Calculus

Problem 29201

Find the tangent line approximation for $f(-2.1)$ given $f(-2)=-3$ and $f'(x)=x-5$ .

See Solution

Problem 29202

Find the tangent line at $x=-3$ for $f$ where $f(-3)=4$ and $f'(x)=4x+4$ . Approximate $f(-3.05)$ .

See Solution

Problem 29203

Approximate $f(-2.95)$ using the tangent line at $x=-3$ where $f(-3)=-5$ and $f'(x)=-x+1$ .

See Solution

Problem 29204

Find the tangent line approximation for $f(-4.05)$ using $f(-4)=-1$ and $f'(x)=3x+3$ .

See Solution

Problem 29205

Find the tangent line approximation for $f(2.15)$ given $f(2)=1$ and $f^{\prime}(2)=-5$ .

See Solution

Problem 29206

Solve the differential equation: $\frac{d y}{d x}=e^{-y} x$ .

See Solution

Problem 29207

Find the tangent line at $x=-4$ for $f$ with $f(-4)=6$ and $f'(-4)=-10$ . Approximate $f(-3.8)$ .

See Solution

Problem 29208

Find the limit of the sequence $a_n = \frac{n^2+n}{2n^2+1}$ without using Theorem 3.4.3 or 3.4.6. Also, provide two divergent sequences $a_n$ and $b_n$ such that the product sequence $(ab)$ diverges.

See Solution

Problem 29209

Calculate the difference quotient for $f(x)=x^{2}+2x$ : $\frac{f(x+h)-f(x)}{h}$ .

See Solution

Problem 29210

Provide two divergent sequences $(a_{n})$ and $(b_{n})$ such that $(a_{n} b_{n})$ converges. Prove all claims.

See Solution

Problem 29211

Find $k$ if $\frac{\mathrm{d} y}{\mathrm{~d} x}=k \mathrm{e}^{2-x}+4 x$ and the curve's gradient is 1 at $(2,10)$ . Then find the curve's equation.

See Solution

Problem 29212

Find the limit: $\lim _{t \rightarrow 2} \frac{\sqrt{t^{2}+3 t-1}-3}{2 t^{2}-8}$ .

See Solution

Problem 29213

Find the limit as $x$ approaches -1 from the left of $\frac{x^{2}+6 x+11}{x^{2}+4 x+3}-\frac{3}{x+1}$ .

See Solution

Problem 29214

Find the limit: $\lim _{t \rightarrow-\infty} \frac{5-3 t}{\sqrt{4 t^{2}-1}-5 t}$ .

See Solution

Problem 29215

Find the maximum IRC of the function $f(x)=3(1.125)^{x}$ for $2 \leq x \leq 9$ .

See Solution

Problem 29216

Find the IRC of the antibiotic concentration $C(t)=\frac{3t}{t^{2}+5}$ at $t=8$ hours.

See Solution

Problem 29217

Find the difference quotient for the curve $f(x)=2 x^{2}+6 x+12$ to estimate the slope of the tangent line at $(x, f(x))$ .

See Solution

Problem 29218

Find the limit $p$ of the sequence $b_{n}=\sqrt[3]{n^{3}+4 n^{2}}-n$ and prove convergence to $p$ .

See Solution

Problem 29219

Find the limits as $x$ approaches $\infty$ and $-\infty$ for $\frac{2x^5 - 3x^2 + 5}{3x^4 + 5x - 4}$ .

See Solution

Problem 29220

Prove that if $\left(x_{n}\right) \rightarrow 3$ , then $\left(\frac{5 x_{n}-1}{2}\right) \rightarrow 7$ .

See Solution

Problem 29221

Bestimme die Ableitung $f^{\prime}(2)$ und die Steigung bei $x_{0}=3$ für $f(x)=-2 x^{2}+7$ mit dem Differenzenquotienten.

See Solution

Problem 29222

Find the particular solution to $\frac{d y}{d x}=\frac{4(1-y)^{2}}{(1-2 x)^{2}}$ given $y(1)=2.$

See Solution

Problem 29223

Find the maximum Instantaneous Rate of Change (IRC) of $f(x)=3(1.125)^{x}$ for $2 \leq x \leq 9$ .

See Solution

Problem 29224

Find the limit: $\lim_{x \to 0^{+}} x^{-5 / \ln x}$ .

See Solution

Problem 29225

Find a simplified expression for the slope of the tangent line at any point $(x, f(x))$ for $f(x)=2 x^{2}+6 x+12$ .

See Solution

Problem 29226

Find the slope of the tangent line for $f(x) = 2x^2 + 6x + 12$ using the difference quotient, then determine the tangent line at $(-3,12)$ .

See Solution

Problem 29227

Bestimmen Sie die Tangentengleichung für $f(x)=4-\frac{1}{2} x^{2}$ bei $x=-2$ . Berechnen Sie auch die Tangenten für die Funktionen in a), b) und c).

See Solution

Problem 29228

Given the function $h(t)=-20 \cos \left[\frac{\pi}{12} t\right]+24$ , find: a) points where IRC is zero, b) ARC from min to max, c) least IRC.

