Calculus

Problem 17401

Find the indefinite integral $\int \frac{x+2}{(4 x-5)^{2}} d x$ using the substitution $u=4 x-5$ . What does the integral become?

See Solution

Problem 17402

Find the limits for $f(x)=\frac{x^{2}-7x-18}{x-9}$ as $x$ approaches 9 from the left: A. $\lim f(x)=\square$ or B. DNE.

See Solution

Problem 17403

Evaluate the integral with substitution: $\int \frac{\sec ^{2}(4 \sqrt{x})}{\sqrt{x}} d x=\square+C$

See Solution

Problem 17404

Find the limits for $f(x)=\frac{x^{2}-7 x-18}{x-9}$ as $x$ approaches 9 from the left and right.

See Solution

Problem 17405

Evaluate the Riemann sum for $f(x)$ from $1$ to $7$ using three equal sub-intervals and left endpoints. Riemann Sum = $\square$ .

See Solution

Problem 17406

Find $F'(x)$ for each: a) $F(x)=\int_{20}^{x} \frac{1}{t} dt$ , b) $F(x)=\int_{x}^{18} \frac{1}{t} dt$ , c) $F(x)=\int_{5}^{x^{3}} \frac{1}{t} dt$ , d) $F(x)=\int_{2+\cos x}^{x^{2}+1} \frac{1}{t} dt$ .

See Solution

Problem 17407

Calculate the integral from 3 to 4 of $\frac{x^{3}+4}{x}$ .

See Solution

Problem 17408

Find critical points of $f(x)=\cos\left(\frac{1}{x}\right)$ for $x \neq 0$ and classify them using the second derivative test.

See Solution

Problem 17409

Is the integral $\int_{6}^{\infty} \frac{2}{(x+4)^{3 / 2}} d x$ convergent or divergent? If convergent, evaluate it.

See Solution

Problem 17410

Use $u$ -substitution to find
$\int_{3}^{4}(4 x-5)^{3} d x = \int_{a}^{b} f(u) d u.$
Determine $u$ , $d u$ , $a$ , $b$ , and $f(u)$ . What is the integral's value?

See Solution

Problem 17411

Find $G^{\prime}(w)$ for $G(w)=\frac{4}{3 w^{5}}+5 \sqrt[4]{w}$ . What is $G^{\prime}(w)=\square$ ?

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Problem 17412

Evaluate the integral $\int_{0}^{1} \frac{-4}{\sqrt{1-x^{2}}} d x$ . If divergent, respond with "D".

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Problem 17413

Find the derivative $G^{\prime}(w)$ of the function $G(w)=\frac{4}{3 w^{5}}+5 \sqrt[4]{w}$ .

See Solution

Problem 17414

Find the derivative $G^{\prime}(w)$ for the function $G(w)=\frac{4}{3 w^{5}}+5 \sqrt[4]{w}$ .

See Solution

Problem 17415

Find the result of the integral $\int e^{3 x} dx$ after the substitution $u=3x$ .

See Solution

Problem 17416

Find $f(-1)$ given that $f^{\prime}(x)=8 x^{3}+12 x+2$ and $f(1)=-4$ .

See Solution

Problem 17417

Evaluate the Riemann sum of $f(x)$ on $[1,7]$ using 3 equal subintervals and left endpoints.

See Solution

Problem 17418

Implicit Differentiation: Verify that tangent lines of the bicorn curve $(x^{2}+8y-16)^{2}=y^{2}(16-x^{2})$ at $(0,4)$ and $(0,4/3)$ are horizontal. Do this by a) expanding and b) not expanding the equation.

See Solution

Problem 17419

Evaluate the integrals and use A/B with C-G for comparison test applicability or just G if not applicable.
1. $\int_{1}^{\infty} \frac{1}{x^{3}+2} d x$
2. $\int_{1}^{\infty} \frac{10+\sin (x)}{\sqrt{x}} d x$
3. $\int_{1}^{\infty} \frac{\cos ^{2}(x)}{x^{2}+2} d x$
4. $\int_{1}^{\infty} \frac{e^{-x}}{x^{2}} d x$

See Solution

Problem 17420

Determine if each integral converges or diverges using the comparison test. Provide answers as A/B and C-G or just G.
1. $\int_{1}^{\infty} \frac{1}{x^{3}+2} dx$
2. $\int_{1}^{\infty} \frac{10+\sin(x)}{\sqrt{x}} dx$
3. $\int_{1}^{\infty} \frac{\cos^2(x)}{x^{2}+2} dx$
4. $\int_{1}^{\infty} \frac{e^{-x}}{x^{2}} dx$

See Solution

Problem 17421

Evaluate the integral $\int \frac{x-3}{(3 x-2)^{2}} d x$ using the substitution $u=3 x-2$ . Which transformed integral is correct?

See Solution

Problem 17422

Approximate $\int_{2}^{6} f(x) dx$ using a midpoint Riemann sum with 2 equal-width rectangles. Use values: $-3$ , $0$ , $1$ , $2$ .

See Solution

Problem 17423

Find $f(x)=x^{4}-8 x^{3}+5$ : (A) Find $f^{\prime}(x)$ . (B) Find the slope at $x=-2$ . (C) Find the tangent line equation at $x=-2$ . (D) Find where the tangent line is horizontal.

See Solution

Problem 17424

Find $f(x)=x^{4}-8 x^{3}+5$ : (A) $f^{\prime}(x)$ , (B) slope at $x=-2$ , (C) tangent line at $x=-2$ , (D) horizontal tangents.

