Calculus

Problem 4601

Evaluate and simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=3x-9$ .

See Solution

Problem 4602

Calculate the average rate of change of $g(x)=4 x^{3}+\frac{6}{x^{4}}$ over the interval $[-2,3]$ .

See Solution

Problem 4603

Evaluate $f(x+h)$ and simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=4x^{2}-4x+5$ .

See Solution

Problem 4604

Calculate the average rate of change of $f(x)=2 x^{2}-7$ from $x=1$ to $x=b$ . Express your answer in terms of $b$ .

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Problem 4605

Find $f(a)$ , $f(a+h)$ , and $\frac{f(a+h)-f(a)}{h}$ for $f(x)=5+5x^{2}$ .

See Solution

Problem 4606

Find the slope $m$ of the tangent line to $y=B(x)$ at $x=12$ where $B(12)=-1$ and $B'(12)=-\frac{1}{2}$ . Then, find the intersection point $(x, y)$ and the tangent line equation.

See Solution

Problem 4607

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=-2x^{2}$ . Steps: 1) Simplify $f(x+h)$ ; 2) Subtract $f(x+h)-f(x)$ ; 3) Divide by $h$ .

See Solution

Problem 4608

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

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Problem 4609

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=x^{2}$ . Step 1: Simplify $f(x+h)$ . Step 2: Simplify $f(x+h)-f(x)$ .

See Solution

Problem 4610

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=x^{2}$ . Steps: 1) Simplify $f(x+h)$ ; 2) Find $f(x+h)-f(x)$ ; 3) Divide by $h$ .

See Solution

Problem 4611

Calculate the integral $\int_{2}^{7} (x^{6}+2x-43) \, dx$ .

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Problem 4612

Find the integrating factor for the linear differential equation: $t^{2} \frac{d x}{d t}=4 t-t^{5} x$ .

See Solution

Problem 4613

Find the time for a sphere to drop from $h=6r$ to $h=2r$ using the equation $\frac{d h}{d t}=-\frac{\alpha}{r} h$ .

See Solution

Problem 4614

Find the slope of the tangent to the graph of $\frac{x^{2}-1}{x^{2}+x+1}$ at (1,0). Give your answer as a reduced fraction.

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Problem 4615

A radioactive substance decays as $P(t)=100(1.2)^{-t}$ . Find the half-life and the decay rate after one half-life.

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Problem 4616

A town's population is decreasing at $1.8\%$ per year, currently at 12000. Model the population and find rates of change in 10 years and at half the population.

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Problem 4617

Find the slope of the tangent to the graph of $\frac{x^{2}-9}{x^{2}+x+1}$ at $(-3,0)$ as a reduced fraction.

See Solution

Problem 4618

A radioactive substance decays exponentially. Given $P(t)=100(1.2)^{-t}$ , find the half-life and the decay rate after one half-life.

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Problem 4619

A tornado's wind speed is given by $S(d)=93 \log d+65$ . Find the average speed change from mile 10 to 100 and the speed change at miles 10 and 100.

See Solution

Problem 4620

The demand function for snack cakes is $p(x)=\frac{15}{2 x^{2}+11 x+5}$ .
a. Find the revenue function. b. Estimate marginal revenue at $x=0.75$ and find it for $x=2$ .

See Solution

Problem 4621

Find the revenue function for snack cakes given $p(x)=\frac{15}{2 x^{2}+11 x+5}$ . Estimate marginal revenue at $x=0.75$ and $x=2$ .

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Problem 4622

Find the slope of the secant line through $(2, f(2))$ and $(3, f(3))$ for $f(x)=2x^{2}-5x-2$ . Also, find the tangent slope at $x=2$ .

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Problem 4623

Find the slope of the tangent to the graph of $\frac{x^{2}-4}{x^{2}+x+1}$ at $(-2,0)$ . Express as a reduced fraction.

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Problem 4624

Is there a time $t$ in $(5,10)$ where the water volume's rate changes from positive to negative? Justify your answer.

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Problem 4625

Find the difference quotient $\frac{f(x)-f(2)}{x-2}$ for $f(x)=2+5x-5x^2$ . Simplify as $x \rightarrow 2$ .

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Problem 4626

Differentiate the function $h(x)=e^{x^{8}+\ln (x)}$ . Find $h^{\prime}(x)=$ .

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Problem 4627

Calculate the area between the curves $x=y^{4}$ , $y=\sqrt{2-x}$ , and the line $y=0$ .

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Problem 4628

Find the derivative $f^{\prime}(1)$ if $f(x)=\cos(\ln(x^{7}))$ .

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Problem 4629

Find the driver's velocity function given their position $x(t)=4 t^{3}-3 t^{2}+2 t-8$ .

