Calculus

Problem 22301

Evaluate the integral: $\int_{0}^{1} \sqrt{\left(\cos \left(t^{2}\right)\right)^{2}+\left(e^{-5 t}\right)^{2}} d t$

See Solution

Problem 22302

Given an even function $f:[-1,1] \rightarrow(0, \infty)$ with $\int_{-1}^{1} f(x) d x=1$ and $f(x)$ increasing on $[-1,0]$ :
(a) Prove $f(x)$ has a maximum at 0. (b) Define $G(r)=\int_{-r}^{r} f(x) d x$ and show $G(r)$ is differentiable on $[-1,1]$ , finding its derivative.

See Solution

Problem 22303

Find the third derivative of the function $g(x)=7 x^{2}-2 \sin x$ , i.e., calculate $g^{\prime \prime \prime}(x)$ .

See Solution

Problem 22304

Find the tangent line to $f(x)=12-x^{2}$ at the point where $x=2$ .

See Solution

Problem 22305

Find the derivative of the function $f(x)=20 \cdot 3^{x}$ . What is $f^{\prime}(x)=?$

See Solution

Problem 22306

Find the derivative of the function $f(r)=3 e^{7 r+6}$ . What is $\frac{d f}{d r}$ ?

See Solution

Problem 22307

Bestimme die Größe der abgeschnittenen Quadrate, um das Volumen einer offenen Schachtel aus 42 cm x 30 cm Pappe zu maximieren.

See Solution

Problem 22308

Find the derivative $\frac{d y}{d x}$ for the function $y=\sqrt[4]{12 e^{x}+8 x^{11}}$ .

See Solution

Problem 22309

Let $f_{0}$ be continuous. Define $f_{n}(x)=\int_{0}^{x} f_{n-1}(t) dt$ . Prove:
(a) $\int_{0}^{x}(x-t)^{m-1} f_{n}(t) dt=\frac{1}{m} \int_{0}^{x}(x-t)^{m} f_{n-1}(t) dt$ .
(b) $f_{n}(x)=\frac{1}{(n-1)!} \int_{0}^{x}(x-t)^{n-1} f_{0}(t) dt$ .

See Solution

Problem 22310

Find the average rate of change of $f(x)=\sin x$ from $x=0$ to $x=2\pi$ .

See Solution

Problem 22311

Berechne die Steigung von $f$ an $x_0$ mit der h-Methode für: a) $f(x)=x^2-x, x_0=1$ ; b) $f(x)=2x^2+1, x_0=-2$ ; c) $f(x)=3x+2, x_0=2$ .

See Solution

Problem 22312

Find the grams of carbon-14 left after 5867 years using the model $A=16 e^{-0.000121 t}$ . Round to the nearest whole number.

See Solution

Problem 22313

Ein Snowboarder bewegt sich nach der Formel $s(t)=1,5t^2$ . Berechne: a) Weg nach 1s und 5s, b) mittlere Geschwindigkeit in 5s, c) Momentangeschwindigkeit nach 5s.

See Solution

Problem 22314

For what values of $P$ is demand elastic for $Q(P) = 5000 e^{0.02 P}$ ?
Also, if price increases by $3\%$ from $40$ €, how does demand change?

See Solution

Problem 22315

Find the max and min of $f(x)=x^{3}-6 x^{2}+12$ on $[1,6]$ .

See Solution

Problem 22316

Finde die Koordinaten des Punktes, wo der Graph von $f(x)=x \cdot e^{2 x}$ eine waagerechte Tangente hat.

See Solution

Problem 22317

Find the derivative of $y=\ln(2 x^{2})$ with respect to $x$ , $t$ , or $\theta$ .

See Solution

Problem 22318

Untersuchen Sie die Wendepunkte der Funktionen: a) $f(x)=\frac{1}{8} x^{3}-\frac{3}{8} x^{2}$ b) $f(x)=\frac{1}{10} x^{5}-x^{2}$ .

See Solution

Problem 22319

Untersuchen Sie $f(x)=a^{2} x^{3}+2 a x^{2}$ auf Wendepunkte bei $x=2$ . Wie muss $a$ gewählt werden? Rechts-links?

See Solution

Problem 22320

Find the average rate of change of $h(x)=-x^{2}-x+5$ over the interval $-6 \leq x \leq 2$ .

See Solution

Problem 22321

Find $h(0) + h'(0)$ given the linear approximation $y = 4x - 5$ at $x = 0$ . Choose: (A) -3 (B) -2 (C) -1 (D) 0.

See Solution

Problem 22322

Find $\frac{d y}{d x}$ for $y=u^{3}+u^{2}-1$ with $u=\frac{1}{1-x}$ at $x=2$ . Show all steps.

See Solution

Problem 22323

Find the local linear approximation of $f(8.1)$ given $f(8)=-5$ and $f^{\prime}(8)=3$ . Choose the correct answer: (A) -4.8 (B) -4.7 (C) -4.6 (D) -4.5.