See Solution

Problem 17425

Given a continuous function $f(x)$ , which statement about Riemann sums $L_{100}, R_{100}, M_{100}$ on $[10,20]$ is FALSE?

See Solution

Problem 17426

For $f(x)=x^{4}-8 x^{3}+5$ , find: (A) $f^{\prime}(x)$ , (B) slope at $x=-2$ , (C) tangent line at $x=-2$ , (D) where tangent is horizontal.

See Solution

Problem 17427

Find the cost of producing the 71st food processor using $C(x)=2000+60x-0.1x^{2}$ and marginal cost.

See Solution

Problem 17428

Find the Cauchy principal value of the integral: $\int_{0}^{\infty} \frac{\cos (9 x)}{(x^{2}+1)^{2}} dx$

See Solution

Problem 17429

Find the indefinite integral $\int \frac{x-3}{(3 x-2)^{2}} d x$ using the substitution $u=3 x-2$ .

See Solution

Problem 17430

Let $f(x)=|x|+\llbracket x \rrbracket$ .
a. Graph $f$ for $x \in [0,2]$ . b. For which $a \in (0,2)$ does $\lim_{x \rightarrow a} f(x)$ exist? c. Where is $f$ continuous in $(0,2)$ ? Explain continuity.

See Solution

Problem 17431

Show that $f(x)=x^{4}+1-\frac{1}{x}$ has a root in $(\frac{1}{2}, 1)$ using the intermediate value theorem, then find it to two decimal places.

See Solution

Problem 17432

Find the limits for the following:
a. $\lim _{h \rightarrow 0}\left[\frac{(3+h)^{-2}-3^{-2}}{h}\right]$
b. $\lim _{x \rightarrow 2} \frac{x^{2}+x-6}{x^{2}-2 x}$
c. $\lim _{x \rightarrow-2} \frac{3 x^{2}-\sqrt{9 x^{4}+x+2}}{x^{3}+2 x^{2}}$
d. $\lim _{x \rightarrow \infty} \frac{2-3 x^{2}}{6 x^{2}+5}$
e. $\lim _{x \rightarrow 0} x^{4} \sin \left(\frac{1}{2 x^{2}+3 x}\right)$

See Solution

Problem 17433

Bestimmen Sie die Ableitung von $f(t) = t^{4} - t + 1$ .

See Solution

Problem 17434

Evaluate the integral $\int \frac{x-3}{(3x-2)^{2}} dx$ using the substitution $u=3x-2$ . Which transformed integral is correct?

See Solution

Problem 17435

Bestimmen Sie die Ableitung von $f(x) = (3+x)(3-x)$ .

See Solution

Problem 17436

Untersuchen Sie die Extrem- und Sattelpunkte von $f(x)=x^{2}-4$ und $g(x)=x^{5}-15x^{3}-30$ .

See Solution

Problem 17437

Untersuchen Sie das Verhalten der Funktionen an den Definitionslücken:
a) $f(x)=\frac{x^{2}-9}{2 x-6}, x_{0}=3$
b) $f(x)=\frac{x+1}{x}, x_{0}=0$
c) $f(x)=\frac{x+1}{x^{2}}, x_{0}=0$

See Solution

Problem 17438

Leite die Funktion $-\frac{5}{x}$ ab.

See Solution

Problem 17439

Gegeben ist die Funktion $f(t)=-0,006 t^{3}+0,18 t^{2}-1,35 t+15$ für $0 \leq t \leq 25$ .
a) Bestimme $f(0)$ und erkläre die Bedeutung. b) Berechne $f(25)$ . c) Finde die Extrempunkte und erkläre ihre Bedeutung. d) Bestimme Intervalle, in denen $f$ fallend ist. e) Finde den Zeitpunkt, an dem die Eisschicht am stärksten zunimmt. f) Interpretiere $\frac{f(15)-f(5)}{15-5}=0,3$ im Kontext der Eisdecke.

See Solution

Problem 17440

Gegeben ist die Ableitungsfunktion $f^{\prime}(x)=x^{2}-2 \cdot x-8$ .
a) Berechne $f'(-4)$ . b) Finde die Nullstellen von $f^{\prime}$ . c) Bestimme die lokalen Extremstellen von $f$ und deren Art anhand der Tabelle. d) Berechne die Nullstellen von $f(x)=x^{2}-6 \cdot x+8$ .

See Solution

Problem 17441

Bestimme die Ableitungen für die Funktionen: a) $f(x)=x^{\frac{1}{3}}$ , b) $f(x)=\sqrt{x}$ , c) $f(x)=\frac{1}{x^{3}}$ , d) $f(x)=x^{\frac{3}{2}}$ , e) $f(x)=\sqrt[4]{x}$ , f) $f(x)=\sqrt[6]{x}$ , g) $f(x)=\frac{1}{\sqrt{x}}-\frac{1}{2} x$ , h) $f(x)=x^{\frac{3}{4}}$ .

See Solution

Problem 17442

Bestimmen Sie die Ableitung von $f_{1}(x)=2 \sin x+8$ .

See Solution

Problem 17443

Find the derivative and integral of $f_{1}(x)=2 \sin x+8$ .

