Calculus

Problem 18401

Find the derivative of $f(x)=-x^{2}+5x+5$ and calculate $f^{\prime}(-4)$ .

See Solution

Problem 18402

Profit function for $x$ BabCo Lounge Chairs is $P(x)=30 x-150-0.2 x^{2}$ .
a. Calculate profit increase from selling 11 vs. 10 chairs.
b. Find marginal profit at $x=10$ .

See Solution

Problem 18403

A company sells $x$ widgets weekly with price-demand $p(x)=18-x$ . Find marginal revenue $R'(x)$ and estimate revenue gain from 5 to 6 widgets. $\$$

See Solution

Problem 18404

Find the marginal cost function for the microwave production cost $C(x)=62,000+40x$ . What is $C'(x)$ ?

See Solution

Problem 18405

Gegeben ist $f: x \mapsto \frac{1}{2} x^{3}-x^{2}-\frac{1}{2} x+1$ a) Zeige, dass -1, 1 und 2 Nullstellen sind b) Berechne die Fläche unter dem Graphen von $f$ und der $x$ -Achse c) Vergleiche mit $\int_{-1}^{2} f(x) d x$

See Solution

Problem 18406

Find the profit function $P(x)$ from price $p=255-\frac{x}{30}$ and cost $C(x)=56000+60x$ . Then, calculate $P^{\prime}(1500)$ .

See Solution

Problem 18407

Acme Office Supplies makes file cabinets.
a. Calculate the extra cost for 11 vs. 10 cabinets: $C(11) - C(10)$ . b. Find the marginal cost function: $C'(x) =$ . c. Use $C'(x)$ to estimate the extra cost for 11 cabinets: $C'(10) \approx \$.

See Solution

Problem 18408

Given the function $f(x)=\frac{2 x}{x^{2}-49}$ , find critical numbers, intervals of decrease, local maxima, minima, inflection points, concavity, and asymptotes.

See Solution

Problem 18409

Find the derivative of the function $f(x)=\left(x^{5}+4 x+1\right)(120-5 x)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 18410

Find the area under the curve $y=\sin x$ from $x=0$ to $x=\frac{\pi}{2}$ .

See Solution

Problem 18411

Find where the function $f(x)=2 x^{3}-18 x^{2}+30 x+14$ is increasing using its derivative $f^{\prime}(x)$ .

See Solution

Problem 18412

Find where the function $f(x)=-4 x^{2}+12 x-7$ is increasing using its derivative $f^{\prime}(x)$ .

See Solution

Problem 18413

Find the position function $s(t)$ of a particle with $a(t)=2 \sin t-\cos t$ , $s(0)=6$ , $v(0)=0$ .

See Solution

Problem 18414

Evaluate the integral: $\int_{-2}^{5}\left|3 x-x^{2}\right| d x$

See Solution

Problem 18415

Find the curvature and any inflection points for the following functions: a) $f(x)=x^{3}+2x$ b) $f(x)=x^{5}+1$ c) $f(x)=x-6x$

See Solution

Problem 18416

Evaluate the derivative: $\frac{d}{d x} \int_{0}^{x}\left(e^{\operatorname{arccot} t}\right) d t$ .

See Solution

Problem 18417

Find the cost change for producing 200 to 300 widescreen TVs using $C(x)=10000+5000x$ . Also, calculate the average rate of change.

See Solution

Problem 18418

Find the derivative of $h(x)=\int_{1}^{e^{x}} 2 \ln t \, dt$ .

See Solution

Problem 18419

Finn krumningen og vendepunkter for funksjonene: a) $f(x)=x^{3}+2 x$ , b) $f(x)=x^{5}+1$ , c) $f(x)=x^{2}-6 x^{2}$ .

See Solution

Problem 18420

Evaluate the derivative $f'(x)$ of $f(x)=\frac{2 x^{2}-3 x+7}{9 x^{2}+x-6}$ at $x=2$ , rounded to 2 decimal places.

See Solution

Problem 18421

Bestem krumningen og vendefunkter for: a) $f(x)=x^{3}+2 x$ , c) $f(x)=x^{4}-6 x^{2}$ , 6) $f(x)=x^{5}+1$ .

See Solution

Problem 18422

Find the derivative of the integral: $\frac{d}{d x} \int_{0}^{x}\left(e^{\operatorname{arccot} t}\right) d t$

See Solution

Problem 18423

Which integrals are true? Choose all that apply:
1) $\int x^{4} dx = \frac{1}{5} x^{5} + A$ 2) $\int x^{-8} dx = -\frac{1}{9} x^{-9} + B$ 3) $\int 1 dx = 1 + C$ 4) $\int \sin(x) dx = \cos(x) + C$ 5) $\int e^{x} dx = e^{x}$ 6) $\int \cos(x) dx = \sin(x) + C$

See Solution

Problem 18424

Given $f(x)=2-x^{2}$ , find: a. $\frac{f(8)-f(2)}{6}=$ , b. $\frac{f(2+h)-f(2)}{h}=$ , c. $\lim_{h \to 0} \frac{f(2+h)-f(2)}{h}=$ .