See Solution

Problem 22324

Sketch a function on $[-8,8]$ with these properties: a) $f'(x)>0$ on $(-1,2)$ and $(4,\infty)$ b) $f'(x)<0$ on $(-\infty,-1)$ and $(2,4)$ c) $f''(x)>0$ on $(-\infty,1)$ and $(3,\infty)$ d) $f''(x)<0$ on $(1,3)$ e) $f'(-1)=f'(2)=f'(4)=0$ f) $f''(x)=0$ at $(1,0)$ and $(3,1)$ .

See Solution

Problem 22325

Find the second partial derivative $\frac{\partial^{2}}{\partial x^{2}}\left(x^{3}+y^{4}+e^{x^{3}+2 y^{2}}\right)$ .

See Solution

Problem 22326

An Egyptian mummy has $\frac{2}{5}$ of carbon-14 compared to living humans. Use $y=y_{0} e^{-0.0001216 t}$ to find $t$ .

See Solution

Problem 22327

An airline limits vertical acceleration to $k=860 \mathrm{mi} / \mathrm{h}^{2}$ . At $35,000 \mathrm{ft}$ and $300 \mathrm{mi} / \mathrm{h}$ , how far from the airport should the pilot start descent? Graph the approach path.

See Solution

Problem 22328

Find the second mixed partial derivative $\frac{\partial^{2}}{\partial x \partial y} e^{x^{2}+3 x y+3 y^{3}}$ .

See Solution

Problem 22329

Find the expression for $\frac{f(x+h)-f(x)}{h}$ given $f(x)=e^{(x-2)}$ .

See Solution

Problem 22330

Find the average price of oil over 12 months given by $P(t)=\frac{1}{6} t^{2}-t+64$ .

See Solution

Problem 22331

Finde die Stammfunktion für $f$ und überprüfe durch Ableiten: a) $f(x)=(3 x+1)^{2}$ , b) $f(x)=\frac{3}{(2 x+7)^{2}}$ , c) $f(x)=\sqrt{6 x-5}$ , d) $f(x)=3 \sin (12 x+7)$ .

See Solution

Problem 22332

A fish population grows at $3\%$ per year, starting at 12 million in 2016.
a) Model the population $t$ years since 2016. b) Project the population in 2030 (to the nearest tenth) and find when it reaches 30 million (to the nearest whole number).

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Problem 22333

Für die Ameisenanzahl $f_{k}(t)=8-2 e^{-k t}$ (in 1000) bestimmen Sie $k$ bei 7000 Ameisen nach 3 Wochen und Änderungsrate 250/Woche. Langfristige Ameisenanzahl und Interpretation von $f_{k}(t+1)-f_{k}(t)=0,1$ und $f_{k}^{\prime}(t)=0,1$ .

See Solution

Problem 22334

Find points $(x, y)$ where $f(x, y)=x^{2}+3y^{2}-y^{3}-4x$ has a relative max, min, or saddle point.

See Solution

Problem 22335

Find the largest profit for a firm with cost $C(q)=q^{3}-18 q^{2}+118 q+105$ and revenue $R(q)=58 q$ .

See Solution

Problem 22336

Find the tangent lines at $x=4$ for $y=f(x) g(x)$ and $y=\frac{f(x)}{g(x)}$ given tangents $y=3x+2$ and $y=6x-5$ .

See Solution

Problem 22337

World population was 7.1 billion in 2013 with a growth rate of $1.1\%$ per year. Find doubling and tripling times in years.

See Solution

Problem 22338

A country has a population of 150 million in 2011. Find the 2039 population if growth is $3\%$ or $2\%$ per year.

See Solution

Problem 22339

Berechnen Sie die Ableitung von $f(x) = (3x + 1)^2$ .

See Solution

Problem 22340

Approximate the integral $\int_{-1}^{1} \sqrt{1+x^3} \, dx$ using the Trapezoidal Rule with $n=4$ . Show all steps.

See Solution

Problem 22341

Differentiate $g(x)=(x^{2}+1)^{3}(7x+2)^{2}$ and simplify the result in factored form.

See Solution

Problem 22342

Differentiate $g(x)=(x^{2}+1)^{3}(7x+2)^{2}$ and simplify the result in factored form.

See Solution

Problem 22343

Find the tangent lines at $x=4$ for $y=f(x)g(x)$ and $y=\frac{f(x)}{g(x)}$ given their tangents at that point.

See Solution

Problem 22344

Berechnen Sie die Ableitung von $f(x)=(3x+1)^2$ .

See Solution

Problem 22345

Finde die Ableitung von $f(x) = \frac{3}{(2x + 7)^2}$ .

See Solution

Problem 22346

A turkey at $185^{\circ} \mathrm{F}$ cools to $155^{\circ} \mathrm{F}$ in 30 min. What's its temp after 45 min? Round to nearest °F.

See Solution

Problem 22347

Find $\frac{d}{d x} \int_{a}^{x} f(t) d t$ and $\frac{d}{d x} \int_{a}^{b} f(t) d t$ , where a and $b$ are constants.

See Solution

Problem 22348

Find the area function $A(x)=\int_{-7}^{x} (t+7) dt$ and verify $A'(x)=(x+7)$ . What is $A(x)$ ?

See Solution

Problem 22349

Find the area function $A(x)=\int_{-7}^{x} (t+7) dt$ , graph it, and show that $A'(x)=f(x)$ . Derivative: $A'(x)=\square$ .

