^3$ .

See Solution

Problem 22682

Calculate and round $e^{-0.41}$ to three decimal places: $e^{-0.41} \approx \square$ .

See Solution

Problem 22683

Calculate $f(a)$ , $f(a+h)$ , and $\frac{f(a+h)-f(a)}{h}$ for $f(x)=7+6x^{2}$ .

See Solution

Problem 22684

Find the limit as $x$ approaches -19 for the piecewise function: $f(x) = x^2 + 38x + 379$ if $x < -19$ and $f(x) = 20x + 398$ if $x \geq -19$ .

See Solution

Problem 22685

Find the limit as $x$ approaches -19 from the left for the function defined as:
$f(x)=\left\{\begin{array}{ll} x^{2}+38 x+379 & \text { if } x<-19 \\ 20 x+398 & \text { if } x \geq-19 \end{array}\right.$

See Solution

Problem 22686

Find the limit as $x$ approaches -19 from the right for the function defined as: $f(x)=\left\{\begin{array}{ll} x^{2}+38 x+379 & \text { if } x<-19 \\ 20 x+398 & \text { if } x \geq-19 \end{array}\right.$

See Solution

Problem 22687

Find the slope of the tangent line for the function $f(x)=2x-8x^{2}-7x^{3}$ at $x=-2$ .

See Solution

Problem 22688

Find the meaning of $f^{\prime}(7)=-1253$ for the function $f(x)=-8 x^{3}-6 x^{2}+7 x+3$ .

See Solution

Problem 22689

Find the derivative of the function using the Product or Quotient Rule: $f(x)=\frac{-9 x^{2}+6}{8 x^{5}-8}$

See Solution

Problem 22690

Given the function $f(x)=x^{3}+3 x$ , find the average rate of change for intervals: $[2,3]$ , $[2,2.5]$ , $[2,2.1]$ , $[2,2.01]$ , $[2,2.001]$ . What value do these rates approach as intervals shrink?

See Solution

Problem 22691

Given $f(x)=x^{3}+3x$ , find the average rate of change for the intervals: $[2,3]$ , $[2,2.5]$ , $[2,2.1]$ , $[2,2.01]$ , $[2,2.001]$ .

See Solution

Problem 22692

Find the slopes of the tangent lines for $f(x)=4-2x-4x^{2}$ at $x=0$ , $x=1$ , and $x=2$ .

See Solution

Problem 22693

A rock creates a ripple with a radius increasing at $4 \mathrm{ft} / \mathrm{sec}$ . Find the area increase speed when radius is 2 ft.

See Solution

Problem 22694

Find the derivative of $\frac{1 - \ln x}{2 x^{2}}$ with respect to $x$ .

See Solution

Problem 22695

Calculate the one-sided limit: $\lim _{x \rightarrow 6^{-}}\left(\frac{x-3}{x-6}\right)$ .

See Solution

Problem 22696

Find the dimensions of a square-based crate with volume $12800 \mathrm{ft}^{3}$ that minimizes material cost.

See Solution

Problem 22697

Find the tangent line equation to $f(x)=\frac{5}{1-3x}$ at point $P(2, f(2))$ .

See Solution

Problem 22698

Find the $x$ -coordinates where the tangent line to $f(x)=x^{3}+x^{2}-x+3$ is horizontal.

See Solution

Problem 22699

Mosquito population grows exponentially. Given $\mathrm{N}(t)=\mathrm{N}_{0} e^{\mathrm{kt}}$ , find $\mathrm{N}(4)$ and time for 70,000 mosquitoes.

See Solution

Problem 22700

Find the maximum value of $y=\frac{2}{3} x^{3}-\frac{1}{2} x^{2}-3 x+2$ on $[0,2]$ .

See Solution

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Calculus

Problem 22601

Find the average rate of change of g(x)=−x2−5x+11g(x)=-x^{2}-5x+11g(x)=−x2−5x+11 from x=−8x=-8x=−8 to x=4x=4x=4.

Problem 22602

Find the function f(x)f(x)f(x) given that f′(x)=0.8e−0.7xf^{\prime}(x)=0.8 e^{-0.7 x}f′(x)=0.8e−0.7x and f(0)=1f(0)=1f(0)=1.