See Solution

Problem 18425

Calculate profit from selling $x$ palm trees using $P(x)=20x-0.01x^2-100$ . Find for 17 trees, average change from 14-17, and instantaneous rate at 17.

See Solution

Problem 18426

Find the revenue change for selling $x=7$ to $x=9$ widgets using $R(x)=140x-0.5x^{2}$ . Also, calculate the average rate of change.

See Solution

Problem 18427

Bestem lerumningen og eventuelle vendefunkter for: a) $f(x)=x^{3}+2x$ , b) $f(x)=x^{5}+1$ , c) $f(x)=x^{4}-6x^{2}$ .

See Solution

Problem 18428

Find the derivatives: a. $f'(2)$ for $f(x)=4x^3$ , b. $g'(3)$ for $g(x)=13x^{1/2}$ , c. $h'(4)$ for $h(x)=8x$ , d. $j'(5)$ for $j(x)=120x^{-1}$ .

See Solution

Problem 18429

Find the derivative of $h(x)=\int_{1}^{\sqrt{x}} \frac{2 z^{2}}{z^{4}+1} d z$ using the Fundamental Theorem of Calculus.

See Solution

Problem 18430

Finn vendepunkter for: a) $f(x)=x^{3}+2x$ , b) $f(x)=x^{5}+1$ , c) $f(x)=x^{2}-6x^{2}$ .

See Solution

Problem 18431

Find the derivative of $g(x)=\int_{4}^{x^{2}} 5 \sqrt{1+t^{2}} dt$ .

See Solution

Problem 18432

Bestimmen Sie den Inhalt der Fläche unter der Funktion $f(x)=-0,1 x^{2}+10$ für die gegebenen Grenzen.

See Solution

Problem 18433

Find the integral of $\sqrt{x}$ with respect to $x$ : $\int \sqrt{x} d x=$

See Solution

Problem 18434

Approximate the mass of air in a 1 m² column, 5 km tall, using densities $f(x)=1.225(0.903)^{x}$ . Compare two methods.

See Solution

Problem 18435

Find the integral of $x^{\frac{1}{2}}$ .

See Solution

Problem 18436

Calculate the integral: $\int 5 x^{-2} d x$ .

See Solution

Problem 18437

Find the indefinite integral of the function: $\int\left(9-8 x^{4}+2 x^{7}\right) d x$

See Solution

Problem 18438

Show that a hemispherical bowl (radius 10 in) is $5 / 16$ full when water is 5 in deep, using $A = \pi(20y - y^2)$ .

See Solution

Problem 18439

Find the indefinite integral of $3 x(4 x^{2}+3)^{7} \, dx$ .

See Solution

Problem 18440

Find the area under the curve of $f(x)=x^{2}-2x+3$ from $x=-1$ to $x=2$ .

See Solution

Problem 18441

Evaluate the integral: $\int -\frac{1}{5} \sin u \, du$

See Solution

Problem 18442

Vis at $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x+\frac{1}{3}$ vokser for alle $x$ , finn minimum vekstfart og likning for vendetangenten.

See Solution

Problem 18443

Find the derivative of $f(x)=\sqrt{x}-\frac{2}{x^{3}}$ . What is $f^{\prime}(x)=?$

See Solution

Problem 18444

Berechne die Stammfunktion von $\int \frac{5}{x^{2}} dx$ .

See Solution

Problem 18445

Evaluate the integral from 0 to 1/4 of $\frac{1}{\sqrt{1-r^{2}}} \, dr$ .

See Solution

Problem 18446

Evaluate the limit of the integral from 0 to π of sin(9x) dx.

See Solution

Problem 18447

Evaluate the integral from 1 to 6: $\int_{1}^{6} \frac{10 x^{2}-7 x+9}{x} d x$ .

See Solution

Problem 18448

Evaluate the integral: $\int 6 e^{\cos x} \sin x \, dx$

See Solution

Problem 18449

Find the integral $\int \tan^{8} x \sec^{2} x \, dx$ using the substitution $u = \tan x$ .

See Solution

Problem 18450

Evaluate the integral using the substitution $u=x^{3}+3$ : $\int x^{2} \sqrt{x^{3}+3} \, dx$ .

See Solution

Problem 18451

Find the integral: $\int x^{-2 / 5} \sqrt{x^{3 / 5}-5} \, dx$

See Solution

Problem 18452

Find the area under the curve $y=\sqrt{2 x+2}$ from $x=0$ to $x=1$ . Round to three decimal places.

