Calculus

Problem 10201

Invest \$30339.86 at 9.6\% for 9 years with continuous compounding. Find the future value, rounded to the nearest cent.

See Solution

Problem 10202

Gegeben ist die Funktion $f(t)=-0.01 t^{3}+0.24 t^{2}+6.84$ für $0 \leq t \leq 24$ in Grad Celsius.
a) Was bedeutet $f(3)$ ? b) Finde die höchste und niedrigste Temperatur in 22 Stunden. c) Wird die Temperatur über 30 Grad Celsius steigen?

See Solution

Problem 10203

Find the derivative of $y=2^{\sin(4x^{3}+e^{5x})}$ .

See Solution

Problem 10204

Find the derivative of $y=2^{\sin(4x^{3}+e^{5x})}$ .

See Solution

Problem 10205

Find the limit: $\lim _{h \rightarrow 0} \frac{x^{3}-2 x^{2}(1+h)-x^{3}-2 x^{2}(1)}{h}$ .

See Solution

Problem 10206

Find the limit as $x$ approaches $x_0$ : $\lim _{x \rightarrow x_{0}} \frac{x^{3}-2 x^{2}-1^{3}-2 \cdot 1^{2}}{x-1}$ .

See Solution

Problem 10207

Find $\left(f^{-1}\right)^{\prime}(0)$ for $f(x)=x^{2}-2x-3$ on $[1, \infty)$ . Is $f^{-1}$ differentiable at $x=-4$ ?

See Solution

Problem 10208

Leiten Sie die Funktion $f$ einmal ab für: a) $f(x)=x \cdot \sin (x)$ , b) $f(x)=3 x \cdot \cos (x)$ , d) $f(x)=\sqrt{x} \cdot(2 x-3)$ , e) $f(x)=\sqrt{x} \cdot \cos (x)$ , g) $f(x)=\frac{2}{x} \cdot \cos (x)$ , h) $f(x)=\sin (x) \cdot \cos (x)$ , k) $f(x)=(x^{2}+3 x) \cdot \sin (x)$ , c) $f(x)=(3 x+2) \cdot \sqrt{x}$ , f) $f(x)=(5-3 x) \cdot \sin (x)$ , i) $f(x)=x^{2} \cdot \sin (x)$ , 1) $f(x)=\sqrt{x} \cdot\left(x^{5}-2 x^{3}\right)$ .

See Solution

Problem 10209

Find the derivative of $y=e^{\sqrt{x}}$ .

See Solution

Problem 10210

Determine the y-intercept and horizontal asymptote of $f(x) = \frac{6}{1+2 e^{-x}}$ .

See Solution

Problem 10211

Find the future value of \$ 1691 invested at 6.1% for 22 years with continuous compounding. Round to the nearest cents.

See Solution

Problem 10212

What is true about the function $f$ on the interval $1<x<3$ if its rate of change is negative and decreasing?

See Solution

Problem 10213

Gegeben ist die Funktion $g(x)$ , bestimme $\lim_{x \to 2} g(x)$ und prüfe die Differenzierbarkeit bei $x_0=2$ .

See Solution

Problem 10214

Invest \$4000 at 8\% interest compounded continuously. (a) Find the account value after 7 years. (b) When will it reach \$50000?

See Solution

Problem 10215

For the polynomial $g$ , if the rate of change decreases for $x<-5$ and increases for $x>-5$ , what must be true?

See Solution

Problem 10216

Find the limit: $\lim \frac{-\left(4^{3}-1\right)^{2}+4-(-3)^{2}+4}{x-3}$ as $x$ approaches 3.

See Solution

Problem 10217

Berechnen Sie den Grenzwert $\lim _{x \rightarrow 2} \frac{-(3-1)^{2}+4-(-3)^{2}+4}{x-3}$ für x > 2.

See Solution

Problem 10218

Find the limit: $\lim_{x \to 3} \frac{-(3-1)^{2}+4-(-3)^{2}+4}{x-3}$ .

See Solution

Problem 10219

Find the limit: $\lim _{x \rightarrow 2} \frac{-(2-1)^{2}+4-(-2)^{2}+4}{x-2}$ .

See Solution

Problem 10220

Gegeben sind $f(x)=0,5 x^{2}+2$ und $g(x)=x^{2}-2 x+2$ . Finde Max/Min der Summe und Differenz der Werte in $[0; 4]$ .