Problem 22603

Find the derivative f′(5)f^{\prime}(5)f′(5) for the function f(x)=2x3−5xf(x) = 2 \sqrt{x^{3}} - \frac{5}{\sqrt{x}}f(x)=2x3​−x​5​.

Problem 22604

Find the derivative f′(5)f^{\prime}(5)f′(5) for the function f(x)=2x3−5xf(x)=2 \sqrt{x^{3}}-\frac{5}{\sqrt{x}}f(x)=2x3​−x​5​.

Problem 22605

Find the function f(x)f(x)f(x) given f′(x)=0.2e−0.3xf'(x) = 0.2 e^{-0.3 x}f′(x)=0.2e−0.3x and f(0)=3.f(0) = 3. f(0)=3.

Problem 22606

Find the function f(x)f(x)f(x) given its derivative f′(x)=0.2e−0.3xf^{\prime}(x) = 0.2 e^{-0.3 x}f′(x)=0.2e−0.3x and f(0)=3f(0) = 3f(0)=3.

Problem 22607

Evaluate the integral ∫(ln⁡(z))61zdz\int(\ln (z))^{6} \frac{1}{z} d z∫(ln(z))6z1​dz using the substitution u=ln⁡(z)u=\ln (z)u=ln(z) and du=1zdzd u=\frac{1}{z} d zdu=z1​dz.

Problem 22608

Find the antiderivative of f′(x)=x4+1f^{\prime}(x)=\sqrt[4]{x}+1f′(x)=4x​+1 with the condition f(1)=0f(1)=0f(1)=0.

Problem 22609

Calculate the area between the curve f(x)=36−x2f(x)=36-x^{2}f(x)=36−x2 and the x-axis from −8-8−8 to 888.

Problem 22610

Find the area between the curve f(x)=e2x−11f(x)=e^{2 x}-11f(x)=e2x−11 and the xxx-axis from 000 to ln⁡112\frac{\ln 11}{2}2ln11​.

Problem 22611

Calculate the area between the curve f(x)=e3x−6f(x)=e^{3 x}-6f(x)=e3x−6 and the xxx-axis from 000 to ln⁡63\frac{\ln 6}{3}3ln6​.

Problem 22612

Problem 22613

Calculate the area between the curves y=4x2−4y=4 x^{2}-4y=4x2−4 and y=−3x2+3y=-3 x^{2}+3y=−3x2+3 for xxx in [−1,1][-1, 1][−1,1].

Problem 22614

Find the derivative of f(x)=∫−2xt3+8dtf(x)=\int_{-2}^{x} \sqrt{t^{3}+8} dtf(x)=∫−2x​t3+8​dt and g(x)=∫5x11+t4dtg(x)=\int_{5}^{x} \frac{1}{1+t^{4}} dtg(x)=∫5x​1+t41​dt.

Problem 22615

Find the area between the curves y=x6y=x^{6}y=x6 and y=8x3y=8x^{3}y=8x3 from their intersection points.

Problem 22616

Find the area between the curves y=−9x2y=-9 x^{2}y=−9x2 and y=−xy=-\sqrt{x}y=−x​ at their intersection points.

Problem 22617

Evaluate the integral from -7 to -2 of 6u2−6u+86u^{2} - 6u + 86u2−6u+8 and round the answer to three decimal places.

Problem 22618

Calculate the average value of f(x)=x2−15f(x) = x^{2} - 15f(x)=x2−15 over the interval from x=0x = 0x=0 to x=9x = 9x=9.

Problem 22619

Calculate the volume of the solid formed by rotating y=36−x2y=\sqrt{36-x^{2}}y=36−x2​ from x=−6x=-6x=−6 to x=6x=6x=6 around the xxx-axis.

Problem 22620

Find the average value of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ from x=112x = \frac{1}{12}x=121​ to x=12x = 12x=12.

Problem 22621

Find the volume of the solid formed by revolving the area between y=3x2y=3 x^{2}y=3x2 and the xxx-axis from x=0x=0x=0 to x=3x=3x=3 around the xxx-axis.