See Solution

Problem 18453

Evaluate the integral: $\int \frac{6+4 x}{\sqrt{7+6 x+2 x^{2}}} d x$

See Solution

Problem 18454

Evaluate the integral: $\int \frac{e^{x}}{e^{x}+7} d x$

See Solution

Problem 18455

Evaluate the integral from 1 to 4 of $\frac{1}{\sqrt{x}}$ dx.

See Solution

Problem 18456

Evaluate the integral: $\int 2 \sin x \cos (\cos x) d x$

See Solution

Problem 18457

Find the particle's position with $v(t)=\sin t-\cos t$ and initial position $s(0)=3$ .

See Solution

Problem 18458

Evaluate the integral: $\int_{-4}^{4} \frac{\sin x}{1+x^{2}} d x$ .

See Solution

Problem 18459

Berechnen Sie die Extrem- und Wendepunkte für die Funktionen: a) $f(x)=(x+2)e^{-x}$ , b) $f(x)=(x-7)e^{x}$ , c) $f(x)=x^2e^{-x}$ , d) $f(x)=(x^2-3)e^{x}$ , e) $f(x)=(-x^2+4)e^{-x}$ , f) $f(x)=e^{-x^2}$ , g) $f(x)=2xe^{2x-7}$ , h) $f(x)=(x-1)e^{2x+1}$ , i) $f(x)=(-x-8)e^{\frac{1}{2}-x}$ .

See Solution

Problem 18460

Hausaufgabe 14: Betrachten Sie die Differentialgleichung $\sin (y) \cdot y^{\prime}-2 x \cdot \cos (y)=0$ für $y \in(0, \pi)$ .
(a) Verifizieren Sie: $\cos (x)=\sin \left(\frac{\pi}{2}-x\right)$ und $\arccos (\cos (x))=x$ für $x \in[0, \pi]$ . Skizzieren Sie den Graphen von $\arccos :[-1,+1] \rightarrow \mathbb{R}$ .
(b) Bestimmen Sie alle konstanten Lösungen der Differentialgleichung.
(c) Bestimmen Sie alle Lösungen der Differentialgleichung.
(d) Lösen Sie das Anfangswertproblem $\sin (y) \cdot y^{\prime}-2 x \cdot \cos (y)=0$ , mit $y(0)=\frac{\pi}{4}$ .

See Solution

Problem 18461

Evaluate the integral from 1 to 4 of $\frac{x^{2}+8}{\sqrt{x}} \, dx$ .

See Solution

Problem 18462

Bestimme die Ableitung von $3 x^{2}$ .

See Solution

Problem 18463

Find the integral: $\int -\frac{1}{5} \sin u \, du$

See Solution

Problem 18464

Find the tangent line equation for $f(x)=\frac{x}{\sqrt{x^{2}+1}}$ that goes through (1,0).

See Solution

Problem 18465

Überprüfe die Geschwindigkeit eines Klippenspringers, der aus 28 m Höhe fällt, mit $h(t)=28-5 t^{2}$ . Ist sie $90 \mathrm{~km/h}$ ?

See Solution

Problem 18466

Bestimme die Uhrzeit mit dem höchsten Zuschauerandrang, die Zuschauerzahl um 18 Uhr und den Durchschnitt von 16 bis 18 Uhr. Funktion: $f(x)=20 x \cdot e^{2-0,05 x}$ .

See Solution

Problem 18467

Freier Fall auf dem Mond:
a) Berechne $s(1)$ und die durchschnittliche Geschwindigkeit. b) Finde die Momentangeschwindigkeit nach 10 Sekunden. c) Bestimme die Aufprallgeschwindigkeit aus 4 m Höhe.

See Solution

Problem 18468

Integrieren Sie die Funktion $f(x)=\sqrt{4-x}$ .

See Solution

Problem 18469

Hausaufgabe 14: Betrachten Sie die Differentialgleichung $\sin (y) \cdot y' - 2 x \cdot \cos (y) = 0$ für $y \in (0, \pi)$ .
(a) Verifizieren Sie die Identitäten $\cos (x)=\sin \left(\frac{\pi}{2}-x\right)$ und $\arccos (\cos (x))=x$ für $x \in [0, \pi]$ . Skizzieren Sie den Graphen von $\arccos$ auf $[-1, 1]$ .
(b) Finden Sie alle konstanten Lösungen der Differentialgleichung.
(c) Bestimmen Sie alle Lösungen der Differentialgleichung.
(d) Lösen Sie das Anfangswertproblem $\sin (y) \cdot y' - 2 x \cdot \cos (y) = 0$ , $y(0)=\frac{\pi}{4}$ .

See Solution

Problem 18470

Solve the integral using substitution: $\int 3(2 t-3)^{-3} d t$ and include the constant $C$ .

See Solution

Problem 18471

Solve the integral using substitution: $\int 2 e^{3 x} d x$ . Include the constant of integration $C$ .

See Solution

Problem 18472

Solve the integral using substitution: $\int \frac{1}{x-6} dx$ . Include the constant $C$ .

