Calculus

Problem 26201

Solve the ODE: $\frac{d y}{d x}=\frac{y(5-6 x)}{x(5-7 y)}$ .

See Solution

Problem 26202

Find the Laplace Transform of $f(t)=\begin{cases}t, & 0 \leq t<1 \\ 0, & t \geq 1\end{cases}$ . What is $L\{f(t)\}$ ?

See Solution

Problem 26203

Show that $\int_{0}^{\pi / 2} \sum_{n=1}^{\infty} \frac{\cos n x}{n^{2}} dx=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n+1)^{3}}$ and prove $\sum_{n=1}^{\infty} f_{n}(x)$ lacks property $\mathscr{C}$ on $[0,1]$ .

See Solution

Problem 26204

Find coefficients $a_n$ and $b_n$ from the series $u(x, t)$ given initial conditions. Which option is correct?

See Solution

Problem 26205

Solve $2 y^{\prime \prime}+y^{\prime}-y=-1$ with $y(0)=1$ and $y^{\prime}(0)=2$ . Find $y^{\prime \prime \prime}(0)$ .

See Solution

Problem 26206

Solve $y^{\prime \prime}-y^{\prime}-2 y=3$ with $y(0)=2$ and $y^{\prime}(0)=1$ . Find $y^{\prime \prime \prime}(0)$ . A. 10 B. -10 C. 13 D. -13

See Solution

Problem 26207

Given the demand function $q=D(p)=526-8p$ , find: a. Domain of $D(p)$ ; b. $D'(p)$ ; c. Elasticity $E(p)$ ; d. For $p=\$45$ , is demand elastic or inelastic? Effect on revenue? e. If price increases by 1\%, how does demand change?

See Solution

Problem 26208

Find $a_{0}, a_{4}, b_{5}$ from the Fourier series $f(x)=\frac{2}{3}+\frac{4}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}} \cos (n \pi x)$ . Options: A, B, C, D, E.

See Solution

Problem 26209

Find the sum of the series $S_{\infty}=20-10+5-\cdots$ . Choices: $\frac{40}{3}$ , $\frac{20}{3}$ , 40, or diverges.

See Solution

Problem 26210

Classify the ODE: $\frac{d y}{d x}=\frac{y(2-6 x)}{x(1-9 y)}$ . Choose: 1st order, linear/homogeneous or non-homogeneous?

See Solution

Problem 26211

Find $a_{0}, a_{4}, b_{5}$ for the Fourier series $f(x)=\frac{2}{3}+\frac{4}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}} \cos (n \pi x)$ .

See Solution

Problem 26212

Find the coefficients $a_{0}, a_{2}, b_{4}$ from the Fourier series $f(x)=\frac{2}{3}+\frac{4}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}} \sin (n \pi x)$ .

See Solution

Problem 26213

Evaluate the integral: $\int \frac{e^{x} d x}{\sqrt{1-e^{2 x}}}$

See Solution

Problem 26214

Calculate the average value of $f(x)=8x$ on the interval $[5,7]$ .

See Solution

Problem 26215

Evaluate the integral using substitution: $\int_{\pi / 3}^{2 \pi} 4 \cos ^{3} x \sin x \, dx$ .

See Solution

Problem 26216

Evaluate the integral: $\int \frac{d x}{x \ln x^{3}}$ .

See Solution

Problem 26217

Calculate the total in a savings account with an investment of \$ 2250 at 9.0\% for 15 years with continuous compounding. Round to two decimals.

See Solution

Problem 26218

Evaluate the integral: $\int \csc \left(z+\frac{\pi}{4}\right) \cot \left(z+\frac{\pi}{4}\right) dz$

See Solution

Problem 26219

Evaluate the integral: $\int \sin 2 x \cos 4 x \, dx$

See Solution

Problem 26220

Find $x$ where $f(x)=\sqrt{x+1}$ equals its average value on the interval $[0,15]$ .

See Solution

Problem 26221

Calculate the area between the curve $y=\frac{6 x}{1+x^{2}}$ and the $x$ -axis for $-4 \leq x \leq 4$ .

See Solution

Problem 26222

Evaluate the integral: $\int x^{2}\left(x^{3}-10\right)^{4} dx$

See Solution

Problem 26223

Evaluate the integral using substitution: $\int_{0}^{\pi / 16} 16 \tan 4 x \, dx$ .

See Solution

Problem 26224

Evaluate the integral from 0 to 1: $\int_{0}^{1} \sqrt{3 x+3} \, dx$ .

See Solution

Problem 26225

Integrate the function $14(2 x-3)^{6}$ from $-1$ to $1.5$ . What is the result?

