Calculus

Problem 13701

Find the derivative of the constant function $f(x)=5$ using first principles.

See Solution

Problem 13702

Finde die Stammfunktionen für:
a) $f(x)=2 x^{4}-3 x+\frac{5}{7}$
b) $f(x)=-3 \cos (x)+e^{2 x}$
c) $f(x)=-(2 x-3)^{5}$

See Solution

Problem 13703

Bestimme das Globalverhalten von $f(x)=x^{2} \cdot e^{-x}$ .

See Solution

Problem 13704

Bestimmen Sie die Tangentengleichung der Funktion $-\frac{1}{2} x^{2}+3$ bei $x_0 = -2$ .

See Solution

Problem 13705

Bestimme das Verhalten von $f(x)=x^{2} \cdot e^{-x}$ für $x \rightarrow-\infty$ und $x \rightarrow\infty$ .

See Solution

Problem 13706

Bestimme das Globalverhalten von $f(x)=(x^{2}-1) \cdot e^{-x}$ für $x \rightarrow -\infty$ und $x \rightarrow +\infty$ .

See Solution

Problem 13707

Welche Aussagen sind korrekt zum Verhalten von $f(x)=x^{2} \cdot e^{-x}$ für $x \rightarrow \pm \infty$ ?

See Solution

Problem 13708

Approximate the displacement of an object with velocity $v=\frac{1}{(4t+1)}$ for $0 \leq t \leq 8$ using 4 subintervals.

See Solution

Problem 13709

Does the right Riemann sum underestimate or overestimate the area under a positive decreasing function? Explain.
A. Underestimate; rectangles fit under the curve. B. Overestimate; rectangles do not fit under the curve. C. Overestimate; rectangles fit under the curve. D. Underestimate; rectangles do not fit under the curve.

See Solution

Problem 13710

Finde die Stellen $x$ , wo der Anstieg von $h(x)=x^{2}-x-1$ gleich 1 ist, und die Gleichung der Normalen in diesem Punkt.

See Solution

Problem 13711

Approximate the displacement of an object with velocity $v=\frac{1}{(4t+1)}$ from $t=0$ to $t=8$ using 4 subintervals. Displacement: $\square$ m.

See Solution

Problem 13712

Gegeben ist die Funktion $f(x)=x^{4}-2 x^{2}+2$ . Bestimme Extrempunkte und Achsenschnittpunkte. Gibt es Wendepunkte?

See Solution

Problem 13713

Bestimme das Globalverhalten von $f(x)=x \cdot e^{-x}$ für $x \rightarrow -\infty$ und $x \rightarrow +\infty$ .

See Solution

Problem 13714

Gegeben ist die Funktion $f(x)=0.5x^2$ .
a) Bestimme die Untersumme $U_{4}$ für das Intervall [0;3] in 4 Teile.
b) Finde den Inhalt der Fläche zwischen $f(x)$ und der x-Achse.

See Solution

Problem 13715

Approximate the displacement of an object with velocity $v=\frac{2}{3} t^{2}+6(f t / s)$ on $0 \leq t \leq 18$ using $n=6$ .

See Solution

Problem 13716

Gegeben ist die Funktion $f(x)=x^{4}-2x^{2}+2$ : a) Bestimme Extrempunkte und Achsenschnittpunkte. b) Gibt es Wendepunkte? c) Finde die Sekanten-Gleichung im Intervall $[0,4]$ und beschreibe, was sie darstellt.

See Solution

Problem 13717

Aufgabe 1: Untersuchen Sie $f(x)=x^{3}-5 x^{2}-6 x$ vollständig (Ableitungen, Nullstellen, Extremstellen, etc.).
Aufgabe 2: a) Bestimmen Sie die Untersumme $\mathrm{U}_{4}$ für [0;3] in 4 Teile. b) Berechnen Sie die Fläche.
Aufgabe 3: Finden Sie eine Stammfunktion für jede Funktion: a) $f(x)=3 x$ , b) $f(x)=2 x-5$ , c) $f(x)=3(x-3)^{3}$ , d) $f(x)=a b^{3} x^{2}$ , e) $f(b)=a b^{3} x^{2}$ , f) $f(x)=x^{n-1}$ , g) $f(x)=(x-14)^{2}$ , h) $f(x)=\frac{5}{x^{4}}$ , i) $f(x)=\sqrt{x}$ , j) $f(x)=\sqrt{9 x}$ , k) $f(x)=\frac{1}{5 \sqrt{x}}$ , l) $f(x)=\frac{1}{3} x+\frac{1}{x^{2}}$ .
Aufgabe 4: a) Finden Sie die Schnittpunkte von $f(x)=-2 x^{2}-5 x+5$ und $g(x)=x^{2}-2 x-1$ . b) Skizzieren Sie die Graphen. c) Berechnen Sie die Fläche zwischen den Graphen.
Aufgabe 5: Bestimmen Sie die Fläche, die vom Graphen $f(x)=x^{3}-4 x^{2}+x+6$ und der X-Achse eingeschlossen wird.

See Solution

Problem 13718

Bestimmen Sie, ob die Funktion $f(x, t) = \frac{1}{x^{2}} \cdot e^{\frac{2 t}{x}}$ integriert, abgeleitet oder grenzwertig untersucht werden soll.