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Problem 4630

Find the derivative of $y=x^{\wedge} 2+4x+3$ at $x=4$ .

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Problem 4631

Find the derivative of the constant function $f(x)=16$ at $x=a$ using the limit definition. $f^{\prime}(a)=$

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Problem 4632

Find the driver's velocity function from the position $x(t)=4 t^{3}-3 t^{2}+2 t-8$ . What is $v(t)$ ?

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Problem 4633

Find the derivative of the function $k(z)=6z+17$ at $z=a$ using the limit definition. What is $k'(a)=?$

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Problem 4634

Find the average velocity of a ball with position $x(t)=30t-3t^{2}$ from $t=0$ to $t=5$ seconds.

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Problem 4635

Calculate the limit: $\lim _{x \rightarrow \infty} e^{x}$ .

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Problem 4636

Find the tangent line equation at $(3, 13)$ with slope $2$ .

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Problem 4637

Find the instantaneous acceleration of the function $f(t) = 0.24t^3 + 2.2t^2$ at $t=1$ second.

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Problem 4638

Find the limit: $\lim _{x \rightarrow \infty} \frac{1}{x}$ .

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Problem 4639

Find the limit as $x$ approaches infinity for the expression $x^{2}-3x+1$ .

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Problem 4640

Find the limit as $x$ approaches infinity for $\frac{x^{2}-3 x+2}{x}$ .

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Problem 4641

Find the bee's instantaneous velocity at $t=2$ seconds for $x(t)=-0.24 t^{3}+2.2 t^{2}$ .

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Problem 4642

Find the average acceleration of a bee from $t=1$ to $t=3$ seconds, given velocities: $5.82$ m/s at $t=1$ , $5.12$ m/s at $t=2$ .

See Solution

Problem 4643

Find the limit: $\lim _{x \rightarrow \infty} \frac{x}{x^{2}-3 x+2}$ .

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Problem 4644

Find the derivative of the function $f(x)=2 \sqrt{x}(x^{3}-6 \sqrt{x}+4)$ . What is $f^{\prime}(x)$ ?

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Problem 4645

Find the derivative of $f(x)=x^{2}-2x+3$ at $a$ .

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Problem 4646

Find the integral of $y=x^{n} 2+4x+3$ from $x=0$ to $x=1$ .

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Problem 4647

Find the integral of $y=x^{\wedge} 2+4x+3$ from $x=0$ to $x=1$ .

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Problem 4648

Find the derivative of $f(x)=8 x^{3}$ at $x=1$ using the limit definition. Provide a whole or exact number.

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Problem 4649

Find the tangent line equation for $f(x)=8 \sqrt{x}-6$ at $(64,58)$ in the form $y=m x+b$ .

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Problem 4650

Find the second derivative $y^{\prime \prime}$ for the equation $x^{2}-5xy+y=8$ .

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Problem 4651

Entscheiden Sie für die Aussagen a) bis f) ob sie immer, nie oder abhängig sind und begründen Sie Ihre Wahl.

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Problem 4652

Find the radius of curvature of $y=x^{2}+4x+3$ at $x=3$ , which is $\sqrt{11} - 4$ .

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Problem 4653

Find the derivative of $y=\sqrt[3]{1+4x}$ .

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Problem 4654

Find the local maximum values and their locations for the continuous function $f$ with points $(-2,0)$ , $(-1,-2)$ , $(2,1)$ , and $(4,-4)$ .

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Problem 4655

Find local minimum values and their points for function $h$ given points (-3,0), (0,4), (3,0), (4,1), (-4,1).

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Problem 4656

Find the derivative of $f(x)=\frac{5 x+4}{2 x-5}$ at $x=0$ and provide the exact decimal answer. $f^{\prime}(0)=$

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Problem 4657

Find the Fourier series of $f(x)=1-x^2$ on $[-1,1]$ and its derivative, $(x + 1) \sin x$ .

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Problem 4658

Evaluate the integral: $\int \frac{2 x^{2}}{1+16 x^{2}} d x$

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Problem 4659

Untersuchen Sie die Funktion $f(x)=\frac{1}{2} x^{3}-\frac{3}{2} x^{2}+3,5$ auf Symmetrie, Steigung, Extrempunkte und Wendepunkt.

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Problem 4660

Find the limit as $x$ approaches infinity for the expression $-3 x^{8}+7 x^{3}-5$ .

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Problem 4661

Bestimmen Sie die Ableitung der Funktion $h(x)=\sqrt[3]{x^{2}}$ .

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Problem 4662

Find $f^{\prime}(-1)$ if $f(x)=x^{12} h(x)$ , with $h(-1)=2$ and $h^{\prime}(-1)=5$ .

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Problem 4663

Find $h^{\prime}(-4)$ for $h(x)=\arcsin(f(x))$ given $f(-4)=-\frac{2}{5}$ and $f^{\prime}(-4)=-4$ .