See Solution

Problem 10221

Leiten Sie die Funktion $f$ mit der Produktregel und der Kettenregel ab: a) $f(x)=x \cdot \sin (3 x)$ b) $f(x)=(2 x-1)^{2} \cdot \sqrt{x}$ c) $f(x)=3 x^{5} \cdot \cos (2 x)$ d) $f(x)=3 x \cdot \sin (4 x-1)$ e) $f(x)=(4-3 x)^{2} \cdot \sin (x)$ f) $f(x)=0,5 x^{2} \cdot \sqrt{4-x}$ g) $f(x)=x^{2} \cdot \cos (1-x)$ h) $f(x)=\sqrt{2 x+3} \cdot x^{2}$ i) $f(x)=(5 x+2)^{7} \cdot \cos (x)$

See Solution

Problem 10222

Find the derivative $f^{\prime}(x)$ for the function $f(x)=10 \sqrt[5]{x^{4}}-\frac{3}{a^{9}}+2$ .

See Solution

Problem 10223

Find $f^{\prime}$ and $f^{\prime \prime}$ for $f(x)=g\left(x^{3}\right)+e^{g(3 x)}$ . Also, find $f^{\prime \prime}$ for $f(x)=e^{x^{2}}+5^{x}$ .

See Solution

Problem 10224

Find the derivative of $f(x) = (5-x)(3+x)$ .

See Solution

Problem 10225

Differentiate $\sqrt{2 x^{2}-3 y}=x^{3} y^{2}+4$ with respect to $x$ using implicit differentiation.

See Solution

Problem 10226

Untersuchen Sie das Verhalten von $f$ für $x \rightarrow \infty$ und $x \rightarrow -\infty$ für die Funktionen: a) $f(x)=\frac{4x-1}{x}$ , $f(x)=\frac{x^{2}-x}{3x^{2}}$ b) $f(x)=\frac{3x^{2}-4}{x^{2}}$ c) $f(x)=\frac{2x+x^{2}}{x^{2}}$ e) $f(x)=\frac{3-x^{3}}{x^{3}}$ f) $f(x)=\frac{x^{2}-1}{x(x-1)}$ Testen Sie Ihre Ergebnisse.

See Solution

Problem 10227

Find a formula for the sequence $S_{t}=\frac{2}{3}+\left(\frac{2}{3}\right)^{2}+\cdots+\left(\frac{2}{3}\right)^{t}$ and its limit as $t$ approaches infinity.

See Solution

Problem 10228

Estimate the population of country $X$ in 25 years, given $N(t)=500 e^{0.02 t}$ .

See Solution

Problem 10229

Differentiate $x^{2} \cos (3 y) - y^{2} \sin (x^{2}) = 5$ with respect to $x$ and $y$ using implicit differentiation.

See Solution

Problem 10230

Differentiate $y=\frac{\left(2 x^{2}-3\right)^{\frac{3}{5}}}{\left(4 x^{2}-3 x+1\right)^{\frac{4}{7}}(6 x+2)^{\frac{1}{2}}}$ using logarithmic differentiation.

See Solution

Problem 10231

A tennis ball is thrown up at $22.5 \mathrm{~m} / \mathrm{s}$ . What is its maximum height? $T=42.6$

See Solution

Problem 10232

A softball is hit upwards with a velocity of $26 \mathrm{~m/s}$ . Find its velocity, position, max height time, and ground strike time.

See Solution

Problem 10233

Klimaanlage Problem: Gegeben ist $T(t)=\frac{200}{4+\frac{6}{0,1 t+1}}$ . Finde: a) Anfangstemperatur, b) Temperatur nach 60 Minuten, c) Grenztemperatur.

See Solution

Problem 10234

Finde den Durchmesser $d$ und die Höhe $h$ einer Regentonne, um das Volumen bei $2 m^{2}$ Material zu maximieren.

See Solution

Problem 10235

An arctic goose dives from a $245 \mathrm{~m}$ cliff. How long to reach the ground and what speed does it land?

See Solution

Problem 10236

A tourist drops a rock from rest. Find its velocity and displacement at $4.0 \mathrm{~s}$ .

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Problem 10237

Find the second derivative of $f(x)=e^{2x}$ .

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Problem 10238

Cost function is $C(x)=920+3x-0.03x^2+0.0006x^3$ . Find marginal cost, $C'(100)$ , and compare with cost of 101st item.

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Problem 10239

Find the limit: $\lim _{k \rightarrow \infty}(k+1)^{\frac{1}{k}}$ .

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Problem 10240

A turkey cools from $85^{\circ} \mathrm{C}$ to $33^{\circ} \mathrm{C}$ over 150 mins. Find average rates for intervals and estimate the instantaneous rate after 1 hour.

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Problem 10241

Solve the initial value problem: $\frac{d y}{d t}=\frac{1}{2^{t}} \sec ^{2}\left(\frac{\pi}{2^{t}}\right)$ , with $y\left(\log _{2} 4\right)=2 / \pi$ .