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Problem 22350

Use Newton's Law of Cooling to find $T(t)$ for body cooling. Given $T(t)=50+48.6 e^{-0.1947 t}$ , find time since death if $T=76^{\circ} \mathrm{F}$ .

See Solution

Problem 22351

Simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=e^{-2 x}$ . Options: a) $\frac{e^{-2 x}(e^{2 h}-1)}{h}$ b) $\frac{e^{-2 x}(e^{-2 h}-1)}{h}$ c) $\frac{e^{-2 x}(e^{2 h}+1)}{h}$ d) $\frac{e^{-2 x}(e^{-2 h}+1)}{h}$

See Solution

Problem 22352

Find the area under the curve $y=x \sqrt[3]{9-x^{2}}$ from $x=0$ to $x=3$ using direct integration.

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Problem 22353

Find the velocity of the particle at $t=1$ for $s(t)=\frac{1}{3} t^{3}-8 t^{2}+28 t$ . When is the particle not moving?

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Problem 22354

Simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=e^{(2 x-1)}$ . Result: $e^{2 x}(e^{2 h}-1)$ .

See Solution

Problem 22355

Simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=e^{-2 x}$ . Choose from: a) $\frac{e^{-2 x}(e^{2 h}-1)}{h}$ , b) $\frac{e^{-2 x}(e^{-2 h}-1)}{h}$ , c) $\frac{e^{-2 x}(e^{2 h}+1)}{h}$ , d) $\frac{e^{-2 x}(e^{-2 h}+1)}{h}$ .

See Solution

Problem 22356

Bestimme Extrem- und Wendepunkt sowie Asymptoten der Funktion $f(x)=3+x \cdot e^{x}$ .

See Solution

Problem 22357

Find the integral using integration by parts: $\int 7 x e^{x} d x$ .

See Solution

Problem 22358

Evaluate the integral from -3 to -1 of $(x^{2}+x+5)$ . Options: 15.17, 14.67, 24.33, 14.5.

See Solution

Problem 22359

Find two methods to determine the derivative of $f(x)=(x^{2}+x-1)(x^{3}+8)$ without actually differentiating.

See Solution

Problem 22360

Find the integral using integration by parts: $\int 9 x \ln x \, dx$ .

See Solution

Problem 22361

Evaluate the integral from 1 to 4 of $x^{1/2} \, dx$ . Options: 4.67, 5.67, 2.8, 3.4.

See Solution

Problem 22362

Find the area between the $x$ -axis and $f(x)=x^{2}-6 x+9$ over the interval $[2,4]$ . Options: $\frac{1}{3}$ , $\frac{4}{3}$ , $\frac{7}{3}$ , $\frac{2}{3}$ .

See Solution

Problem 22363

Find the horizontal asymptotes of the function $f(x)=\frac{3(x+7)(x-7)}{2 x(x+7)}$ .

See Solution

Problem 22364

Find two methods to determine the derivative of $y=\frac{\left(x^{2}-4\right)^{4}}{\left(x^{2}+1\right)^{5}}$ without differentiating.

See Solution

Problem 22365

Determine the horizontal asymptotes of the function $f(x)=\frac{3(2 x+7)(x+8)}{x+1}$ .

See Solution

Problem 22366

Find the area between the $x$ -axis and $f(x)=x^{2}+1$ over $[0,1]$ using the definite integral. Options: $\frac{1}{3}$ , $\frac{4}{3}$ , $\frac{2}{3}$ , $\frac{5}{3}$ .

See Solution

Problem 22367

Bestimmen Sie die Extrem- und Wendepunkte der Funktion $f(x)=\frac{1}{27}(3-4 x)^{3}+4 x$ .

See Solution

Problem 22368

Find the consumer's surplus for the demand function $D(x)=30-x^{2}$ at equilibrium $x=4$ . Options: $\frac{64}{3}$ , $\frac{128}{3}$ , 64.

See Solution

Problem 22369

Evaluate the integral $\int_{1}^{\infty} x^{5} e^{-2 x^{6}} \, dx$ .

See Solution

Problem 22370

Find the local max and min of $g(x)=e^{x}+e^{-2x}$ , and the $x$ values where they occur, rounded to two decimals.

See Solution

Problem 22371

Simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=e^{-3x}$ .

See Solution

Problem 22372

Evaluate the integral $\int_{0}^{2} \frac{x}{\sqrt{16-x^{4}}} d x$ .

See Solution

Problem 22373

Find the derivative of $y$ with respect to $x$ for $y=(x^{2}-2x+3)e^{x}$ .

See Solution

Problem 22374

1- For $S(t)=(t^{2}-1)^{3}$ , find max/min points and inflection points. 2- Determine critical values and their nature for $f(x)=\frac{2 x^{5}}{5}-\frac{x^{4}}{4}-5 x^{3}$ . 3- Given $p(q)=180-2q$ and $c(q)=q^{3}+5q+162$ , find total revenue, quantity for max profit, and quantity for min cost.

See Solution

Problem 22375

Bestimmen Sie die Ableitung und klammern Sie, wenn möglich, aus für die Funktionen a) bis f).