See Solution

Problem 18473

Solve the integral using substitution: $\int \frac{-2}{\sqrt{7 y+3}} d y$ , and include the constant $C$ .

See Solution

Problem 18474

Find the tangent line equation for the function $f(x)=-7 x^{2}+9 x+4$ at the point where $x=4$ .

See Solution

Problem 18475

Solve the integral using substitution: $\int \frac{1}{x+1} d x$ . Use $C$ for the constant of integration.

See Solution

Problem 18476

Find the tangent line equation for $f(x)=x^{2}-12$ at $x=-5$ .

See Solution

Problem 18477

Solve the integral using substitution: $\int\left(3 x^{8}+6 x-4\right)^{\frac{1}{3}}\left(24 x^{7}+6\right) d x$ and include constant $C$ .

See Solution

Problem 18478

Evaluate the integral: $\int 5 e^{-5 t}\left(1+4 e^{-5 t}\right)^{3} d t$

See Solution

Problem 18479

Solve the integral using substitution: $\int e^{2 x^{8}+7} \cdot 16 x^{7} d x$ and include constant $C$ .

See Solution

Problem 18480

Solve the integral using substitution: $\int (x-6)^{4} \, dx$ and include the constant $C$ .

See Solution

Problem 18481

Find the tangent line equation for the function $f(x)=x^{2}+2$ at the point where $x=10$ .

See Solution

Problem 18482

Solve the integral using substitution: $\int 2 e^{4 y}\left(2+3 e^{4 y}\right)^{3} d y$ . Use $C$ for the constant.

See Solution

Problem 18483

Find the limit as $x$ approaches 2 for $\frac{x^{2}+x-6}{|x-2|}$ . Also, determine where $f$ is discontinuous.

See Solution

Problem 18484

Find the limit: $\lim _{+\infty} \frac{e^{x}+\sin 4 x}{e^{x}-\cos x}$

See Solution

Problem 18485

Find the first and second derivatives of $f(x)=\frac{1}{\ln (x)}$ .

See Solution

Problem 18486

Find where $f(x)=8 x^{2}-2 x^{4}$ is increasing, decreasing, and the $x$ -coordinates of relative maxima and minima.

See Solution

Problem 18487

Show that the function $f(x)=\begin{cases} x^{4} \sin \left(\frac{1}{x}\right) & \text{if } x \neq 0 \\ 0 & \text{if } x=0 \end{cases}$ is continuous at $x=0$ .

See Solution

Problem 18488

Find the first and second derivatives of $f(x)=\frac{\ln (\sqrt{x})}{\ln (x)}$ for positive real $x$ .

See Solution

Problem 18489

Find the derivative of these functions: a) $y=x \cos 2 x$ , b) $f(x)=-x^{2} \sin (3 x-\pi)$ , c) $y=2 \sin \theta \cos \theta$ .

See Solution

Problem 18490

Find the average rate of change of $f(x)=2x^{3}+1$ from $x=2$ to $x=4$ and include the calculated value.

See Solution

Problem 18491

Find the average rate of change of $f(x)=2 x^{3}+1$ from $x=2$ to $x=4$ and include the value in your response.

See Solution

Problem 18492

Find local max and min of $f(x)=x+\sqrt{1-x}$ using first and second derivative tests. Which method do you prefer?

See Solution

Problem 18493

Find the derivative $\left(f^{-1}\right)^{\prime}(0)$ for the function $f(x)=x^{3}-8$ . First, solve for $a$ where $f(a)=0$ .

See Solution

Problem 18494

Bestimme den Definitionsbereich von $f(x)=x-(x-1) \cdot \ln (x-1)$ und die Ableitung $f^{\prime}(x)$ . Zeige, dass $f$ nur einen Extrempunkt bei $H(2 \mid 2)$ hat.

See Solution

Problem 18495

Find the derivative of $y=\tan ^{-1}(\sqrt{x})$ with respect to $x$ : compute $d y / d x$ .

See Solution

Problem 18496

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\tan ^{3}(\sqrt{x})$ .

See Solution

Problem 18497

Differentiate $y=\left(\frac{(x+1) \cos x}{x^{2}}\right)^{3}$ using logarithmic differentiation to find $\frac{d y}{d x}$ .

See Solution

Problem 18498

Determine the horizontal asymptotes of the function $f(x)=\frac{8-2 x^{2}}{x^{2}+5 x+6}$ .

See Solution

Problem 18499

Find the derivative of the composition $(f \circ f)^{\prime}(x)$ for $f(x)=\ln x^{3}, \quad x>0$ .

See Solution

Problem 18500

Find the derivative of $y=8^{\tan x}$ with respect to $x$ . What is $\frac{dy}{dx}$ ?