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Problem 10242

Find the stationary points of $f(x)=x^{3}-9x^{2}+16x+8$ and determine their nature.

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Problem 10243

Create a value table for $f(x)=\frac{x-4}{\sqrt{x}-2}$ near $x=4$ and find $\lim _{x \rightarrow 4} f(x)$ .

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Problem 10244

Find the derivative of $f(x)=(12-x^{2}) \cdot \frac{1}{3x+3}$ .

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Problem 10245

Solve Legendre's equation for $-1<t<1$ : $(1-t^{2}) y'' - 2t y' + 2y = 0$ . Hint: $y=t$ is a solution.

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Problem 10246

Untersuchen Sie die Grenzwerte: a) $\lim _{x \rightarrow 4} \frac{x^{2}-16}{x-4}$ b) $\lim _{x \rightarrow -1} \frac{x^{3}-x}{x+1}$ c) $\lim _{x \rightarrow 3} \frac{3-x}{2 x^{2}-6 x}$ d) $\lim _{x \rightarrow 2} \frac{x^{4}-16}{x-2}$

See Solution

Problem 10247

A projectile is launched horizontally at $15 \mathrm{~m/s}$ from a 30 m cliff. Find its range.

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Problem 10248

Find the derivative dy/dx for the function $y=(6 x+2) \cos x$ .

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Problem 10249

Un peintre doit peindre la surface entre les courbes $x=9-y^{2}$ , $x=y-3$ et $x=-\frac{y}{2}-\frac{3}{2}$ . A-t-il assez de peinture pour $40 \, m^{2}$ ? Trouvez les intersections et calculez l'aire.

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Problem 10250

Find the instantaneous rate of change of $f(x)=\sqrt{x+2}$ at the left endpoint of the interval $[-1,2]$ .

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Problem 10251

Find the derivative of $f(x)=(4-2x)^{2} \cdot \frac{6}{x^{3}}$ .

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Problem 10252

Determine the highest point of a projectile launched at 128 ft/s at an unknown angle.

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Problem 10253

Find $\frac{d y}{d t}$ when $y=x^{2}+4$ , $\frac{d x}{d t}=2$ , and $x=5$ . $\frac{d y}{d t}=\square$ .

See Solution

Problem 10254

Find the derivative of $y=\ln \left(\frac{\sqrt{4+x^{2}}}{x}\right)$ .

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Problem 10255

Identify which sequences diverge: i. $t_{n}=0.8^{n}$ ii. $t_{n}=\left(\frac{5}{2}\right)^{n}$ iii. $t_{n}=3 n-7$ iv. $t_{n}=\frac{1}{n}$ .

See Solution

Problem 10256

Find the derivatives of these functions: (a) $f(x)=\sin x \cos x$ , (b) $g(x)=\frac{1+\tan x}{\sin x}$ .

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Problem 10257

Bestimmen Sie die Tangentengleichung im Wendepunkt der Funktionen: a) $f(x)=\frac{1}{6} x^{3}-\frac{3}{4} x^{2}+2$ , b) $g(x)=9 x \cdot\left(\frac{1}{6} x-1\right)^{2}$ , c) $h(t)=\left(t^{2}-t\right) \cdot(2 t-1)$ .

See Solution

Problem 10258

Determine which statements are true for $\lim _{x \rightarrow a} f(x)=L$ : I, II, III, IV.

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Problem 10259

Differentiate the function $f(x)=\frac{x^{6} e^{x}}{x^{6}+e^{x}}$ .

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Problem 10260

Quel est le profit total pour 10 unités, sachant que $P_{m}=\frac{80}{\sqrt{4 q+1}}$ et que le profit total pour 6 unités est 230 \$?

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Problem 10261

Find the derivative of $f(x)=3 x^{3} \cdot \sqrt{4 x^{2}-4 x+1}$ and simplify it.

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Problem 10262

Finde die drei Ableitungen von $h(t) = (t^{2}-t)(2t-1)$ .

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Problem 10263

Differentiate the function $F(x)=\frac{1}{3 x^{3}-8 x^{2}+7}$ .

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Problem 10264

Calculate the arc length of $x=6 y^{3/2}$ from $y=1$ to $y=4$ . Provide the exact answer or round to 3 decimal places.

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Problem 10265

Invest \$250000 at 5% interest compounded continuously. (a) Find account value after 5 years. (b) When will it reach \$50000?

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Problem 10266

Bearbeite die Funktion $f(t)=e^{3 t}$ : Finde die Ableitung, das Integral oder die allgemeine Lösung.

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Problem 10267

Invest \$25000 at 3\% interest compounded continuously. (a) Find the value after 10 years. (b) When will it reach \$50000?

