Calculus

Problem 25201

Déterminez les intervalles de convexité de la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ avec $f^{\prime \prime}(x)=-\frac{72 x^{2}+96}{\left(x^{2}-4\right)^{3}}$ .

See Solution

Problem 25202

Soit la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ . Trouvez les intervalles de convexité et concavité de $f$ .

See Solution

Problem 25203

Find the change in kinetic energy of a $7.5 \mathrm{~kg}$ projectile fired at $11.8 \mathrm{~m/s}$ from $20.9 \mathrm{~m}$ height. Use $g = 9.81 \mathrm{~m/s}^{2}$ . Answer in $J$ .

See Solution

Problem 25204

Calculer $\left.\frac{d Q}{d K}\right|_{i=\frac{\pi}{2}}$ , déterminer quand $\frac{d K}{d t}=0$ et $\frac{d Q}{d t}$ à $t=\frac{3 \pi}{2}$ .

See Solution

Problem 25205

Find the tangent line equation for $f(x)=\frac{4}{x}$ at $x=9$ .

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Problem 25206

Find the limits: (a) $\lim_{x \to \infty} \left(\sqrt{x^2 - 5x + 1} - x\right)$ , (b) $\lim_{x \to -\infty} \left(\sqrt{x^2 - 5x + 1} - x\right)$ .

See Solution

Problem 25207

Find the volume of the solid formed by revolving region $R$ (bounded by $x=y^{\frac{2}{3}}$ and $y=x^{\frac{1}{3}}$ ) around $y=0$ using the disc method.

See Solution

Problem 25208

Déterminez les intervalles de concavité de la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ avec $f^{\prime \prime}(x)=-\frac{72 x^{2}+96}{\left(x^{2}-4\right)^{3}}$ .

See Solution

Problem 25209

Déterminez les intervalles de concavité de la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ avec $f^{\prime \prime}(x)=-\frac{72 x^{2}+96}{\left(x^{2}-4\right)^{3}}$ .

See Solution

Problem 25210

Déterminez les points d'inflexion de la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ en utilisant $f^{\prime}(x)$ et $f^{\prime \prime}(x)$ .

See Solution

Problem 25211

Differentiate the function: $10(\sin^{2}(t) + \cos(t)) + 30$ with respect to $t$ .

See Solution

Problem 25212

Tracez la courbe de la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ avec $f^{\prime}(x)$ et $f^{\prime \prime}(x)$ donnés.

See Solution

Problem 25213

Calculate $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=x^{2}-3x+5$ .

See Solution

Problem 25214

Find the average velocity of a train with location $s(t)=\frac{50}{t}$ from $t=3$ to $t=8$ hours.

See Solution

Problem 25215

Un fil de 48 cm est plié pour former une boîte. Trouvez les dimensions pour un volume maximal $V = l \times L \times h$ avec $4l + 4L + 4h = 48$ .

See Solution

Problem 25216

Find the horizontal asymptote of the function $f(x)=\frac{3 x^{2}-5 x+2}{2 x^{2}-8 x}$ .

See Solution

Problem 25217

Find the horizontal asymptote of each function: a. $f(x)=\frac{3 x^{2}-5 x+2}{2 x^{2}-8 x}$ ; answer: $y=\frac{3}{2}$ . b. $g(x)=\frac{x^{3}-5 x^{6}+1}{8 x^{3}}$ .

See Solution

Problem 25218

Approximate area under $f(x)=\sqrt{x}$ from $a=5$ to $b=8$ using 6 rectangles. Then find exact area via integral.

See Solution

Problem 25219

Deriver funksjonen $a(x)=32 x^{2}-\sqrt{x}$ .

See Solution

Problem 25220

Find the function $f$ from the limit of Riemann sums: $\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(x_{k}^{*}\right)^{6} \Delta x_{k} ;[3,11]$ and express it as a definite integral.

See Solution

Problem 25221

Find the production level $x$ that minimizes the average cost given by $c(x)=x^{3}-20x^{2}+20,000x$ .

See Solution

Problem 25222

Calculate the integral $\int_{0}^{4} f(x) dx$ for the piecewise function: $f(x)=5$ if $x \leq 2$ , $f(x)=4x-3$ if $x>2$ .

See Solution

Problem 25223

Find the derivative of $f(x)=x^{4 x}$ using logarithmic differentiation. What is $f^{\prime}(x)=?$

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Problem 25224

Soit $f(x)=\frac{1}{x-1}$ . Trouvez les équations des deux tangentes de pente -1 et le point de tangence avec $y=\frac{-x-3}{4}$ .

See Solution

Problem 25225

Trouvez l'équation de la tangente à $f(x)=x^{3}-3 x^{2}-3$ au point où $x=-3$ .

See Solution

Problem 25226

A farmer uses 60 m of fencing for 3 sides of a corral. Find dimensions for maximum area.

See Solution

Problem 25227

Find the horizontal asymptotes of the curve $y=\frac{16 x}{(x^{4}+1)^{2}}$ , denoted as $y_{1}$ and $y_{2}$ .

See Solution

Problem 25228

Find the dimensions for a rectangular field with $600 \mathrm{~m}$ of fencing that maximizes the area and calculate that area.