See Solution

Problem 13719

Find $\frac{d y}{d x}$ using first principles for: a) $y=8 x$ , b) $y=3 x^{2}-2 x$ , c) $y=7-x^{2}$ , d) $y=x(4 x+5)$ .

See Solution

Problem 13720

Find $\frac{d y}{d x}$ using first principles for: a) $y=8 x$ , b) $y=3 x^{2}-2 x$ , c) $y=7-x^{2}$ , d) $y=x(4 x+5)$ , e) $y=(2 x-1)^{2}$ .

See Solution

Problem 13721

Solve the differential equation $y^{\prime}=3 e^{x}+x^{2}-4$ with the initial condition $y(0)=5$ .

See Solution

Problem 13722

Find the derivative $f^{\prime}(x)$ using first principles for the function $y=(2x-1)^{2}$ .

See Solution

Problem 13723

Find the general antiderivative of $f(x)=\frac{1}{5+5 x^{2}}$ using $C$ as an arbitrary constant.

See Solution

Problem 13724

Find the general antiderivative of $f(x)=(2-3x^{2})^{2}$ on interval $I$ , using $C$ for the constant of integration.

See Solution

Problem 13725

Expand $(x+h)^{4}$ and differentiate $y=x^{4}$ , $y=2x^{4}$ , $y=3x^{4}$ using first principles. Identify and predict patterns.

See Solution

Problem 13726

Bestimme das höchste Fieber und den Zeitpunkt, wann es auftritt. Finde den Zeitpunkt des stärksten Fieberabfalls und die durchschnittliche Temperatur in 12 Stunden.

See Solution

Problem 13727

Find $\frac{d y}{d x}$ for the curve $x(t)=e^{-t}$ and $y(t)=e^{2 t}-1$ at $t=\frac{1}{3}$ .

See Solution

Problem 13728

Berechne die Ableitung von $f^{\prime}(x)=\frac{\sqrt{7 x-7}}{2 x-2}$ unter Verwendung der Ketten- und Quotientenregel.

See Solution

Problem 13729

Find the derivative of $y$ with respect to $x$ using logarithmic differentiation for $y = x^{9} \sin x$ .

See Solution

Problem 13730

Find the integral expression for the volume of the solid formed by rotating the region bounded by $y=x^{5}$ , $x=-1$ , $x=2$ , and the $y$ -axis around the $y$ -axis.

See Solution

Problem 13731

A 30 ft ladder slides down a wall at 10 ft/s. How fast is the bottom sliding out when the top is 18 ft high?

See Solution

Problem 13732

Find the integral for the volume of the solid formed by rotating the region bounded by $y=x^{5}$ , $x=-1$ , $x=2$ , and the $y$ -axis around the $y$ -axis.

See Solution

Problem 13733

A cone filter is 20 in tall and 5 in radius. Liquid flows out at 4 in³/s. Find depth and surface area change at 12 in deep.

See Solution

Problem 13734

Find the derivative of $f(x)=3 x^{-4}-2 x^{7}$ .

See Solution

Problem 13735

Determine if the integral $\int_{4}^{\infty} \frac{\sqrt[3]{x^{4}+1}}{x^{2}} d x$ converges or diverges using comparison or Big-O notation.

See Solution

Problem 13736

Given $xy=2$ , find $d y / d t$ when $x=2$ , $d x / d t=10$ and $d x / d t$ when $x=1$ , $d y / d t=-8$ .

See Solution

Problem 13737

Given $y=\sqrt{x}$ , find $d y / d t$ when $x=1$ and $d x / d t=6$ , and $d x / d t$ when $x=49$ and $d y / d t=5$ .

See Solution

Problem 13738

Bestimme $a$ so, dass die Fläche unter $f(x)=x^{2}$ von $0$ bis $4$ im Verhältnis $1:7$ durch $x=a$ geteilt wird.

See Solution

Problem 13739

Calculez la dérivée de $S(t)=3.1 \ln [1210(t+10)]$ pour trouver le taux de variation en 2005 ( $t=3$ ).

See Solution

Problem 13740

Profit annuel $P(q)=500 \sqrt{q}+10 q$ avec $q(n)=10000 n-100 n^{2}-n^{3}$ . Trouvez $\frac{d P}{d n}$ et évaluez-le pour $n=25$ . Justifiez l'embauche de travailleurs.

See Solution

Problem 13741

Find the volume of the solid formed by rotating the area between $y=\sin(x)$ and the $x$ -axis ( $\frac{\pi}{2} \leq x \leq \pi$ ) around $y=-1$ .

See Solution

Problem 13742

Calculez le profit marginal $\frac{d P}{d n}$ à partir de $P(q)=500 \sqrt{q}+10 q$ et $q(n)=10000 n-100 n^{2}-n^{3}$ .

See Solution

Problem 13743

Find the derivative of $f(x)=\frac{\sin x}{3 e^{x}}$ .

See Solution

Problem 13744

Calculez le profit marginal $P'(q)$ pour 25 travailleurs, sachant que $P(q)=500 \sqrt{q}+10 q$ et $q(n)=10000 n-100 n^{2}-n^{3}$ .