See Solution

Problem 26226

Differentiate $\sin^{4}(3x)$ with respect to $x$ .

See Solution

Problem 26227

Evaluate the integral using substitutions: $\int \frac{(2 r-5) \cos \sqrt{3(2 r-5)^{2}+10}}{\sqrt{3(2 r-5)^{2}+10}} d r$

See Solution

Problem 26228

Evaluate the integral from 0 to 1: $\int_{0}^{1} \sqrt{3 x+3} \, dx$ .

See Solution

Problem 26229

Evaluate the integral: $\int \frac{8 d x}{\sqrt{25-64 x^{2}}}$ .

See Solution

Problem 26230

Find the integral of $\sin 2x \cos 4x$ with respect to $x$ : $\int \sin 2x \cos 4x \, dx$ .

See Solution

Problem 26231

Calculate the integral $\int \tan ^{5} x \sec ^{3} x \, dx$ .

See Solution

Problem 26232

Evaluate the integral: $\int \frac{8 d x}{\sqrt{25-64 x^{2}}}$

See Solution

Problem 26233

Evaluate the integral $\int \sin 2x \cos 4x \, dx$ .

See Solution

Problem 26234

Calculate the integral $\int_{0}^{\frac{\pi}{2}} \cos ^{3} x \, dx$ .

See Solution

Problem 26235

Evaluate the integral $\int \cot^{2} x \csc^{4} x \, dx$ .

See Solution

Problem 26236

Evaluate the integral $\int \frac{d x}{x^{2}-4 x+13}$ .

See Solution

Problem 26237

Evaluate the integral: $\int x^{3} \sqrt{x^{4}+9} \, dx$

See Solution

Problem 26238

Calcula el integral definido usando el Teorema Fundamental del Cálculo: $\int_{1}^{3} \frac{2 x+3}{x+1} d x$

See Solution

Problem 26239

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{0}^{\frac{\pi}{2}} \cos ^{3} x \, dx$ .

See Solution

Problem 26240

Calcula el integral definido usando el Teorema Fundamental del Cálculo: $\int_{0}^{\frac{\pi}{2}} \cos ^{3} x \, dx$

See Solution

Problem 26241

Find the value(s) of $c$ satisfying the Mean Value Theorem for $f(x) = \tan^{-1}x$ on $[-\sqrt{3}, \sqrt{3}]$ .

See Solution

Problem 26242

Evaluate the integral using substitution: $\int_{1}^{4} \frac{3-\sqrt{x}}{\sqrt{x}} d x$ .

See Solution

Problem 26243

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{0}^{\frac{\pi}{3}} \frac{\tan ^{3} x}{\sec x} d x$ .

See Solution

Problem 26244

Calculate the area between the curve $y=\frac{8}{x^{2}+4}$ , the $x$ -axis, $y$ -axis, and line $x=2$ .

See Solution

Problem 26245

Evaluate the integral $\int_{0}^{1} \frac{d x}{e^{x}+e^{-x}}$ using the Fundamental Theorem of Calculus.

See Solution

Problem 26246

Find the acceleration of an object with position $s(t) = 8t^3 - 4t^2$ m at time $t = 4$ s.

See Solution

Problem 26247

A basketball is thrown up at $5.0 \frac{\mathrm{m}}{\mathrm{s}}$ . What is its maximum height without air resistance?

See Solution

Problem 26248

Find the integral of $\cot^{2} x \csc^{4} x$ with respect to $x$ : $\int \cot^{2} x \csc^{4} x \, dx$ .

See Solution

Problem 26249

Solve the equation $y^{\prime \prime}-4 y^{\prime}+3 y=1$ for $0 \leq t<6$ and $0$ for $t \geq 6$ , with $y(0)=y^{\prime}(0)=0$ . Find $f(t)$ and $g(t)$ .

See Solution

Problem 26250

Find the maximum of $f(x)=-x^{3}+2 x^{2}+2$ for $-2 \leq x \leq 3$ .

See Solution

Problem 26251

Calculate the integral: $\int \frac{d x}{x^{2}-4 x+13}$ .

See Solution

Problem 26252

Find the acceleration of the object at $t=4$ s given the position function $s(t)=8t^{3}-4t^{2}$ .

See Solution

Problem 26253

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

See Solution

Problem 26254

Find the absolute extreme values of $f(x)$ given the behavior of its derivative $f'(x)$ as described.

See Solution

Problem 26255

Evaluate the integral $\int_{0}^{\frac{\pi}{3}} \frac{\tan ^{3} x}{\sec x} d x$ .

See Solution

Problem 26256

Evaluate the integral $\int_{0}^{1} \frac{d x}{e^{x}+e^{-x}}$ .