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Problem 4664

Find the derivative of $F(t)=(3t-1)^{4}(2t+1)^{-3}$ .

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Problem 4665

Calculate the integral $\int_{-1}^{2}(4 x^{3}-1) dx$ .

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Problem 4666

Find the derivatives:
1. $f(x)=5 \sin ^{-1}(6 x)$
2. $f(x)=3 \cos ^{-1}(8 x+11)$
3. $f(x)=6 \cdot \tan ^{-1}(\sin (x))$
4. $f(x)=8 \cot ^{-1}(x^{5})$
5. $f(x)=5 \sec ^{-1}(6 \ln (x))$
6. $f(x)=\csc ^{-1}(e^{x})$

See Solution

Problem 4667

Untersuchen Sie die Abweichungen der Modellfunktion $f(t)$ von Messwerten, bestimmen Sie Maximalwerte und Änderungszeiten. Prüfen Sie Fahrverbote und vergleichen Sie 2019 und 2023.

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Problem 4668

Find the velocity of a ground hog at $t=0$ given its position function $x(t)=t^{3}-6 t^{2}+9 t+12$ .

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Problem 4669

Calculate the integral: $\int \frac{2 x^{2}-15 x-4}{5 \sqrt{x}} d x$

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Problem 4670

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{3} \sqrt{x+1}$ .

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Problem 4671

Kostenfunktion $K(x)=0,1 x^{3}-2,4 x^{2}+30 x+640$ : a) Finde Betriebsminimum. b) Wo sind Grenzkosten minimal? c) Betriebsoptimum bei 20? d) Grenzkosten = Durchschnittskosten im Optimum?

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Problem 4672

Berechnen Sie die Niederschlagsmenge $f(1)$ , $f(2)$ , $f(3)$ , $f^{\prime}(1)$ und den Zeitpunkt des stärksten Regens.

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Problem 4673

Gegeben ist die Kostenfunktion $K(x)=0,1 x^{3}-2,4 x^{2}+30 x+640$ . Zeigen Sie, dass im Optimum $K'(x)=\frac{K(x)}{x}$ gilt.

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Problem 4674

Find the second derivative $D_{x}^{2} y$ for $y=\frac{1-x}{x-3}$ . Options: A. 0 B. $\frac{-8}{(x-3)^{3}}$ C. $\frac{-4}{(x-2)^{3}}$ D. $\frac{-4}{(x-3)^{3}}$

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Problem 4675

Finde den Wendepunkt der Funktion $f(x)=3 x^{4}-4 x^{3}$ .

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Problem 4676

Feinstaubmessungen in Hamburg 2019: Untersuche $f(t)=\frac{4}{1000} t^{4}-\frac{1}{5} t^{3}+2 t^{2}+45$ . Vergleiche mit Messwerten, finde Höchstwert, Steig-/Fallzeiten, prüfe Fahrverbote und vergleiche mit 2023.

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Problem 4677

Find $\frac{d y}{d x}$ if $y=\frac{x}{x+y}$ . Options: A. $\frac{y}{(x+y)^{2}+x}$ B. $\frac{1-y}{x+2 y}$ C. $\frac{-1+\sqrt{5} x}{2}$ D. $\frac{1}{1+y}$

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Problem 4678

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{3} \sqrt{x+1}$ .

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Problem 4679

Bestimme den größtmöglichen Wert von $c$ , sodass die Funktion $v(x) = \frac{1}{12} x^{4}-\frac{1}{3} x^{3}-\frac{3}{2} x^{2}+2 x+\frac{1}{2}$ für $0<x<c$ rechtsgekrümmt ist.

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Problem 4680

Bestätige, dass vor der Einnahme eines Medikaments ( $t=0$ ) die Konzentration $f(0)=10 \cdot 0 \cdot e^{0}=0$ mg/L ist.

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Problem 4681

Bestimme den Zeitpunkt der höchsten Medikamentenkonzentration $f(t)=10 \cdot t \cdot e^{-0,5 t}$ und die Konzentration. Zeige, dass der Abbau nach 4 Stunden am schnellsten ist und berechne die Geschwindigkeit.

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Problem 4682

Nach Einnahme eines Medikaments beschreibt die Funktion $f(t)=10 \cdot t \cdot e^{-0,5 t}$ die Konzentration.
a) Zeige, dass vor der Einnahme die Konzentration 0 ist. b) Bestimme graphisch, wann die Konzentration 6 mg/L erreicht wird.