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Problem 10268

Compute $F^{\prime}(0)$ for $F(x)=(f \circ g)(x)$ given $f^{\prime}(x)=-3$ and $g^{\prime}(x)=7$ .

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Problem 10269

Differentiate the function: $J(u)=\left(\frac{3}{u}+\frac{3}{u^{2}}\right)\left(2 u+\frac{2}{u}\right)$ .

See Solution

Problem 10270

Find the derivative of $y=x^{-1/x}$ using logarithmic differentiation.

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Problem 10271

Estimate $\Delta y$ for $y=\sin(3x)$ at $x=0$ with $\Delta x=0.2$ , then find the percentage error rounded to 1 decimal place.

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Problem 10272

Raylin invests \ $15,000 at 3.5% interest compounded continuously. Find the balance after 30 years using$ A = Pe^{rt}$.

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Problem 10273

Find $g^{\prime}(1)$ where $f(x)=\sin x+2x+1$ and $g$ is the inverse of $f$ . Choices: (A) $\frac{1}{3}$ (B) 1 (C) 3 (D) $\frac{1}{2+\cos 1}$ (E) $2+\cos 1$ .

See Solution

Problem 10274

Verify the first and second derivatives of $f(x) = x^{2}e^{-x}$ and find its critical points.

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Problem 10275

Calculez la durée de vie moyenne d'un tube au néon avec $f(t)=0,0002 e^{-0,0002 t}$ pour $t \geq 0$ . Utilisez $\mu=\int_{0}^{\infty} t f(t) d t$ .

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Problem 10276

Find tangent line equations to $y=\frac{x-1}{x+1}$ parallel to $x-2y=4$ . List answers as comma-separated equations.

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Problem 10277

Bestimmen Sie die Ableitung von $f(x)=4 e^{x}+\frac{x}{2}^{3}$ .

See Solution

Problem 10278

Calculate the surface area from revolving $y=\frac{x^{3}}{9}$ around the $x$ -axis for $0 \leq x \leq 2$ .

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Problem 10279

Calculate the integrals $\int x(x+1)^{5/2} \mathrm{~d} x$ and $\int \frac{1}{1+e^{x}} \mathrm{~d} x$ . Find critical points.

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Problem 10280

Find the limit using L'Hôpital's rule: $\lim _{x \rightarrow 8} \frac{x^{2}-x-56}{x-8}$ and verify with simplification.

See Solution

Problem 10281

Switch the order of integration and compute the integral: $\int_{-1}^{1}\left(\int_{-2}^{2}(4 x+5 y+8) d x\right) d y$

See Solution

Problem 10282

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x^{4}+y^{5}=-7$ .

See Solution

Problem 10283

Find the slope $m$ of the tangent line to the curve $x^{2}-3xy+3y^{3}=103$ at the point $(-2,3)$ .

See Solution

Problem 10284

Find the limit using L'Hôpital's rule: $\lim _{x \rightarrow-\infty} \frac{x^{8}-57}{x-4}$

See Solution

Problem 10285

Calculate the integral: $\int \frac{1}{1+e^{x}} d x$

See Solution

Problem 10286

Find critical points of $f(x, y)=2 x^{2}-x+y^{3}+3 x y^{2}-1$ and classify each as min, max, or saddle point.

See Solution

Problem 10287

Find the limit using I'Hôpital's rule: $\lim _{x \rightarrow 0} \frac{e^{4 x}-1}{3 x}$ .

See Solution

Problem 10288

Find $\frac{d g}{d t}$ for $g(t)=(t^{3}-5)e^{t}$ .

See Solution

Problem 10289

Find $\frac{d y}{d x}$ for the equation $-10 x^{8}+7 x^{20} y+y^{3}=-2$ . Then, find the tangent line at $(1,1)$ in $y=m x+b$ format.

See Solution

Problem 10290

Evaluate the integral $\int_{0}^{2}\left(x^{3}-4 x\right) dx$ .

See Solution

Problem 10291

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x^{6}+y^{2}=e^{y}$ .

See Solution

Problem 10292

Find $\left.\frac{d y}{d x}\right|_{x=4, y=-14}$ for $3 x^{2}+3 x+x y=4$ using implicit differentiation. $\left.\frac{d y}{d x}\right|_{x=4, y=-14}=$

See Solution

Problem 10293

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $e^{x} y^{5}+\cos (x)=e^{y}$ .

See Solution

Problem 10294

Find $\left.\frac{d y}{d x}\right|_{x=4}$ for $\sqrt{x}+\sqrt{y}=7$ given the point $(4,25)$ .

See Solution

Problem 10295

Find the slope of the tangent line to the curve $x^{2}+4xy+4y^{3}=-119$ at the point $(1,-3)$ .

See Solution

Problem 10296

Calculate the integral from 1 to 2 of $(x^{2}-3)^{2}$ .