See Solution

Problem 22376

Leiten Sie die Funktionen ab und klammern Sie aus: a) $f(x)=2 x \cdot e^{3 x}$ b) $g(x)=(x-3) \cdot e^{2 x}$ c) $f(t)=t^{2} \cdot e^{\frac{1}{2} t}+5$ d) $g(t)=(t^{3}-2 t^{2}) \cdot e^{-2 t}$ e) $f(x)=(2 x-4) \cdot e^{0,5 x}$ $h(x)=(x^{2}-x) \cdot e^{-0,01 x}$

See Solution

Problem 22377

Finde die Stammfunktion von $f(x) = \frac{2}{x^2}$ .

See Solution

Problem 22378

Find the derivative $s^{\prime}(t)$ of the function $s(t)=3 t^{\frac{1}{3}}+2 t^{2}$ .

See Solution

Problem 22379

Bestimme die Tangentengleichung an $A(2, e^2)$ der Exponentialfunktion und ihre Schnittpunkte mit den Achsen. Beschreibe die Graphen von $f(x)=-e^{x}+5$ , $g(x)=e^{x-3}$ und $h(x)=-e^{-x}+5$ .

See Solution

Problem 22380

Find the derivative $\frac{d y}{d x}$ for $y=2 x^{3}-3 x+1$ . Choose the correct option: a) None b) $6 x^{2}-3 x$ c) $6 x^{2}-3$ d) $x^{2}-3$ e) $3 x^{2}-3$ .

See Solution

Problem 22381

Finde die Stammfunktion von $f(x) = \frac{1}{\sqrt{x}} - x$ .

See Solution

Problem 22382

Find the absolute maximum and minimum of $f(x)=x^{3}-9 x^{2}+24 x-10$ on the interval $[0,4]$ .

See Solution

Problem 22383

Finde die Stammfunktion von $f(x)=\frac{2}{x^{2}}$ .

See Solution

Problem 22384

Finde die Stammfunktion von $f(x) = -3x^2 + 2x$ .

See Solution

Problem 22385

Finde die Stammfunktion von $f(x) = -2x + 2$ .

See Solution

Problem 22386

Which of these does not cause a derivative to fail: a) cusp, b) horizontal tangent, c) discontinuity, d) vertical tangent, e) None?

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Problem 22387

Berechne die Tangente an $f$ bei $x=-2$ und die Nullstelle der Tangente. Bestimme den Flächeninhalt des Dreiecks. Funktion: $f(x) = x^3 + 6x + 12,5x + 10$ .

See Solution

Problem 22388

Evaluate the integral $\int \frac{x^{2}}{(x^{3}-7)^{3}} dx$ using the substitution $u=x^{3}-7$ , $du=3x^{2} dx$ .

See Solution

Problem 22389

1. Given $p(q)=180-2q$ and $c(q)=q^{3}+5q+162$ , find total revenue, max profit quantity, and min cost quantity.
2. For $f(x)=6x^{4}-8x^{3}+1$ , find all points of minima or maxima.
3. Determine the concavity and inflection points for $y=3x^{3}-5x^{2}+6x+15$ .

See Solution

Problem 22390

Gegeben ist die Funktion $f(x)=x^{3}+6x^{2}+12.5x+10$ .
a) Zeige, dass $f$ keine Extremstellen hat.
b) Finde die Tangentengleichung bei $x=-2$ und berechne deren Nullstelle sowie den Flächeninhalt des Dreiecks mit den Achsen.
c) (1) Bestimme die Steigungswinkel der Tangente und der Funktion bei ihrem Schnitt mit der y-Achse.
(2) Finde die Stellen, an denen $f$ den Steigungswinkel $45^{\circ}$ hat.
d) Gib eine Funktion an, die die momentane Zuwachsrate von $f$ beschreibt.

See Solution

Problem 22391

Gegeben ist die Funktion $f(x) = x^{3} + 6x^{2} + 12.5x + 10$ .
a) Zeige, dass $f$ keine Extremstellen hat.
b) Finde die Tangentengleichung bei $x = -2$ und berechne deren Nullstelle sowie den Flächeninhalt des Dreiecks mit den Achsen.
c) (1) Bestimme die Steigungswinkel der Tangente und von $f$ an der y-Achse. (2) Finde die Stellen, wo $f$ einen Steigungswinkel von $45^{\circ}$ hat.
d) Gib eine Funktion für die momentane Zuwachsrate von $f$ an und zeige, dass es ein $x$ mit extremer Zuwachsrate gibt.

See Solution

Problem 22392

Given $f(x)=\sin^{-1} x$ on $[-1,1]$ , check if the Mean Value Theorem applies and find values of $c$ in $[-1,1]$ . If not, enter DNE.

See Solution

Problem 22393

Zeigen Sie, dass $A'(t) = e^{\frac{t}{4}}\left(\frac{1}{4} t^{2} + \frac{3}{2} t - \frac{7}{4}\right)$ die Ableitung von $A(t) = (t^2 - 2t + 1)e^{\frac{1}{4}t}$ ist.

See Solution

Problem 22394

Find error bounds for $\int_{0}^{6} \frac{1}{x+1} dx$ using trapezoid and Simpson's rule with $n=6$ .