See Solution

Problem 29229

Bestimme die Tangente für $f$ an $x$ : a) $f(x)=3 x^{2}+2$ , $x=1$ ; b) $f(x)=2 x^{2}+4 x-1$ , $x=-3$ ; c) $f(x)=\frac{1}{9} x^{3}-x^{2}$ , $x=3$ .
Ein Zoowärter wird von einer Giftschlange gebissen. Die Konzentration $c(t)=1,5 t^{3}-60 t^{2}+600 t$ beschreibt die Giftkonzentration. Berechne, wann das Gift nach $t=12$ Stunden vollständig abgebaut ist.

See Solution

Problem 29230

Define the limits as $x$ approaches 2 for $f(x)=\left\{\begin{array}{l}2+3 x \text { if } x \neq 2 \\ 2 \text { if } x=2\end{array}\right.$ and 1 for $g(x)=\frac{2}{x}+1$ . Check continuity at those points.

See Solution

Problem 29231

Bestimme den Schnittpunkt der Tangente an $f$ bei $B(1 \mid f(1))$ mit der $x$ -Achse für die Funktionen: a) $f(x)=-\frac{1}{2} x^{3}+\frac{1}{2} x+2$ b) $f(x)=4 \sqrt{x}$

See Solution

Problem 29232

Zeigen Sie, dass die Ableitung $f^{\prime}(x)=\chi \cdot(x+2) \cdot e^{x}$ für $f(x)=x^{2} \cdot \mathrm{e}^{x}$ gilt.

See Solution

Problem 29233

Bestimme die Extremstellen und deren Art für die Funktion $f(x)=x^{2} \cdot e^{x}, x \in \mathbb{R}$ .

See Solution

Problem 29234

Find the derivative of $y$ where $y = (2x - 4)^{98}$ .

See Solution

Problem 29235

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{x^{3}+9}{x^{2}+x}$ .

See Solution

Problem 29236

Find the value of the function $f(x)=\frac{\sin (2 x)}{5 x}$ at $x=0$ , i.e., compute $f(0)$ .

See Solution

Problem 29237

Show that $f(x)=9 x^{4}-2 x^{2}+5 x-1$ has a zero in $[0,1]$ using the Intermediate Value Theorem. Approximate it to 2 decimal places.

See Solution

Problem 29238

Show that $f(x)=9 x^{4}-2 x^{2}+5 x-1$ has a zero in $[0,1]$ using the Intermediate Value Theorem. Approximate the zero to two decimal places.

See Solution

Problem 29239

Mathew Bernard plc produces turkeys with price $p=-2q^2+40q+200$ . Find the output level for max profit and check if MR=MC. What’s the selling price?

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Problem 29240

Given the values of $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ , determine the truth of these statements: I. No extrema at $x=1$ . II. Maximum at $x=1$ . III. Maximum at $x=4$ .

See Solution

Problem 29241

If your salary is \ $41,600 and grows at$ 2\%$ annually, what will it be after 40 years with continuous compounding?

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Problem 29242

Given values of $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ , determine the truth of these statements about $f$ : I. No min/max at $x=2$ II. Max at $x=2$ III. Max at $x=8$

See Solution

Problem 29243

Which condition ensures $f$ has a relative minimum at $x=c$ if $f$ is twice-differentiable? Options: (A) $f'(c)=0$ (B) $f'(c)=0$ and $f'(c)<0$ (C) $f'(c)=0$ and $f'(c)>0$ (D) $f'(x)>0$ for $x<c$ and $f'(x)<0$ for $x>c$ .

See Solution

Problem 29244

Given that $g^{\prime}(2)=0$ and $g^{\prime \prime}(2)<0$ , what can we conclude about $g$ at $x=2$ ?

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Problem 29245

Given the function $f(x)=\frac{\sin (2 x)}{5 x}$ , predict $f(0)$ using the graph and table data.

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Problem 29246

Untersuche das Verhalten der Funktionen für $x \rightarrow+\infty$ und $x \rightarrow-\infty$ bezüglich Konvergenz und Divergenz.

See Solution

Problem 29247

Find a limit that equals the integral $\int_{0}^{4}(x^{2}+1) \, dx$ .

See Solution

Problem 29248

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{x}-1}{3 x}$ .

See Solution

Problem 29249

Find the limit: $\lim _{t \rightarrow 7} \frac{\frac{1}{t}-\frac{1}{7}}{t-7}$ .

See Solution

Problem 29250

Find the average rate of change of the function $f(x)=e^{x} \cos x$ on the interval $0 \leq x \leq \pi$ .

See Solution

Problem 29251

Berechnen Sie die Ableitung $f^{\prime}(x)$ für die Funktion $f(x)=x^{3}+x^{2}+x+6$ .

See Solution

Problem 29252

Finde die erste Ableitung von $f(x) = 3 - 2x$ .

See Solution

Problem 29253

Find the limit: $\lim _{x \rightarrow 0} \frac{1-e^{x}}{x^{2}-x}$ .

See Solution

Problem 29254

Finde $k$ , sodass $\int_{0}^{k}(x^{2}+1) \, \mathrm{d} x=\frac{4}{3}$ .