See Solution

Problem 10297

Use substitution to find $\alpha \in \mathbb{R}$ for $F(x)=\alpha(4x+5)^{-6}+c$ where $f(x)=\frac{1}{(4x+5)^{7}}$ . Answer: $\alpha=$

See Solution

Problem 10298

Find the tangent line to the curve $x y^{3}+x y=14$ at $(7,1)$ in the form $y=m x+b$ .

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Problem 10299

Approximate $\int_{0}^{1} e^{t^{2}} dt$ using the degree 3 Taylor polynomial of $f(x)=\int_{0}^{x} e^{t^{2}} dt$ .

See Solution

Problem 10300

Bestimmen Sie die Ableitung der Funktion $f(x)=3^{x}+3$ .

See Solution

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Calculus

Problem 10201

Invest \$30339.86 at 9.6\% for 9 years with continuous compounding. Find the future value, rounded to the nearest cent.

Problem 10202

Problem 10203

Find the derivative of y=2sin⁡(4x3+e5x)y=2^{\sin(4x^{3}+e^{5x})}y=2sin(4x3+e5x).

Problem 10204

Find the derivative of y=2sin⁡(4x3+e5x)y=2^{\sin(4x^{3}+e^{5x})}y=2sin(4x3+e5x).

Problem 10205

Find the limit: lim⁡h→0x3−2x2(1+h)−x3−2x2(1)h\lim _{h \rightarrow 0} \frac{x^{3}-2 x^{2}(1+h)-x^{3}-2 x^{2}(1)}{h}limh→0​hx3−2x2(1+h)−x3−2x2(1)​.

Problem 10206

Find the limit as xxx approaches x0x_0x0​: lim⁡x→x0x3−2x2−13−2⋅12x−1\lim _{x \rightarrow x_{0}} \frac{x^{3}-2 x^{2}-1^{3}-2 \cdot 1^{2}}{x-1}limx→x0​​x−1x3−2x2−13−2⋅12​.

Problem 10207

Find (f−1)′(0)\left(f^{-1}\right)^{\prime}(0)(f−1)′(0) for f(x)=x2−2x−3f(x)=x^{2}-2x-3f(x)=x2−2x−3 on [1,∞)[1, \infty)[1,∞). Is f−1f^{-1}f−1 differentiable at x=−4x=-4x=−4?

Problem 10208

Problem 10209

Find the derivative of y=exy=e^{\sqrt{x}}y=ex​.

Problem 10210

Determine the y-intercept and horizontal asymptote of f(x)=61+2e−xf(x) = \frac{6}{1+2 e^{-x}}f(x)=1+2e−x6​.

Problem 10211

Find the future value of \$ 1691 invested at 6.1% for 22 years with continuous compounding. Round to the nearest cents.

Problem 10212

What is true about the function fff on the interval 1<x<31<x<31<x<3 if its rate of change is negative and decreasing?

Problem 10213

Gegeben ist die Funktion g(x)g(x)g(x), bestimme lim⁡x→2g(x)\lim_{x \to 2} g(x)limx→2​g(x) und prüfe die Differenzierbarkeit bei x0=2x_0=2x0​=2.

Problem 10214

Invest \$4000 at 8\% interest compounded continuously. (a) Find the account value after 7 years. (b) When will it reach \$50000?

Problem 10215

For the polynomial ggg, if the rate of change decreases for x<−5x<-5x<−5 and increases for x>−5x>-5x>−5, what must be true?

Problem 10216

Find the limit: lim⁡−(43−1)2+4−(−3)2+4x−3\lim \frac{-\left(4^{3}-1\right)^{2}+4-(-3)^{2}+4}{x-3}limx−3−(43−1)2+4−(−3)2+4​ as xxx approaches 3.

Problem 10217

Berechnen Sie den Grenzwert lim⁡x→2−(3−1)2+4−(−3)2+4x−3\lim _{x \rightarrow 2} \frac{-(3-1)^{2}+4-(-3)^{2}+4}{x-3}limx→2​x−3−(3−1)2+4−(−3)2+4​ für x > 2.

Problem 10218

Find the limit: lim⁡x→3−(3−1)2+4−(−3)2+4x−3\lim_{x \to 3} \frac{-(3-1)^{2}+4-(-3)^{2}+4}{x-3}limx→3​x−3−(3−1)2+4−(−3)2+4​.

Problem 10219

Find the limit: lim⁡x→2−(2−1)2+4−(−2)2+4x−2\lim _{x \rightarrow 2} \frac{-(2-1)^{2}+4-(-2)^{2}+4}{x-2}limx→2​x−2−(2−1)2+4−(−2)2+4​.