See Solution

Problem 13745

Find the tangent line equation for $f(x)=\frac{5 x^{2}}{(3-2 x)^{3}}$ at $x=2$ : $y=$

See Solution

Problem 13746

Find the second derivative $d^{2} y / d x^{2}$ for the function $y=x \cos (5 x)-\sin ^{2}(x)$ .

See Solution

Problem 13747

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(8x-5)^{3}(4x^{2}+2)^{2}$ .

See Solution

Problem 13748

Find the population $N(9)$ of a city after 9 years if $\frac{dN}{dt}=900+600\sqrt{t}$ and $N(0)=5000$ .

See Solution

Problem 13749

Find the total cost function $M(x)$ for drilling an oil well, given $M^{\prime}(x)=2000+80x$ and $M(0)=40000$ . $M(x)=\square$

See Solution

Problem 13750

Find the population $N(16)$ after 16 years if $\frac{d N}{d t}=600+300 \sqrt{t}$ and $N(0)=3000$ .

See Solution

Problem 13751

Find the total cost function $M(x)$ for drilling an oil well, given $\mathrm{M}^{\prime}(x)=2000+80x$ and fixed cost $\$ 40,000$ .

See Solution

Problem 13752

Fonction $f(x)=x e^{1-2 x^{2}}$ : a) Trouver $\lim_{x \to -\infty} f(x)$ et $\lim_{x \to \infty} f(x)$ . b) Identifier et classifier les extremums de $f$ .

See Solution

Problem 13753

Find $g(x)$ if $\int g(x) dx = (2x^2 + 9)^4 + C$ .

See Solution

Problem 13754

Find $f(x)$ if $\int f(x) d x=(x+3)^{3} e^{x}+\ln (|x|)+C$ .

See Solution

Problem 13755

Find $f(x)$ if $\int f(x) dx = (x+3) \ln (|x-3|) + C$ .

See Solution

Problem 13756

Find the company's worth in 2000 if its growth rate is given by $f^{\prime}(t)=1870-8t^{2}$ and $f(0)=75000$ .

See Solution

Problem 13757

Find the integral of $\frac{1}{(4 x-9)^{5}}$ with respect to $x$ .

See Solution

Problem 13758

Find the integral $\int(1+2 x^{2}) e^{x^{2}} dx$ . Which option is correct? A, B, or C?

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Problem 13759

Find the total cost for producing 22 units given the marginal cost $C^{\prime}(q)=3 q^{2}-18 q+78$ and $C(0)=490$ .

See Solution

Problem 13760

Find $f(3)$ given that $f^{\prime}(t)=3 t^{2}+e^{t}$ and $f(0)=4$ .

See Solution

Problem 13761

A bike manufacturer has a marginal cost function $C'(q)=\frac{700}{0.8 q+6}$ .
a) With fixed costs of \$3400, find total cost for 25 bikes.
b) If selling price is \$300 each, what is profit/loss for 25 bikes?
c) What is the marginal profit for bike 26?

See Solution

Problem 13762

Find the distance traveled by a car with velocity $v(t)=10 t+\left(\frac{3}{64}\right) t^{2}$ from $t=0$ to $t=4$ .

See Solution

Problem 13763

Una moneda cae desde una mesa y tarda 9.1 segundos en llegar al suelo. ¿Cuál es su velocidad de caída? (5 puntos)

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Problem 13764

A 25 ft ladder leans against a wall. Base moves away at 2 ft/sec. Find top's velocity at 7, 15, and 20 ft from the wall. Also, rate of angle change at 15 ft.

See Solution

Problem 13765

Find the profit formula $P(x)$ given $P^{\prime}(x)=300+0.6x$ and $P(0)=-3000$ .

See Solution

Problem 13766

A bike manufacturer has a marginal cost function $C^{\prime}(q)=\frac{700}{0.8 q+6}$ .
a) Total cost for 25 bikes with fixed cost \$ 3400?
b) Profit/loss on first 25 bikes sold at \$ 300 each?
c) Marginal profit on bike 26?

See Solution

Problem 13767

Find the limit of $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{1}{x+1}$ .

See Solution

Problem 13768

Find the rates of change of area $A = xy$ and diagonal $d = \sqrt{x^2 + y^2}$ for given $x$ and $y$ values.

See Solution

Problem 13769

Find the rate of change of area $A = xy$ when $x=3 \mathrm{~cm}$ , $y=4 \mathrm{~cm}$ , with rates $dx/dt = -3$ and $dy/dt = 2$ .

See Solution

Problem 13770

The radius $r$ is shrinking at $2 \mathrm{~cm/sec}$ , and height $h$ is growing at $1 \mathrm{~cm/sec}$ . Find $A'(t)$ for $A=2 \pi r h$ when $r=1 \mathrm{~cm}$ and $h=3 \mathrm{~cm}$ .

See Solution

Problem 13771

Evaluate the limit as $x$ approaches 2 from the right: $\lim _{x \rightarrow 2^{+}}\left(\frac{\ln (x-1)}{(x-2)^{2}}\right)$ .

See Solution

Problem 13772

Show that $\lim _{x \rightarrow 0}\left(\frac{\sin (x)}{x}\right)=1$ using l'Hopital's Rule.