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Problem 4683

Zeigen Sie, dass die Folgen a) bis d) konvergent sind, indem Sie Monotonie und Beschränktheit nachweisen. Bestimmen Sie den Grenzwert. a) $a_{n}=\frac{n+1}{5 n}$ b) $a_{n}=\frac{\sqrt{5 n}}{\sqrt{n+1}}$ c) $a_{n}=\frac{n \sqrt{n}+10}{n^{2}}$ d) $a_{n}=\frac{n}{n^{2}+1}$

See Solution

Problem 4684

Bestimme den Zeitpunkt und die Konzentration der höchsten Medikamentenkonzentration $f(t)=10 \cdot t \cdot e^{-0,5 t}$ . Zeige, dass nach 4 Stunden der Abbau am schnellsten ist und berechne die Abbaugeschwindigkeit.

See Solution

Problem 4685

Bestimme den Zeitpunkt und die Konzentration der höchsten Medikamentenmenge $f(t)=10 \cdot t \cdot e^{-0,5 t}$ . Zeige, dass nach 4 Stunden der Abbau am schnellsten ist und berechne die Abbaugeschwindigkeit.

See Solution

Problem 4686

Bestimme die lineare Funktion $w$ , die die Abbaukonzentration des Medikaments nach 4 Stunden beschreibt.

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Problem 4687

Gegeben ist die Funktion $\mathrm{f}$ für die Medikamentenkonzentration im Blut.
a) Zeigen Sie, dass das Medikament vor der Einnahme nicht nachweisbar ist.
b) Bestimmen Sie graphisch, wann die Konzentration 6 mg/L beträgt.
c) Finden Sie den Zeitpunkt und die Konzentration der höchsten Medikamentenmenge.
d) Berechnen Sie, dass das Medikament nach 4 Stunden am schnellsten abgebaut wird und bestimmen Sie die Abbaugeschwindigkeit.

See Solution

Problem 4688

Gegeben ist die Funktion $f(x)=x^{3}-6 x^{2}+8 x$ .
a) Berechne $\int_{0}^{4} f(x) d x$ . b) Bestimme den Flächeninhalt im 4. Quadranten. c) Flächeninhalt im 1. Quadranten, begründe ohne Rechnung.
Für $g(x)=-0,5 x+1$ : d) Bestimme den Flächeninhalt zwischen $f$ und $g$ . e) Finde $b>0$ so, dass $\int_{2}^{b} g(x) d x=-4$ .

See Solution

Problem 4689

Berechne die unbestimmten Integrale: a) $\int x^{6} dx$ , b) $\int 6 x^{2} dx$ , e) $\int(2 x^{3}-4 x+1) dx$ , f) $\int(a x^{2}+6 x) dx$ , i) $\int(x+\frac{3}{x^{2}}) dx$ , j) $\int e^{x} \cdot e^{x+2} dx$ , c) $\int n \cdot x^{2 n-1} dx$ , d) $\int(4 x^{2}+2 x) dx$ , g) $\int 3 x^{-2} dx$ , h) $\int(2 x+\frac{1}{x}) \cdot x dx$ , k) $\int \frac{4}{e^{x}} dx$ , l) $\int(\sin x+2 \cos x) dx$ .

See Solution

Problem 4690

Zeige, dass $F(t)=-20 \cdot(t+2) \cdot e^{-0,5 t}$ eine Stammfunktion von $f$ ist. Berechne $\frac{1}{8} \int_{0}^{8} f(t) d t$ .

See Solution

Problem 4691

Aufgabe: Analysiere die Bestellrate $f(t)=-4,98 t^{4}+126 t^{3}-1020 t^{2}+2700 t+1440$ für $t$ von 0 bis 12. Bestimme $f(1)$ , $f'(1)$ , max. Bestellungen und Gesamtbestellungen bis 12:00 Uhr. Ab 14:00 Uhr gilt $g(t)=-30 t^{2}+300 t+1320$ . Finde die Schnittstellen von $f$ und $g$ und die verkauften Karten bis 18:00 Uhr.

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Problem 4692

Untersuchen Sie die Funktion $f(x)=-5 e^{-0.5 x}+6 e^{-3 x}+6$ auf Extrempunkte.

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Problem 4693

Untersuchen Sie das Flussbett mit der Funktion $f(x)=-2 x^{2} \cdot e^{x}$ für $x \in(-7 ; 0)$ .
1. a) Bestimmen Sie $f(-7)$ und erläutern Sie die Bedeutung. b) Finden Sie den tiefsten Punkt des Flussbetts. c) Überprüfen Sie, ob der Winkel von $45^{\circ}$ erreicht wird.

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Problem 4694

If $\arcsin (x)=\ln (y)$ , find $\frac{d y}{d x}$ .

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Problem 4695

Find the limit as $x$ approaches $-\infty$ for the function $f(x)=2 x e^{2 x}$ .