See Solution

Problem 22395

Find the min/max points of $f(x)=6 x^{4}-8 x^{3}+1$ . Also, analyze concavity and inflection for $y=3 x^{3}-5 x^{2}+6 x+15$ . Lastly, for revenue $R(x)=-x^{3}+3.5 x^{2}+6 x+500$ , find units for max revenue and the max revenue value.

See Solution

Problem 22396

Determine the concavity and inflection points for the function $y=3 x^{3}-5 x^{2}+6 x+15$ .

See Solution

Problem 22397

Find the smallest $n$ for which the error in $\int_{0}^{10.5} \sin(3x) \, dx$ is less than $10^{-4}$ using trapezoid and Simpson's rules.

See Solution

Problem 22398

Calculate the distance an object falls in 5.23 seconds under gravity. Use the formula $d = \frac{1}{2} g t^2$ .

See Solution

Problem 22399

Estimate $f(0.3)$ and $f(0.4)$ using $f(x) \approx x$ with $n=2$ ; find the maximum error for both.

See Solution

Problem 22400

Find the velocity of an object at $t=1$ for $s(t)=5 t(t-2)^{2}$ . Options: a) 0 m/s b) None c) 15 m/s d) -5 m/s e) 5 m/s.

See Solution

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Calculus

Problem 22301

Evaluate the integral: ∫01(cos⁡(t2))2+(e−5t)2dt\int_{0}^{1} \sqrt{\left(\cos \left(t^{2}\right)\right)^{2}+\left(e^{-5 t}\right)^{2}} d t∫01​(cos(t2))2+(e−5t)2​dt

Problem 22302

Problem 22303

Find the third derivative of the function g(x)=7x2−2sin⁡xg(x)=7 x^{2}-2 \sin xg(x)=7x2−2sinx, i.e., calculate g′′′(x)g^{\prime \prime \prime}(x)g′′′(x).

Problem 22304

Find the tangent line to f(x)=12−x2f(x)=12-x^{2}f(x)=12−x2 at the point where x=2x=2x=2.

Problem 22305

Find the derivative of the function f(x)=20⋅3xf(x)=20 \cdot 3^{x}f(x)=20⋅3x. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 22306

Find the derivative of the function f(r)=3e7r+6f(r)=3 e^{7 r+6}f(r)=3e7r+6. What is dfdr\frac{d f}{d r}drdf​?

Problem 22307

Bestimme die Größe der abgeschnittenen Quadrate, um das Volumen einer offenen Schachtel aus 42 cm x 30 cm Pappe zu maximieren.

Problem 22308

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=12ex+8x114y=\sqrt[4]{12 e^{x}+8 x^{11}}y=412ex+8x11​.

Problem 22309

Problem 22310

Find the average rate of change of f(x)=sin⁡xf(x)=\sin xf(x)=sinx from x=0x=0x=0 to x=2πx=2\pix=2π.

Problem 22311

Berechne die Steigung von fff an x0x_0x0​ mit der h-Methode für: a) f(x)=x2−x,x0=1f(x)=x^2-x, x_0=1f(x)=x2−x,x0​=1; b) f(x)=2x2+1,x0=−2f(x)=2x^2+1, x_0=-2f(x)=2x2+1,x0​=−2; c) f(x)=3x+2,x0=2f(x)=3x+2, x_0=2f(x)=3x+2,x0​=2.

Problem 22312

Find the grams of carbon-14 left after 5867 years using the model A=16e−0.000121tA=16 e^{-0.000121 t}A=16e−0.000121t. Round to the nearest whole number.

Problem 22313

Ein Snowboarder bewegt sich nach der Formel s(t)=1,5t2s(t)=1,5t^2s(t)=1,5t2. Berechne: a) Weg nach 1s und 5s, b) mittlere Geschwindigkeit in 5s, c) Momentangeschwindigkeit nach 5s.

Problem 22314

For what values of P P P is demand elastic for Q(P)=5000e0.02P Q(P) = 5000 e^{0.02 P} Q(P)=5000e0.02P? Also, if price increases by 3% 3\% 3% from 40 40 40€, how does demand change?

Problem 22315

Find the max and min of f(x)=x3−6x2+12f(x)=x^{3}-6 x^{2}+12f(x)=x3−6x2+12 on [1,6][1,6][1,6].

Problem 22316

Finde die Koordinaten des Punktes, wo der Graph von f(x)=x⋅e2xf(x)=x \cdot e^{2 x}f(x)=x⋅e2x eine waagerechte Tangente hat.

Problem 22317

Find the derivative of y=ln⁡(2x2)y=\ln(2 x^{2})y=ln(2x2) with respect to xxx, ttt, or θ\thetaθ.

Problem 22318

Untersuchen Sie die Wendepunkte der Funktionen: a) f(x)=18x3−38x2f(x)=\frac{1}{8} x^{3}-\frac{3}{8} x^{2}f(x)=81​x3−83​x2 b) f(x)=110x5−x2f(x)=\frac{1}{10} x^{5}-x^{2}f(x)=101​x5−x2.