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Problem 17444

Bestimmen Sie eine Stammfunktion für die folgenden Funktionen:
1. a) $f(x)=4 x^{3}+2 x^{2}-1$ b) $f(x)=x^{5}-4 x^{3}$ c) $f(x)=-x^{3}-2 x+7$ d) $f(x)=\frac{3}{4} x^{2}+\frac{8}{7} x^{3}$ e) $f(x)=0,24 x^{7}-5,4 x$ f) $f(x)=-25 x^{4}+33 x^{2}$ g) $f(x)=2 x^{3}-15 x^{2}$ h) $f(x)=162 x^{8}-54 x^{5}$
2. a) $f(x)=\frac{1}{4} x^{-2}$ b) $f(x)=\frac{3}{x^{4}}$ c) $f(x)=0,3 x^{-10}$ d) $f(x)=0,15 x^{-6}$ e) $f(x)=\frac{-5}{x^{6}}$ f) $f(x)=\frac{x^{2}+2 x}{x^{5}}$ g) $f(x)=\frac{x(x+1)}{x^{4}}$ h) $f(x)=\frac{6 x^{2}-21 x}{3 x^{7}}$
3. a) $f(x)=2 \sin (x)$ b) $f(x)=0,5 \cos (x)$ c) $f(x)=x+2 \sin (x)$ d) $f(x)=\cos (x)-\pi$

See Solution

Problem 17445

Given $f(x)=\frac{2-x}{x+4}$ , find intercepts, asymptotes, increasing/decreasing intervals, concavity, and sketch the graph.

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Problem 17446

Hitung biaya marjinal (MC) dari persamaan total biaya $T C=2 Q^{2}-24 Q+102$ saat produksi meningkat 1 unit.

See Solution

Problem 17447

Hitung biaya marjinal (MC) dari persamaan $T C=2 Q^{2}-24 Q+102$ jika produksinya ditingkatkan 1 unit.

See Solution

Problem 17448

Cari unit $X$ yang harus dikonsumsi agar kepuasan maksimum tercapai, dengan fungsi total utilitas $TU = 100x - 5x^{2}$ dan harga Rp50 per unit.

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Problem 17449

Bestimme $u$ , sodass der Flächeninhalt des Rechtecks, definiert durch $P(u|f(u))$ , maximal ist. Was passiert, wenn $u \to \infty$ ?

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Problem 17450

Ein Flugobjekt wird in 10 Sekunden von $150 \mathrm{~km/h}$ auf $96 \mathrm{~km/h}$ abgebremst. Bestimme die Bremsstrecke mit $v(t)=\frac{3}{20} t^{2}-3 t+41,6$ für $t \in[0;10]$ . Berechne die Länge der Bremsstrecke und zeichne den Graphen. Wie viele Teilintervalle sind nötig, damit die Differenz zwischen Unter- und Obersumme klein bleibt?

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Problem 17451

Ein Flugobjekt bremst von $150 \mathrm{~km/h}$ auf $96 \mathrm{~km/h}$ in 10 Sekunden. Berechnet die Bremsstrecke und die Teilintervalle.

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Problem 17452

Find the derivative of $y=\frac{1}{\sqrt{x}}-\frac{24}{x}$ and evaluate it at $x=2$ , rounding to two decimal places.

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Problem 17453

Find the value of the function $y=\frac{x^{3}}{3}-x^{2}-3 x+\frac{11}{3}$ at its inflection point.

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Problem 17454

Find the stationary point of the function $y=3x^{2}-3x+20$ and determine if it's a max, min, or inflection point.

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Problem 17455

Find the derivative of $y=\left(x^{3}+1\right)\left(x^{0.5}-1/x\right)$ at $x=3$ and round to two decimal places.

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Problem 17456

Find the derivative of $y=\frac{40\left(x^{0.4}+2\right)}{1-x^{2}}$ at $x=6$ and round to two decimal places.

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Problem 17457

Determine the correctness of these statements for the function $\frac{x^{3}}{3}-x^{2}-3 x+9$ : a) max at $x=3$ , b) inflection at $x=1$ , c) decreasing for $x>3$ , d) min at $x=9$ .

See Solution

Problem 17458

Substitute $h=0$ into $nt=\frac{1.5h}{0.1+0.1h}$ and $y(h)=\frac{1.5h^2}{0.1+0.1h}$ . Find values rounded to two decimals. Check stability using the slope criterion.

See Solution

Problem 17459

Evaluate the integral $\int_{0}^{2}|x-1| e^{x} d x$ using the piecewise definition of $|x-1|$ .

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Problem 17460

Calculate the area between the parabola $y=3x^2+20x+5$ and the line $y=2x-10$ for $x \in[-10,0]$ .

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Problem 17461

Evaluate the integral $\int \frac{\mathrm{e}^{1 / \mathrm{x}^{2}}}{\mathrm{x}^{3}} \mathrm{dx}$ .

See Solution

Problem 17462

Find the integral $\int \frac{\tan ^{3} x}{\sqrt{\sec x}} d x$ .

See Solution

Problem 17463

Evaluate the Riemann sum of $f(x)$ on $[1,7]$ using three equal subintervals and left endpoints for given values.

See Solution

Problem 17464

Approximate $\int_{0}^{6} f(x) dx$ using a Riemann sum with 3 equal-width rectangles and midpoints.

See Solution

Problem 17465

Compute $\int_{-2}^{5} f(x) d x$ given $\int_{-2}^{7}(3 f(x)) d x=12$ and $\int_{5}^{7} f(x) d x=2$ .

See Solution

Problem 17466

Given a continuous function $f(x)$ on $[10,20]$ , which statement about Riemann sums $L_{100}, R_{100}, M_{100}$ is FALSE?

See Solution

Problem 17467

Find inflection points and concavity intervals for the function $f(x)=x e^{x-1}$ .