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Problem 4696

Evaluate the limit: $\lim _{x \rightarrow 2} \frac{x-2}{\sqrt{x-2}}$

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Problem 4697

Check if the function $f$ is differentiable at $x=-1$ , where $f(x)=\begin{cases} \arcsin (\sqrt{x+1}) & \text{if } x<-1 \\ \arccos (\sqrt{x+2}) & \text{if } x \geq -1 \end{cases}$ .

See Solution

Problem 4698

Calculate the arc length of $y=e^{x}$ from $x=1$ to $x=2$ , rounding your answer to 3 decimal places.

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Problem 4699

Calculate the arc length of $x=5 y^{\frac{3}{2}}$ from $y=0$ to $y=4$ , rounding to 3 decimal places.

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Problem 4700

Find the instantaneous rate of change of $y=(2 x-1)^{2}+x y$ at $x=0$ .

See Solution

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Calculus

Problem 4601

Evaluate and simplify f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=3x−9f(x)=3x-9f(x)=3x−9.

Problem 4602

Calculate the average rate of change of g(x)=4x3+6x4g(x)=4 x^{3}+\frac{6}{x^{4}}g(x)=4x3+x46​ over the interval [−2,3][-2,3][−2,3].

Problem 4603

Evaluate f(x+h)f(x+h)f(x+h) and simplify f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=4x2−4x+5f(x)=4x^{2}-4x+5f(x)=4x2−4x+5.

Problem 4604

Calculate the average rate of change of f(x)=2x2−7f(x)=2 x^{2}-7f(x)=2x2−7 from x=1x=1x=1 to x=bx=bx=b. Express your answer in terms of bbb.

Problem 4605

Find f(a)f(a)f(a), f(a+h)f(a+h)f(a+h), and f(a+h)−f(a)h\frac{f(a+h)-f(a)}{h}hf(a+h)−f(a)​ for f(x)=5+5x2f(x)=5+5x^{2}f(x)=5+5x2.

Problem 4606

Find the slope mmm of the tangent line to y=B(x)y=B(x)y=B(x) at x=12x=12x=12 where B(12)=−1B(12)=-1B(12)=−1 and B′(12)=−12B'(12)=-\frac{1}{2}B′(12)=−21​. Then, find the intersection point (x,y)(x, y)(x,y) and the tangent line equation.

Problem 4607

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=−2x2f(x)=-2x^{2}f(x)=−2x2. Steps: 1) Simplify f(x+h)f(x+h)f(x+h); 2) Subtract f(x+h)−f(x)f(x+h)-f(x)f(x+h)−f(x); 3) Divide by hhh.

Problem 4608

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=−2x2f(x)=-2 x^{2}f(x)=−2x2. Simplify f(x+h)f(x+h)f(x+h) and f(x+h)−f(x)f(x+h)-f(x)f(x+h)−f(x).

Problem 4609

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=x2f(x)=x^{2}f(x)=x2. Step 1: Simplify f(x+h)f(x+h)f(x+h). Step 2: Simplify f(x+h)−f(x)f(x+h)-f(x)f(x+h)−f(x).

Problem 4610

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=x2f(x)=x^{2}f(x)=x2. Steps: 1) Simplify f(x+h)f(x+h)f(x+h); 2) Find f(x+h)−f(x)f(x+h)-f(x)f(x+h)−f(x); 3) Divide by hhh.

Problem 4611

Calculate the integral ∫27(x6+2x−43) dx\int_{2}^{7} (x^{6}+2x-43) \, dx∫27​(x6+2x−43)dx.

Problem 4612

Find the integrating factor for the linear differential equation: t2dxdt=4t−t5xt^{2} \frac{d x}{d t}=4 t-t^{5} xt2dtdx​=4t−t5x.

Problem 4613

Find the time for a sphere to drop from h=6rh=6rh=6r to h=2rh=2rh=2r using the equation dhdt=−αrh\frac{d h}{d t}=-\frac{\alpha}{r} hdtdh​=−rα​h.

Problem 4614

Find the slope of the tangent to the graph of x2−1x2+x+1\frac{x^{2}-1}{x^{2}+x+1}x2+x+1x2−1​ at (1,0). Give your answer as a reduced fraction.

Problem 4615

A radioactive substance decays as P(t)=100(1.2)−tP(t)=100(1.2)^{-t}P(t)=100(1.2)−t. Find the half-life and the decay rate after one half-life.

Problem 4616

A town's population is decreasing at 1.8%1.8\%1.8% per year, currently at 12000. Model the population and find rates of change in 10 years and at half the population.

Problem 4617

Find the slope of the tangent to the graph of x2−9x2+x+1\frac{x^{2}-9}{x^{2}+x+1}x2+x+1x2−9​ at (−3,0)(-3,0)(−3,0) as a reduced fraction.