Problem 22622

Prove that if f(x)=xnf(x)=x^{n}f(x)=xn, then f′(x)=nxn−1f^{\prime}(x)=n x^{n-1}f′(x)=nxn−1 using implicit or logarithmic differentiation.

Problem 22623

Find the volume of the solid formed by revolving y=xy=\sqrt{x}y=x​ from x=0x=0x=0 to x=14x=14x=14 around the xxx-axis.

Problem 22624

Prove the Quotient Rule: If f(x)=u(x)v(x)f(x)=\frac{u(x)}{v(x)}f(x)=v(x)u(x)​, show that f′(x)=u′v−uv′v2f^{\prime}(x)=\frac{u^{\prime} v - u v^{\prime}}{v^{2}}f′(x)=v2u′v−uv′​.

Problem 22625

Find the area between the curves x=y2+1x=y^{2}+1x=y2+1 and x=5x=5x=5 by integrating with respect to yyy. Provide the exact answer.

Problem 22626

Calculate the integral: ∫(x4/3−3x5/2)dx\int\left(x^{4 / 3}-3 x^{5 / 2}\right) d x∫(x4/3−3x5/2)dx

Problem 22627

Find the sum: ∑k=1∞1k2+1\sum_{k=1}^{\infty} \frac{1}{\sqrt{k^{2}+1}}k=1∑∞​k2+1​1​.

Problem 22628

Calculate the integral: ∫7x−3dx\int 7 x^{-3} d x∫7x−3dx

Problem 22629

Find the bounds aaa and bbb for the integral ∫abf(x)dx\int_{a}^{b} f(x) dx∫ab​f(x)dx given that a=2a=2a=2 and b=11b=11b=11.

Problem 22630

Evaluate the integral: ∫(tt+e3t)dt\int\left(t^{t}+e^{3 t}\right) d t∫(tt+e3t)dt

Problem 22631

Calculate the integral: ∫(t4+e3t)dt\int\left(t^{4}+e^{3 t}\right) d t∫(t4+e3t)dt

Problem 22632

Find the tangent line to y=f(x)y=f(x)y=f(x) where f(x)=2csc⁡2(x)f(x)=2 \csc ^{2}(x)f(x)=2csc2(x) at x=2π3x=\frac{2 \pi}{3}x=32π​.

Problem 22633

Evaluate the integral: ∫xdx(7x2+3)5\int \frac{x d x}{\left(7 x^{2}+3\right)^{5}}∫(7x2+3)5xdx​

Problem 22634

Find the tangent line equation for y=−3sec⁡2(x)y=-3 \sec ^{2}(x)y=−3sec2(x) at x=0x=0x=0.

Problem 22635

Find C(x)C(x)C(x) given C′(x)=5x2−7x+4C'(x)=5 x^{2}-7 x+4C′(x)=5x2−7x+4 and C(6)=260C(6)=260C(6)=260.

Problem 22636

Problem 22637

Ein Bauer will mit einem 200 m langen Zaun die größte rechteckige Weide am Kanal abstecken. Bestimme die Zielfunktion und das Maximum.

Problem 22638

Bestimmen Sie die Flächeninhalte der Funktion f(x)=4f(x)=4f(x)=4 im Intervall [0;3][0 ; 3][0;3].

Problem 22639

Problem 22640

Problem 22641

Ein Bauer möchte mit einem 200 m langen Zaun eine rechteckige Weide am Kanal abstecken. Bestimmen Sie die maximalen Maße der Weide.

Problem 22642

Find the average rate of change of $g(x)=-x^{2}-5x+11$ from $x=-8$ to $x=4$ .

Find the function $f(x)$ given that $f^{\prime}(x)=0.8 e^{-0.7 x}$ and $f(0)=1$ .

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

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

Find the function $f(x)$ given $f'(x) = 0.2 e^{-0.3 x}$ and $f(0) = 3.$

Find the function $f(x)$ given its derivative $f^{\prime}(x) = 0.2 e^{-0.3 x}$ and $f(0) = 3$ .

Evaluate the integral $\int(\ln (z))^{6} \frac{1}{z} d z$ using the substitution $u=\ln (z)$ and $d u=\frac{1}{z} d z$ .