See Solution

Problem 29255

Find the limits: 1) $\lim _{x \rightarrow+\infty} x+1+\frac{7}{x+1}+\frac{-3}{(x+1)^{2}}$ and y) $\lim _{x \rightarrow-\infty} x+1+\frac{7}{x+1}+\frac{-3}{(x+1)^{2}}$ .

See Solution

Problem 29256

Étudiez l'illénionbilité de $f(x) = x + 1 + \frac{7}{x+1} - \frac{3}{(x+1)^{2}}$ et le signe de sa dérivée.

See Solution

Problem 29257

Berechne die Integrale: a) $\int_{2}^{5} x d x$ , b) $\int_{-1}^{1}(2 x+1) d x$ , c) $\int_{-1}^{2}-2 t d t$ , d) $\int_{0}^{4}-2 d x$ .

See Solution

Problem 29258

Analysez la dérivabilité de $f(x)$ et déterminez le signe de sa dérivée pour $f(x)=x+1+\frac{7}{x^{2}+1}+\frac{-3}{(x+1)^{2}}$ .

See Solution

Problem 29259

Autofahrt: 1) Berechne die Distanz zwischen den Ampeln für $s(t)=-\frac{1}{64} t^3 + \frac{15}{16} t^2$ bei $t=40$ . 2) Bestimme die mittlere Geschwindigkeit. 3) Finde die momentane Geschwindigkeit $s'(t)$ . 4) Berechne $s'(15)$ in km/h. 5) Zeige, dass das Auto nach 40s stoppt. 6) Erstelle die Beschleunigungsfunktion $s''(t)$ .

See Solution

Problem 29260

Aufgabe: Autofahrt zwischen Ampeln.
1) Bestimme die Entfernung der Ampeln bei 40 Sekunden Fahrt. 2) Berechne die mittlere Geschwindigkeit in diesen 40 Sekunden. 3) Finde die Funktion für die momentane Geschwindigkeit. 4) Berechne die Geschwindigkeit bei $t=15 \mathrm{~s}$ in $\mathrm{km/h}$ . 5) Zeige, dass das Auto nach 40 Sekunden stoppt. 6) Erstelle die Formel für die Beschleunigung des Autos.

See Solution

Problem 29261

Étudiez la dérivabilité de $f(x)=x+1+\frac{7}{x+1}-\frac{3}{(x+1)^{2}}$ sur $]-\infty;-1[ \cup ]-1;+\infty[$ .

See Solution

Problem 29262

A ball is dropped from a bridge. After $1.55 \mathrm{~s}$ , how far has it fallen? Find $\Delta y=[?] \mathrm{m}$ .

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Problem 29263

Find the secant slope function, tangent slope function, slope at point Q(0,-1), and tangent line equation for $f(x)=2 x^{2}-3 x-1$ .

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Problem 29264

Bestimme die steilste Stelle des Hangs für $f(x)=-\frac{1}{400} x^{3}+\frac{3}{25} x^{2}-\frac{6}{5} x+\frac{18}{5}$ und den Zeitpunkt des stärksten Temperaturanstiegs für $f(t)=-\frac{1}{72} t^{3}+\frac{5}{12} t^{2}-\frac{8}{3} t+\frac{188}{9}$ .

See Solution

Problem 29265

Bestimme den Wendepunkt und die Wendetangente der Funktion $f(x)=x^{3}+3 x^{2}-4$ . Berechne die Länge von $\overline{PQ}$ und den Flächeninhalt des Dreiecks.

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Problem 29266

Étudiez la limite de $f(x)$ quand $x$ tend vers $-\infty$ et $+\infty$ . Analysez la dérivabilité et le signe de $f'(x)$ sur $]-\infty:-3[u]-2:+\infty[$ avec $f(x)=\frac{1}{2} x^{2}-2+\frac{-4}{x+3}$ .

See Solution

Problem 29267

Calculez les limites de la fonction $f(x)=\frac{1}{2} x^{2}-2+\frac{-4}{x+3}$ quand $x$ tend vers $-\infty$ et $+\infty$ .

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Problem 29268

Déterminez si $f(x)$ est dérivable et le signe de sa dérivée sur $]-\infty, -3[$ et $]-3, +\infty[$ avec $f(x) = \frac{1}{2} x^{2} - 2 - \frac{4}{x+3}$ .

See Solution

Problem 29269

Find the tripling time for the deer population modeled by $P(t)=(267) 3^{\frac{t}{4}}$ .

See Solution

Problem 29270

Find critical points and use the second derivative test for: a. $y=x^{3}-6 x^{2}-15 x+10$ , b. $y=\frac{25}{x^{2}+48}$ , c. $s=t+t^{-1}$ , d. $y=(x-3)^{3}+8$ .

See Solution

Problem 29271

Find the drug concentration $C(6)$ after 6 minutes using $C(t)=0.05(1-e^{-0.2 t})$ . Round to three decimal places.

See Solution

Problem 29272

Find the midpoint Riemann sum for $\int_{3}^{5} 3 \ln (x) \, dx$ using 4 equal subintervals. Round to the nearest thousandth.