See Solution

Problem 25229

Gitt funksjonen $f(x)=-x^{3}+3 x$ : Finn nullpunkt, topp/bunnpunkter, og vendetangenter.

See Solution

Problem 25230

Prove that the function $g(x)=\frac{d-x}{\sqrt{b^{2}+(d-x)^{2}}}$ is decreasing in $x$ .

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Problem 25231

Prove that the function defined by $\frac{d t}{d x}=\frac{x}{c_{1} \sqrt{a^{2}+x^{2}}}-\frac{d-x}{c_{2} \sqrt{b^{2}+(d-x)^{2}}}$ is increasing in $x$ .

See Solution

Problem 25232

Find the derivative of the function given by $P^{\prime}(t)=\left(t^{2}-\frac{4 t^{5 / 2}}{5}\right)^{\prime}$ .

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Problem 25233

Calculate the fetal growth rate $\frac{dH}{dt}$ using the formula $H=-30.56+1.856 t^{2}-0.5741 t^{2} \log t$ .

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Problem 25234

Given $s(t)=4 t^{2}+4 t$ , find: a) $v(t)$ , b) $a(t)$ , c) velocity and acceleration at $t=4$ sec.

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Problem 25235

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\ln \sqrt{\frac{6 x-4}{7 x+4}}$ .

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Problem 25236

Find where the tangent line of $y=13x-x^{2}$ has a slope. Choose from: A. $(1,12)$ B. $(6,42)$ C. $(6.5,42.25)$ D. $(12,84)$ .

See Solution

Problem 25237

Find the derivative of the function $(5 x^{2}+7 x+3)(2 x)$ .

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Problem 25238

Find the inflection points of the function $f(x)=\frac{18 x}{1.5 x^{2}+2}$ using its derivatives. Lowest =, Middle =, Highest =.

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Problem 25239

Find $\frac{d y}{d u}$ , $\frac{d u}{d x}$ , and $\frac{d y}{d x}$ for $y=u^{45}$ and $u=5 x^{3}+3 x^{2}$ .

See Solution

Problem 25240

Differentiate $f(x)=(7 x^{2}-2 x+6)(4 x^{2}+3 x-6)$ . Which option shows how to find $f'(x)$ ?

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Problem 25241

Find $\lim _{n \rightarrow \infty} \sqrt[n]{\left(\frac{4}{3}\right)^{n}+\left(\frac{8}{3}\right)^{n}}$ .

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Problem 25242

Find the volume of the solid formed by rotating the area between $y=x^{2}$ and $y=x^{3}$ in the first quadrant around $x=1$ .

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Problem 25243

A wrench is dropped from an $80.0 \mathrm{~m}$ tower. What is its velocity upon hitting the ground?

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Problem 25244

A city’s population grows from 200,000 to $P(t)=200,000+5000t^{2}$ . Find: a) $\frac{\mathrm{dP}}{\mathrm{dt}}$ ; b) $P(20)$ ; c) Growth rate at $t=20$ ; d) Meaning of part (c).

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Problem 25245

Find the area between the curves $y = 6 \sin x$ and $y = 6 \cos x$ over the interval $[0, \frac{\pi}{2}]$ .

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Problem 25246

Find the limit: $\lim _{n \rightarrow \infty} \frac{\sqrt{4+6 n+5 n^{2}}}{3 n+2}$ .

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Problem 25247

Given the function $f(x)=\frac{x^{2}-49}{x-7}$ , find where it's not differentiable and compute $f^{\prime}(-1)$ .

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Problem 25248

Bestimme die Durchschnittsgeschwindigkeiten für $[0; 3]$ und $[3; 7]$ der Funktion $s(t)=6t^{2}-0,4t^{3}$ . Finde die Momentangeschwindigkeit bei $t=3$ und $t=12$ . Wann erreicht der Ballon den Boden?

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Problem 25249

Find the volume of the solid formed by rotating the area between $y=x^{2}$ and $y=x^{3}$ in the first quadrant around $x=1$ .

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Problem 25250

Find $x$ where the tangent line to $y=x^{2}+2x-3$ is horizontal. Options: A. $\frac{1}{2}$ B. 0 C. 1 D. -1

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Problem 25251

Find the derivative of $f(x)=\frac{8}{x}$ and evaluate it at $x=-1$ . What is $f^{\prime}(-1)$ ?

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Problem 25252

Find the tangent line equation for the curve $y=x-x^{2}$ at the point $(2,-2)$ . Choose from the options.

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Problem 25253

Berechnen Sie die Ableitung $f^{\prime}(2)$ für die Funktion $f(x) = 5x^{2} + 1$ .

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Problem 25254

Finn topp- og bunnpunkter samt vendepunkt for funksjonen $p(x)=\ln \left(x^{2}-1\right)$ der $x<-1$ eller $x>1$ .

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Problem 25255

Find the velocity $v(t)$ and acceleration $a(t)$ for $s(t)=5 t^{2}+15 t$ . Calculate both at $t=2$ sec.

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Problem 25256

Differentiate $y=\frac{4 x^{2}-7}{2 x^{3}+3}$ . Which option shows the correct derivative formula?