See Solution

Problem 13773

Find the horizontal asymptote of the function $f(x)=\frac{3 x^{4}+5}{5 x^{4}-2}$ .

See Solution

Problem 13774

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(x^{2}-e^{x}\right)$ .

See Solution

Problem 13775

Trouvez l'équation de la tangente à $y=\frac{x^{2}+2}{x-1}$ au point $x=0$ et tracez-la sur le graphique.

See Solution

Problem 13776

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

See Solution

Problem 13777

Déterminez $\lim _{x \rightarrow-\infty} x e^{1-2 x^{2}}$ et $\lim_{x \rightarrow \infty} x e^{1-2 x^{2}}$ .

See Solution

Problem 13778

A 700 g isotope decays as $A(t)=700 e^{-0.045 t}$ . Find: (a) amount left after 30 years, (b) time to half decay.

See Solution

Problem 13779

Pour la fonction $f(x)=x e^{1-2 x^{2}}$ , calculez les limites $\lim _{x \rightarrow-\infty} f(x)$ et $\lim _{x \rightarrow \infty} f(x)$ . Ensuite, trouvez et classez les extremums de $f$ .

See Solution

Problem 13780

A dinosaur's bones weigh 170 lbs and are 9 ft long. Show $A=A_{0} e^{-0.50228 t}$ using decay rate $k$ .

See Solution

Problem 13781

Calculate the left and right Riemann sums for $f(x)=\frac{2}{x}+3$ on $[1,5]$ with $n=4$ .

See Solution

Problem 13782

Invest \ $18,755 at 6.9\% interest, compounded continuously. Find the function for amount after time$ t$, and balances for 1, 2, 5, 10 years, plus doubling time.

See Solution

Problem 13783

Calculate the left and right Riemann sums for $f(x)=\frac{2}{x}+3$ on $[1,5]$ with $n=4$ .

See Solution

Problem 13784

Find the area between the curves $y=2 x^{3}+18$ and $y=2 x^{3}+x^{2}-3 x$ at their intersection points.

See Solution

Problem 13785

Calculate the area under the curve $y=12 x^{2}$ from $x=1$ to $x=2$ .

See Solution

Problem 13786

Calculate the area between the curves $y=x^{3}+6$ and $y=x^{3}+3 x^{2}-3 x$ where they intersect.

See Solution

Problem 13787

Find the derivative of the function $f(x)=\sqrt{2x+1}$ . What is $f'(x)$ ?

See Solution

Problem 13788

Find the value(s) of $c$ from the Mean Value Theorem for Integrals for $f(x)=\cos x$ on $[-\frac{\pi}{3}, \frac{\pi}{3}]$ . Round to four decimal places.

See Solution

Problem 13789

Find $c$ from the Mean Value Theorem for Integrals for $f(x)=\cos x$ over $\left[-\frac{\pi}{3}, \frac{\pi}{3}\right]$ .

See Solution

Problem 13790

Calculate the left and right Riemann sums for $f(x)=\frac{3}{x}+2$ on $[1,5]$ with $n=4$ .

See Solution

Problem 13791

Find $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ . When is the curve concave upward for $x=t^{2}+a t+a^{2}$ , $y=\frac{1}{3} t^{3}-2 a t^{2}$ ?

See Solution

Problem 13792

Find the derivative of $f(x)=\frac{1}{8} e^{x}+14 x^{6}+\frac{x}{8} e^{x}$ .

See Solution

Problem 13793

Find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ . For which $t$ is the curve concave upward? Given $x=t^2+at+a^2$ , $y=\frac{1}{3}t^3-2at^2$ ( $a>0$ ).

See Solution

Problem 13794

Analyze the function $y=\frac{x}{x^{2}+49}$ . Find intercepts, extrema, inflection points, and asymptotes.

See Solution

Problem 13795

Find values of $t$ where the curve of the function $\frac{(2 t-4 a)(2 t+a)-\left(t^{2}-4 a b(2)\right)}{(2 t+a)^{3}}$ is increasing.

See Solution

Problem 13796

Analyze the function $y=\frac{x}{x^{2}+49}$ . Find intercepts, extrema, inflection points, and asymptotes.

See Solution

Problem 13797

Determine if the curve defined by $x=c^{2 \cos a t}, y=\sin^{2} a t$ has no horizontal tangent for $a>0$ . True or false?

See Solution

Problem 13798

How long does it take for a basketball dropped from $24 \mathrm{~m}$ to hit the ground?

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Problem 13799

Find the rate of change for $f(x)=3x+1$ and $g(x)=3^x+1$ over $[1,4]$ and determine which grows faster.

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Problem 13800

Berechne die Halbwertszeit von Cäsium, wenn nach 1 Jahr 2,3\% zerfallen sind. Bestimme auch den Rest nach 60 Jahren. Jod mit 8 Tagen Halbwertszeit: Wie lange bis 0,1\% übrig?

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Calculus

Problem 13701

Find the derivative of the constant function f(x)=5f(x)=5f(x)=5 using first principles.