See Solution

Problem 29273

Find the midpoint Riemann sum for $\int_{1}^{7} 6 \ln (x) \, dx$ using 3 equal subintervals. Round to the nearest thousandth.

See Solution

Problem 29274

Find the right Riemann sum for $f(x)=\frac{6}{x}$ over $[2, 3.5]$ using 6 equal subintervals. Round to the nearest thousandth.

See Solution

Problem 29275

Find the left Riemann sum for $\int_{0}^{6} x^{2} dx$ using 3 equal subintervals. Round to the nearest thousandth.

See Solution

Problem 29276

Find the first and second derivatives of $y=\frac{25}{x^{2}+48}$ , given that a critical number is 7.

See Solution

Problem 29277

Find the right Riemann sum for $\int_{3}^{9} 5 \sqrt{x} dx$ using 6 equal subintervals. Round to the nearest thousandth.

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Problem 29278

Aufgabe 1: Finde die Stammfunktion und die Ableitung für $f(x)=\frac{7}{x^{3}}$ und $g(x)=\sqrt{16x}$ .
Aufgabe 2: Bestimme den größten Abstand zwischen $f(x)=(3-x)e^{x}$ und $g(x)=-e^{x}$ sowie die Fläche mit der $y$ -Achse.

See Solution

Problem 29279

Find the trapezoidal sum for $\int_{1}^{2} x^{5} dx$ using 4 equal subintervals. Round to the nearest thousandth.

See Solution

Problem 29280

Find the derivative of the function $y=-\frac{50 x}{(x^{2}+48)^{2}}$ .

See Solution

Problem 29281

Find the right Riemann sum for $f(x)=\frac{5}{x}$ over $[3,5]$ using 5 equal subintervals. Round to the nearest thousandth.

See Solution

Problem 29282

Find the midpoint Riemann sum for $f(x)=2 \sqrt{x}$ over $[1,3]$ using 4 equal subintervals. Round to three decimals.

See Solution

Problem 29283

Find the critical number of the function $y=\frac{25}{x^{2}+48}$ .

See Solution

Problem 29284

Find the right Riemann sum for $f(x)=x^{2}$ over $[3, 4.5]$ using 3 equal subintervals. Round to the nearest thousandth.

See Solution

Problem 29285

Calculate the average rate of change of $h(x)=5 x^{2}-2$ over the interval $[0,5]$ .

See Solution

Problem 29286

Find the average rate of change of $g(x)=2 x^{3}$ on the interval $[-1,2]$ .

See Solution

Problem 29287

Find the limit: $\lim _{x \rightarrow 18} \frac{\sqrt{x-2}-4}{18 x-x^{2}}$

See Solution

Problem 29288

Find the limit: $\lim _{x \rightarrow 0^{+}}(1+\sinh 2 x)^{\operatorname{coth} x}$ .

See Solution

Problem 29289

Calculate the average rate of change of $f(x)=4x-5$ over the interval $[x, x+h]$ .

See Solution

Problem 29290

Find the average rate of change of $h(x)=-x^{2}+5x+8$ from $x=0$ to $x=7$ .

See Solution

Problem 29291

Find the average rate of change of $f(x)=x^{2}+10 x+22$ on the interval $-9 \leq x \leq -2$ .

See Solution

Problem 29292

Find the average rate of change of $f(x)$ from $x=-3$ to $x=-2$ using points $(-3,-1)$ and $(-2,1)$ .

See Solution

Problem 29293

Calculate the limit: $\lim _{x \rightarrow+\infty}\left(e^{-x}+1\right)^{e^{x}}$ .

See Solution

Problem 29294

Find the limit: $\lim _{x \rightarrow+\infty}\left(e^{-x}+1\right)^{e^{x}}$ .

See Solution

Problem 29295

Find the derivative $f'(0)$ where $f(x) = \int_{x^{2}+x}^{0} \cos^{-1}(t^{2}) dt$ .

See Solution

Problem 29296

Find critical points and use the second derivative test for: a. $y=x^{3}-6 x^{2}-15 x+10$ , b. $y=\frac{25}{x^{2}+48}$ , c. $s=t+t^{-1}$ , d. $y=(x-3)^{3}+8$ . Also, find points of inflection and test the second derivative's sign change.

See Solution

Problem 29297

Evaluate the integral: $\int_{1}^{e^{2}} \frac{x^{-1}}{3^{\ln x}} d x$

See Solution

Problem 29298

Find the integral of $(10 x - 18)^{13}$ and identify $u$ .
$u =$

See Solution

Problem 29299

A quantity starts at 180 and decays at $70\%$ per minute. Find its value after 333 seconds, rounded to the nearest hundredth.

See Solution

Problem 29300

A quantity starts at 5500 and grows at $0.95\%$ daily. Find its value after 6 weeks, rounded to the nearest hundredth.

See Solution

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Calculus

Problem 29201

Find the tangent line approximation for f(−2.1)f(-2.1)f(−2.1) given f(−2)=−3f(-2)=-3f(−2)=−3 and f′(x)=x−5f'(x)=x-5f′(x)=x−5.