See Solution

Problem 26257

Calculate the area between the curve $y=\frac{8}{x^{2}+4}$ , the $x$ -axis, $y$ -axis, and the line $x=2$ .

See Solution

Problem 26258

Find the final speed of an object dropped from a height of $193.9 \mathrm{~m}$ .

See Solution

Problem 26259

A lighthouse is 2 km from the shore. When $\theta=\pi/4$ , find the speed of light spot $B$ on the shore. Round to 1 decimal.

See Solution

Problem 26260

What are the horizontal asymptotes for the function $F(x)=\frac{2x^2 -5}{x^2}$ ? Choose A, B, or C.

See Solution

Problem 26261

Find the coordinates of the relative maximum. The relative max occurs at the point $(\square)$ .

See Solution

Problem 26262

Verify if (1,3) is on the curve $4 x^{2}+2 x y+y^{2}=19$ and find the tangent line at that point.

See Solution

Problem 26263

Evaluate the integral $\int \sin ^{4} \theta \cos \theta d \theta$ and verify by differentiating your result.

See Solution

Problem 26264

Find $\frac{f(x+h)-f(x)}{h}$ for $f(x)=2x$ . What is the result? A) 1 B) 2 C) $h$ D) $2h$

See Solution

Problem 26265

Calculate the integral: $\int\left(\frac{3}{x^{2}}+\frac{8}{5 x}\right) d x$

See Solution

Problem 26266

Evaluate the integral: $\int \frac{1-\sin ^{2} x}{\cos x} d x$

See Solution

Problem 26267

Given differentiable functions $f(x)$ and $g(x)$ with $f(-2)=g(-2)$ and $f'(x)<g'(x)$ for all $x$ , which statements are true?

See Solution

Problem 26268

Find the derivative of the function $f(x) = \frac{\sqrt{x}}{x}$ .

See Solution

Problem 26269

Find the definite integrals to calculate the volume of a soap dish rotated around the $y$ -axis with $f_{m}(x)=x^{m}+1$ , $y=2$ , $x=2$ .

See Solution

Problem 26270

10. For the function $f(x)$ , find the units of $f^{\prime}(x)$ and determine if you should "speed up" or "slow down" at $f^{\prime}(101)=0.74$ .

See Solution

Problem 26271

Estimate the temperature of tea after 48 seconds if it starts at $70^{\circ} \mathrm{C}$ using $\frac{d T}{d t}=-0.05(T-20)$ .

See Solution

Problem 26272

Calculate the volume of a soap dish with a circular base radius of $2 \mathrm{~cm}$ and total volume $7.5 \pi \mathrm{cm}^{3}$ . Use $f_{m}(x)=x^{m}+1$ and $y=2$ for the cross-section. Find the value of $m$ .

See Solution

Problem 26273

Find the function $f(x)$ such that $\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(4+\frac{k}{n}\right)^{3} \frac{3}{n}=\int_{12}^{15} f(x) d x$ . Choices: $\frac{3}{16} x^{3}$ , $x^{3}$ , $3 x^{3}$ , $\frac{1}{27} x^{3}$ , $4 x^{3}$ .

See Solution

Problem 26274

Bestimme die Ableitungsfunktion von $f$ : a) $f(x)=2 x+x^{3}$ , b) $f(x)=5 x$ , c) $f(x)=a x^{2}$ , d) $f(x)=a x^{n}$ , e) $f(x)=2 x^{2}+4 x$ , f) $f(x)=\frac{1}{2} x^{2}+5$ , g) $f(x)=2 x^{3}-3 x^{2}+2$ , h) $f(x)=a x^{3}+b x+c$ .

See Solution

Problem 26275

Find the derivative of the function $f(x) = \frac{x}{\sqrt{x}-1}$ and explain how to calculate it.

See Solution

Problem 26276

Compressed air escapes a container with pressure $P$ . The rate of decrease is given by $\frac{\mathrm{d} P}{\mathrm{~d} t}=-k \sqrt{ }(P-A)$ . Solve for $P$ given $P=5A$ at $t=0$ and $P=2A$ at $t=2$ . Find $t$ when $P=A$ and express $P$ in terms of $A$ and $t$ .

See Solution

Problem 26277

Find the derivative of the function $f(x) = \frac{\sin x}{x^{2}}$ . What is $f'(x)$ ?

See Solution

Problem 26278

Find the derivative of $f(x)=\frac{x}{\sqrt{x}-1}$ and explain the process.

See Solution

Problem 26279

Soit la fonction $f : [2,+\infty[ \rightarrow B$ définie par $f(x) = \frac{1}{x}$ . 1- Trouvez Df. 2- Montrez que $f$ est décroissante sur $[2,+\infty[$ .