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Problem 25257

Find the difference quotient for $f(x)=9x-2$ at $x=5$ for $h=2, 1, 0.1, 0.01$ .

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Problem 25258

Calculate the sum: $\sum_{n=0}^{\infty} \frac{\left(\frac{2}{3}\right)^{n}(1.5)^{n}}{n !}$ .

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Problem 25259

Find the limit: $\lim _{n \rightarrow \infty}\left(\sqrt{3+6 n+n^{2}}-\sqrt{3+13 n+n^{2}}\right)$ .

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Problem 25260

Find points on the graph of $y=13x-x^{2}$ where the tangent line has a slope of 1.

See Solution

Problem 25261

Calculate the average value of $y=9 e^{-x}$ over the interval $[0,1]$ . The average value is $\square$ .

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Problem 25262

A particle's speed is $v(t)=-0.2 t^{2}+10 t$ . Find distances traveled in the first and second 4 seconds.

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Problem 25263

Gegeben ist die Funktion $f$ : (I) $f(x) = x^{2} + 1$ , (II) $f(x) = x^{3}$ , (III) $f(x) = 0,5 x^{2}$ , (IV) $f(x) = -x^{3}$ .
a) Zeichne den Graphen und bestimme die Ableitung an $x_{0} = -1$ .
b) Berechne die Ableitung an $x_{0} = -1$ mit einem kleinen $\Delta x$ .

See Solution

Problem 25264

Find the velocity function $v(t)$ given acceleration $a(t)=8t$ and initial condition $v(0)=15$ .

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Problem 25265

Find the total cost of producing 220 dresses with marginal cost $C^{\prime}(x)=-\frac{4}{25} x+50$ . Total cost is \$\square.

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Problem 25266

Calculate the area under the curve of $f(x)$ from $x = -2$ to $x = 4$ , where $f(x) = x^{2} + 6$ for $x \leq 2$ and $f(x) = 5x$ for $x > 2$ . The area is $\square$ .

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Problem 25267

Determine the center, radius, and interval of convergence for the series: $\sum_{n=0}^{\infty} \frac{\left(\frac{2}{3}\right)^{n}(x-4)^{n}}{n !}$

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Problem 25268

Find the limit as $x$ approaches 6 for $\frac{[f(x)]^{2}-6 f(x)+2}{[g(x)]^{2}-49}$ given $f(6)=10$ , $f'(6)=-1$ , $g(6)=-8$ , $g'(6)=4$ .

See Solution

Problem 25269

At what minimum times is $A^{\prime}(t)=0$ for the function $A(t)$ , given $0 \leq t \leq 8$ ? Options: a. 0, b. 2, c. 3, d. 4, e. 5.

See Solution

Problem 25270

Differentiate $h(t)=\frac{7}{2} t^{2}-8 t+5$ with respect to $t$ .

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Problem 25271

Find the sum of values of $c$ that satisfy the Mean Value Theorem for $f(x)=2 \cos x-4 \cos 2x$ on $[0,\pi]$ .

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Problem 25272

Find the distance the particle travels to the right given its position $s(t)=3t-2t^{2}$ for $t \geq 0$ .

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Problem 25273

Evaluate $f(x+h)$ and find $\frac{f(x+h)-f(x)}{h}$ for $f(x)=4x^{2}+2x$ .

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Problem 25274

Find the rate of area increase of a circular plate when the radius is 50 cm and increases at 0.01 cm/s.

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Problem 25275

Evaluate $\int_{2}^{6} 9 f(x) d x$ given $\int_{2}^{7} f(x) d x=14$ and $\int_{6}^{7} f(x) d x=8$ .

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Problem 25276

Calculate the average rate of change of $f(x)=\frac{1}{x-2}$ from $x=-4$ to $x=0$ .

See Solution

Problem 25277

Evaluate the integrals given:
a. $\int_{2}^{6} 9 f(x) d x$
b. $\int_{2}^{7}(f(x)-g(x)) d x$

See Solution

Problem 25278

Given continuous functions $f$ and $g$ , define $h(x)=f(g(x))-x$ .
a. Show there exists $t$ in $(1,4)$ where $h'(t)=3$ .
b. Show there exists $t$ in $(1,4)$ where $h(t)=-1$ .

See Solution

Problem 25279

Bestimmen Sie alle $\alpha \in \mathbb{R}$ , für die die Reihe $\sum_{n=0}^{\infty} \frac{3^{n}}{\sqrt{n}+2}(\alpha+1)^{n}$ konvergiert.

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Problem 25280

Evaluate $\int_{2}^{6} 9 f(x) d x$ given $\int_{2}^{7} f(x) d x=14$ and $\int_{2}^{6} g(x) d x=4$ .

See Solution

Problem 25281

Find the rate of change of Kleenexes $K$ when there are 52 left and sick students $s$ are increasing at 3 per day.

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Problem 25282

Find the simplified difference quotient for $f(x)=-x^{2}$ and complete the table for $x=5$ and various $h$ values.