Find the antiderivative of $f^{\prime}(x)=\sqrt[4]{x}+1$ with the condition $f(1)=0$ .

Calculate the area between the curve $f(x)=36-x^{2}$ and the x-axis from $-8$ to $8$ .

Find the area between the curve $f(x)=e^{2 x}-11$ and the $x$ -axis from $0$ to $\frac{\ln 11}{2}$ .

Calculate the area between the curve $f(x)=e^{3 x}-6$ and the $x$ -axis from $0$ to $\frac{\ln 6}{3}$ .

Calculate the area between the curves $y=4 x^{2}-4$ and $y=-3 x^{2}+3$ for $x$ in $[-1, 1]$ .

Find the derivative of $f(x)=\int_{-2}^{x} \sqrt{t^{3}+8} dt$ and $g(x)=\int_{5}^{x} \frac{1}{1+t^{4}} dt$ .

Find the area between the curves $y=x^{6}$ and $y=8x^{3}$ from their intersection points.

Find the area between the curves $y=-9 x^{2}$ and $y=-\sqrt{x}$ at their intersection points.

Evaluate the integral from -7 to -2 of $6u^{2} - 6u + 8$ and round the answer to three decimal places.

Calculate the average value of $f(x) = x^{2} - 15$ over the interval from $x = 0$ to $x = 9$ .

Calculate the volume of the solid formed by rotating $y=\sqrt{36-x^{2}}$ from $x=-6$ to $x=6$ around the $x$ -axis.

Find the average value of $f(x) = \frac{1}{x}$ from $x = \frac{1}{12}$ to $x = 12$ .

Find the volume of the solid formed by revolving the area between $y=3 x^{2}$ and the $x$ -axis from $x=0$ to $x=3$ around the $x$ -axis.

Prove that if $f(x)=x^{n}$ , then $f^{\prime}(x)=n x^{n-1}$ using implicit or logarithmic differentiation.

Find the volume of the solid formed by revolving $y=\sqrt{x}$ from $x=0$ to $x=14$ around the $x$ -axis.

Prove the Quotient Rule: If $f(x)=\frac{u(x)}{v(x)}$ , show that $f^{\prime}(x)=\frac{u^{\prime} v - u v^{\prime}}{v^{2}}$ .

Find the area between the curves $x=y^{2}+1$ and $x=5$ by integrating with respect to $y$ . Provide the exact answer.

Calculate the integral: $\int\left(x^{4 / 3}-3 x^{5 / 2}\right) d x$

Find the sum: $\sum_{k=1}^{\infty} \frac{1}{\sqrt{k^{2}+1}}$ .

Calculate the integral: $\int 7 x^{-3} d x$

Find the bounds $a$ and $b$ for the integral $\int_{a}^{b} f(x) dx$ given that $a=2$ and $b=11$ .

Evaluate the integral: $\int\left(t^{t}+e^{3 t}\right) d t$

Calculate the integral: $\int\left(t^{4}+e^{3 t}\right) d t$

Find the tangent line to $y=f(x)$ where $f(x)=2 \csc ^{2}(x)$ at $x=\frac{2 \pi}{3}$ .

Evaluate the integral: $\int \frac{x d x}{\left(7 x^{2}+3\right)^{5}}$

Find the tangent line equation for $y=-3 \sec ^{2}(x)$ at $x=0$ .

Find $C(x)$ given $C'(x)=5 x^{2}-7 x+4$ and $C(6)=260$ .

Bestimmen Sie die Flächeninhalte der Funktion $f(x)=4$ im Intervall $[0 ; 3]$ .

Find the position function $s(t)$ if $a(t)=4t+4$ , $v(0)=4$ , and $s(0)=w$ .

Find the position function $s(t)$ for an object with acceleration $a(t)=4t+4$ , initial velocity $v(0)=4$ , and $s(0)=10$ .

Calculate the average rate of change of $f(x)=\sqrt{x}$ from $x_{1}=16$ to $x_{2}=36$ . The answer is $\square$ .

Find the marginal cost of producing the 9,001st cell phone using $M(n)=0.0000015 n^{2}+0.0001 n+4$ . Answer in dollars.