See Solution

Problem 26280

Find the curve where the gradient at $(x, y)$ is proportional to $\frac{y}{\sqrt{x+1}}$ and passes through $(0,1)$ and $(3, e)$ . Express $y$ as $y=\exp (\sqrt{x+1}-1)$ .

See Solution

Problem 26281

Solve the equation $e^{2 x} \frac{d y}{d x}=4 x y^{2}$ with $y=1$ at $x=0$ , finding $y$ in terms of $x$ .

See Solution

Problem 26282

Solve the equation $e^{3 t} \frac{d x}{d t}=\cos ^{2} 2 x$ with $x(0)=0$ for $x$ in terms of $t$ .

See Solution

Problem 26283

Solve the differential equation $\frac{\mathrm{d} x}{\mathrm{~d} t}=x^{2}(1+2 x)$ with $x=1$ at $t=0$ for $t$ in terms of $x$ .

See Solution

Problem 26284

Solve the differential equation $x \ln x + t \frac{\mathrm{d} x}{\mathrm{~d} t} = 0$ with $x=\mathrm{e}$ at $t=2$ . Find $x(t)$ .

See Solution

Problem 26285

Express $y$ in terms of $x$ . Given $76 \ln x + t \frac{dx}{dt} = 0$ and $x=e$ when $t=2$ , solve for $x$ in terms of $t$ .

See Solution

Problem 26286

A plantation of area $20 \mathrm{~km}^{2}$ is infected by a disease. Show that $x$ and $t$ satisfy $\frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{19 x}{20-x}$ . Find $k=19$ . Solve for $x$ and show $x=\mathrm{e}^{0.9+0.05 x}$ when $t=1$ .

See Solution

Problem 26287

A water tank is a hemisphere with radius $r$ . Water depth $h$ decreases at rate proportional to $\sqrt{h}$ . Show $h$ and $t$ satisfy:
$\frac{\mathrm{d} h}{\mathrm{~d} t}=-\frac{B}{2 r h^{\frac{1}{2}}-h^{\frac{3}{2}}}.$
Then, solve for $t$ in terms of $h$ and $r$ .

See Solution

Problem 26288

Find the derivative of the function $f(x) = \frac{1}{\sqrt{x}}$ .

See Solution

Problem 26289

Gjeni derivatën e funksionit $f(x)=(x-3)(x+1)$ në lidhje me $x$ .

See Solution

Problem 26290

Find the derivative of $f(x)=\frac{x^{2}}{\sqrt{x+2}}$ using $g(x)=x^2$ and $h(x)=\sqrt{x+2}$ .

See Solution

Problem 26291

Find the limit: $\lim _{x \rightarrow 0} \frac{2 \sin (x)-\sin (2 x)}{x-\sin (x)}$ .

See Solution

Problem 26292

Find the derivative of $f(x) = x \sqrt{1 - x^2}$ and explain each step clearly.

See Solution

Problem 26293

Find the limit: $\lim _{x \rightarrow 0} \frac{\ln \left(e^{-x}+x e^{-x}\right)}{x^{2}}=$ ?

See Solution

Problem 26294

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

See Solution

Problem 26295

Find the derivative of $f(t) = \sqrt{\frac{1}{t^{2} - 2}}$ and explain each step in detail.

See Solution

Problem 26296

Find the derivative of $f(x)=\left(\frac{x+5}{x^{2}+2}\right)^{2}$ and explain each step.

See Solution

Problem 26297

Find $F(x)=\int_{0}^{x} \frac{e^{\tan^{-1} t}}{1+t^{2}} dt$ given $\frac{d}{dx} \tan^{-1} x=\frac{1}{1+x^{2}}$ .

See Solution

Problem 26298

Determine the truth of these statements about the function $f(x)=\frac{x^{3}}{1-e^{x}}$ for $x \neq 0$ and $f(0)=0$ .

See Solution

Problem 26299

求 $K_{P}$ 和 $K_{I}$ 的单位，给定粒子速度 $v(t)$ 满足方程 $v^{\prime}(t)=\frac{1}{m} f(v(t))-K_{P} v(t)+K_{I} \int_{0}^{t} v(s) d s$ 。

See Solution

Problem 26300

Evaluate the integral: $\int\left(\sin ^{3}(x)-3 \sin ^{2}(x)\right)\left(\sin ^{4}(x)-4 \sin ^{3}(x)\right)^{6} \cos (x) d x$ .