See Solution

Problem 17468

Gegeben ist die Funktion $f(x)=x^{3}-6 x^{2}+11 x-6$ .
a) Zeigen Sie, dass der Wendepunkt auf der Geraden $y=x-2$ liegt.
b) Nach Verschiebung hat der Punkt (2,10) die Koordinaten (3,2). Bestimmen Sie die Gleichung der Funktion $h$ .

See Solution

Problem 17469

Determine the horizontal asymptote(s) of the function $f(x)=-\frac{8}{x-2}$ .

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Problem 17470

Oblicz pochodną funkcji $y'=\left(\sqrt[4]{1+\cos ^{2} x}\right)$ .

See Solution

Problem 17471

Rozwiąż równanie różniczkowe $y'=\sqrt[4]{1+\sin^2 x}$ .

See Solution

Problem 17472

Une entreprise a un revenu $R(q)=9000 \ln (0,02 q+1)$ et des coûts $C(q)=7000 e^{0,0012 q}$ .
a) Calculer le profit pour 1000 unités. b) Trouver $q$ pour $R(q)=21597 \$. c) Trouver$ q $pour$ C(q)=29545 \ $. d) Pour$ q=800 $, déterminer la variation des coûts et du revenu. e) Évaluer$ \mathrm{C}_{\text {marginal }}(400) $et$ \mathrm{R}_{\text {marginal }}(400)$.

See Solution

Problem 17473

Find the elasticity function for the demand function $p=D(x)=196-3.5x$ .

See Solution

Problem 17474

Find where $f(x)=2x+\frac{8}{x}$ is increasing/decreasing and the $x$ -coordinates of relative maxima/minima.

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Problem 17475

Find the units $x$ and $y$ for two plants to minimize costs given $C_{1}(x)=\frac{x^{2}}{20}-10x+400$ and $C_{2}(y)=\frac{y^{3}}{147}-y+400$ .

See Solution

Problem 17476

1. La relation entre le prix $p$ en \ $et la quantité$ q $en milliers est donnée par$ 10 p^{2}+p q+\frac{q^{2}}{2}=C $. a) Trouver$ \frac{d p}{d q} $. b) Si$ C=1000 $,$ p=8\$$, et $q=20$, calculez $\frac{d p}{d q}$. c) Calculez $\frac{d p}{d q}$ pour $p=10\$$.

See Solution

Problem 17477

Find the volume of the solid formed by revolving the area under $y=4x-x^2$ (above $x$ -axis) from $x=1$ to $x=4$ around $x=1$ .

See Solution

Problem 17478

Find the volume of the solid formed by revolving the area under $y=4x-x^2$ above the $x$ -axis around $x=1$ .

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Problem 17479

Bestimme $k$ für $h(t)=0,02 \cdot e^{kt}$ mit $h(6)=0,4$ . Finde $t$ für $h(t)=3$ und Wachstumsrate $0,3$ . Wann $h$ steigt um $0,43$ ?

See Solution

Problem 17480

Find the volume of the solid formed by revolving the area under $y=4x-x^2$ (from $x=1$ to $x=4$ ) around $x=1$ .

See Solution

Problem 17481

Show that $\int_{-\infty}^{\infty} \frac{\sin x}{x^{2}+4 x+5} dx = -\frac{\pi \sin 2}{e}$ using residues.

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Problem 17482

A bag leaks sand modeled by $S(t)=K\left(1-\frac{t^{2}}{\mu}\right)^{3}$ . Answer the following:
1. (a) Amount of sand at $t=0$ ? (b) Rate of leak at time $t$ ? (c) At $t=2$ with $\mu=6$ , is the leak speeding up or slowing down?
2. (a) Time to empty the bag? (b) Check if $\lim _{t \rightarrow T^{-}} \frac{d S}{d t}=0$ makes sense.
3. Find $\left(S^{-1}\right)^{\prime}(10)$ for $K=80$ , $\mu=8$ .
4. Laila studies $S_{1}(\mu)=K\left(1-\frac{1}{\mu}\right)^{3}$ at $t=1$ : (a) Find $\frac{d S_{1}}{d \mu}$ . (b) Does larger $\mu$ mean more sand leaked at $t=1$ ? Explain.

See Solution

Problem 17483

Evaluate the integral: $I = \int \sin(3x) \cos^2(x) \, dx$

See Solution

Problem 17484

Differentiate the expression $10 p^{2} + p \cdot q + \frac{1}{2} q^{2}$ with respect to $q$ and set it to 0.

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Problem 17485

Déterminez les dimensions d'un cylindre inscrit dans un cône de hauteur $H=4$ et de rayon $R=7.5$ pour volume maximal.

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Problem 17486

Find dimensions $x$ and $y$ for a cylinder from a rectangle with perimeter $27 \mathrm{~cm}$ for max volume.

See Solution

Problem 17487

Find the critical numbers of the function $f$ given that its derivative $f^{\prime}$ passes through (0,0) and (2,0).

See Solution

Problem 17488

Find points of inflection for $f(x)=x^{2}+6x+11$ . Options: $(-4,1)$ , $(-2,3)$ , $(-3,2)$ , or none.

See Solution

Problem 17489

Find the point of diminishing returns for profit $P(x)=-x^{3}+36 x^{2}+14 x-30$ where $0 \leq x \leq 20$ .

See Solution

Problem 17490

Minimize the surface area $A$ of a cylinder with volume $V=500 \, \text{cm}^3$ . Find $r$ for minimum $A$ .

See Solution

Problem 17491

Find the relative max and min of $f(x)=2x^{3}+3x^{2}-36x+6$ using its derivative. Calculate max and min values.