Problem 22319

Untersuchen Sie f(x)=a2x3+2ax2f(x)=a^{2} x^{3}+2 a x^{2}f(x)=a2x3+2ax2 auf Wendepunkte bei x=2x=2x=2. Wie muss aaa gewählt werden? Rechts-links?

Problem 22320

Find the average rate of change of h(x)=−x2−x+5h(x)=-x^{2}-x+5h(x)=−x2−x+5 over the interval −6≤x≤2-6 \leq x \leq 2−6≤x≤2.

Problem 22321

Find h(0)+h′(0)h(0) + h'(0)h(0)+h′(0) given the linear approximation y=4x−5y = 4x - 5y=4x−5 at x=0x = 0x=0. Choose: (A) -3 (B) -2 (C) -1 (D) 0.

Problem 22322

Find dydx\frac{d y}{d x}dxdy​ for y=u3+u2−1y=u^{3}+u^{2}-1y=u3+u2−1 with u=11−xu=\frac{1}{1-x}u=1−x1​ at x=2x=2x=2. Show all steps.

Problem 22323

Find the local linear approximation of f(8.1)f(8.1)f(8.1) given f(8)=−5f(8)=-5f(8)=−5 and f′(8)=3f^{\prime}(8)=3f′(8)=3. Choose the correct answer: (A) -4.8 (B) -4.7 (C) -4.6 (D) -4.5.

Problem 22324

Problem 22325

Find the second partial derivative ∂2∂x2(x3+y4+ex3+2y2)\frac{\partial^{2}}{\partial x^{2}}\left(x^{3}+y^{4}+e^{x^{3}+2 y^{2}}\right)∂x2∂2​(x3+y4+ex3+2y2).

Problem 22326

An Egyptian mummy has 25\frac{2}{5}52​ of carbon-14 compared to living humans. Use y=y0e−0.0001216ty=y_{0} e^{-0.0001216 t}y=y0​e−0.0001216t to find ttt.

Problem 22327

An airline limits vertical acceleration to k=860mi/h2k=860 \mathrm{mi} / \mathrm{h}^{2}k=860mi/h2. At 35,000ft35,000 \mathrm{ft}35,000ft and 300mi/h300 \mathrm{mi} / \mathrm{h}300mi/h, how far from the airport should the pilot start descent? Graph the approach path.

Problem 22328

Find the second mixed partial derivative ∂2∂x∂yex2+3xy+3y3\frac{\partial^{2}}{\partial x \partial y} e^{x^{2}+3 x y+3 y^{3}}∂x∂y∂2​ex2+3xy+3y3.

Problem 22329

Find the expression for f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ given f(x)=e(x−2)f(x)=e^{(x-2)}f(x)=e(x−2).

Problem 22330

Find the average price of oil over 12 months given by P(t)=16t2−t+64P(t)=\frac{1}{6} t^{2}-t+64P(t)=61​t2−t+64.

Problem 22331

Finde die Stammfunktion für fff und überprüfe durch Ableiten: a) f(x)=(3x+1)2f(x)=(3 x+1)^{2}f(x)=(3x+1)2, b) f(x)=3(2x+7)2f(x)=\frac{3}{(2 x+7)^{2}}f(x)=(2x+7)23​, c) f(x)=6x−5f(x)=\sqrt{6 x-5}f(x)=6x−5​, d) f(x)=3sin⁡(12x+7)f(x)=3 \sin (12 x+7)f(x)=3sin(12x+7).

Problem 22332

A fish population grows at 3%3\%3% per year, starting at 12 million in 2016. a) Model the population ttt years since 2016. b) Project the population in 2030 (to the nearest tenth) and find when it reaches 30 million (to the nearest whole number).

Problem 22333

Problem 22334

Find points (x,y)(x, y)(x,y) where f(x,y)=x2+3y2−y3−4xf(x, y)=x^{2}+3y^{2}-y^{3}-4xf(x,y)=x2+3y2−y3−4x has a relative max, min, or saddle point.

Problem 22335

Find the largest profit for a firm with cost C(q)=q3−18q2+118q+105C(q)=q^{3}-18 q^{2}+118 q+105C(q)=q3−18q2+118q+105 and revenue R(q)=58qR(q)=58 qR(q)=58q.

Problem 22336

Find the tangent lines at x=4x=4x=4 for y=f(x)g(x)y=f(x) g(x)y=f(x)g(x) and y=f(x)g(x)y=\frac{f(x)}{g(x)}y=g(x)f(x)​ given tangents y=3x+2y=3x+2y=3x+2 and y=6x−5y=6x-5y=6x−5.

Problem 22337

World population was 7.1 billion in 2013 with a growth rate of 1.1%1.1\%1.1% per year. Find doubling and tripling times in years.

Problem 22338

A country has a population of 150 million in 2011. Find the 2039 population if growth is 3%3\%3% or 2%2\%2% per year.

Problem 22339

Berechnen Sie die Ableitung von f(x)=(3x+1)2f(x) = (3x + 1)^2f(x)=(3x+1)2.

Problem 22340

Approximate the integral ∫−111+x3 dx\int_{-1}^{1} \sqrt{1+x^3} \, dx∫−11​1+x3​dx using the Trapezoidal Rule with n=4n=4n=4. Show all steps.