Problem 4618

A radioactive substance decays exponentially. Given P(t)=100(1.2)−tP(t)=100(1.2)^{-t}P(t)=100(1.2)−t, find the half-life and the decay rate after one half-life.

Problem 4619

A tornado's wind speed is given by S(d)=93log⁡d+65S(d)=93 \log d+65S(d)=93logd+65. Find the average speed change from mile 10 to 100 and the speed change at miles 10 and 100.

Problem 4620

The demand function for snack cakes is p(x)=152x2+11x+5p(x)=\frac{15}{2 x^{2}+11 x+5}p(x)=2x2+11x+515​. a. Find the revenue function. b. Estimate marginal revenue at x=0.75x=0.75x=0.75 and find it for x=2x=2x=2.

Problem 4621

Find the revenue function for snack cakes given p(x)=152x2+11x+5p(x)=\frac{15}{2 x^{2}+11 x+5}p(x)=2x2+11x+515​. Estimate marginal revenue at x=0.75x=0.75x=0.75 and x=2x=2x=2.

Problem 4622

Find the slope of the secant line through (2,f(2))(2, f(2))(2,f(2)) and (3,f(3))(3, f(3))(3,f(3)) for f(x)=2x2−5x−2f(x)=2x^{2}-5x-2f(x)=2x2−5x−2. Also, find the tangent slope at x=2x=2x=2.

Problem 4623

Find the slope of the tangent to the graph of x2−4x2+x+1\frac{x^{2}-4}{x^{2}+x+1}x2+x+1x2−4​ at (−2,0)(-2,0)(−2,0). Express as a reduced fraction.

Problem 4624

Is there a time ttt in (5,10)(5,10)(5,10) where the water volume's rate changes from positive to negative? Justify your answer.

Problem 4625

Find the difference quotient f(x)−f(2)x−2\frac{f(x)-f(2)}{x-2}x−2f(x)−f(2)​ for f(x)=2+5x−5x2f(x)=2+5x-5x^2f(x)=2+5x−5x2. Simplify as x→2x \rightarrow 2x→2.

Problem 4626

Differentiate the function h(x)=ex8+ln⁡(x)h(x)=e^{x^{8}+\ln (x)}h(x)=ex8+ln(x). Find h′(x)=h^{\prime}(x)=h′(x)=.

Problem 4627

Calculate the area between the curves x=y4x=y^{4}x=y4, y=2−xy=\sqrt{2-x}y=2−x​, and the line y=0y=0y=0.

Problem 4628

Find the derivative f′(1)f^{\prime}(1)f′(1) if f(x)=cos⁡(ln⁡(x7))f(x)=\cos(\ln(x^{7}))f(x)=cos(ln(x7)).

Problem 4629

Find the driver's velocity function given their position x(t)=4t3−3t2+2t−8x(t)=4 t^{3}-3 t^{2}+2 t-8x(t)=4t3−3t2+2t−8.

Problem 4630

Find the derivative of y=x∧2+4x+3y=x^{\wedge} 2+4x+3y=x∧2+4x+3 at x=4x=4x=4.

Problem 4631

Find the derivative of the constant function f(x)=16f(x)=16f(x)=16 at x=ax=ax=a using the limit definition. f′(a)=f^{\prime}(a)= f′(a)=

Problem 4632

Find the driver's velocity function from the position x(t)=4t3−3t2+2t−8x(t)=4 t^{3}-3 t^{2}+2 t-8x(t)=4t3−3t2+2t−8. What is v(t)v(t)v(t)?

Problem 4633

Find the derivative of the function k(z)=6z+17k(z)=6z+17k(z)=6z+17 at z=az=az=a using the limit definition. What is k′(a)=?k'(a)=?k′(a)=?

Problem 4634

Find the average velocity of a ball with position x(t)=30t−3t2x(t)=30t-3t^{2}x(t)=30t−3t2 from t=0t=0t=0 to t=5t=5t=5 seconds.

Problem 4635

Calculate the limit: lim⁡x→∞ex\lim _{x \rightarrow \infty} e^{x}limx→∞​ex.

Problem 4636

Find the tangent line equation at (3,13)(3, 13)(3,13) with slope 222.

Problem 4637

Find the instantaneous acceleration of the function f(t)=0.24t3+2.2t2f(t) = 0.24t^3 + 2.2t^2f(t)=0.24t3+2.2t2 at t=1t=1t=1 second.

Problem 4638

Find the limit: lim⁡x→∞1x\lim _{x \rightarrow \infty} \frac{1}{x}limx→∞​x1​.

Problem 4639

Find the limit as xxx approaches infinity for the expression x2−3x+1x^{2}-3x+1x2−3x+1.

Problem 4640

Evaluate and simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=3x-9$ .

Calculate the average rate of change of $g(x)=4 x^{3}+\frac{6}{x^{4}}$ over the interval $[-2,3]$ .