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Problem 17492

Untersuchen Sie die Folgen auf Konvergenz, Grenzwert, Häufungspunkte, Limes superior und Limes inferior:
(a) $a_{n}=\frac{36 n^{3}-9}{24 n^{2}+12 \sqrt{n}}$ (b) $a_{n}=(-1)^{n} \sqrt[n]{\frac{2 n}{3^{n^{2}}}}+\frac{\cos (n)}{n}$ (c) $a_{n}=\operatorname{Re}\left(2^{-n}(-1-\mathrm{i} \sqrt{3})^{n}\right)$ (d) $a_{n}=\sum_{k=0}^{n} \frac{2^{k}+(-1)^{k}}{5^{k}}$

See Solution

Problem 17493

Show that for any numbers $a$ and $b$ , the inequality $|\tan^{-1}(b) - \tan^{-1}(a)| \leq |b - a|$ holds using the Mean Value Theorem.

See Solution

Problem 17494

Show that if $f(x)=10^{x}$ , then $\frac{f(x+h)-f(x)}{h}=10^{x}\left(\frac{10^{h}-1}{h}\right)$ .

See Solution

Problem 17495

Une entreprise a un revenu $R(q)=9000 \ln (0,02 q+1)$ et des coûts $C(q)=7000 e^{0,0012 q}$ .
a) Quel est le profit pour 1000 unités vendues ? b) Combien d'unités pour un revenu de 21597 \ $? c) Combien d'unités pour des coûts de 29545 \$ ? d) Pour$ q=800 $, calculez la variation des coûts et du revenu. e) Évaluez$ \mathrm{C}_{\text {marginal }}(400) $et$ \mathrm{R}_{\text {marginal }}(400)$. f) Quel est le revenu lorsque la hausse est de 30 \$/unité ? g) Quelle est la variation des coûts à 23240,82 \$ ? h) Combien d'unités pour maximiser le profit et quel est ce profit maximal ?

See Solution

Problem 17496

Une entreprise a un revenu $R(q)=9000 \ln (0,02 q+1)$ et des coûts $C(q)=7000 e^{0,0012 q}$ .
a) Quel est le profit pour 1000 unités vendues ? b) Pour quel $q$ le revenu est 21597 \ $? c) Pour quel$ q $les coûts sont 29545 \$ ? d) À$ q=800 $, quelle est la variation des coûts et du revenu ? e) Évaluer : i.$ C_{\text{marginal}}(400) $ii.$ R_{\text{marginal}}(400) $f) À quel revenu total correspond une augmentation de 30 \$/unité ? g) Quel est le rythme de variation des coûts à 23240,82 \$ ? h) Quel$ q$ maximise le profit et quel est ce profit maximal ?

See Solution

Problem 17497

Find critical points and intervals of increase/decrease for $f(x)=-4 e^{-x} \cos (x)$ in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ .

See Solution

Problem 17498

Evaluate the integral: $\int \sec^{2}(\theta) \tan^{3}(\theta) d\theta$ (use $C$ for the constant).

See Solution

Problem 17499

Find the indefinite integral and include the constant of integration $C$ : $\int(5-2 x)^{11} d x$

See Solution

Problem 17500

Calculate the average value $g_{\text {ave }}$ of the function $g(x)=3 \cos (x)$ over the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ .

See Solution

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Calculus

Problem 17401

Find the indefinite integral ∫x+2(4x−5)2dx\int \frac{x+2}{(4 x-5)^{2}} d x∫(4x−5)2x+2​dx using the substitution u=4x−5u=4 x-5u=4x−5. What does the integral become?

Problem 17402

Find the limits for f(x)=x2−7x−18x−9f(x)=\frac{x^{2}-7x-18}{x-9}f(x)=x−9x2−7x−18​ as xxx approaches 9 from the left: A. lim⁡f(x)=□\lim f(x)=\squarelimf(x)=□ or B. DNE.

Problem 17403

Evaluate the integral with substitution: ∫sec⁡2(4x)xdx=□+C\int \frac{\sec ^{2}(4 \sqrt{x})}{\sqrt{x}} d x=\square+C∫x​sec2(4x​)​dx=□+C

Problem 17404

Find the limits for f(x)=x2−7x−18x−9f(x)=\frac{x^{2}-7 x-18}{x-9}f(x)=x−9x2−7x−18​ as xxx approaches 9 from the left and right.

Problem 17405

Evaluate the Riemann sum for f(x)f(x)f(x) from 111 to 777 using three equal sub-intervals and left endpoints. Riemann Sum = □\square□.

Problem 17406

Problem 17407

Calculate the integral from 3 to 4 of x3+4x\frac{x^{3}+4}{x}xx3+4​.

Problem 17408

Find critical points of f(x)=cos⁡(1x)f(x)=\cos\left(\frac{1}{x}\right)f(x)=cos(x1​) for x≠0x \neq 0x=0 and classify them using the second derivative test.

Problem 17409

Is the integral ∫6∞2(x+4)3/2dx\int_{6}^{\infty} \frac{2}{(x+4)^{3 / 2}} d x∫6∞​(x+4)3/22​dx convergent or divergent? If convergent, evaluate it.

Problem 17410

Use uuu-substitution to find ∫34(4x−5)3dx=∫abf(u)du. \int_{3}^{4}(4 x-5)^{3} d x = \int_{a}^{b} f(u) d u. ∫34​(4x−5)3dx=∫ab​f(u)du. Determine uuu, dud udu, aaa, bbb, and f(u)f(u)f(u). What is the integral's value?