Problem 10220

Gegeben sind f(x)=0,5x2+2f(x)=0,5 x^{2}+2f(x)=0,5x2+2 und g(x)=x2−2x+2g(x)=x^{2}-2 x+2g(x)=x2−2x+2. Finde Max/Min der Summe und Differenz der Werte in [0;4][0; 4][0;4].

Problem 10221

Problem 10222

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=10x45−3a9+2f(x)=10 \sqrt[5]{x^{4}}-\frac{3}{a^{9}}+2f(x)=105x4​−a93​+2.

Problem 10223

Find f′f^{\prime}f′ and f′′f^{\prime \prime}f′′ for f(x)=g(x3)+eg(3x)f(x)=g\left(x^{3}\right)+e^{g(3 x)}f(x)=g(x3)+eg(3x). Also, find f′′f^{\prime \prime}f′′ for f(x)=ex2+5xf(x)=e^{x^{2}}+5^{x}f(x)=ex2+5x.

Problem 10224

Find the derivative of f(x)=(5−x)(3+x)f(x) = (5-x)(3+x)f(x)=(5−x)(3+x).

Problem 10225

Differentiate 2x2−3y=x3y2+4\sqrt{2 x^{2}-3 y}=x^{3} y^{2}+42x2−3y​=x3y2+4 with respect to xxx using implicit differentiation.

Problem 10226

Problem 10227

Find a formula for the sequence St=23+(23)2+⋯+(23)tS_{t}=\frac{2}{3}+\left(\frac{2}{3}\right)^{2}+\cdots+\left(\frac{2}{3}\right)^{t}St​=32​+(32​)2+⋯+(32​)t and its limit as ttt approaches infinity.

Problem 10228

Estimate the population of country XXX in 25 years, given N(t)=500e0.02tN(t)=500 e^{0.02 t}N(t)=500e0.02t.

Problem 10229

Differentiate x2cos⁡(3y)−y2sin⁡(x2)=5x^{2} \cos (3 y) - y^{2} \sin (x^{2}) = 5x2cos(3y)−y2sin(x2)=5 with respect to xxx and yyy using implicit differentiation.

Problem 10230

Differentiate y=(2x2−3)35(4x2−3x+1)47(6x+2)12y=\frac{\left(2 x^{2}-3\right)^{\frac{3}{5}}}{\left(4 x^{2}-3 x+1\right)^{\frac{4}{7}}(6 x+2)^{\frac{1}{2}}}y=(4x2−3x+1)74​(6x+2)21​(2x2−3)53​​ using logarithmic differentiation.

Problem 10231

A tennis ball is thrown up at 22.5 m/s22.5 \mathrm{~m} / \mathrm{s}22.5 m/s. What is its maximum height? T=42.6T=42.6T=42.6

Problem 10232

A softball is hit upwards with a velocity of 26 m/s26 \mathrm{~m/s}26 m/s. Find its velocity, position, max height time, and ground strike time.

Problem 10233

Klimaanlage Problem: Gegeben ist T(t)=2004+60,1t+1T(t)=\frac{200}{4+\frac{6}{0,1 t+1}}T(t)=4+0,1t+16​200​. Finde: a) Anfangstemperatur, b) Temperatur nach 60 Minuten, c) Grenztemperatur.

Problem 10234

Finde den Durchmesser ddd und die Höhe hhh einer Regentonne, um das Volumen bei 2m22 m^{2}2m2 Material zu maximieren.

Problem 10235

An arctic goose dives from a 245 m245 \mathrm{~m}245 m cliff. How long to reach the ground and what speed does it land?

Problem 10236

A tourist drops a rock from rest. Find its velocity and displacement at 4.0 s4.0 \mathrm{~s}4.0 s.

Problem 10237

Find the second derivative of f(x)=e2xf(x)=e^{2x}f(x)=e2x.

Problem 10238

Cost function is C(x)=920+3x−0.03x2+0.0006x3C(x)=920+3x-0.03x^2+0.0006x^3C(x)=920+3x−0.03x2+0.0006x3. Find marginal cost, C′(100)C'(100)C′(100), and compare with cost of 101st item.

Problem 10239

Find the limit: lim⁡k→∞(k+1)1k\lim _{k \rightarrow \infty}(k+1)^{\frac{1}{k}}limk→∞​(k+1)k1​.

Problem 10240

A turkey cools from 85∘C85^{\circ} \mathrm{C}85∘C to 33∘C33^{\circ} \mathrm{C}33∘C over 150 mins. Find average rates for intervals and estimate the instantaneous rate after 1 hour.