See Solution

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Calculus

Problem 26201

Solve the ODE: dydx=y(5−6x)x(5−7y)\frac{d y}{d x}=\frac{y(5-6 x)}{x(5-7 y)}dxdy​=x(5−7y)y(5−6x)​.

Problem 26202

Find the Laplace Transform of f(t)={t,0≤t<10,t≥1f(t)=\begin{cases}t, & 0 \leq t<1 \\ 0, & t \geq 1\end{cases}f(t)={t,0,​0≤t<1t≥1​. What is L{f(t)}L\{f(t)\}L{f(t)}?

Problem 26203

Problem 26204

Find coefficients ana_nan​ and bnb_nbn​ from the series u(x,t)u(x, t)u(x,t) given initial conditions. Which option is correct?

Problem 26205

Solve 2y′′+y′−y=−12 y^{\prime \prime}+y^{\prime}-y=-12y′′+y′−y=−1 with y(0)=1y(0)=1y(0)=1 and y′(0)=2y^{\prime}(0)=2y′(0)=2. Find y′′′(0)y^{\prime \prime \prime}(0)y′′′(0).

Problem 26206

Solve y′′−y′−2y=3y^{\prime \prime}-y^{\prime}-2 y=3y′′−y′−2y=3 with y(0)=2y(0)=2y(0)=2 and y′(0)=1y^{\prime}(0)=1y′(0)=1. Find y′′′(0)y^{\prime \prime \prime}(0)y′′′(0). A. 10 B. -10 C. 13 D. -13

Problem 26207

Given the demand function q=D(p)=526−8pq=D(p)=526-8pq=D(p)=526−8p, find: a. Domain of D(p)D(p)D(p); b. D′(p)D'(p)D′(p); c. Elasticity E(p)E(p)E(p); d. For p=$45p=\$45p=$45, is demand elastic or inelastic? Effect on revenue? e. If price increases by 1\%, how does demand change?

Problem 26208

Find a0,a4,b5a_{0}, a_{4}, b_{5}a0​,a4​,b5​ from the Fourier series f(x)=23+4π2∑n=1∞(−1)n+1n2cos⁡(nπx)f(x)=\frac{2}{3}+\frac{4}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}} \cos (n \pi x)f(x)=32​+π24​∑n=1∞​n2(−1)n+1​cos(nπx). Options: A, B, C, D, E.

Problem 26209

Find the sum of the series S∞=20−10+5−⋯S_{\infty}=20-10+5-\cdotsS∞​=20−10+5−⋯. Choices: 403\frac{40}{3}340​, 203\frac{20}{3}320​, 40, or diverges.

Problem 26210

Classify the ODE: dydx=y(2−6x)x(1−9y)\frac{d y}{d x}=\frac{y(2-6 x)}{x(1-9 y)}dxdy​=x(1−9y)y(2−6x)​. Choose: 1st order, linear/homogeneous or non-homogeneous?

Problem 26211

Find a0,a4,b5a_{0}, a_{4}, b_{5}a0​,a4​,b5​ for the Fourier series f(x)=23+4π2∑n=1∞(−1)n+1n2cos⁡(nπx)f(x)=\frac{2}{3}+\frac{4}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}} \cos (n \pi x)f(x)=32​+π24​∑n=1∞​n2(−1)n+1​cos(nπx).

Problem 26212

Find the coefficients a0,a2,b4a_{0}, a_{2}, b_{4}a0​,a2​,b4​ from the Fourier series f(x)=23+4π2∑n=1∞(−1)nn2sin⁡(nπx)f(x)=\frac{2}{3}+\frac{4}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}} \sin (n \pi x)f(x)=32​+π24​∑n=1∞​n2(−1)n​sin(nπx).

Problem 26213

Evaluate the integral: ∫exdx1−e2x\int \frac{e^{x} d x}{\sqrt{1-e^{2 x}}}∫1−e2x​exdx​

Problem 26214

Calculate the average value of f(x)=8xf(x)=8xf(x)=8x on the interval [5,7][5,7][5,7].

Problem 26215

Evaluate the integral using substitution: ∫π/32π4cos⁡3xsin⁡x dx\int_{\pi / 3}^{2 \pi} 4 \cos ^{3} x \sin x \, dx∫π/32π​4cos3xsinxdx.

Problem 26216

Evaluate the integral: ∫dxxln⁡x3\int \frac{d x}{x \ln x^{3}}∫xlnx3dx​.

Problem 26217

Calculate the total in a savings account with an investment of \$ 2250 at 9.0\% for 15 years with continuous compounding. Round to two decimals.