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Problem 25283

Given $u(1)=2$ , $u'(1)=0$ , $v(1)=5$ , $v'(1)=-1$ , find the derivatives at $x=1$ :
a. $\frac{d}{dx}(uv)$ , b. $\frac{d}{dx}(\frac{u}{v})$ , c. $\frac{d}{dx}(\frac{v}{u})$ , d. $\frac{d}{dx}(7v-2u)$ .

See Solution

Problem 25284

Given $u(1)=2$ , $u'(1)=0$ , $w(1)=5$ , and $v'(1)=-1$ , find the derivatives at $x=1$ :
a. $\frac{d}{dx}(uv)$ , b. $\frac{d}{dx}\left(\frac{u}{v}\right)$ , c. $\frac{d}{dx}\left(\frac{x}{u}\right)$ , d. $\frac{d}{dx}(7v-2u)$ .

See Solution

Problem 25285

Evaluate the integrals given $\int_{2}^{7} f(x) d x=14$ , $\int_{2}^{7} g(x) d x=6$ :
a. $\int_{2}^{6} 9 f(x) d x$ b. $\int_{2}^{7}(f(x)-g(x)) d x$

See Solution

Problem 25286

A balloon inflates at $100 \pi$ cubic feet/min. Find the rate of radius increase when the radius is 2 feet.

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Problem 25287

Find the derivative of $f(x)=2 x^{2}-6 x+2$ and calculate $f^{\prime}(1)$ .

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Problem 25288

Evaluate the integrals given:
a. $\int_{2}^{6} 9 f(x) d x$
b. $\int_{2}^{7}(f(x)-g(x)) d x$
c. $\int_{2}^{6}(f(x)-g(x)) d x=\square$

See Solution

Problem 25289

Given the piecewise function $f(x) = \begin{cases} x^2 + 7 & \text{if } x \leq 3 \\ 3x + 7 & \text{if } x > 3 \end{cases}$ , check: (a) Is $f$ continuous at $x=3$ ? (b) Is $f$ differentiable at $x=3$ ?

See Solution

Problem 25290

Find the tangent line equation to $f(x)=x^{2}-3$ at the point $(-5,22)$ . $y=\square$

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Problem 25291

A researcher found antacid level $L(t)=\frac{6t}{t^2+2t+1}$ . Find $t$ for $L'(t)=0$ , then $L(t)$ , graph $L(t)$ , and analyze levels at $1$ min and $2 \leq t \leq 8$ .

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Problem 25292

Find where the tangent line to $y=\frac{1}{3} x^{3}-7 x+5$ is horizontal. State if none exist.

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Problem 25293

Find $\lim _{x \rightarrow 3} f(x)$ for the piecewise function: $f(x)=\frac{x+15}{3}$ (if $x<3$ ), $3$ (if $x=3$ ), $x^{2}-11 x+30$ (if $x>3$ ).

See Solution

Problem 25294

A sociologist finds $N(t)=20t-t^{2}$ vocabulary terms learned after $t$ hours.
a. How many terms from $t=2$ to $t=3$ ? b. Rate of learning at $t=2$ ? c. Maximum learning rate?

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Problem 25295

Find the point(s) where the tangent line of $y=\frac{1}{3} x^{3}-4 x^{2}+16 x+25$ has a slope of 4.

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Problem 25296

A company reduces production of protein bars by 100 cases/day. Given the demand function $p(x)=30-\frac{x}{200}$ , find the revenue's rate of change when production is 900 cases.

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Problem 25297

Calculate the integral $\int_{-1}^{5}\left[4 y - (y^{2} - 5)\right] dy$ .

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Problem 25298

Evaluate the integral from -9 to 0 of $\left(\frac{y}{9}+\sqrt{y+10}\right)$ .

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Problem 25299

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

See Solution

Problem 25300

Evaluate $\int_{2}^{5} 8 f(x) d x$ given $\int_{2}^{7} f(x) dx=11$ , $\int_{5}^{7} f(x) dx=5$ .

See Solution

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Calculus

Problem 25201

Déterminez les intervalles de convexité de la fonction f(x)=−3x2x2−4f(x)=\frac{-3 x^{2}}{x^{2}-4}f(x)=x2−4−3x2​ avec f′′(x)=−72x2+96(x2−4)3f^{\prime \prime}(x)=-\frac{72 x^{2}+96}{\left(x^{2}-4\right)^{3}}f′′(x)=−(x2−4)372x2+96​.

Problem 25202

Soit la fonction f(x)=−3x2x2−4f(x)=\frac{-3 x^{2}}{x^{2}-4}f(x)=x2−4−3x2​. Trouvez les intervalles de convexité et concavité de fff.

Problem 25203

Find the change in kinetic energy of a 7.5 kg7.5 \mathrm{~kg}7.5 kg projectile fired at 11.8 m/s11.8 \mathrm{~m/s}11.8 m/s from 20.9 m20.9 \mathrm{~m}20.9 m height. Use g=9.81 m/s2g = 9.81 \mathrm{~m/s}^{2}g=9.81 m/s2. Answer in JJJ.

Problem 25204

Calculer dQdK∣i=π2\left.\frac{d Q}{d K}\right|_{i=\frac{\pi}{2}}dKdQ​∣∣​i=2π​​, déterminer quand dKdt=0\frac{d K}{d t}=0dtdK​=0 et dQdt\frac{d Q}{d t}dtdQ​ à t=3π2t=\frac{3 \pi}{2}t=23π​.