Evaluate $f(x+h)$ and simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=4x^{2}-4x+5$ .

Calculate the average rate of change of $f(x)=2 x^{2}-7$ from $x=1$ to $x=b$ . Express your answer in terms of $b$ .

Find $f(a)$ , $f(a+h)$ , and $\frac{f(a+h)-f(a)}{h}$ for $f(x)=5+5x^{2}$ .

Find the slope $m$ of the tangent line to $y=B(x)$ at $x=12$ where $B(12)=-1$ and $B'(12)=-\frac{1}{2}$ . Then, find the intersection point $(x, y)$ and the tangent line equation.

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=-2x^{2}$ . Steps: 1) Simplify $f(x+h)$ ; 2) Subtract $f(x+h)-f(x)$ ; 3) Divide by $h$ .

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

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=x^{2}$ . Step 1: Simplify $f(x+h)$ . Step 2: Simplify $f(x+h)-f(x)$ .

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=x^{2}$ . Steps: 1) Simplify $f(x+h)$ ; 2) Find $f(x+h)-f(x)$ ; 3) Divide by $h$ .

Calculate the integral $\int_{2}^{7} (x^{6}+2x-43) \, dx$ .

Find the integrating factor for the linear differential equation: $t^{2} \frac{d x}{d t}=4 t-t^{5} x$ .

Find the time for a sphere to drop from $h=6r$ to $h=2r$ using the equation $\frac{d h}{d t}=-\frac{\alpha}{r} h$ .

Find the slope of the tangent to the graph of $\frac{x^{2}-1}{x^{2}+x+1}$ at (1,0). Give your answer as a reduced fraction.

A radioactive substance decays as $P(t)=100(1.2)^{-t}$ . Find the half-life and the decay rate after one half-life.

A town's population is decreasing at $1.8\%$ per year, currently at 12000. Model the population and find rates of change in 10 years and at half the population.

Find the slope of the tangent to the graph of $\frac{x^{2}-9}{x^{2}+x+1}$ at $(-3,0)$ as a reduced fraction.

A radioactive substance decays exponentially. Given $P(t)=100(1.2)^{-t}$ , find the half-life and the decay rate after one half-life.

A tornado's wind speed is given by $S(d)=93 \log d+65$ . Find the average speed change from mile 10 to 100 and the speed change at miles 10 and 100.

The demand function for snack cakes is $p(x)=\frac{15}{2 x^{2}+11 x+5}$ .
a. Find the revenue function. b. Estimate marginal revenue at $x=0.75$ and find it for $x=2$ .

Find the revenue function for snack cakes given $p(x)=\frac{15}{2 x^{2}+11 x+5}$ . Estimate marginal revenue at $x=0.75$ and $x=2$ .

Find the slope of the secant line through $(2, f(2))$ and $(3, f(3))$ for $f(x)=2x^{2}-5x-2$ . Also, find the tangent slope at $x=2$ .

Find the slope of the tangent to the graph of $\frac{x^{2}-4}{x^{2}+x+1}$ at $(-2,0)$ . Express as a reduced fraction.

Is there a time $t$ in $(5,10)$ where the water volume's rate changes from positive to negative? Justify your answer.

Find the difference quotient $\frac{f(x)-f(2)}{x-2}$ for $f(x)=2+5x-5x^2$ . Simplify as $x \rightarrow 2$ .

Differentiate the function $h(x)=e^{x^{8}+\ln (x)}$ . Find $h^{\prime}(x)=$ .

Calculate the area between the curves $x=y^{4}$ , $y=\sqrt{2-x}$ , and the line $y=0$ .

Find the derivative $f^{\prime}(1)$ if $f(x)=\cos(\ln(x^{7}))$ .

Find the driver's velocity function given their position $x(t)=4 t^{3}-3 t^{2}+2 t-8$ .

Find the derivative of $y=x^{\wedge} 2+4x+3$ at $x=4$ .

Find the derivative of the constant function $f(x)=16$ at $x=a$ using the limit definition. $f^{\prime}(a)=$

Find the driver's velocity function from the position $x(t)=4 t^{3}-3 t^{2}+2 t-8$ . What is $v(t)$ ?

Find the derivative of the function $k(z)=6z+17$ at $z=a$ using the limit definition. What is $k'(a)=?$

Find the average velocity of a ball with position $x(t)=30t-3t^{2}$ from $t=0$ to $t=5$ seconds.

Calculate the limit: $\lim _{x \rightarrow \infty} e^{x}$ .

Find the tangent line equation at $(3, 13)$ with slope $2$ .

Find the instantaneous acceleration of the function $f(t) = 0.24t^3 + 2.2t^2$ at $t=1$ second.

Find the limit: $\lim _{x \rightarrow \infty} \frac{1}{x}$ .