Problem 26218

Evaluate the integral: ∫csc⁡(z+π4)cot⁡(z+π4)dz\int \csc \left(z+\frac{\pi}{4}\right) \cot \left(z+\frac{\pi}{4}\right) dz∫csc(z+4π​)cot(z+4π​)dz

Problem 26219

Evaluate the integral: ∫sin⁡2xcos⁡4x dx\int \sin 2 x \cos 4 x \, dx∫sin2xcos4xdx

Problem 26220

Find xxx where f(x)=x+1f(x)=\sqrt{x+1}f(x)=x+1​ equals its average value on the interval [0,15][0,15][0,15].

Problem 26221

Calculate the area between the curve y=6x1+x2y=\frac{6 x}{1+x^{2}}y=1+x26x​ and the xxx-axis for −4≤x≤4-4 \leq x \leq 4−4≤x≤4.

Problem 26222

Evaluate the integral: ∫x2(x3−10)4dx\int x^{2}\left(x^{3}-10\right)^{4} dx∫x2(x3−10)4dx

Problem 26223

Evaluate the integral using substitution: ∫0π/1616tan⁡4x dx\int_{0}^{\pi / 16} 16 \tan 4 x \, dx∫0π/16​16tan4xdx.

Problem 26224

Evaluate the integral from 0 to 1: ∫013x+3 dx\int_{0}^{1} \sqrt{3 x+3} \, dx∫01​3x+3​dx.

Problem 26225

Integrate the function 14(2x−3)614(2 x-3)^{6}14(2x−3)6 from −1-1−1 to 1.51.51.5. What is the result?

Problem 26226

Differentiate sin⁡4(3x)\sin^{4}(3x)sin4(3x) with respect to xxx.

Problem 26227

Evaluate the integral using substitutions: ∫(2r−5)cos⁡3(2r−5)2+103(2r−5)2+10dr\int \frac{(2 r-5) \cos \sqrt{3(2 r-5)^{2}+10}}{\sqrt{3(2 r-5)^{2}+10}} d r∫3(2r−5)2+10​(2r−5)cos3(2r−5)2+10​​dr

Problem 26228

Evaluate the integral from 0 to 1: ∫013x+3 dx\int_{0}^{1} \sqrt{3 x+3} \, dx∫01​3x+3​dx.

Problem 26229

Evaluate the integral: ∫8dx25−64x2\int \frac{8 d x}{\sqrt{25-64 x^{2}}}∫25−64x2​8dx​.

Problem 26230

Find the integral of sin⁡2xcos⁡4x\sin 2x \cos 4xsin2xcos4x with respect to xxx: ∫sin⁡2xcos⁡4x dx\int \sin 2x \cos 4x \, dx∫sin2xcos4xdx.

Problem 26231

Calculate the integral ∫tan⁡5xsec⁡3x dx\int \tan ^{5} x \sec ^{3} x \, dx∫tan5xsec3xdx.

Problem 26232

Evaluate the integral: ∫8dx25−64x2\int \frac{8 d x}{\sqrt{25-64 x^{2}}}∫25−64x2​8dx​

Problem 26233

Evaluate the integral ∫sin⁡2xcos⁡4x dx\int \sin 2x \cos 4x \, dx∫sin2xcos4xdx.

Problem 26234

Calculate the integral ∫0π2cos⁡3x dx\int_{0}^{\frac{\pi}{2}} \cos ^{3} x \, dx∫02π​​cos3xdx.

Problem 26235

Evaluate the integral ∫cot⁡2xcsc⁡4x dx\int \cot^{2} x \csc^{4} x \, dx∫cot2xcsc4xdx.

Problem 26236

Evaluate the integral ∫dxx2−4x+13\int \frac{d x}{x^{2}-4 x+13}∫x2−4x+13dx​.

Problem 26237

Evaluate the integral: ∫x3x4+9 dx\int x^{3} \sqrt{x^{4}+9} \, dx∫x3x4+9​dx

Problem 26238

Calcula el integral definido usando el Teorema Fundamental del Cálculo: ∫132x+3x+1dx\int_{1}^{3} \frac{2 x+3}{x+1} d x∫13​x+12x+3​dx

Problem 26239

Evaluate the integral using the Fundamental Theorem of Calculus: ∫0π2cos⁡3x dx\int_{0}^{\frac{\pi}{2}} \cos ^{3} x \, dx∫02π​​cos3xdx.

Problem 26240

Calcula el integral definido usando el Teorema Fundamental del Cálculo: ∫0π2cos⁡3x dx\int_{0}^{\frac{\pi}{2}} \cos ^{3} x \, dx∫02π​​cos3xdx

Solve the ODE: $\frac{d y}{d x}=\frac{y(5-6 x)}{x(5-7 y)}$ .