Problem 13702

Finde die Stammfunktionen für:a) f(x)=2x4−3x+57f(x)=2 x^{4}-3 x+\frac{5}{7}f(x)=2x4−3x+75​b) f(x)=−3cos⁡(x)+e2xf(x)=-3 \cos (x)+e^{2 x}f(x)=−3cos(x)+e2xc) f(x)=−(2x−3)5f(x)=-(2 x-3)^{5}f(x)=−(2x−3)5

Problem 13703

Bestimme das Globalverhalten von f(x)=x2⋅e−xf(x)=x^{2} \cdot e^{-x}f(x)=x2⋅e−x.

Problem 13704

Bestimmen Sie die Tangentengleichung der Funktion −12x2+3-\frac{1}{2} x^{2}+3−21​x2+3 bei x0=−2x_0 = -2x0​=−2.

Problem 13705

Bestimme das Verhalten von f(x)=x2⋅e−xf(x)=x^{2} \cdot e^{-x}f(x)=x2⋅e−x für x→−∞x \rightarrow-\inftyx→−∞ und x→∞x \rightarrow\inftyx→∞.

Problem 13706

Bestimme das Globalverhalten von f(x)=(x2−1)⋅e−xf(x)=(x^{2}-1) \cdot e^{-x}f(x)=(x2−1)⋅e−x für x→−∞x \rightarrow -\inftyx→−∞ und x→+∞x \rightarrow +\inftyx→+∞.

Problem 13707

Welche Aussagen sind korrekt zum Verhalten von f(x)=x2⋅e−xf(x)=x^{2} \cdot e^{-x}f(x)=x2⋅e−x für x→±∞x \rightarrow \pm \inftyx→±∞?

Problem 13708

Approximate the displacement of an object with velocity v=1(4t+1)v=\frac{1}{(4t+1)}v=(4t+1)1​ for 0≤t≤80 \leq t \leq 80≤t≤8 using 4 subintervals.

Problem 13709

Problem 13710

Finde die Stellen xxx, wo der Anstieg von h(x)=x2−x−1h(x)=x^{2}-x-1h(x)=x2−x−1 gleich 1 ist, und die Gleichung der Normalen in diesem Punkt.

Problem 13711

Approximate the displacement of an object with velocity v=1(4t+1)v=\frac{1}{(4t+1)}v=(4t+1)1​ from t=0t=0t=0 to t=8t=8t=8 using 4 subintervals. Displacement: □\square□ m.

Problem 13712

Gegeben ist die Funktion f(x)=x4−2x2+2f(x)=x^{4}-2 x^{2}+2f(x)=x4−2x2+2. Bestimme Extrempunkte und Achsenschnittpunkte. Gibt es Wendepunkte?

Problem 13713

Bestimme das Globalverhalten von f(x)=x⋅e−xf(x)=x \cdot e^{-x}f(x)=x⋅e−x für x→−∞x \rightarrow -\inftyx→−∞ und x→+∞x \rightarrow +\inftyx→+∞.

Problem 13714

Gegeben ist die Funktion f(x)=0.5x2f(x)=0.5x^2f(x)=0.5x2. a) Bestimme die Untersumme U4U_{4}U4​ für das Intervall [0;3] in 4 Teile.b) Finde den Inhalt der Fläche zwischen f(x)f(x)f(x) und der x-Achse.

Problem 13715

Approximate the displacement of an object with velocity v=23t2+6(ft/s)v=\frac{2}{3} t^{2}+6(f t / s)v=32​t2+6(ft/s) on 0≤t≤180 \leq t \leq 180≤t≤18 using n=6n=6n=6.

Problem 13716

Gegeben ist die Funktion f(x)=x4−2x2+2f(x)=x^{4}-2x^{2}+2f(x)=x4−2x2+2: a) Bestimme Extrempunkte und Achsenschnittpunkte. b) Gibt es Wendepunkte? c) Finde die Sekanten-Gleichung im Intervall [0,4][0,4][0,4] und beschreibe, was sie darstellt.

Problem 13717

Problem 13718

Bestimmen Sie, ob die Funktion f(x,t)=1x2⋅e2txf(x, t) = \frac{1}{x^{2}} \cdot e^{\frac{2 t}{x}}f(x,t)=x21​⋅ex2t​ integriert, abgeleitet oder grenzwertig untersucht werden soll.

Problem 13719

Find dydx\frac{d y}{d x}dxdy​ using first principles for: a) y=8xy=8 xy=8x, b) y=3x2−2xy=3 x^{2}-2 xy=3x2−2x, c) y=7−x2y=7-x^{2}y=7−x2, d) y=x(4x+5)y=x(4 x+5)y=x(4x+5).

Problem 13720

Find dydx\frac{d y}{d x}dxdy​ using first principles for: a) y=8xy=8 xy=8x, b) y=3x2−2xy=3 x^{2}-2 xy=3x2−2x, c) y=7−x2y=7-x^{2}y=7−x2, d) y=x(4x+5)y=x(4 x+5)y=x(4x+5), e) y=(2x−1)2y=(2 x-1)^{2}y=(2x−1)2.

Problem 13721

Solve the differential equation y′=3ex+x2−4y^{\prime}=3 e^{x}+x^{2}-4y′=3ex+x2−4 with the initial condition y(0)=5y(0)=5y(0)=5.