Problem 29202

Find the tangent line at x=−3x=-3x=−3 for fff where f(−3)=4f(-3)=4f(−3)=4 and f′(x)=4x+4f'(x)=4x+4f′(x)=4x+4. Approximate f(−3.05)f(-3.05)f(−3.05).

Problem 29203

Approximate f(−2.95)f(-2.95)f(−2.95) using the tangent line at x=−3x=-3x=−3 where f(−3)=−5f(-3)=-5f(−3)=−5 and f′(x)=−x+1f'(x)=-x+1f′(x)=−x+1.

Problem 29204

Find the tangent line approximation for f(−4.05)f(-4.05)f(−4.05) using f(−4)=−1f(-4)=-1f(−4)=−1 and f′(x)=3x+3f'(x)=3x+3f′(x)=3x+3.

Problem 29205

Find the tangent line approximation for f(2.15)f(2.15)f(2.15) given f(2)=1f(2)=1f(2)=1 and f′(2)=−5f^{\prime}(2)=-5f′(2)=−5.

Problem 29206

Solve the differential equation: dydx=e−yx\frac{d y}{d x}=e^{-y} xdxdy​=e−yx.

Problem 29207

Find the tangent line at x=−4x=-4x=−4 for fff with f(−4)=6f(-4)=6f(−4)=6 and f′(−4)=−10f'(-4)=-10f′(−4)=−10. Approximate f(−3.8)f(-3.8)f(−3.8).

Problem 29208

Find the limit of the sequence an=n2+n2n2+1a_n = \frac{n^2+n}{2n^2+1}an​=2n2+1n2+n​ without using Theorem 3.4.3 or 3.4.6. Also, provide two divergent sequences ana_nan​ and bnb_nbn​ such that the product sequence (ab)(ab)(ab) diverges.

Problem 29209

Calculate the difference quotient for f(x)=x2+2xf(x)=x^{2}+2xf(x)=x2+2x: f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​.

Problem 29210

Provide two divergent sequences (an)(a_{n})(an​) and (bn)(b_{n})(bn​) such that (anbn)(a_{n} b_{n})(an​bn​) converges. Prove all claims.

Problem 29211

Find kkk if dy dx=ke2−x+4x\frac{\mathrm{d} y}{\mathrm{~d} x}=k \mathrm{e}^{2-x}+4 x dxdy​=ke2−x+4x and the curve's gradient is 1 at (2,10)(2,10)(2,10). Then find the curve's equation.

Problem 29212

Find the limit: lim⁡t→2t2+3t−1−32t2−8\lim _{t \rightarrow 2} \frac{\sqrt{t^{2}+3 t-1}-3}{2 t^{2}-8}limt→2​2t2−8t2+3t−1​−3​.

Problem 29213

Find the limit as xxx approaches -1 from the left of x2+6x+11x2+4x+3−3x+1\frac{x^{2}+6 x+11}{x^{2}+4 x+3}-\frac{3}{x+1}x2+4x+3x2+6x+11​−x+13​.

Problem 29214

Find the limit: lim⁡t→−∞5−3t4t2−1−5t\lim _{t \rightarrow-\infty} \frac{5-3 t}{\sqrt{4 t^{2}-1}-5 t}limt→−∞​4t2−1​−5t5−3t​.

Problem 29215

Find the maximum IRC of the function f(x)=3(1.125)xf(x)=3(1.125)^{x}f(x)=3(1.125)x for 2≤x≤92 \leq x \leq 92≤x≤9.

Problem 29216

Find the IRC of the antibiotic concentration C(t)=3tt2+5C(t)=\frac{3t}{t^{2}+5}C(t)=t2+53t​ at t=8t=8t=8 hours.

Problem 29217

Find the difference quotient for the curve f(x)=2x2+6x+12f(x)=2 x^{2}+6 x+12f(x)=2x2+6x+12 to estimate the slope of the tangent line at (x,f(x))(x, f(x))(x,f(x)).

Problem 29218

Find the limit ppp of the sequence bn=n3+4n23−nb_{n}=\sqrt[3]{n^{3}+4 n^{2}}-nbn​=3n3+4n2​−n and prove convergence to ppp.

Problem 29219

Find the limits as xxx approaches ∞\infty∞ and −∞-\infty−∞ for 2x5−3x2+53x4+5x−4\frac{2x^5 - 3x^2 + 5}{3x^4 + 5x - 4}3x4+5x−42x5−3x2+5​.

Problem 29220

Prove that if (xn)→3\left(x_{n}\right) \rightarrow 3(xn​)→3, then (5xn−12)→7\left(\frac{5 x_{n}-1}{2}\right) \rightarrow 7(25xn​−1​)→7.

Problem 29221

Bestimme die Ableitung f′(2)f^{\prime}(2)f′(2) und die Steigung bei x0=3x_{0}=3x0​=3 für f(x)=−2x2+7f(x)=-2 x^{2}+7f(x)=−2x2+7 mit dem Differenzenquotienten.