Problem 25205

Find the tangent line equation for f(x)=4xf(x)=\frac{4}{x}f(x)=x4​ at x=9x=9x=9.

Problem 25206

Find the limits: (a) lim⁡x→∞(x2−5x+1−x)\lim_{x \to \infty} \left(\sqrt{x^2 - 5x + 1} - x\right)limx→∞​(x2−5x+1​−x), (b) lim⁡x→−∞(x2−5x+1−x)\lim_{x \to -\infty} \left(\sqrt{x^2 - 5x + 1} - x\right)limx→−∞​(x2−5x+1​−x).

Problem 25207

Find the volume of the solid formed by revolving region RRR (bounded by x=y23x=y^{\frac{2}{3}}x=y32​ and y=x13y=x^{\frac{1}{3}}y=x31​) around y=0y=0y=0 using the disc method.

Problem 25208

Déterminez les intervalles de concavité de la fonction f(x)=−3x2x2−4f(x)=\frac{-3 x^{2}}{x^{2}-4}f(x)=x2−4−3x2​ avec f′′(x)=−72x2+96(x2−4)3f^{\prime \prime}(x)=-\frac{72 x^{2}+96}{\left(x^{2}-4\right)^{3}}f′′(x)=−(x2−4)372x2+96​.

Problem 25209

Déterminez les intervalles de concavité de la fonction f(x)=−3x2x2−4f(x)=\frac{-3 x^{2}}{x^{2}-4}f(x)=x2−4−3x2​ avec f′′(x)=−72x2+96(x2−4)3f^{\prime \prime}(x)=-\frac{72 x^{2}+96}{\left(x^{2}-4\right)^{3}}f′′(x)=−(x2−4)372x2+96​.

Problem 25210

Déterminez les points d'inflexion de la fonction f(x)=−3x2x2−4f(x)=\frac{-3 x^{2}}{x^{2}-4}f(x)=x2−4−3x2​ en utilisant f′(x)f^{\prime}(x)f′(x) et f′′(x)f^{\prime \prime}(x)f′′(x).

Problem 25211

Differentiate the function: 10(sin⁡2(t)+cos⁡(t))+3010(\sin^{2}(t) + \cos(t)) + 3010(sin2(t)+cos(t))+30 with respect to ttt.

Problem 25212

Tracez la courbe de la fonction f(x)=−3x2x2−4f(x)=\frac{-3 x^{2}}{x^{2}-4}f(x)=x2−4−3x2​ avec f′(x)f^{\prime}(x)f′(x) et f′′(x)f^{\prime \prime}(x)f′′(x) donnés.

Problem 25213

Calculate f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for the function f(x)=x2−3x+5f(x)=x^{2}-3x+5f(x)=x2−3x+5.

Problem 25214

Find the average velocity of a train with location s(t)=50ts(t)=\frac{50}{t}s(t)=t50​ from t=3t=3t=3 to t=8t=8t=8 hours.

Problem 25215

Un fil de 48 cm est plié pour former une boîte. Trouvez les dimensions pour un volume maximal V=l×L×hV = l \times L \times hV=l×L×h avec 4l+4L+4h=484l + 4L + 4h = 484l+4L+4h=48.

Problem 25216

Find the horizontal asymptote of the function f(x)=3x2−5x+22x2−8xf(x)=\frac{3 x^{2}-5 x+2}{2 x^{2}-8 x}f(x)=2x2−8x3x2−5x+2​.

Problem 25217

Find the horizontal asymptote of each function: a. f(x)=3x2−5x+22x2−8xf(x)=\frac{3 x^{2}-5 x+2}{2 x^{2}-8 x}f(x)=2x2−8x3x2−5x+2​; answer: y=32y=\frac{3}{2}y=23​. b. g(x)=x3−5x6+18x3g(x)=\frac{x^{3}-5 x^{6}+1}{8 x^{3}}g(x)=8x3x3−5x6+1​.

Problem 25218

Approximate area under f(x)=xf(x)=\sqrt{x}f(x)=x​ from a=5a=5a=5 to b=8b=8b=8 using 6 rectangles. Then find exact area via integral.

Problem 25219

Deriver funksjonen a(x)=32x2−xa(x)=32 x^{2}-\sqrt{x}a(x)=32x2−x​.

Problem 25220

Find the function fff from the limit of Riemann sums: lim⁡Δ→0∑k=1n(xk∗)6Δxk;[3,11]\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(x_{k}^{*}\right)^{6} \Delta x_{k} ;[3,11]Δ→0lim​k=1∑n​(xk∗​)6Δxk​;[3,11] and express it as a definite integral.

Problem 25221

Find the production level xxx that minimizes the average cost given by c(x)=x3−20x2+20,000xc(x)=x^{3}-20x^{2}+20,000xc(x)=x3−20x2+20,000x.

Problem 25222

Calculate the integral ∫04f(x)dx\int_{0}^{4} f(x) dx∫04​f(x)dx for the piecewise function: f(x)=5f(x)=5f(x)=5 if x≤2x \leq 2x≤2, f(x)=4x−3f(x)=4x-3f(x)=4x−3 if x>2x>2x>2.