See Solution

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Calculus

Problem 18401

Find the derivative of f(x)=−x2+5x+5f(x)=-x^{2}+5x+5f(x)=−x2+5x+5 and calculate f′(−4)f^{\prime}(-4)f′(−4).

Problem 18402

Profit function for xxx BabCo Lounge Chairs is P(x)=30x−150−0.2x2P(x)=30 x-150-0.2 x^{2}P(x)=30x−150−0.2x2. a. Calculate profit increase from selling 11 vs. 10 chairs. b. Find marginal profit at x=10x=10x=10.

Problem 18403

A company sells xxx widgets weekly with price-demand p(x)=18−xp(x)=18-xp(x)=18−x. Find marginal revenue R′(x)R'(x)R′(x) and estimate revenue gain from 5 to 6 widgets. $\$$

Problem 18404

Find the marginal cost function for the microwave production cost C(x)=62,000+40xC(x)=62,000+40xC(x)=62,000+40x. What is C′(x)C'(x)C′(x)?

Problem 18405

Problem 18406

Find the profit function P(x)P(x)P(x) from price p=255−x30p=255-\frac{x}{30}p=255−30x​ and cost C(x)=56000+60xC(x)=56000+60xC(x)=56000+60x. Then, calculate P′(1500)P^{\prime}(1500)P′(1500).

Problem 18407

Acme Office Supplies makes file cabinets. a. Calculate the extra cost for 11 vs. 10 cabinets: C(11)−C(10)C(11) - C(10)C(11)−C(10). b. Find the marginal cost function: C′(x)=C'(x) = C′(x)=. c. Use C′(x)C'(x)C′(x) to estimate the extra cost for 11 cabinets: $C'(10) \approx \$.

Problem 18408

Given the function f(x)=2xx2−49f(x)=\frac{2 x}{x^{2}-49}f(x)=x2−492x​, find critical numbers, intervals of decrease, local maxima, minima, inflection points, concavity, and asymptotes.

Problem 18409

Find the derivative of the function f(x)=(x5+4x+1)(120−5x)f(x)=\left(x^{5}+4 x+1\right)(120-5 x)f(x)=(x5+4x+1)(120−5x). What is f′(x)f^{\prime}(x)f′(x)?

Problem 18410

Find the area under the curve y=sin⁡xy=\sin xy=sinx from x=0x=0x=0 to x=π2x=\frac{\pi}{2}x=2π​.

Problem 18411

Find where the function f(x)=2x3−18x2+30x+14f(x)=2 x^{3}-18 x^{2}+30 x+14f(x)=2x3−18x2+30x+14 is increasing using its derivative f′(x)f^{\prime}(x)f′(x).

Problem 18412

Find where the function f(x)=−4x2+12x−7f(x)=-4 x^{2}+12 x-7f(x)=−4x2+12x−7 is increasing using its derivative f′(x)f^{\prime}(x)f′(x).

Problem 18413

Find the position function s(t)s(t)s(t) of a particle with a(t)=2sin⁡t−cos⁡ta(t)=2 \sin t-\cos ta(t)=2sint−cost, s(0)=6s(0)=6s(0)=6, v(0)=0v(0)=0v(0)=0.

Problem 18414

Evaluate the integral: ∫−25∣3x−x2∣dx\int_{-2}^{5}\left|3 x-x^{2}\right| d x∫−25​∣∣​3x−x2∣∣​dx

Problem 18415

Find the curvature and any inflection points for the following functions: a) f(x)=x3+2xf(x)=x^{3}+2xf(x)=x3+2x b) f(x)=x5+1f(x)=x^{5}+1f(x)=x5+1 c) f(x)=x−6xf(x)=x-6xf(x)=x−6x

Problem 18416

Evaluate the derivative: ddx∫0x(earccot⁡t)dt\frac{d}{d x} \int_{0}^{x}\left(e^{\operatorname{arccot} t}\right) d tdxd​∫0x​(earccott)dt.

Problem 18417

Find the cost change for producing 200 to 300 widescreen TVs using C(x)=10000+5000xC(x)=10000+5000xC(x)=10000+5000x. Also, calculate the average rate of change.

Problem 18418

Find the derivative of h(x)=∫1ex2ln⁡t dth(x)=\int_{1}^{e^{x}} 2 \ln t \, dth(x)=∫1ex​2lntdt.

Problem 18419

Finn krumningen og vendepunkter for funksjonene: a) f(x)=x3+2xf(x)=x^{3}+2 xf(x)=x3+2x, b) f(x)=x5+1f(x)=x^{5}+1f(x)=x5+1, c) f(x)=x2−6x2f(x)=x^{2}-6 x^{2}f(x)=x2−6x2.

Problem 18420

Evaluate the derivative f′(x)f'(x)f′(x) of f(x)=2x2−3x+79x2+x−6f(x)=\frac{2 x^{2}-3 x+7}{9 x^{2}+x-6}f(x)=9x2+x−62x2−3x+7​ at x=2x=2x=2, rounded to 2 decimal places.