Find the Laplace Transform of $f(t)=\begin{cases}t, & 0 \leq t<1 \\ 0, & t \geq 1\end{cases}$ . What is $L\{f(t)\}$ ?

Find coefficients $a_n$ and $b_n$ from the series $u(x, t)$ given initial conditions. Which option is correct?

Solve $2 y^{\prime \prime}+y^{\prime}-y=-1$ with $y(0)=1$ and $y^{\prime}(0)=2$ . Find $y^{\prime \prime \prime}(0)$ .

Solve $y^{\prime \prime}-y^{\prime}-2 y=3$ with $y(0)=2$ and $y^{\prime}(0)=1$ . Find $y^{\prime \prime \prime}(0)$ . A. 10 B. -10 C. 13 D. -13

Given the demand function $q=D(p)=526-8p$ , find: a. Domain of $D(p)$ ; b. $D'(p)$ ; c. Elasticity $E(p)$ ; d. For $p=\$45$ , is demand elastic or inelastic? Effect on revenue? e. If price increases by 1\%, how does demand change?

Find $a_{0}, a_{4}, b_{5}$ from the Fourier series $f(x)=\frac{2}{3}+\frac{4}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}} \cos (n \pi x)$ . Options: A, B, C, D, E.

Find the sum of the series $S_{\infty}=20-10+5-\cdots$ . Choices: $\frac{40}{3}$ , $\frac{20}{3}$ , 40, or diverges.

Classify the ODE: $\frac{d y}{d x}=\frac{y(2-6 x)}{x(1-9 y)}$ . Choose: 1st order, linear/homogeneous or non-homogeneous?

Find $a_{0}, a_{4}, b_{5}$ for the Fourier series $f(x)=\frac{2}{3}+\frac{4}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}} \cos (n \pi x)$ .

Find the coefficients $a_{0}, a_{2}, b_{4}$ from the Fourier series $f(x)=\frac{2}{3}+\frac{4}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}} \sin (n \pi x)$ .

Evaluate the integral: $\int \frac{e^{x} d x}{\sqrt{1-e^{2 x}}}$

Calculate the average value of $f(x)=8x$ on the interval $[5,7]$ .

Evaluate the integral using substitution: $\int_{\pi / 3}^{2 \pi} 4 \cos ^{3} x \sin x \, dx$ .

Evaluate the integral: $\int \frac{d x}{x \ln x^{3}}$ .

Evaluate the integral: $\int \csc \left(z+\frac{\pi}{4}\right) \cot \left(z+\frac{\pi}{4}\right) dz$

Evaluate the integral: $\int \sin 2 x \cos 4 x \, dx$

Find $x$ where $f(x)=\sqrt{x+1}$ equals its average value on the interval $[0,15]$ .

Calculate the area between the curve $y=\frac{6 x}{1+x^{2}}$ and the $x$ -axis for $-4 \leq x \leq 4$ .

Evaluate the integral: $\int x^{2}\left(x^{3}-10\right)^{4} dx$

Evaluate the integral using substitution: $\int_{0}^{\pi / 16} 16 \tan 4 x \, dx$ .

Evaluate the integral from 0 to 1: $\int_{0}^{1} \sqrt{3 x+3} \, dx$ .

Integrate the function $14(2 x-3)^{6}$ from $-1$ to $1.5$ . What is the result?

Differentiate $\sin^{4}(3x)$ with respect to $x$ .

Evaluate the integral using substitutions: $\int \frac{(2 r-5) \cos \sqrt{3(2 r-5)^{2}+10}}{\sqrt{3(2 r-5)^{2}+10}} d r$

Evaluate the integral from 0 to 1: $\int_{0}^{1} \sqrt{3 x+3} \, dx$ .

Evaluate the integral: $\int \frac{8 d x}{\sqrt{25-64 x^{2}}}$ .

Find the integral of $\sin 2x \cos 4x$ with respect to $x$ : $\int \sin 2x \cos 4x \, dx$ .

Calculate the integral $\int \tan ^{5} x \sec ^{3} x \, dx$ .

Evaluate the integral: $\int \frac{8 d x}{\sqrt{25-64 x^{2}}}$

Evaluate the integral $\int \sin 2x \cos 4x \, dx$ .

Calculate the integral $\int_{0}^{\frac{\pi}{2}} \cos ^{3} x \, dx$ .

Evaluate the integral $\int \cot^{2} x \csc^{4} x \, dx$ .

Evaluate the integral $\int \frac{d x}{x^{2}-4 x+13}$ .

Evaluate the integral: $\int x^{3} \sqrt{x^{4}+9} \, dx$

Calcula el integral definido usando el Teorema Fundamental del Cálculo: $\int_{1}^{3} \frac{2 x+3}{x+1} d x$

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{0}^{\frac{\pi}{2}} \cos ^{3} x \, dx$ .