Problem 17411

Find G′(w)G^{\prime}(w)G′(w) for G(w)=43w5+5w4G(w)=\frac{4}{3 w^{5}}+5 \sqrt[4]{w}G(w)=3w54​+54w​. What is G′(w)=□G^{\prime}(w)=\squareG′(w)=□?

Problem 17412

Evaluate the integral ∫01−41−x2dx\int_{0}^{1} \frac{-4}{\sqrt{1-x^{2}}} d x∫01​1−x2​−4​dx. If divergent, respond with "D".

Problem 17413

Find the derivative G′(w)G^{\prime}(w)G′(w) of the function G(w)=43w5+5w4G(w)=\frac{4}{3 w^{5}}+5 \sqrt[4]{w}G(w)=3w54​+54w​.

Problem 17414

Find the derivative G′(w)G^{\prime}(w)G′(w) for the function G(w)=43w5+5w4G(w)=\frac{4}{3 w^{5}}+5 \sqrt[4]{w}G(w)=3w54​+54w​.

Problem 17415

Find the result of the integral ∫e3xdx\int e^{3 x} dx∫e3xdx after the substitution u=3xu=3xu=3x.

Problem 17416

Find f(−1)f(-1)f(−1) given that f′(x)=8x3+12x+2f^{\prime}(x)=8 x^{3}+12 x+2f′(x)=8x3+12x+2 and f(1)=−4f(1)=-4f(1)=−4.

Problem 17417

Evaluate the Riemann sum of f(x)f(x)f(x) on [1,7][1,7][1,7] using 3 equal subintervals and left endpoints.

Problem 17418

Implicit Differentiation: Verify that tangent lines of the bicorn curve (x2+8y−16)2=y2(16−x2)(x^{2}+8y-16)^{2}=y^{2}(16-x^{2})(x2+8y−16)2=y2(16−x2) at (0,4)(0,4)(0,4) and (0,4/3)(0,4/3)(0,4/3) are horizontal. Do this by a) expanding and b) not expanding the equation.

Problem 17419

Problem 17420

Problem 17421

Evaluate the integral ∫x−3(3x−2)2dx\int \frac{x-3}{(3 x-2)^{2}} d x∫(3x−2)2x−3​dx using the substitution u=3x−2u=3 x-2u=3x−2. Which transformed integral is correct?

Problem 17422

Approximate ∫26f(x)dx\int_{2}^{6} f(x) dx∫26​f(x)dx using a midpoint Riemann sum with 2 equal-width rectangles. Use values: −3-3−3, 000, 111, 222.

Problem 17423

Find f(x)=x4−8x3+5f(x)=x^{4}-8 x^{3}+5f(x)=x4−8x3+5: (A) Find f′(x)f^{\prime}(x)f′(x). (B) Find the slope at x=−2x=-2x=−2. (C) Find the tangent line equation at x=−2x=-2x=−2. (D) Find where the tangent line is horizontal.

Problem 17424

Find f(x)=x4−8x3+5f(x)=x^{4}-8 x^{3}+5f(x)=x4−8x3+5: (A) f′(x)f^{\prime}(x)f′(x), (B) slope at x=−2x=-2x=−2, (C) tangent line at x=−2x=-2x=−2, (D) horizontal tangents.

Problem 17425

Given a continuous function f(x)f(x)f(x), which statement about Riemann sums L100,R100,M100L_{100}, R_{100}, M_{100}L100​,R100​,M100​ on [10,20][10,20][10,20] is FALSE?

Problem 17426

For f(x)=x4−8x3+5f(x)=x^{4}-8 x^{3}+5f(x)=x4−8x3+5, find: (A) f′(x)f^{\prime}(x)f′(x), (B) slope at x=−2x=-2x=−2, (C) tangent line at x=−2x=-2x=−2, (D) where tangent is horizontal.

Problem 17427

Find the cost of producing the 71st food processor using C(x)=2000+60x−0.1x2C(x)=2000+60x-0.1x^{2}C(x)=2000+60x−0.1x2 and marginal cost.

Problem 17428

Find the Cauchy principal value of the integral: ∫0∞cos⁡(9x)(x2+1)2dx\int_{0}^{\infty} \frac{\cos (9 x)}{(x^{2}+1)^{2}} dx∫0∞​(x2+1)2cos(9x)​dx

Problem 17429

Find the indefinite integral ∫x−3(3x−2)2dx\int \frac{x-3}{(3 x-2)^{2}} d x∫(3x−2)2x−3​dx using the substitution u=3x−2u=3 x-2u=3x−2.

Problem 17430

Problem 17431

Show that f(x)=x4+1−1xf(x)=x^{4}+1-\frac{1}{x}f(x)=x4+1−x1​ has a root in (12,1)(\frac{1}{2}, 1)(21​,1) using the intermediate value theorem, then find it to two decimal places.

Problem 17432

Problem 17433

Bestimmen Sie die Ableitung von f(t)=t4−t+1f(t) = t^{4} - t + 1f(t)=t4−t+1.

Problem 17434

Evaluate the integral ∫x−3(3x−2)2dx\int \frac{x-3}{(3x-2)^{2}} dx∫(3x−2)2x−3​dx using the substitution u=3x−2u=3x-2u=3x−2. Which transformed integral is correct?

Problem 17435

Bestimmen Sie die Ableitung von f(x)=(3+x)(3−x)f(x) = (3+x)(3-x)f(x)=(3+x)(3−x).