Problem 18421

Bestem krumningen og vendefunkter for: a) f(x)=x3+2xf(x)=x^{3}+2 xf(x)=x3+2x, c) f(x)=x4−6x2f(x)=x^{4}-6 x^{2}f(x)=x4−6x2, 6) f(x)=x5+1f(x)=x^{5}+1f(x)=x5+1.

Problem 18422

Find the derivative of the integral: ddx∫0x(earccot⁡t)dt\frac{d}{d x} \int_{0}^{x}\left(e^{\operatorname{arccot} t}\right) d tdxd​∫0x​(earccott)dt

Problem 18423

Problem 18424

Given f(x)=2−x2f(x)=2-x^{2}f(x)=2−x2, find: a. f(8)−f(2)6=\frac{f(8)-f(2)}{6}=6f(8)−f(2)​=, b. f(2+h)−f(2)h=\frac{f(2+h)-f(2)}{h}=hf(2+h)−f(2)​=, c. lim⁡h→0f(2+h)−f(2)h=\lim_{h \to 0} \frac{f(2+h)-f(2)}{h}=limh→0​hf(2+h)−f(2)​=.

Problem 18425

Calculate profit from selling xxx palm trees using P(x)=20x−0.01x2−100P(x)=20x-0.01x^2-100P(x)=20x−0.01x2−100. Find for 17 trees, average change from 14-17, and instantaneous rate at 17.

Problem 18426

Find the revenue change for selling x=7x=7x=7 to x=9x=9x=9 widgets using R(x)=140x−0.5x2R(x)=140x-0.5x^{2}R(x)=140x−0.5x2. Also, calculate the average rate of change.

Problem 18427

Bestem lerumningen og eventuelle vendefunkter for: a) f(x)=x3+2xf(x)=x^{3}+2xf(x)=x3+2x, b) f(x)=x5+1f(x)=x^{5}+1f(x)=x5+1, c) f(x)=x4−6x2f(x)=x^{4}-6x^{2}f(x)=x4−6x2.

Problem 18428

Find the derivatives: a. f′(2)f'(2)f′(2) for f(x)=4x3f(x)=4x^3f(x)=4x3, b. g′(3)g'(3)g′(3) for g(x)=13x1/2g(x)=13x^{1/2}g(x)=13x1/2, c. h′(4)h'(4)h′(4) for h(x)=8xh(x)=8xh(x)=8x, d. j′(5)j'(5)j′(5) for j(x)=120x−1j(x)=120x^{-1}j(x)=120x−1.

Problem 18429

Find the derivative of h(x)=∫1x2z2z4+1dzh(x)=\int_{1}^{\sqrt{x}} \frac{2 z^{2}}{z^{4}+1} d zh(x)=∫1x​​z4+12z2​dz using the Fundamental Theorem of Calculus.

Problem 18430

Finn vendepunkter for: a) f(x)=x3+2xf(x)=x^{3}+2xf(x)=x3+2x, b) f(x)=x5+1f(x)=x^{5}+1f(x)=x5+1, c) f(x)=x2−6x2f(x)=x^{2}-6x^{2}f(x)=x2−6x2.

Problem 18431

Find the derivative of g(x)=∫4x251+t2dtg(x)=\int_{4}^{x^{2}} 5 \sqrt{1+t^{2}} dtg(x)=∫4x2​51+t2​dt.

Problem 18432

Bestimmen Sie den Inhalt der Fläche unter der Funktion f(x)=−0,1x2+10f(x)=-0,1 x^{2}+10f(x)=−0,1x2+10 für die gegebenen Grenzen.

Problem 18433

Find the integral of x\sqrt{x}x​ with respect to xxx: ∫xdx=\int \sqrt{x} d x=∫x​dx=

Problem 18434

Approximate the mass of air in a 1 m² column, 5 km tall, using densities f(x)=1.225(0.903)xf(x)=1.225(0.903)^{x}f(x)=1.225(0.903)x. Compare two methods.

Problem 18435

Find the integral of x12x^{\frac{1}{2}}x21​.

Problem 18436

Calculate the integral: ∫5x−2dx\int 5 x^{-2} d x∫5x−2dx.

Problem 18437

Find the indefinite integral of the function: ∫(9−8x4+2x7)dx\int\left(9-8 x^{4}+2 x^{7}\right) d x∫(9−8x4+2x7)dx

Problem 18438

Show that a hemispherical bowl (radius 10 in) is 5/165 / 165/16 full when water is 5 in deep, using A=π(20y−y2)A = \pi(20y - y^2)A=π(20y−y2).

Problem 18439

Find the indefinite integral of 3x(4x2+3)7 dx3 x(4 x^{2}+3)^{7} \, dx3x(4x2+3)7dx.