Problem 10241

Solve the initial value problem: dydt=12tsec⁡2(π2t)\frac{d y}{d t}=\frac{1}{2^{t}} \sec ^{2}\left(\frac{\pi}{2^{t}}\right)dtdy​=2t1​sec2(2tπ​), with y(log⁡24)=2/πy\left(\log _{2} 4\right)=2 / \piy(log2​4)=2/π.

Problem 10242

Find the derivative of $y=2^{\sin(4x^{3}+e^{5x})}$ .

Find the derivative of $y=2^{\sin(4x^{3}+e^{5x})}$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{x^{3}-2 x^{2}(1+h)-x^{3}-2 x^{2}(1)}{h}$ .

Find the limit as $x$ approaches $x_0$ : $\lim _{x \rightarrow x_{0}} \frac{x^{3}-2 x^{2}-1^{3}-2 \cdot 1^{2}}{x-1}$ .

Find $\left(f^{-1}\right)^{\prime}(0)$ for $f(x)=x^{2}-2x-3$ on $[1, \infty)$ . Is $f^{-1}$ differentiable at $x=-4$ ?

Find the derivative of $y=e^{\sqrt{x}}$ .

Determine the y-intercept and horizontal asymptote of $f(x) = \frac{6}{1+2 e^{-x}}$ .

What is true about the function $f$ on the interval $1<x<3$ if its rate of change is negative and decreasing?

Gegeben ist die Funktion $g(x)$ , bestimme $\lim_{x \to 2} g(x)$ und prüfe die Differenzierbarkeit bei $x_0=2$ .

For the polynomial $g$ , if the rate of change decreases for $x<-5$ and increases for $x>-5$ , what must be true?

Find the limit: $\lim \frac{-\left(4^{3}-1\right)^{2}+4-(-3)^{2}+4}{x-3}$ as $x$ approaches 3.

Berechnen Sie den Grenzwert $\lim _{x \rightarrow 2} \frac{-(3-1)^{2}+4-(-3)^{2}+4}{x-3}$ für x > 2.

Find the limit: $\lim_{x \to 3} \frac{-(3-1)^{2}+4-(-3)^{2}+4}{x-3}$ .

Find the limit: $\lim _{x \rightarrow 2} \frac{-(2-1)^{2}+4-(-2)^{2}+4}{x-2}$ .

Gegeben sind $f(x)=0,5 x^{2}+2$ und $g(x)=x^{2}-2 x+2$ . Finde Max/Min der Summe und Differenz der Werte in $[0; 4]$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=10 \sqrt[5]{x^{4}}-\frac{3}{a^{9}}+2$ .

Find $f^{\prime}$ and $f^{\prime \prime}$ for $f(x)=g\left(x^{3}\right)+e^{g(3 x)}$ . Also, find $f^{\prime \prime}$ for $f(x)=e^{x^{2}}+5^{x}$ .

Find the derivative of $f(x) = (5-x)(3+x)$ .

Differentiate $\sqrt{2 x^{2}-3 y}=x^{3} y^{2}+4$ with respect to $x$ using implicit differentiation.

Find a formula for the sequence $S_{t}=\frac{2}{3}+\left(\frac{2}{3}\right)^{2}+\cdots+\left(\frac{2}{3}\right)^{t}$ and its limit as $t$ approaches infinity.

Estimate the population of country $X$ in 25 years, given $N(t)=500 e^{0.02 t}$ .

Differentiate $x^{2} \cos (3 y) - y^{2} \sin (x^{2}) = 5$ with respect to $x$ and $y$ using implicit differentiation.

Differentiate $y=\frac{\left(2 x^{2}-3\right)^{\frac{3}{5}}}{\left(4 x^{2}-3 x+1\right)^{\frac{4}{7}}(6 x+2)^{\frac{1}{2}}}$ using logarithmic differentiation.

A tennis ball is thrown up at $22.5 \mathrm{~m} / \mathrm{s}$ . What is its maximum height? $T=42.6$

A softball is hit upwards with a velocity of $26 \mathrm{~m/s}$ . Find its velocity, position, max height time, and ground strike time.

Klimaanlage Problem: Gegeben ist $T(t)=\frac{200}{4+\frac{6}{0,1 t+1}}$ . Finde: a) Anfangstemperatur, b) Temperatur nach 60 Minuten, c) Grenztemperatur.

Finde den Durchmesser $d$ und die Höhe $h$ einer Regentonne, um das Volumen bei $2 m^{2}$ Material zu maximieren.

An arctic goose dives from a $245 \mathrm{~m}$ cliff. How long to reach the ground and what speed does it land?

A tourist drops a rock from rest. Find its velocity and displacement at $4.0 \mathrm{~s}$ .

Find the second derivative of $f(x)=e^{2x}$ .

Cost function is $C(x)=920+3x-0.03x^2+0.0006x^3$ . Find marginal cost, $C'(100)$ , and compare with cost of 101st item.