Problem 17436

Untersuchen Sie die Extrem- und Sattelpunkte von f(x)=x2−4f(x)=x^{2}-4f(x)=x2−4 und g(x)=x5−15x3−30g(x)=x^{5}-15x^{3}-30g(x)=x5−15x3−30.

Problem 17437

Untersuchen Sie das Verhalten der Funktionen an den Definitionslücken: a) f(x)=x2−92x−6,x0=3f(x)=\frac{x^{2}-9}{2 x-6}, x_{0}=3f(x)=2x−6x2−9​,x0​=3 b) f(x)=x+1x,x0=0f(x)=\frac{x+1}{x}, x_{0}=0f(x)=xx+1​,x0​=0 c) f(x)=x+1x2,x0=0f(x)=\frac{x+1}{x^{2}}, x_{0}=0f(x)=x2x+1​,x0​=0

Problem 17438

Leite die Funktion −5x-\frac{5}{x}−x5​ ab.

Problem 17439

Problem 17440

Problem 17441

Problem 17442

Bestimmen Sie die Ableitung von f1(x)=2sin⁡x+8f_{1}(x)=2 \sin x+8f1​(x)=2sinx+8.

Problem 17443

Find the derivative and integral of f1(x)=2sin⁡x+8f_{1}(x)=2 \sin x+8f1​(x)=2sinx+8.

Problem 17444

Find the indefinite integral $\int \frac{x+2}{(4 x-5)^{2}} d x$ using the substitution $u=4 x-5$ . What does the integral become?

Find the limits for $f(x)=\frac{x^{2}-7x-18}{x-9}$ as $x$ approaches 9 from the left: A. $\lim f(x)=\square$ or B. DNE.

Evaluate the integral with substitution: $\int \frac{\sec ^{2}(4 \sqrt{x})}{\sqrt{x}} d x=\square+C$

Find the limits for $f(x)=\frac{x^{2}-7 x-18}{x-9}$ as $x$ approaches 9 from the left and right.

Evaluate the Riemann sum for $f(x)$ from $1$ to $7$ using three equal sub-intervals and left endpoints. Riemann Sum = $\square$ .

Calculate the integral from 3 to 4 of $\frac{x^{3}+4}{x}$ .

Find critical points of $f(x)=\cos\left(\frac{1}{x}\right)$ for $x \neq 0$ and classify them using the second derivative test.

Is the integral $\int_{6}^{\infty} \frac{2}{(x+4)^{3 / 2}} d x$ convergent or divergent? If convergent, evaluate it.

Use $u$ -substitution to find
$\int_{3}^{4}(4 x-5)^{3} d x = \int_{a}^{b} f(u) d u.$
Determine $u$ , $d u$ , $a$ , $b$ , and $f(u)$ . What is the integral's value?

Find $G^{\prime}(w)$ for $G(w)=\frac{4}{3 w^{5}}+5 \sqrt[4]{w}$ . What is $G^{\prime}(w)=\square$ ?

Evaluate the integral $\int_{0}^{1} \frac{-4}{\sqrt{1-x^{2}}} d x$ . If divergent, respond with "D".

Find the derivative $G^{\prime}(w)$ of the function $G(w)=\frac{4}{3 w^{5}}+5 \sqrt[4]{w}$ .

Find the derivative $G^{\prime}(w)$ for the function $G(w)=\frac{4}{3 w^{5}}+5 \sqrt[4]{w}$ .

Find the result of the integral $\int e^{3 x} dx$ after the substitution $u=3x$ .

Find $f(-1)$ given that $f^{\prime}(x)=8 x^{3}+12 x+2$ and $f(1)=-4$ .

Evaluate the Riemann sum of $f(x)$ on $[1,7]$ using 3 equal subintervals and left endpoints.

Implicit Differentiation: Verify that tangent lines of the bicorn curve $(x^{2}+8y-16)^{2}=y^{2}(16-x^{2})$ at $(0,4)$ and $(0,4/3)$ are horizontal. Do this by a) expanding and b) not expanding the equation.

Evaluate the integral $\int \frac{x-3}{(3 x-2)^{2}} d x$ using the substitution $u=3 x-2$ . Which transformed integral is correct?

Approximate $\int_{2}^{6} f(x) dx$ using a midpoint Riemann sum with 2 equal-width rectangles. Use values: $-3$ , $0$ , $1$ , $2$ .

Find $f(x)=x^{4}-8 x^{3}+5$ : (A) Find $f^{\prime}(x)$ . (B) Find the slope at $x=-2$ . (C) Find the tangent line equation at $x=-2$ . (D) Find where the tangent line is horizontal.

Find $f(x)=x^{4}-8 x^{3}+5$ : (A) $f^{\prime}(x)$ , (B) slope at $x=-2$ , (C) tangent line at $x=-2$ , (D) horizontal tangents.

Given a continuous function $f(x)$ , which statement about Riemann sums $L_{100}, R_{100}, M_{100}$ on $[10,20]$ is FALSE?

For $f(x)=x^{4}-8 x^{3}+5$ , find: (A) $f^{\prime}(x)$ , (B) slope at $x=-2$ , (C) tangent line at $x=-2$ , (D) where tangent is horizontal.

Find the cost of producing the 71st food processor using $C(x)=2000+60x-0.1x^{2}$ and marginal cost.

Find the Cauchy principal value of the integral: $\int_{0}^{\infty} \frac{\cos (9 x)}{(x^{2}+1)^{2}} dx$

Find the indefinite integral $\int \frac{x-3}{(3 x-2)^{2}} d x$ using the substitution $u=3 x-2$ .