Bestimme die Ableitung von $f(t)=1,5 \cdot t \cdot e^{-0,5 t+1}+37$ .

Bestimmen Sie die lokale Änderungsrate von $f$ an den Stellen a mit der h-Methode für die Funktionen a) bis f) und vergleichen Sie die Grenzwerte mit dem Differenzenquotienten für $h=0,000001$ .

Milch bei $6^{\circ} \mathrm{C}$ erwärmt sich im Raum mit $25^{\circ} \mathrm{C}$ . Bestimme die Erwärmungsgeschwindigkeit und die Temperatur in Abhängigkeit von der Zeit.

Berechne den Flächeninhalt unter $f(x)=3 \cdot x^{2}$ im Intervall $[0; 2]$ mit $n=16$ gleich großen Teilintervallen.

Berechne den Flächeninhalt unter $f(x)=3 \cdot x^{2}$ im Intervall $[0 ; 2]$ mit $n=8$ Untersummen. $U_{8} =$

Berechne den Flächeninhalt zwischen dem Graphen von $f(x)=3 \cdot x^{2}$ und der $x$ -Achse im Intervall $[0;2]$ für $n=16$ .

Izračunajte limitu: $\lim _{n \rightarrow \infty} \frac{2^{n}-3^{n}}{1+3^{n}}$ .

Find the derivative $\frac{d y}{d x}$ for the polar function $r=2 \sin (3 \theta)$ .

Find the slope of the tangent line to the polar curve $r=2 \theta$ at $\theta=\frac{\pi}{2}$ .

Bestimme die Ableitung von $f(x)=2 \sin(x) + x^{-3}$ .

Estimate the area under $g(x) = 5x^2 + 5$ on $[1,3]$ using 8 rectangles. Find the area bounds.

Graph the function $f(x)=3x+2$ and estimate the area under it from $x=0$ to $x=3$ using rectangles.

Evaluate the definite integral using the limit definition: $\int_{1}^{6} 4 \, dx$ .

Find the absolute minimum of $f(x)=x^{4}-8 x^{2}+10$ on the interval $[-3,1]$ .

Find the absolute minimum value of the function $f(x)=x^{4}-8 x^{2}+10$ on the interval $[-3,1]$ .

Find intervals where the function $f$ is decreasing given that $f^{\prime}(x)=-x^{3}+18 x^{2}-72 x$ .

Find the intervals where the function $f(x)=x^{4}-16 x^{3}$ is decreasing.

Find the intervals where the function $f(x)=2 x^{3}-3 x^{2}-12 x+20$ is increasing.

Berechnen Sie die Ableitungen von $f$ für die folgenden Funktionen: a) $f(x)=x^{3}+x^{2}$ b) $f(x)=1-x^{4}$ c) $f(x)=x^{3}+x^{5}+x+2$

Find the absolute maximum of the function $f(x)=x^{3}-9 x^{2}+27 x-5$ on the interval $[2,4]$ .

Find the second derivative of $f$ where $f'(x)=-2+x+3 e^{-\cos(4x)}$ and analyze $f$ on the interval $0 < x < ?$ .

Find the first derivative of $f(x) = \frac{5}{x^3} - \frac{x^3}{5}$ .

Find the horizontal asymptote of the function $f(x)=\frac{9 x^{3}+3 x^{2}-2 x}{9 x^{2}+3 x-2}$ .

Bestimme die erste Ableitung von $f(x) = (x + 4)^2$ .

Find the second derivative of the function $f$ given that $f^{\prime}(x)=-2+x+3 e^{-\cos (4 x)}$ .

Find the value of $x$ that minimizes the average cost function for $C(x)=489888+1.7x+42x^{2}$ , where $1 \leq x \leq 285$ .

Berechnen Sie die erste Ableitung von $f(x) = \sqrt[4]{x} + \frac{1}{x^{2}}$ .

The drug concentration $C(t)=\frac{1}{2t^{2}+1}$ :
a. Find the horizontal asymptote and describe concentration behavior as $t$ increases. b. Graph $C(t)$ ; state the domain and range with reasoning. c. Find when the concentration is highest.