Problem 13722

Find the derivative f′(x)f^{\prime}(x)f′(x) using first principles for the function y=(2x−1)2y=(2x-1)^{2}y=(2x−1)2.

Problem 13723

Find the general antiderivative of f(x)=15+5x2f(x)=\frac{1}{5+5 x^{2}}f(x)=5+5x21​ using CCC as an arbitrary constant.

Problem 13724

Find the general antiderivative of f(x)=(2−3x2)2f(x)=(2-3x^{2})^{2}f(x)=(2−3x2)2 on interval III, using CCC for the constant of integration.

Problem 13725

Expand (x+h)4(x+h)^{4}(x+h)4 and differentiate y=x4y=x^{4}y=x4, y=2x4y=2x^{4}y=2x4, y=3x4y=3x^{4}y=3x4 using first principles. Identify and predict patterns.

Problem 13726

Bestimme das höchste Fieber und den Zeitpunkt, wann es auftritt. Finde den Zeitpunkt des stärksten Fieberabfalls und die durchschnittliche Temperatur in 12 Stunden.

Problem 13727

Find dydx\frac{d y}{d x}dxdy​ for the curve x(t)=e−tx(t)=e^{-t}x(t)=e−t and y(t)=e2t−1y(t)=e^{2 t}-1y(t)=e2t−1 at t=13t=\frac{1}{3}t=31​.

Problem 13728

Berechne die Ableitung von f′(x)=7x−72x−2f^{\prime}(x)=\frac{\sqrt{7 x-7}}{2 x-2}f′(x)=2x−27x−7​​ unter Verwendung der Ketten- und Quotientenregel.

Problem 13729

Find the derivative of yyy with respect to xxx using logarithmic differentiation for y=x9sin⁡xy = x^{9} \sin xy=x9sinx.

Problem 13730

Find the integral expression for the volume of the solid formed by rotating the region bounded by y=x5y=x^{5}y=x5, x=−1x=-1x=−1, x=2x=2x=2, and the yyy-axis around the yyy-axis.

Problem 13731

A 30 ft ladder slides down a wall at 10 ft/s. How fast is the bottom sliding out when the top is 18 ft high?

Problem 13732

Find the integral for the volume of the solid formed by rotating the region bounded by y=x5y=x^{5}y=x5, x=−1x=-1x=−1, x=2x=2x=2, and the yyy-axis around the yyy-axis.

Problem 13733

A cone filter is 20 in tall and 5 in radius. Liquid flows out at 4 in³/s. Find depth and surface area change at 12 in deep.

Problem 13734

Find the derivative of f(x)=3x−4−2x7f(x)=3 x^{-4}-2 x^{7}f(x)=3x−4−2x7.

Problem 13735

Determine if the integral ∫4∞x4+13x2dx\int_{4}^{\infty} \frac{\sqrt[3]{x^{4}+1}}{x^{2}} d x∫4∞​x23x4+1​​dx converges or diverges using comparison or Big-O notation.

Problem 13736

Given xy=2xy=2xy=2, find dy/dtd y / d tdy/dt when x=2x=2x=2, dx/dt=10d x / d t=10dx/dt=10 and dx/dtd x / d tdx/dt when x=1x=1x=1, dy/dt=−8d y / d t=-8dy/dt=−8.

Problem 13737

Given y=xy=\sqrt{x}y=x​, find dy/dtd y / d tdy/dt when x=1x=1x=1 and dx/dt=6d x / d t=6dx/dt=6, and dx/dtd x / d tdx/dt when x=49x=49x=49 and dy/dt=5d y / d t=5dy/dt=5.

Problem 13738

Bestimme aaa so, dass die Fläche unter f(x)=x2f(x)=x^{2}f(x)=x2 von 000 bis 444 im Verhältnis 1:71:71:7 durch x=ax=ax=a geteilt wird.

Problem 13739

Calculez la dérivée de S(t)=3.1ln⁡[1210(t+10)]S(t)=3.1 \ln [1210(t+10)]S(t)=3.1ln[1210(t+10)] pour trouver le taux de variation en 2005 (t=3t=3t=3).

Problem 13740

Profit annuel P(q)=500q+10qP(q)=500 \sqrt{q}+10 qP(q)=500q​+10q avec q(n)=10000n−100n2−n3q(n)=10000 n-100 n^{2}-n^{3}q(n)=10000n−100n2−n3. Trouvez dPdn\frac{d P}{d n}dndP​ et évaluez-le pour n=25n=25n=25. Justifiez l'embauche de travailleurs.

Problem 13741

Find the derivative of the constant function $f(x)=5$ using first principles.

Finde die Stammfunktionen für:
a) $f(x)=2 x^{4}-3 x+\frac{5}{7}$
b) $f(x)=-3 \cos (x)+e^{2 x}$
c) $f(x)=-(2 x-3)^{5}$

Bestimme das Globalverhalten von $f(x)=x^{2} \cdot e^{-x}$ .

Bestimmen Sie die Tangentengleichung der Funktion $-\frac{1}{2} x^{2}+3$ bei $x_0 = -2$ .