Problem 29222

Find the particular solution to dydx=4(1−y)2(1−2x)2\frac{d y}{d x}=\frac{4(1-y)^{2}}{(1-2 x)^{2}}dxdy​=(1−2x)24(1−y)2​ given y(1)=2.y(1)=2.y(1)=2.

Problem 29223

Find the maximum Instantaneous Rate of Change (IRC) of f(x)=3(1.125)xf(x)=3(1.125)^{x}f(x)=3(1.125)x for 2≤x≤92 \leq x \leq 92≤x≤9.

Problem 29224

Find the limit: lim⁡x→0+x−5/ln⁡x\lim_{x \to 0^{+}} x^{-5 / \ln x}limx→0+​x−5/lnx.

Problem 29225

Find a simplified expression for the slope of the tangent line at any point (x,f(x))(x, f(x))(x,f(x)) for f(x)=2x2+6x+12f(x)=2 x^{2}+6 x+12f(x)=2x2+6x+12.

Problem 29226

Find the slope of the tangent line for f(x)=2x2+6x+12f(x) = 2x^2 + 6x + 12f(x)=2x2+6x+12 using the difference quotient, then determine the tangent line at (−3,12)(-3,12)(−3,12).

Problem 29227

Bestimmen Sie die Tangentengleichung für f(x)=4−12x2f(x)=4-\frac{1}{2} x^{2}f(x)=4−21​x2 bei x=−2x=-2x=−2. Berechnen Sie auch die Tangenten für die Funktionen in a), b) und c).

Problem 29228

Given the function h(t)=−20cos⁡[π12t]+24h(t)=-20 \cos \left[\frac{\pi}{12} t\right]+24h(t)=−20cos[12π​t]+24, find: a) points where IRC is zero, b) ARC from min to max, c) least IRC.

Problem 29229

Problem 29230

Define the limits as xxx approaches 2 for f(x)={2+3x if x≠22 if x=2f(x)=\left\{\begin{array}{l}2+3 x \text { if } x \neq 2 \\ 2 \text { if } x=2\end{array}\right.f(x)={2+3x if x=22 if x=2​ and 1 for g(x)=2x+1g(x)=\frac{2}{x}+1g(x)=x2​+1. Check continuity at those points.

Problem 29231

Bestimme den Schnittpunkt der Tangente an fff bei B(1∣f(1))B(1 \mid f(1))B(1∣f(1)) mit der xxx-Achse für die Funktionen: a) f(x)=−12x3+12x+2f(x)=-\frac{1}{2} x^{3}+\frac{1}{2} x+2f(x)=−21​x3+21​x+2 b) f(x)=4xf(x)=4 \sqrt{x}f(x)=4x​

Problem 29232

Zeigen Sie, dass die Ableitung f′(x)=χ⋅(x+2)⋅exf^{\prime}(x)=\chi \cdot(x+2) \cdot e^{x}f′(x)=χ⋅(x+2)⋅ex für f(x)=x2⋅exf(x)=x^{2} \cdot \mathrm{e}^{x}f(x)=x2⋅ex gilt.

Problem 29233

Bestimme die Extremstellen und deren Art für die Funktion f(x)=x2⋅ex,x∈Rf(x)=x^{2} \cdot e^{x}, x \in \mathbb{R}f(x)=x2⋅ex,x∈R.

Problem 29234

Find the derivative of yyy where y=(2x−4)98y = (2x - 4)^{98}y=(2x−4)98.

Problem 29235

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=x3+9x2+xy=\frac{x^{3}+9}{x^{2}+x}y=x2+xx3+9​.

Problem 29236

Find the value of the function f(x)=sin⁡(2x)5xf(x)=\frac{\sin (2 x)}{5 x}f(x)=5xsin(2x)​ at x=0x=0x=0, i.e., compute f(0)f(0)f(0).

Problem 29237

Show that f(x)=9x4−2x2+5x−1f(x)=9 x^{4}-2 x^{2}+5 x-1f(x)=9x4−2x2+5x−1 has a zero in [0,1][0,1][0,1] using the Intermediate Value Theorem. Approximate it to 2 decimal places.

Problem 29238

Show that f(x)=9x4−2x2+5x−1f(x)=9 x^{4}-2 x^{2}+5 x-1f(x)=9x4−2x2+5x−1 has a zero in [0,1][0,1][0,1] using the Intermediate Value Theorem. Approximate the zero to two decimal places.

Problem 29239

Mathew Bernard plc produces turkeys with price p=−2q2+40q+200p=-2q^2+40q+200p=−2q2+40q+200. Find the output level for max profit and check if MR=MC. What’s the selling price?

Problem 29240

Given the values of f′(x)f^{\prime}(x)f′(x) and f′′(x)f^{\prime \prime}(x)f′′(x), determine the truth of these statements: I. No extrema at x=1x=1x=1. II. Maximum at x=1x=1x=1. III. Maximum at x=4x=4x=4.

Find the tangent line approximation for $f(-2.1)$ given $f(-2)=-3$ and $f'(x)=x-5$ .

Find the tangent line at $x=-3$ for $f$ where $f(-3)=4$ and $f'(x)=4x+4$ . Approximate $f(-3.05)$ .