Problem 25223

Find the derivative of f(x)=x4xf(x)=x^{4 x}f(x)=x4x using logarithmic differentiation. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 25224

Soit f(x)=1x−1f(x)=\frac{1}{x-1}f(x)=x−11​. Trouvez les équations des deux tangentes de pente -1 et le point de tangence avec y=−x−34y=\frac{-x-3}{4}y=4−x−3​.

Problem 25225

Trouvez l'équation de la tangente à f(x)=x3−3x2−3f(x)=x^{3}-3 x^{2}-3f(x)=x3−3x2−3 au point où x=−3x=-3x=−3.

Problem 25226

A farmer uses 60 m of fencing for 3 sides of a corral. Find dimensions for maximum area.

Problem 25227

Find the horizontal asymptotes of the curve y=16x(x4+1)2y=\frac{16 x}{(x^{4}+1)^{2}}y=(x4+1)216x​, denoted as y1y_{1}y1​ and y2y_{2}y2​.

Problem 25228

Find the dimensions for a rectangular field with 600 m600 \mathrm{~m}600 m of fencing that maximizes the area and calculate that area.

Problem 25229

Gitt funksjonen f(x)=−x3+3xf(x)=-x^{3}+3 xf(x)=−x3+3x: Finn nullpunkt, topp/bunnpunkter, og vendetangenter.

Problem 25230

Prove that the function g(x)=d−xb2+(d−x)2g(x)=\frac{d-x}{\sqrt{b^{2}+(d-x)^{2}}}g(x)=b2+(d−x)2​d−x​ is decreasing in xxx.

Problem 25231

Prove that the function defined by dtdx=xc1a2+x2−d−xc2b2+(d−x)2\frac{d t}{d x}=\frac{x}{c_{1} \sqrt{a^{2}+x^{2}}}-\frac{d-x}{c_{2} \sqrt{b^{2}+(d-x)^{2}}}dxdt​=c1​a2+x2​x​−c2​b2+(d−x)2​d−x​ is increasing in xxx.

Problem 25232

Find the derivative of the function given by P′(t)=(t2−4t5/25)′P^{\prime}(t)=\left(t^{2}-\frac{4 t^{5 / 2}}{5}\right)^{\prime}P′(t)=(t2−54t5/2​)′.

Problem 25233

Calculate the fetal growth rate dHdt\frac{dH}{dt}dtdH​ using the formula H=−30.56+1.856t2−0.5741t2log⁡tH=-30.56+1.856 t^{2}-0.5741 t^{2} \log tH=−30.56+1.856t2−0.5741t2logt.

Problem 25234

Given s(t)=4t2+4ts(t)=4 t^{2}+4 ts(t)=4t2+4t, find: a) v(t)v(t)v(t), b) a(t)a(t)a(t), c) velocity and acceleration at t=4t=4t=4 sec.

Problem 25235

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=ln⁡6x−47x+4f(x)=\ln \sqrt{\frac{6 x-4}{7 x+4}}f(x)=ln7x+46x−4​​.

Problem 25236

Find where the tangent line of y=13x−x2y=13x-x^{2}y=13x−x2 has a slope. Choose from: A. (1,12)(1,12)(1,12) B. (6,42)(6,42)(6,42) C. (6.5,42.25)(6.5,42.25)(6.5,42.25) D. (12,84)(12,84)(12,84).

Problem 25237

Find the derivative of the function (5x2+7x+3)(2x)(5 x^{2}+7 x+3)(2 x)(5x2+7x+3)(2x).

Problem 25238

Find the inflection points of the function f(x)=18x1.5x2+2f(x)=\frac{18 x}{1.5 x^{2}+2}f(x)=1.5x2+218x​ using its derivatives. Lowest =, Middle =, Highest =.

Problem 25239

Find dydu\frac{d y}{d u}dudy​, dudx\frac{d u}{d x}dxdu​, and dydx\frac{d y}{d x}dxdy​ for y=u45y=u^{45}y=u45 and u=5x3+3x2u=5 x^{3}+3 x^{2}u=5x3+3x2.

Problem 25240

Déterminez les intervalles de convexité de la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ avec $f^{\prime \prime}(x)=-\frac{72 x^{2}+96}{\left(x^{2}-4\right)^{3}}$ .

Soit la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ . Trouvez les intervalles de convexité et concavité de $f$ .

Find the change in kinetic energy of a $7.5 \mathrm{~kg}$ projectile fired at $11.8 \mathrm{~m/s}$ from $20.9 \mathrm{~m}$ height. Use $g = 9.81 \mathrm{~m/s}^{2}$ . Answer in $J$ .

Calculer $\left.\frac{d Q}{d K}\right|_{i=\frac{\pi}{2}}$ , déterminer quand $\frac{d K}{d t}=0$ et $\frac{d Q}{d t}$ à $t=\frac{3 \pi}{2}$ .

Find the tangent line equation for $f(x)=\frac{4}{x}$ at $x=9$ .