Bestimme das Verhalten von $f(x)=x^{2} \cdot e^{-x}$ für $x \rightarrow-\infty$ und $x \rightarrow\infty$ .

Bestimme das Globalverhalten von $f(x)=(x^{2}-1) \cdot e^{-x}$ für $x \rightarrow -\infty$ und $x \rightarrow +\infty$ .

Welche Aussagen sind korrekt zum Verhalten von $f(x)=x^{2} \cdot e^{-x}$ für $x \rightarrow \pm \infty$ ?

Approximate the displacement of an object with velocity $v=\frac{1}{(4t+1)}$ for $0 \leq t \leq 8$ using 4 subintervals.

Finde die Stellen $x$ , wo der Anstieg von $h(x)=x^{2}-x-1$ gleich 1 ist, und die Gleichung der Normalen in diesem Punkt.

Approximate the displacement of an object with velocity $v=\frac{1}{(4t+1)}$ from $t=0$ to $t=8$ using 4 subintervals. Displacement: $\square$ m.

Gegeben ist die Funktion $f(x)=x^{4}-2 x^{2}+2$ . Bestimme Extrempunkte und Achsenschnittpunkte. Gibt es Wendepunkte?

Bestimme das Globalverhalten von $f(x)=x \cdot e^{-x}$ für $x \rightarrow -\infty$ und $x \rightarrow +\infty$ .

Gegeben ist die Funktion $f(x)=0.5x^2$ .
a) Bestimme die Untersumme $U_{4}$ für das Intervall [0;3] in 4 Teile.
b) Finde den Inhalt der Fläche zwischen $f(x)$ und der x-Achse.

Approximate the displacement of an object with velocity $v=\frac{2}{3} t^{2}+6(f t / s)$ on $0 \leq t \leq 18$ using $n=6$ .

Gegeben ist die Funktion $f(x)=x^{4}-2x^{2}+2$ : a) Bestimme Extrempunkte und Achsenschnittpunkte. b) Gibt es Wendepunkte? c) Finde die Sekanten-Gleichung im Intervall $[0,4]$ und beschreibe, was sie darstellt.

Bestimmen Sie, ob die Funktion $f(x, t) = \frac{1}{x^{2}} \cdot e^{\frac{2 t}{x}}$ integriert, abgeleitet oder grenzwertig untersucht werden soll.

Find $\frac{d y}{d x}$ using first principles for: a) $y=8 x$ , b) $y=3 x^{2}-2 x$ , c) $y=7-x^{2}$ , d) $y=x(4 x+5)$ .

Find $\frac{d y}{d x}$ using first principles for: a) $y=8 x$ , b) $y=3 x^{2}-2 x$ , c) $y=7-x^{2}$ , d) $y=x(4 x+5)$ , e) $y=(2 x-1)^{2}$ .

Solve the differential equation $y^{\prime}=3 e^{x}+x^{2}-4$ with the initial condition $y(0)=5$ .

Find the derivative $f^{\prime}(x)$ using first principles for the function $y=(2x-1)^{2}$ .

Find the general antiderivative of $f(x)=\frac{1}{5+5 x^{2}}$ using $C$ as an arbitrary constant.

Find the general antiderivative of $f(x)=(2-3x^{2})^{2}$ on interval $I$ , using $C$ for the constant of integration.

Expand $(x+h)^{4}$ and differentiate $y=x^{4}$ , $y=2x^{4}$ , $y=3x^{4}$ using first principles. Identify and predict patterns.

Find $\frac{d y}{d x}$ for the curve $x(t)=e^{-t}$ and $y(t)=e^{2 t}-1$ at $t=\frac{1}{3}$ .

Berechne die Ableitung von $f^{\prime}(x)=\frac{\sqrt{7 x-7}}{2 x-2}$ unter Verwendung der Ketten- und Quotientenregel.

Find the derivative of $y$ with respect to $x$ using logarithmic differentiation for $y = x^{9} \sin x$ .

Find the integral expression for the volume of the solid formed by rotating the region bounded by $y=x^{5}$ , $x=-1$ , $x=2$ , and the $y$ -axis around the $y$ -axis.

Find the integral for the volume of the solid formed by rotating the region bounded by $y=x^{5}$ , $x=-1$ , $x=2$ , and the $y$ -axis around the $y$ -axis.

Find the derivative of $f(x)=3 x^{-4}-2 x^{7}$ .

Determine if the integral $\int_{4}^{\infty} \frac{\sqrt[3]{x^{4}+1}}{x^{2}} d x$ converges or diverges using comparison or Big-O notation.

Given $xy=2$ , find $d y / d t$ when $x=2$ , $d x / d t=10$ and $d x / d t$ when $x=1$ , $d y / d t=-8$ .

Given $y=\sqrt{x}$ , find $d y / d t$ when $x=1$ and $d x / d t=6$ , and $d x / d t$ when $x=49$ and $d y / d t=5$ .

Bestimme $a$ so, dass die Fläche unter $f(x)=x^{2}$ von $0$ bis $4$ im Verhältnis $1:7$ durch $x=a$ geteilt wird.

Calculez la dérivée de $S(t)=3.1 \ln [1210(t+10)]$ pour trouver le taux de variation en 2005 ( $t=3$ ).