Problem 22341

Differentiate g(x)=(x2+1)3(7x+2)2g(x)=(x^{2}+1)^{3}(7x+2)^{2}g(x)=(x2+1)3(7x+2)2 and simplify the result in factored form.

Problem 22342

Evaluate the integral: $\int_{0}^{1} \sqrt{\left(\cos \left(t^{2}\right)\right)^{2}+\left(e^{-5 t}\right)^{2}} d t$

Find the third derivative of the function $g(x)=7 x^{2}-2 \sin x$ , i.e., calculate $g^{\prime \prime \prime}(x)$ .

Find the tangent line to $f(x)=12-x^{2}$ at the point where $x=2$ .

Find the derivative of the function $f(x)=20 \cdot 3^{x}$ . What is $f^{\prime}(x)=?$

Find the derivative of the function $f(r)=3 e^{7 r+6}$ . What is $\frac{d f}{d r}$ ?

Find the derivative $\frac{d y}{d x}$ for the function $y=\sqrt[4]{12 e^{x}+8 x^{11}}$ .

Find the average rate of change of $f(x)=\sin x$ from $x=0$ to $x=2\pi$ .

Berechne die Steigung von $f$ an $x_0$ mit der h-Methode für: a) $f(x)=x^2-x, x_0=1$ ; b) $f(x)=2x^2+1, x_0=-2$ ; c) $f(x)=3x+2, x_0=2$ .

Find the grams of carbon-14 left after 5867 years using the model $A=16 e^{-0.000121 t}$ . Round to the nearest whole number.

Ein Snowboarder bewegt sich nach der Formel $s(t)=1,5t^2$ . Berechne: a) Weg nach 1s und 5s, b) mittlere Geschwindigkeit in 5s, c) Momentangeschwindigkeit nach 5s.

For what values of $P$ is demand elastic for $Q(P) = 5000 e^{0.02 P}$ ?
Also, if price increases by $3\%$ from $40$ €, how does demand change?

Find the max and min of $f(x)=x^{3}-6 x^{2}+12$ on $[1,6]$ .

Finde die Koordinaten des Punktes, wo der Graph von $f(x)=x \cdot e^{2 x}$ eine waagerechte Tangente hat.

Find the derivative of $y=\ln(2 x^{2})$ with respect to $x$ , $t$ , or $\theta$ .

Untersuchen Sie die Wendepunkte der Funktionen: a) $f(x)=\frac{1}{8} x^{3}-\frac{3}{8} x^{2}$ b) $f(x)=\frac{1}{10} x^{5}-x^{2}$ .

Untersuchen Sie $f(x)=a^{2} x^{3}+2 a x^{2}$ auf Wendepunkte bei $x=2$ . Wie muss $a$ gewählt werden? Rechts-links?

Find the average rate of change of $h(x)=-x^{2}-x+5$ over the interval $-6 \leq x \leq 2$ .

Find $h(0) + h'(0)$ given the linear approximation $y = 4x - 5$ at $x = 0$ . Choose: (A) -3 (B) -2 (C) -1 (D) 0.

Find $\frac{d y}{d x}$ for $y=u^{3}+u^{2}-1$ with $u=\frac{1}{1-x}$ at $x=2$ . Show all steps.

Find the local linear approximation of $f(8.1)$ given $f(8)=-5$ and $f^{\prime}(8)=3$ . Choose the correct answer: (A) -4.8 (B) -4.7 (C) -4.6 (D) -4.5.

Find the second partial derivative $\frac{\partial^{2}}{\partial x^{2}}\left(x^{3}+y^{4}+e^{x^{3}+2 y^{2}}\right)$ .

An Egyptian mummy has $\frac{2}{5}$ of carbon-14 compared to living humans. Use $y=y_{0} e^{-0.0001216 t}$ to find $t$ .

An airline limits vertical acceleration to $k=860 \mathrm{mi} / \mathrm{h}^{2}$ . At $35,000 \mathrm{ft}$ and $300 \mathrm{mi} / \mathrm{h}$ , how far from the airport should the pilot start descent? Graph the approach path.

Find the second mixed partial derivative $\frac{\partial^{2}}{\partial x \partial y} e^{x^{2}+3 x y+3 y^{3}}$ .

Find the expression for $\frac{f(x+h)-f(x)}{h}$ given $f(x)=e^{(x-2)}$ .

Find the average price of oil over 12 months given by $P(t)=\frac{1}{6} t^{2}-t+64$ .

Finde die Stammfunktion für $f$ und überprüfe durch Ableiten: a) $f(x)=(3 x+1)^{2}$ , b) $f(x)=\frac{3}{(2 x+7)^{2}}$ , c) $f(x)=\sqrt{6 x-5}$ , d) $f(x)=3 \sin (12 x+7)$ .

A fish population grows at $3\%$ per year, starting at 12 million in 2016.
a) Model the population $t$ years since 2016. b) Project the population in 2030 (to the nearest tenth) and find when it reaches 30 million (to the nearest whole number).

Find points $(x, y)$ where $f(x, y)=x^{2}+3y^{2}-y^{3}-4x$ has a relative max, min, or saddle point.