Calcula el integral definido usando el Teorema Fundamental del Cálculo: $\int_{0}^{\frac{\pi}{2}} \cos ^{3} x \, dx$

Find the value(s) of $c$ satisfying the Mean Value Theorem for $f(x) = \tan^{-1}x$ on $[-\sqrt{3}, \sqrt{3}]$ .

Evaluate the integral using substitution: $\int_{1}^{4} \frac{3-\sqrt{x}}{\sqrt{x}} d x$ .

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{0}^{\frac{\pi}{3}} \frac{\tan ^{3} x}{\sec x} d x$ .

Calculate the area between the curve $y=\frac{8}{x^{2}+4}$ , the $x$ -axis, $y$ -axis, and line $x=2$ .

Evaluate the integral $\int_{0}^{1} \frac{d x}{e^{x}+e^{-x}}$ using the Fundamental Theorem of Calculus.

Find the acceleration of an object with position $s(t) = 8t^3 - 4t^2$ m at time $t = 4$ s.

A basketball is thrown up at $5.0 \frac{\mathrm{m}}{\mathrm{s}}$ . What is its maximum height without air resistance?

Find the integral of $\cot^{2} x \csc^{4} x$ with respect to $x$ : $\int \cot^{2} x \csc^{4} x \, dx$ .

Solve the equation $y^{\prime \prime}-4 y^{\prime}+3 y=1$ for $0 \leq t<6$ and $0$ for $t \geq 6$ , with $y(0)=y^{\prime}(0)=0$ . Find $f(t)$ and $g(t)$ .

Find the maximum of $f(x)=-x^{3}+2 x^{2}+2$ for $-2 \leq x \leq 3$ .

Calculate the integral: $\int \frac{d x}{x^{2}-4 x+13}$ .

Find the acceleration of the object at $t=4$ s given the position function $s(t)=8t^{3}-4t^{2}$ .

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

Find the absolute extreme values of $f(x)$ given the behavior of its derivative $f'(x)$ as described.

Evaluate the integral $\int_{0}^{\frac{\pi}{3}} \frac{\tan ^{3} x}{\sec x} d x$ .

Evaluate the integral $\int_{0}^{1} \frac{d x}{e^{x}+e^{-x}}$ .

Calculate the area between the curve $y=\frac{8}{x^{2}+4}$ , the $x$ -axis, $y$ -axis, and the line $x=2$ .

Find the final speed of an object dropped from a height of $193.9 \mathrm{~m}$ .

A lighthouse is 2 km from the shore. When $\theta=\pi/4$ , find the speed of light spot $B$ on the shore. Round to 1 decimal.

What are the horizontal asymptotes for the function $F(x)=\frac{2x^2 -5}{x^2}$ ? Choose A, B, or C.

Find the coordinates of the relative maximum. The relative max occurs at the point $(\square)$ .

Verify if (1,3) is on the curve $4 x^{2}+2 x y+y^{2}=19$ and find the tangent line at that point.

Evaluate the integral $\int \sin ^{4} \theta \cos \theta d \theta$ and verify by differentiating your result.

Find $\frac{f(x+h)-f(x)}{h}$ for $f(x)=2x$ . What is the result? A) 1 B) 2 C) $h$ D) $2h$

Calculate the integral: $\int\left(\frac{3}{x^{2}}+\frac{8}{5 x}\right) d x$

Evaluate the integral: $\int \frac{1-\sin ^{2} x}{\cos x} d x$

Given differentiable functions $f(x)$ and $g(x)$ with $f(-2)=g(-2)$ and $f'(x)<g'(x)$ for all $x$ , which statements are true?

Find the derivative of the function $f(x) = \frac{\sqrt{x}}{x}$ .

Find the definite integrals to calculate the volume of a soap dish rotated around the $y$ -axis with $f_{m}(x)=x^{m}+1$ , $y=2$ , $x=2$ .

10. For the function $f(x)$ , find the units of $f^{\prime}(x)$ and determine if you should "speed up" or "slow down" at $f^{\prime}(101)=0.74$ .

Estimate the temperature of tea after 48 seconds if it starts at $70^{\circ} \mathrm{C}$ using $\frac{d T}{d t}=-0.05(T-20)$ .

Calculate the volume of a soap dish with a circular base radius of $2 \mathrm{~cm}$ and total volume $7.5 \pi \mathrm{cm}^{3}$ . Use $f_{m}(x)=x^{m}+1$ and $y=2$ for the cross-section. Find the value of $m$ .