Profit annuel $P(q)=500 \sqrt{q}+10 q$ avec $q(n)=10000 n-100 n^{2}-n^{3}$ . Trouvez $\frac{d P}{d n}$ et évaluez-le pour $n=25$ . Justifiez l'embauche de travailleurs.

Find the volume of the solid formed by rotating the area between $y=\sin(x)$ and the $x$ -axis ( $\frac{\pi}{2} \leq x \leq \pi$ ) around $y=-1$ .

Calculez le profit marginal $\frac{d P}{d n}$ à partir de $P(q)=500 \sqrt{q}+10 q$ et $q(n)=10000 n-100 n^{2}-n^{3}$ .

Find the derivative of $f(x)=\frac{\sin x}{3 e^{x}}$ .

Calculez le profit marginal $P'(q)$ pour 25 travailleurs, sachant que $P(q)=500 \sqrt{q}+10 q$ et $q(n)=10000 n-100 n^{2}-n^{3}$ .

Find the tangent line equation for $f(x)=\frac{5 x^{2}}{(3-2 x)^{3}}$ at $x=2$ : $y=$

Find the second derivative $d^{2} y / d x^{2}$ for the function $y=x \cos (5 x)-\sin ^{2}(x)$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(8x-5)^{3}(4x^{2}+2)^{2}$ .

Find the population $N(9)$ of a city after 9 years if $\frac{dN}{dt}=900+600\sqrt{t}$ and $N(0)=5000$ .

Find the total cost function $M(x)$ for drilling an oil well, given $M^{\prime}(x)=2000+80x$ and $M(0)=40000$ . $M(x)=\square$

Find the population $N(16)$ after 16 years if $\frac{d N}{d t}=600+300 \sqrt{t}$ and $N(0)=3000$ .

Find the total cost function $M(x)$ for drilling an oil well, given $\mathrm{M}^{\prime}(x)=2000+80x$ and fixed cost $\$ 40,000$ .

Fonction $f(x)=x e^{1-2 x^{2}}$ : a) Trouver $\lim_{x \to -\infty} f(x)$ et $\lim_{x \to \infty} f(x)$ . b) Identifier et classifier les extremums de $f$ .

Find $g(x)$ if $\int g(x) dx = (2x^2 + 9)^4 + C$ .

Find $f(x)$ if $\int f(x) d x=(x+3)^{3} e^{x}+\ln (|x|)+C$ .

Find $f(x)$ if $\int f(x) dx = (x+3) \ln (|x-3|) + C$ .

Find the company's worth in 2000 if its growth rate is given by $f^{\prime}(t)=1870-8t^{2}$ and $f(0)=75000$ .

Find the integral of $\frac{1}{(4 x-9)^{5}}$ with respect to $x$ .

Find the integral $\int(1+2 x^{2}) e^{x^{2}} dx$ . Which option is correct? A, B, or C?

Find the total cost for producing 22 units given the marginal cost $C^{\prime}(q)=3 q^{2}-18 q+78$ and $C(0)=490$ .

Find $f(3)$ given that $f^{\prime}(t)=3 t^{2}+e^{t}$ and $f(0)=4$ .

A bike manufacturer has a marginal cost function $C'(q)=\frac{700}{0.8 q+6}$ .
a) With fixed costs of \$3400, find total cost for 25 bikes.
b) If selling price is \$300 each, what is profit/loss for 25 bikes?
c) What is the marginal profit for bike 26?

Find the distance traveled by a car with velocity $v(t)=10 t+\left(\frac{3}{64}\right) t^{2}$ from $t=0$ to $t=4$ .

Find the profit formula $P(x)$ given $P^{\prime}(x)=300+0.6x$ and $P(0)=-3000$ .

A bike manufacturer has a marginal cost function $C^{\prime}(q)=\frac{700}{0.8 q+6}$ .
a) Total cost for 25 bikes with fixed cost \$ 3400?
b) Profit/loss on first 25 bikes sold at \$ 300 each?
c) Marginal profit on bike 26?

Find the limit of $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{1}{x+1}$ .

Find the rates of change of area $A = xy$ and diagonal $d = \sqrt{x^2 + y^2}$ for given $x$ and $y$ values.

Find the rate of change of area $A = xy$ when $x=3 \mathrm{~cm}$ , $y=4 \mathrm{~cm}$ , with rates $dx/dt = -3$ and $dy/dt = 2$ .

The radius $r$ is shrinking at $2 \mathrm{~cm/sec}$ , and height $h$ is growing at $1 \mathrm{~cm/sec}$ . Find $A'(t)$ for $A=2 \pi r h$ when $r=1 \mathrm{~cm}$ and $h=3 \mathrm{~cm}$ .

Evaluate the limit as $x$ approaches 2 from the right: $\lim _{x \rightarrow 2^{+}}\left(\frac{\ln (x-1)}{(x-2)^{2}}\right)$ .

Show that $\lim _{x \rightarrow 0}\left(\frac{\sin (x)}{x}\right)=1$ using l'Hopital's Rule.

Find the horizontal asymptote of the function $f(x)=\frac{3 x^{4}+5}{5 x^{4}-2}$ .

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(x^{2}-e^{x}\right)$ .