Find the limit: $\lim _{k \rightarrow \infty}(k+1)^{\frac{1}{k}}$ .

A turkey cools from $85^{\circ} \mathrm{C}$ to $33^{\circ} \mathrm{C}$ over 150 mins. Find average rates for intervals and estimate the instantaneous rate after 1 hour.

Solve the initial value problem: $\frac{d y}{d t}=\frac{1}{2^{t}} \sec ^{2}\left(\frac{\pi}{2^{t}}\right)$ , with $y\left(\log _{2} 4\right)=2 / \pi$ .

Find the stationary points of $f(x)=x^{3}-9x^{2}+16x+8$ and determine their nature.

Create a value table for $f(x)=\frac{x-4}{\sqrt{x}-2}$ near $x=4$ and find $\lim _{x \rightarrow 4} f(x)$ .

Find the derivative of $f(x)=(12-x^{2}) \cdot \frac{1}{3x+3}$ .

Solve Legendre's equation for $-1<t<1$ : $(1-t^{2}) y'' - 2t y' + 2y = 0$ . Hint: $y=t$ is a solution.

A projectile is launched horizontally at $15 \mathrm{~m/s}$ from a 30 m cliff. Find its range.

Find the derivative dy/dx for the function $y=(6 x+2) \cos x$ .

Un peintre doit peindre la surface entre les courbes $x=9-y^{2}$ , $x=y-3$ et $x=-\frac{y}{2}-\frac{3}{2}$ . A-t-il assez de peinture pour $40 \, m^{2}$ ? Trouvez les intersections et calculez l'aire.

Find the instantaneous rate of change of $f(x)=\sqrt{x+2}$ at the left endpoint of the interval $[-1,2]$ .

Find the derivative of $f(x)=(4-2x)^{2} \cdot \frac{6}{x^{3}}$ .

Find $\frac{d y}{d t}$ when $y=x^{2}+4$ , $\frac{d x}{d t}=2$ , and $x=5$ . $\frac{d y}{d t}=\square$ .

Find the derivative of $y=\ln \left(\frac{\sqrt{4+x^{2}}}{x}\right)$ .

Identify which sequences diverge: i. $t_{n}=0.8^{n}$ ii. $t_{n}=\left(\frac{5}{2}\right)^{n}$ iii. $t_{n}=3 n-7$ iv. $t_{n}=\frac{1}{n}$ .

Find the derivatives of these functions: (a) $f(x)=\sin x \cos x$ , (b) $g(x)=\frac{1+\tan x}{\sin x}$ .

Determine which statements are true for $\lim _{x \rightarrow a} f(x)=L$ : I, II, III, IV.

Differentiate the function $f(x)=\frac{x^{6} e^{x}}{x^{6}+e^{x}}$ .

Quel est le profit total pour 10 unités, sachant que $P_{m}=\frac{80}{\sqrt{4 q+1}}$ et que le profit total pour 6 unités est 230 \$?

Find the derivative of $f(x)=3 x^{3} \cdot \sqrt{4 x^{2}-4 x+1}$ and simplify it.

Finde die drei Ableitungen von $h(t) = (t^{2}-t)(2t-1)$ .

Differentiate the function $F(x)=\frac{1}{3 x^{3}-8 x^{2}+7}$ .

Calculate the arc length of $x=6 y^{3/2}$ from $y=1$ to $y=4$ . Provide the exact answer or round to 3 decimal places.

Bearbeite die Funktion $f(t)=e^{3 t}$ : Finde die Ableitung, das Integral oder die allgemeine Lösung.

Compute $F^{\prime}(0)$ for $F(x)=(f \circ g)(x)$ given $f^{\prime}(x)=-3$ and $g^{\prime}(x)=7$ .

Differentiate the function: $J(u)=\left(\frac{3}{u}+\frac{3}{u^{2}}\right)\left(2 u+\frac{2}{u}\right)$ .

Find the derivative of $y=x^{-1/x}$ using logarithmic differentiation.

Estimate $\Delta y$ for $y=\sin(3x)$ at $x=0$ with $\Delta x=0.2$ , then find the percentage error rounded to 1 decimal place.

Raylin invests \ $15,000 at 3.5% interest compounded continuously. Find the balance after 30 years using$ A = Pe^{rt}$.

Find $g^{\prime}(1)$ where $f(x)=\sin x+2x+1$ and $g$ is the inverse of $f$ . Choices: (A) $\frac{1}{3}$ (B) 1 (C) 3 (D) $\frac{1}{2+\cos 1}$ (E) $2+\cos 1$ .

Verify the first and second derivatives of $f(x) = x^{2}e^{-x}$ and find its critical points.