Approximate $f(-2.95)$ using the tangent line at $x=-3$ where $f(-3)=-5$ and $f'(x)=-x+1$ .

Find the tangent line approximation for $f(-4.05)$ using $f(-4)=-1$ and $f'(x)=3x+3$ .

Find the tangent line approximation for $f(2.15)$ given $f(2)=1$ and $f^{\prime}(2)=-5$ .

Solve the differential equation: $\frac{d y}{d x}=e^{-y} x$ .

Find the tangent line at $x=-4$ for $f$ with $f(-4)=6$ and $f'(-4)=-10$ . Approximate $f(-3.8)$ .

Find the limit of the sequence $a_n = \frac{n^2+n}{2n^2+1}$ without using Theorem 3.4.3 or 3.4.6. Also, provide two divergent sequences $a_n$ and $b_n$ such that the product sequence $(ab)$ diverges.

Calculate the difference quotient for $f(x)=x^{2}+2x$ : $\frac{f(x+h)-f(x)}{h}$ .

Provide two divergent sequences $(a_{n})$ and $(b_{n})$ such that $(a_{n} b_{n})$ converges. Prove all claims.

Find $k$ if $\frac{\mathrm{d} y}{\mathrm{~d} x}=k \mathrm{e}^{2-x}+4 x$ and the curve's gradient is 1 at $(2,10)$ . Then find the curve's equation.

Find the limit: $\lim _{t \rightarrow 2} \frac{\sqrt{t^{2}+3 t-1}-3}{2 t^{2}-8}$ .

Find the limit as $x$ approaches -1 from the left of $\frac{x^{2}+6 x+11}{x^{2}+4 x+3}-\frac{3}{x+1}$ .

Find the limit: $\lim _{t \rightarrow-\infty} \frac{5-3 t}{\sqrt{4 t^{2}-1}-5 t}$ .

Find the maximum IRC of the function $f(x)=3(1.125)^{x}$ for $2 \leq x \leq 9$ .

Find the IRC of the antibiotic concentration $C(t)=\frac{3t}{t^{2}+5}$ at $t=8$ hours.

Find the difference quotient for the curve $f(x)=2 x^{2}+6 x+12$ to estimate the slope of the tangent line at $(x, f(x))$ .

Find the limit $p$ of the sequence $b_{n}=\sqrt[3]{n^{3}+4 n^{2}}-n$ and prove convergence to $p$ .

Find the limits as $x$ approaches $\infty$ and $-\infty$ for $\frac{2x^5 - 3x^2 + 5}{3x^4 + 5x - 4}$ .

Prove that if $\left(x_{n}\right) \rightarrow 3$ , then $\left(\frac{5 x_{n}-1}{2}\right) \rightarrow 7$ .

Bestimme die Ableitung $f^{\prime}(2)$ und die Steigung bei $x_{0}=3$ für $f(x)=-2 x^{2}+7$ mit dem Differenzenquotienten.

Find the particular solution to $\frac{d y}{d x}=\frac{4(1-y)^{2}}{(1-2 x)^{2}}$ given $y(1)=2.$

Find the maximum Instantaneous Rate of Change (IRC) of $f(x)=3(1.125)^{x}$ for $2 \leq x \leq 9$ .

Find the limit: $\lim_{x \to 0^{+}} x^{-5 / \ln x}$ .

Find a simplified expression for the slope of the tangent line at any point $(x, f(x))$ for $f(x)=2 x^{2}+6 x+12$ .

Find the slope of the tangent line for $f(x) = 2x^2 + 6x + 12$ using the difference quotient, then determine the tangent line at $(-3,12)$ .

Bestimmen Sie die Tangentengleichung für $f(x)=4-\frac{1}{2} x^{2}$ bei $x=-2$ . Berechnen Sie auch die Tangenten für die Funktionen in a), b) und c).

Given the function $h(t)=-20 \cos \left[\frac{\pi}{12} t\right]+24$ , find: a) points where IRC is zero, b) ARC from min to max, c) least IRC.

Define the limits as $x$ approaches 2 for $f(x)=\left\{\begin{array}{l}2+3 x \text { if } x \neq 2 \\ 2 \text { if } x=2\end{array}\right.$ and 1 for $g(x)=\frac{2}{x}+1$ . Check continuity at those points.

Bestimme den Schnittpunkt der Tangente an $f$ bei $B(1 \mid f(1))$ mit der $x$ -Achse für die Funktionen: a) $f(x)=-\frac{1}{2} x^{3}+\frac{1}{2} x+2$ b) $f(x)=4 \sqrt{x}$

Zeigen Sie, dass die Ableitung $f^{\prime}(x)=\chi \cdot(x+2) \cdot e^{x}$ für $f(x)=x^{2} \cdot \mathrm{e}^{x}$ gilt.

Bestimme die Extremstellen und deren Art für die Funktion $f(x)=x^{2} \cdot e^{x}, x \in \mathbb{R}$ .

Find the derivative of $y$ where $y = (2x - 4)^{98}$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{x^{3}+9}{x^{2}+x}$ .