Problem 18440

Find the area under the curve of f(x)=x2−2x+3f(x)=x^{2}-2x+3f(x)=x2−2x+3 from x=−1x=-1x=−1 to x=2x=2x=2.

Problem 18441

Find the derivative of $f(x)=-x^{2}+5x+5$ and calculate $f^{\prime}(-4)$ .

Profit function for $x$ BabCo Lounge Chairs is $P(x)=30 x-150-0.2 x^{2}$ .
a. Calculate profit increase from selling 11 vs. 10 chairs.
b. Find marginal profit at $x=10$ .

A company sells $x$ widgets weekly with price-demand $p(x)=18-x$ . Find marginal revenue $R'(x)$ and estimate revenue gain from 5 to 6 widgets. $\$$

Find the marginal cost function for the microwave production cost $C(x)=62,000+40x$ . What is $C'(x)$ ?

Find the profit function $P(x)$ from price $p=255-\frac{x}{30}$ and cost $C(x)=56000+60x$ . Then, calculate $P^{\prime}(1500)$ .

Acme Office Supplies makes file cabinets.
a. Calculate the extra cost for 11 vs. 10 cabinets: $C(11) - C(10)$ . b. Find the marginal cost function: $C'(x) =$ . c. Use $C'(x)$ to estimate the extra cost for 11 cabinets: $C'(10) \approx \$.

Given the function $f(x)=\frac{2 x}{x^{2}-49}$ , find critical numbers, intervals of decrease, local maxima, minima, inflection points, concavity, and asymptotes.

Find the derivative of the function $f(x)=\left(x^{5}+4 x+1\right)(120-5 x)$ . What is $f^{\prime}(x)$ ?

Find the area under the curve $y=\sin x$ from $x=0$ to $x=\frac{\pi}{2}$ .

Find where the function $f(x)=2 x^{3}-18 x^{2}+30 x+14$ is increasing using its derivative $f^{\prime}(x)$ .

Find where the function $f(x)=-4 x^{2}+12 x-7$ is increasing using its derivative $f^{\prime}(x)$ .

Find the position function $s(t)$ of a particle with $a(t)=2 \sin t-\cos t$ , $s(0)=6$ , $v(0)=0$ .

Evaluate the integral: $\int_{-2}^{5}\left|3 x-x^{2}\right| d x$

Find the curvature and any inflection points for the following functions: a) $f(x)=x^{3}+2x$ b) $f(x)=x^{5}+1$ c) $f(x)=x-6x$

Evaluate the derivative: $\frac{d}{d x} \int_{0}^{x}\left(e^{\operatorname{arccot} t}\right) d t$ .

Find the cost change for producing 200 to 300 widescreen TVs using $C(x)=10000+5000x$ . Also, calculate the average rate of change.

Find the derivative of $h(x)=\int_{1}^{e^{x}} 2 \ln t \, dt$ .

Finn krumningen og vendepunkter for funksjonene: a) $f(x)=x^{3}+2 x$ , b) $f(x)=x^{5}+1$ , c) $f(x)=x^{2}-6 x^{2}$ .

Evaluate the derivative $f'(x)$ of $f(x)=\frac{2 x^{2}-3 x+7}{9 x^{2}+x-6}$ at $x=2$ , rounded to 2 decimal places.

Bestem krumningen og vendefunkter for: a) $f(x)=x^{3}+2 x$ , c) $f(x)=x^{4}-6 x^{2}$ , 6) $f(x)=x^{5}+1$ .

Find the derivative of the integral: $\frac{d}{d x} \int_{0}^{x}\left(e^{\operatorname{arccot} t}\right) d t$

Given $f(x)=2-x^{2}$ , find: a. $\frac{f(8)-f(2)}{6}=$ , b. $\frac{f(2+h)-f(2)}{h}=$ , c. $\lim_{h \to 0} \frac{f(2+h)-f(2)}{h}=$ .

Calculate profit from selling $x$ palm trees using $P(x)=20x-0.01x^2-100$ . Find for 17 trees, average change from 14-17, and instantaneous rate at 17.

Find the revenue change for selling $x=7$ to $x=9$ widgets using $R(x)=140x-0.5x^{2}$ . Also, calculate the average rate of change.

Bestem lerumningen og eventuelle vendefunkter for: a) $f(x)=x^{3}+2x$ , b) $f(x)=x^{5}+1$ , c) $f(x)=x^{4}-6x^{2}$ .

Find the derivatives: a. $f'(2)$ for $f(x)=4x^3$ , b. $g'(3)$ for $g(x)=13x^{1/2}$ , c. $h'(4)$ for $h(x)=8x$ , d. $j'(5)$ for $j(x)=120x^{-1}$ .

Find the derivative of $h(x)=\int_{1}^{\sqrt{x}} \frac{2 z^{2}}{z^{4}+1} d z$ using the Fundamental Theorem of Calculus.