Find the limits: (a) $\lim_{x \to \infty} \left(\sqrt{x^2 - 5x + 1} - x\right)$ , (b) $\lim_{x \to -\infty} \left(\sqrt{x^2 - 5x + 1} - x\right)$ .

Find the volume of the solid formed by revolving region $R$ (bounded by $x=y^{\frac{2}{3}}$ and $y=x^{\frac{1}{3}}$ ) around $y=0$ using the disc method.

Déterminez les intervalles de concavité de la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ avec $f^{\prime \prime}(x)=-\frac{72 x^{2}+96}{\left(x^{2}-4\right)^{3}}$ .

Déterminez les intervalles de concavité de la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ avec $f^{\prime \prime}(x)=-\frac{72 x^{2}+96}{\left(x^{2}-4\right)^{3}}$ .

Déterminez les points d'inflexion de la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ en utilisant $f^{\prime}(x)$ et $f^{\prime \prime}(x)$ .

Differentiate the function: $10(\sin^{2}(t) + \cos(t)) + 30$ with respect to $t$ .

Tracez la courbe de la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ avec $f^{\prime}(x)$ et $f^{\prime \prime}(x)$ donnés.

Calculate $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=x^{2}-3x+5$ .

Find the average velocity of a train with location $s(t)=\frac{50}{t}$ from $t=3$ to $t=8$ hours.

Un fil de 48 cm est plié pour former une boîte. Trouvez les dimensions pour un volume maximal $V = l \times L \times h$ avec $4l + 4L + 4h = 48$ .

Find the horizontal asymptote of the function $f(x)=\frac{3 x^{2}-5 x+2}{2 x^{2}-8 x}$ .

Find the horizontal asymptote of each function: a. $f(x)=\frac{3 x^{2}-5 x+2}{2 x^{2}-8 x}$ ; answer: $y=\frac{3}{2}$ . b. $g(x)=\frac{x^{3}-5 x^{6}+1}{8 x^{3}}$ .

Approximate area under $f(x)=\sqrt{x}$ from $a=5$ to $b=8$ using 6 rectangles. Then find exact area via integral.

Deriver funksjonen $a(x)=32 x^{2}-\sqrt{x}$ .

Find the function $f$ from the limit of Riemann sums: $\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(x_{k}^{*}\right)^{6} \Delta x_{k} ;[3,11]$ and express it as a definite integral.

Find the production level $x$ that minimizes the average cost given by $c(x)=x^{3}-20x^{2}+20,000x$ .

Calculate the integral $\int_{0}^{4} f(x) dx$ for the piecewise function: $f(x)=5$ if $x \leq 2$ , $f(x)=4x-3$ if $x>2$ .

Find the derivative of $f(x)=x^{4 x}$ using logarithmic differentiation. What is $f^{\prime}(x)=?$

Soit $f(x)=\frac{1}{x-1}$ . Trouvez les équations des deux tangentes de pente -1 et le point de tangence avec $y=\frac{-x-3}{4}$ .

Trouvez l'équation de la tangente à $f(x)=x^{3}-3 x^{2}-3$ au point où $x=-3$ .

Find the horizontal asymptotes of the curve $y=\frac{16 x}{(x^{4}+1)^{2}}$ , denoted as $y_{1}$ and $y_{2}$ .

Find the dimensions for a rectangular field with $600 \mathrm{~m}$ of fencing that maximizes the area and calculate that area.

Gitt funksjonen $f(x)=-x^{3}+3 x$ : Finn nullpunkt, topp/bunnpunkter, og vendetangenter.

Prove that the function $g(x)=\frac{d-x}{\sqrt{b^{2}+(d-x)^{2}}}$ is decreasing in $x$ .

Prove that the function defined by $\frac{d t}{d x}=\frac{x}{c_{1} \sqrt{a^{2}+x^{2}}}-\frac{d-x}{c_{2} \sqrt{b^{2}+(d-x)^{2}}}$ is increasing in $x$ .

Find the derivative of the function given by $P^{\prime}(t)=\left(t^{2}-\frac{4 t^{5 / 2}}{5}\right)^{\prime}$ .

Calculate the fetal growth rate $\frac{dH}{dt}$ using the formula $H=-30.56+1.856 t^{2}-0.5741 t^{2} \log t$ .

Given $s(t)=4 t^{2}+4 t$ , find: a) $v(t)$ , b) $a(t)$ , c) velocity and acceleration at $t=4$ sec.

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\ln \sqrt{\frac{6 x-4}{7 x+4}}$ .

Find where the tangent line of $y=13x-x^{2}$ has a slope. Choose from: A. $(1,12)$ B. $(6,42)$ C. $(6.5,42.25)$ D. $(12,84)$ .

Find the derivative of the function $(5 x^{2}+7 x+3)(2 x)$ .

Find the inflection points of the function $f(x)=\frac{18 x}{1.5 x^{2}+2}$ using its derivatives. Lowest =, Middle =, Highest =.

Find $\frac{d y}{d u}$ , $\frac{d u}{d x}$ , and $\frac{d y}{d x}$ for $y=u^{45}$ and $u=5 x^{3}+3 x^{2}$ .