Find the largest profit for a firm with cost $C(q)=q^{3}-18 q^{2}+118 q+105$ and revenue $R(q)=58 q$ .

Find the tangent lines at $x=4$ for $y=f(x) g(x)$ and $y=\frac{f(x)}{g(x)}$ given tangents $y=3x+2$ and $y=6x-5$ .

World population was 7.1 billion in 2013 with a growth rate of $1.1\%$ per year. Find doubling and tripling times in years.

A country has a population of 150 million in 2011. Find the 2039 population if growth is $3\%$ or $2\%$ per year.

Berechnen Sie die Ableitung von $f(x) = (3x + 1)^2$ .

Approximate the integral $\int_{-1}^{1} \sqrt{1+x^3} \, dx$ using the Trapezoidal Rule with $n=4$ . Show all steps.

Differentiate $g(x)=(x^{2}+1)^{3}(7x+2)^{2}$ and simplify the result in factored form.

Differentiate $g(x)=(x^{2}+1)^{3}(7x+2)^{2}$ and simplify the result in factored form.

Find the tangent lines at $x=4$ for $y=f(x)g(x)$ and $y=\frac{f(x)}{g(x)}$ given their tangents at that point.

Berechnen Sie die Ableitung von $f(x)=(3x+1)^2$ .

Finde die Ableitung von $f(x) = \frac{3}{(2x + 7)^2}$ .

A turkey at $185^{\circ} \mathrm{F}$ cools to $155^{\circ} \mathrm{F}$ in 30 min. What's its temp after 45 min? Round to nearest °F.

Find $\frac{d}{d x} \int_{a}^{x} f(t) d t$ and $\frac{d}{d x} \int_{a}^{b} f(t) d t$ , where a and $b$ are constants.

Find the area function $A(x)=\int_{-7}^{x} (t+7) dt$ and verify $A'(x)=(x+7)$ . What is $A(x)$ ?

Find the area function $A(x)=\int_{-7}^{x} (t+7) dt$ , graph it, and show that $A'(x)=f(x)$ . Derivative: $A'(x)=\square$ .

Use Newton's Law of Cooling to find $T(t)$ for body cooling. Given $T(t)=50+48.6 e^{-0.1947 t}$ , find time since death if $T=76^{\circ} \mathrm{F}$ .

Find the area under the curve $y=x \sqrt[3]{9-x^{2}}$ from $x=0$ to $x=3$ using direct integration.

Find the velocity of the particle at $t=1$ for $s(t)=\frac{1}{3} t^{3}-8 t^{2}+28 t$ . When is the particle not moving?

Simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=e^{(2 x-1)}$ . Result: $e^{2 x}(e^{2 h}-1)$ .

Bestimme Extrem- und Wendepunkt sowie Asymptoten der Funktion $f(x)=3+x \cdot e^{x}$ .

Find the integral using integration by parts: $\int 7 x e^{x} d x$ .

Evaluate the integral from -3 to -1 of $(x^{2}+x+5)$ . Options: 15.17, 14.67, 24.33, 14.5.

Find two methods to determine the derivative of $f(x)=(x^{2}+x-1)(x^{3}+8)$ without actually differentiating.

Find the integral using integration by parts: $\int 9 x \ln x \, dx$ .

Evaluate the integral from 1 to 4 of $x^{1/2} \, dx$ . Options: 4.67, 5.67, 2.8, 3.4.

Find the area between the $x$ -axis and $f(x)=x^{2}-6 x+9$ over the interval $[2,4]$ . Options: $\frac{1}{3}$ , $\frac{4}{3}$ , $\frac{7}{3}$ , $\frac{2}{3}$ .

Find the horizontal asymptotes of the function $f(x)=\frac{3(x+7)(x-7)}{2 x(x+7)}$ .

Find two methods to determine the derivative of $y=\frac{\left(x^{2}-4\right)^{4}}{\left(x^{2}+1\right)^{5}}$ without differentiating.

Determine the horizontal asymptotes of the function $f(x)=\frac{3(2 x+7)(x+8)}{x+1}$ .

Find the area between the $x$ -axis and $f(x)=x^{2}+1$ over $[0,1]$ using the definite integral. Options: $\frac{1}{3}$ , $\frac{4}{3}$ , $\frac{2}{3}$ , $\frac{5}{3}$ .

Bestimmen Sie die Extrem- und Wendepunkte der Funktion $f(x)=\frac{1}{27}(3-4 x)^{3}+4 x$ .

Find the consumer's surplus for the demand function $D(x)=30-x^{2}$ at equilibrium $x=4$ . Options: $\frac{64}{3}$ , $\frac{128}{3}$ , 64.

Evaluate the integral $\int_{1}^{\infty} x^{5} e^{-2 x^{6}} \, dx$ .

Find the local max and min of $g(x)=e^{x}+e^{-2x}$ , and the $x$ values where they occur, rounded to two decimals.

Simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=e^{-3x}$ .

Evaluate the integral $\int_{0}^{2} \frac{x}{\sqrt{16-x^{4}}} d x$ .

Find the derivative of $y$ with respect to $x$ for $y=(x^{2}-2x+3)e^{x}$ .