Find the limit as $x$ approaches infinity for the expression $x^{2}-3x+1$ .

Find the limit as $x$ approaches infinity for $\frac{x^{2}-3 x+2}{x}$ .

Find the bee's instantaneous velocity at $t=2$ seconds for $x(t)=-0.24 t^{3}+2.2 t^{2}$ .

Find the average acceleration of a bee from $t=1$ to $t=3$ seconds, given velocities: $5.82$ m/s at $t=1$ , $5.12$ m/s at $t=2$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{x}{x^{2}-3 x+2}$ .

Find the derivative of the function $f(x)=2 \sqrt{x}(x^{3}-6 \sqrt{x}+4)$ . What is $f^{\prime}(x)$ ?

Find the derivative of $f(x)=x^{2}-2x+3$ at $a$ .

Find the integral of $y=x^{n} 2+4x+3$ from $x=0$ to $x=1$ .

Find the integral of $y=x^{\wedge} 2+4x+3$ from $x=0$ to $x=1$ .

Find the derivative of $f(x)=8 x^{3}$ at $x=1$ using the limit definition. Provide a whole or exact number.

Find the tangent line equation for $f(x)=8 \sqrt{x}-6$ at $(64,58)$ in the form $y=m x+b$ .

Find the second derivative $y^{\prime \prime}$ for the equation $x^{2}-5xy+y=8$ .

Find the radius of curvature of $y=x^{2}+4x+3$ at $x=3$ , which is $\sqrt{11} - 4$ .

Find the derivative of $y=\sqrt[3]{1+4x}$ .

Find the local maximum values and their locations for the continuous function $f$ with points $(-2,0)$ , $(-1,-2)$ , $(2,1)$ , and $(4,-4)$ .

Find local minimum values and their points for function $h$ given points (-3,0), (0,4), (3,0), (4,1), (-4,1).

Find the derivative of $f(x)=\frac{5 x+4}{2 x-5}$ at $x=0$ and provide the exact decimal answer. $f^{\prime}(0)=$

Find the Fourier series of $f(x)=1-x^2$ on $[-1,1]$ and its derivative, $(x + 1) \sin x$ .

Evaluate the integral: $\int \frac{2 x^{2}}{1+16 x^{2}} d x$

Untersuchen Sie die Funktion $f(x)=\frac{1}{2} x^{3}-\frac{3}{2} x^{2}+3,5$ auf Symmetrie, Steigung, Extrempunkte und Wendepunkt.

Find the limit as $x$ approaches infinity for the expression $-3 x^{8}+7 x^{3}-5$ .

Bestimmen Sie die Ableitung der Funktion $h(x)=\sqrt[3]{x^{2}}$ .

Find $f^{\prime}(-1)$ if $f(x)=x^{12} h(x)$ , with $h(-1)=2$ and $h^{\prime}(-1)=5$ .

Find $h^{\prime}(-4)$ for $h(x)=\arcsin(f(x))$ given $f(-4)=-\frac{2}{5}$ and $f^{\prime}(-4)=-4$ .

Find the derivative of $F(t)=(3t-1)^{4}(2t+1)^{-3}$ .

Calculate the integral $\int_{-1}^{2}(4 x^{3}-1) dx$ .

Untersuchen Sie die Abweichungen der Modellfunktion $f(t)$ von Messwerten, bestimmen Sie Maximalwerte und Änderungszeiten. Prüfen Sie Fahrverbote und vergleichen Sie 2019 und 2023.

Find the velocity of a ground hog at $t=0$ given its position function $x(t)=t^{3}-6 t^{2}+9 t+12$ .

Calculate the integral: $\int \frac{2 x^{2}-15 x-4}{5 \sqrt{x}} d x$

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{3} \sqrt{x+1}$ .

Kostenfunktion $K(x)=0,1 x^{3}-2,4 x^{2}+30 x+640$ : a) Finde Betriebsminimum. b) Wo sind Grenzkosten minimal? c) Betriebsoptimum bei 20? d) Grenzkosten = Durchschnittskosten im Optimum?

Berechnen Sie die Niederschlagsmenge $f(1)$ , $f(2)$ , $f(3)$ , $f^{\prime}(1)$ und den Zeitpunkt des stärksten Regens.

Gegeben ist die Kostenfunktion $K(x)=0,1 x^{3}-2,4 x^{2}+30 x+640$ . Zeigen Sie, dass im Optimum $K'(x)=\frac{K(x)}{x}$ gilt.

Find the second derivative $D_{x}^{2} y$ for $y=\frac{1-x}{x-3}$ . Options: A. 0 B. $\frac{-8}{(x-3)^{3}}$ C. $\frac{-4}{(x-2)^{3}}$ D. $\frac{-4}{(x-3)^{3}}$