Finn vendepunkter for: a) $f(x)=x^{3}+2x$ , b) $f(x)=x^{5}+1$ , c) $f(x)=x^{2}-6x^{2}$ .

Find the derivative of $g(x)=\int_{4}^{x^{2}} 5 \sqrt{1+t^{2}} dt$ .

Bestimmen Sie den Inhalt der Fläche unter der Funktion $f(x)=-0,1 x^{2}+10$ für die gegebenen Grenzen.

Find the integral of $\sqrt{x}$ with respect to $x$ : $\int \sqrt{x} d x=$

Approximate the mass of air in a 1 m² column, 5 km tall, using densities $f(x)=1.225(0.903)^{x}$ . Compare two methods.

Find the integral of $x^{\frac{1}{2}}$ .

Calculate the integral: $\int 5 x^{-2} d x$ .

Find the indefinite integral of the function: $\int\left(9-8 x^{4}+2 x^{7}\right) d x$

Show that a hemispherical bowl (radius 10 in) is $5 / 16$ full when water is 5 in deep, using $A = \pi(20y - y^2)$ .

Find the indefinite integral of $3 x(4 x^{2}+3)^{7} \, dx$ .

Find the area under the curve of $f(x)=x^{2}-2x+3$ from $x=-1$ to $x=2$ .

Evaluate the integral: $\int -\frac{1}{5} \sin u \, du$

Vis at $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x+\frac{1}{3}$ vokser for alle $x$ , finn minimum vekstfart og likning for vendetangenten.

Find the derivative of $f(x)=\sqrt{x}-\frac{2}{x^{3}}$ . What is $f^{\prime}(x)=?$

Berechne die Stammfunktion von $\int \frac{5}{x^{2}} dx$ .

Evaluate the integral from 0 to 1/4 of $\frac{1}{\sqrt{1-r^{2}}} \, dr$ .

Evaluate the integral from 1 to 6: $\int_{1}^{6} \frac{10 x^{2}-7 x+9}{x} d x$ .

Evaluate the integral: $\int 6 e^{\cos x} \sin x \, dx$

Find the integral $\int \tan^{8} x \sec^{2} x \, dx$ using the substitution $u = \tan x$ .

Evaluate the integral using the substitution $u=x^{3}+3$ : $\int x^{2} \sqrt{x^{3}+3} \, dx$ .

Find the integral: $\int x^{-2 / 5} \sqrt{x^{3 / 5}-5} \, dx$

Find the area under the curve $y=\sqrt{2 x+2}$ from $x=0$ to $x=1$ . Round to three decimal places.

Evaluate the integral: $\int \frac{6+4 x}{\sqrt{7+6 x+2 x^{2}}} d x$

Evaluate the integral: $\int \frac{e^{x}}{e^{x}+7} d x$

Evaluate the integral from 1 to 4 of $\frac{1}{\sqrt{x}}$ dx.

Evaluate the integral: $\int 2 \sin x \cos (\cos x) d x$

Find the particle's position with $v(t)=\sin t-\cos t$ and initial position $s(0)=3$ .

Evaluate the integral: $\int_{-4}^{4} \frac{\sin x}{1+x^{2}} d x$ .

Evaluate the integral from 1 to 4 of $\frac{x^{2}+8}{\sqrt{x}} \, dx$ .

Bestimme die Ableitung von $3 x^{2}$ .

Find the integral: $\int -\frac{1}{5} \sin u \, du$

Find the tangent line equation for $f(x)=\frac{x}{\sqrt{x^{2}+1}}$ that goes through (1,0).

Überprüfe die Geschwindigkeit eines Klippenspringers, der aus 28 m Höhe fällt, mit $h(t)=28-5 t^{2}$ . Ist sie $90 \mathrm{~km/h}$ ?

Bestimme die Uhrzeit mit dem höchsten Zuschauerandrang, die Zuschauerzahl um 18 Uhr und den Durchschnitt von 16 bis 18 Uhr. Funktion: $f(x)=20 x \cdot e^{2-0,05 x}$ .

Freier Fall auf dem Mond:
a) Berechne $s(1)$ und die durchschnittliche Geschwindigkeit. b) Finde die Momentangeschwindigkeit nach 10 Sekunden. c) Bestimme die Aufprallgeschwindigkeit aus 4 m Höhe.

Integrieren Sie die Funktion $f(x)=\sqrt{4-x}$ .

Solve the integral using substitution: $\int 3(2 t-3)^{-3} d t$ and include the constant $C$ .

Solve the integral using substitution: $\int 2 e^{3 x} d x$ . Include the constant of integration $C$ .

Solve the integral using substitution: $\int \frac{1}{x-6} dx$ . Include the constant $C$ .

Solve the integral using substitution: $\int \frac{-2}{\sqrt{7 y+3}} d y$ , and include the constant $C$ .

Find the tangent line equation for the function $f(x)=-7 x^{2}+9 x+4$ at the point where $x=4$ .