Differentiate $f(x)=(7 x^{2}-2 x+6)(4 x^{2}+3 x-6)$ . Which option shows how to find $f'(x)$ ?

Find $\lim _{n \rightarrow \infty} \sqrt[n]{\left(\frac{4}{3}\right)^{n}+\left(\frac{8}{3}\right)^{n}}$ .

Find the volume of the solid formed by rotating the area between $y=x^{2}$ and $y=x^{3}$ in the first quadrant around $x=1$ .

A wrench is dropped from an $80.0 \mathrm{~m}$ tower. What is its velocity upon hitting the ground?

A city’s population grows from 200,000 to $P(t)=200,000+5000t^{2}$ . Find: a) $\frac{\mathrm{dP}}{\mathrm{dt}}$ ; b) $P(20)$ ; c) Growth rate at $t=20$ ; d) Meaning of part (c).

Find the area between the curves $y = 6 \sin x$ and $y = 6 \cos x$ over the interval $[0, \frac{\pi}{2}]$ .

Find the limit: $\lim _{n \rightarrow \infty} \frac{\sqrt{4+6 n+5 n^{2}}}{3 n+2}$ .

Given the function $f(x)=\frac{x^{2}-49}{x-7}$ , find where it's not differentiable and compute $f^{\prime}(-1)$ .

Bestimme die Durchschnittsgeschwindigkeiten für $[0; 3]$ und $[3; 7]$ der Funktion $s(t)=6t^{2}-0,4t^{3}$ . Finde die Momentangeschwindigkeit bei $t=3$ und $t=12$ . Wann erreicht der Ballon den Boden?

Find the volume of the solid formed by rotating the area between $y=x^{2}$ and $y=x^{3}$ in the first quadrant around $x=1$ .

Find $x$ where the tangent line to $y=x^{2}+2x-3$ is horizontal. Options: A. $\frac{1}{2}$ B. 0 C. 1 D. -1

Find the derivative of $f(x)=\frac{8}{x}$ and evaluate it at $x=-1$ . What is $f^{\prime}(-1)$ ?

Find the tangent line equation for the curve $y=x-x^{2}$ at the point $(2,-2)$ . Choose from the options.

Berechnen Sie die Ableitung $f^{\prime}(2)$ für die Funktion $f(x) = 5x^{2} + 1$ .

Finn topp- og bunnpunkter samt vendepunkt for funksjonen $p(x)=\ln \left(x^{2}-1\right)$ der $x<-1$ eller $x>1$ .

Find the velocity $v(t)$ and acceleration $a(t)$ for $s(t)=5 t^{2}+15 t$ . Calculate both at $t=2$ sec.

Differentiate $y=\frac{4 x^{2}-7}{2 x^{3}+3}$ . Which option shows the correct derivative formula?

Find the difference quotient for $f(x)=9x-2$ at $x=5$ for $h=2, 1, 0.1, 0.01$ .

Calculate the sum: $\sum_{n=0}^{\infty} \frac{\left(\frac{2}{3}\right)^{n}(1.5)^{n}}{n !}$ .

Find the limit: $\lim _{n \rightarrow \infty}\left(\sqrt{3+6 n+n^{2}}-\sqrt{3+13 n+n^{2}}\right)$ .

Find points on the graph of $y=13x-x^{2}$ where the tangent line has a slope of 1.

Calculate the average value of $y=9 e^{-x}$ over the interval $[0,1]$ . The average value is $\square$ .

A particle's speed is $v(t)=-0.2 t^{2}+10 t$ . Find distances traveled in the first and second 4 seconds.

Find the velocity function $v(t)$ given acceleration $a(t)=8t$ and initial condition $v(0)=15$ .

Find the total cost of producing 220 dresses with marginal cost $C^{\prime}(x)=-\frac{4}{25} x+50$ . Total cost is \$\square.

Calculate the area under the curve of $f(x)$ from $x = -2$ to $x = 4$ , where $f(x) = x^{2} + 6$ for $x \leq 2$ and $f(x) = 5x$ for $x > 2$ . The area is $\square$ .

Determine the center, radius, and interval of convergence for the series: $\sum_{n=0}^{\infty} \frac{\left(\frac{2}{3}\right)^{n}(x-4)^{n}}{n !}$

Find the limit as $x$ approaches 6 for $\frac{[f(x)]^{2}-6 f(x)+2}{[g(x)]^{2}-49}$ given $f(6)=10$ , $f'(6)=-1$ , $g(6)=-8$ , $g'(6)=4$ .

At what minimum times is $A^{\prime}(t)=0$ for the function $A(t)$ , given $0 \leq t \leq 8$ ? Options: a. 0, b. 2, c. 3, d. 4, e. 5.

Differentiate $h(t)=\frac{7}{2} t^{2}-8 t+5$ with respect to $t$ .

Find the sum of values of $c$ that satisfy the Mean Value Theorem for $f(x)=2 \cos x-4 \cos 2x$ on $[0,\pi]$ .

Find the distance the particle travels to the right given its position $s(t)=3t-2t^{2}$ for $t \geq 0$ .

Evaluate $f(x+h)$ and find $\frac{f(x+h)-f(x)}{h}$ for $f(x)=4x^{2}+2x$ .