Find the value of the function $f(x)=\frac{\sin (2 x)}{5 x}$ at $x=0$ , i.e., compute $f(0)$ .

Show that $f(x)=9 x^{4}-2 x^{2}+5 x-1$ has a zero in $[0,1]$ using the Intermediate Value Theorem. Approximate it to 2 decimal places.

Show that $f(x)=9 x^{4}-2 x^{2}+5 x-1$ has a zero in $[0,1]$ using the Intermediate Value Theorem. Approximate the zero to two decimal places.

Mathew Bernard plc produces turkeys with price $p=-2q^2+40q+200$ . Find the output level for max profit and check if MR=MC. What’s the selling price?

Given the values of $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ , determine the truth of these statements: I. No extrema at $x=1$ . II. Maximum at $x=1$ . III. Maximum at $x=4$ .

If your salary is \ $41,600 and grows at$ 2\%$ annually, what will it be after 40 years with continuous compounding?

Given values of $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ , determine the truth of these statements about $f$ : I. No min/max at $x=2$ II. Max at $x=2$ III. Max at $x=8$

Given that $g^{\prime}(2)=0$ and $g^{\prime \prime}(2)<0$ , what can we conclude about $g$ at $x=2$ ?

Given the function $f(x)=\frac{\sin (2 x)}{5 x}$ , predict $f(0)$ using the graph and table data.

Untersuche das Verhalten der Funktionen für $x \rightarrow+\infty$ und $x \rightarrow-\infty$ bezüglich Konvergenz und Divergenz.

Find a limit that equals the integral $\int_{0}^{4}(x^{2}+1) \, dx$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{x}-1}{3 x}$ .

Find the limit: $\lim _{t \rightarrow 7} \frac{\frac{1}{t}-\frac{1}{7}}{t-7}$ .

Find the average rate of change of the function $f(x)=e^{x} \cos x$ on the interval $0 \leq x \leq \pi$ .

Berechnen Sie die Ableitung $f^{\prime}(x)$ für die Funktion $f(x)=x^{3}+x^{2}+x+6$ .

Finde die erste Ableitung von $f(x) = 3 - 2x$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{1-e^{x}}{x^{2}-x}$ .

Finde $k$ , sodass $\int_{0}^{k}(x^{2}+1) \, \mathrm{d} x=\frac{4}{3}$ .

Étudiez l'illénionbilité de $f(x) = x + 1 + \frac{7}{x+1} - \frac{3}{(x+1)^{2}}$ et le signe de sa dérivée.

Berechne die Integrale: a) $\int_{2}^{5} x d x$ , b) $\int_{-1}^{1}(2 x+1) d x$ , c) $\int_{-1}^{2}-2 t d t$ , d) $\int_{0}^{4}-2 d x$ .

Analysez la dérivabilité de $f(x)$ et déterminez le signe de sa dérivée pour $f(x)=x+1+\frac{7}{x^{2}+1}+\frac{-3}{(x+1)^{2}}$ .

Étudiez la dérivabilité de $f(x)=x+1+\frac{7}{x+1}-\frac{3}{(x+1)^{2}}$ sur $]-\infty;-1[ \cup ]-1;+\infty[$ .

A ball is dropped from a bridge. After $1.55 \mathrm{~s}$ , how far has it fallen? Find $\Delta y=[?] \mathrm{m}$ .

Find the secant slope function, tangent slope function, slope at point Q(0,-1), and tangent line equation for $f(x)=2 x^{2}-3 x-1$ .

Bestimme den Wendepunkt und die Wendetangente der Funktion $f(x)=x^{3}+3 x^{2}-4$ . Berechne die Länge von $\overline{PQ}$ und den Flächeninhalt des Dreiecks.

Calculez les limites de la fonction $f(x)=\frac{1}{2} x^{2}-2+\frac{-4}{x+3}$ quand $x$ tend vers $-\infty$ et $+\infty$ .

Déterminez si $f(x)$ est dérivable et le signe de sa dérivée sur $]-\infty, -3[$ et $]-3, +\infty[$ avec $f(x) = \frac{1}{2} x^{2} - 2 - \frac{4}{x+3}$ .

Find the tripling time for the deer population modeled by $P(t)=(267) 3^{\frac{t}{4}}$ .

Find critical points and use the second derivative test for: a. $y=x^{3}-6 x^{2}-15 x+10$ , b. $y=\frac{25}{x^{2}+48}$ , c. $s=t+t^{-1}$ , d. $y=(x-3)^{3}+8$ .

Find the drug concentration $C(6)$ after 6 minutes using $C(t)=0.05(1-e^{-0.2 t})$ . Round to three decimal places.

Find the midpoint Riemann sum for $\int_{3}^{5} 3 \ln (x) \, dx$ using 4 equal subintervals. Round to the nearest thousandth.

Find the midpoint Riemann sum for $\int_{1}^{7} 6 \ln (x) \, dx$ using 3 equal subintervals. Round to the nearest thousandth.

Find the right Riemann sum for $f(x)=\frac{6}{x}$ over $[2, 3.5]$ using 6 equal subintervals. Round to the nearest thousandth.