Show that $f(x)=x^{4}+1-\frac{1}{x}$ has a root in $(\frac{1}{2}, 1)$ using the intermediate value theorem, then find it to two decimal places.

Bestimmen Sie die Ableitung von $f(t) = t^{4} - t + 1$ .

Evaluate the integral $\int \frac{x-3}{(3x-2)^{2}} dx$ using the substitution $u=3x-2$ . Which transformed integral is correct?

Bestimmen Sie die Ableitung von $f(x) = (3+x)(3-x)$ .

Untersuchen Sie die Extrem- und Sattelpunkte von $f(x)=x^{2}-4$ und $g(x)=x^{5}-15x^{3}-30$ .

Untersuchen Sie das Verhalten der Funktionen an den Definitionslücken:
a) $f(x)=\frac{x^{2}-9}{2 x-6}, x_{0}=3$
b) $f(x)=\frac{x+1}{x}, x_{0}=0$
c) $f(x)=\frac{x+1}{x^{2}}, x_{0}=0$

Leite die Funktion $-\frac{5}{x}$ ab.

Bestimmen Sie die Ableitung von $f_{1}(x)=2 \sin x+8$ .

Find the derivative and integral of $f_{1}(x)=2 \sin x+8$ .

Given $f(x)=\frac{2-x}{x+4}$ , find intercepts, asymptotes, increasing/decreasing intervals, concavity, and sketch the graph.

Hitung biaya marjinal (MC) dari persamaan total biaya $T C=2 Q^{2}-24 Q+102$ saat produksi meningkat 1 unit.

Hitung biaya marjinal (MC) dari persamaan $T C=2 Q^{2}-24 Q+102$ jika produksinya ditingkatkan 1 unit.

Cari unit $X$ yang harus dikonsumsi agar kepuasan maksimum tercapai, dengan fungsi total utilitas $TU = 100x - 5x^{2}$ dan harga Rp50 per unit.

Bestimme $u$ , sodass der Flächeninhalt des Rechtecks, definiert durch $P(u|f(u))$ , maximal ist. Was passiert, wenn $u \to \infty$ ?

Ein Flugobjekt bremst von $150 \mathrm{~km/h}$ auf $96 \mathrm{~km/h}$ in 10 Sekunden. Berechnet die Bremsstrecke und die Teilintervalle.

Find the derivative of $y=\frac{1}{\sqrt{x}}-\frac{24}{x}$ and evaluate it at $x=2$ , rounding to two decimal places.

Find the value of the function $y=\frac{x^{3}}{3}-x^{2}-3 x+\frac{11}{3}$ at its inflection point.

Find the stationary point of the function $y=3x^{2}-3x+20$ and determine if it's a max, min, or inflection point.

Find the derivative of $y=\left(x^{3}+1\right)\left(x^{0.5}-1/x\right)$ at $x=3$ and round to two decimal places.

Find the derivative of $y=\frac{40\left(x^{0.4}+2\right)}{1-x^{2}}$ at $x=6$ and round to two decimal places.

Determine the correctness of these statements for the function $\frac{x^{3}}{3}-x^{2}-3 x+9$ : a) max at $x=3$ , b) inflection at $x=1$ , c) decreasing for $x>3$ , d) min at $x=9$ .

Substitute $h=0$ into $nt=\frac{1.5h}{0.1+0.1h}$ and $y(h)=\frac{1.5h^2}{0.1+0.1h}$ . Find values rounded to two decimals. Check stability using the slope criterion.

Evaluate the integral $\int_{0}^{2}|x-1| e^{x} d x$ using the piecewise definition of $|x-1|$ .

Calculate the area between the parabola $y=3x^2+20x+5$ and the line $y=2x-10$ for $x \in[-10,0]$ .

Evaluate the integral $\int \frac{\mathrm{e}^{1 / \mathrm{x}^{2}}}{\mathrm{x}^{3}} \mathrm{dx}$ .

Find the integral $\int \frac{\tan ^{3} x}{\sqrt{\sec x}} d x$ .

Evaluate the Riemann sum of $f(x)$ on $[1,7]$ using three equal subintervals and left endpoints for given values.

Approximate $\int_{0}^{6} f(x) dx$ using a Riemann sum with 3 equal-width rectangles and midpoints.

Compute $\int_{-2}^{5} f(x) d x$ given $\int_{-2}^{7}(3 f(x)) d x=12$ and $\int_{5}^{7} f(x) d x=2$ .

Given a continuous function $f(x)$ on $[10,20]$ , which statement about Riemann sums $L_{100}, R_{100}, M_{100}$ is FALSE?

Find inflection points and concavity intervals for the function $f(x)=x e^{x-1}$ .

Gegeben ist die Funktion $f(x)=x^{3}-6 x^{2}+11 x-6$ .
a) Zeigen Sie, dass der Wendepunkt auf der Geraden $y=x-2$ liegt.
b) Nach Verschiebung hat der Punkt (2,10) die Koordinaten (3,2). Bestimmen Sie die Gleichung der Funktion $h$ .

Determine the horizontal asymptote(s) of the function $f(x)=-\frac{8}{x-2}$ .

Oblicz pochodną funkcji $y'=\left(\sqrt[4]{1+\cos ^{2} x}\right)$ .

Rozwiąż równanie różniczkowe $y'=\sqrt[4]{1+\sin^2 x}$ .

Find the elasticity function for the demand function $p=D(x)=196-3.5x$ .

Find where $f(x)=2x+\frac{8}{x}$ is increasing/decreasing and the $x$ -coordinates of relative maxima/minima.