Calculus

Problem 26901

Given the biomass change $B(t)$ with $\frac{d}{dt} B(t)=2t+1$ for $t \in [0,6]$ and $B(0)=10$ , find: a. cumulative change, b. solution, c. average biomass.

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Problem 26902

Compute the average value of $f(x)=\cos (\pi x)+3^{x}$ over the interval $[2,4]$ .

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Problem 26903

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{x+4}-2}{x}$ .

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Problem 26904

Find the limit: $\lim _{x \rightarrow 0} \frac{\frac{4}{x+10}-\frac{2}{5}}{x}$ .

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Problem 26905

Find the limit as $x$ approaches 2 for the expression $\frac{5 x^{2}+15 x}{5 x^{2}+12 x+4}$ .

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Problem 26906

Die Funktion $p(t)=0,2 t e^{-0,0625 t}+5$ beschreibt Plastikmüll in Mio. Tonnen. Analysieren Sie $p(0)$ , $p(10)-p(5)$ und $p^{\prime}(16)$ .

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Problem 26907

Aufgabe:
a) Bestimme den Grenzwert von $h(x)$ für große $x$ . b) Ab welcher Stückzahl ist das Kunststoffleitwerk günstiger als das Metallleitwerk?

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Problem 26908

Find the integral of $(\sqrt[6]{x})^{3}$ with respect to $x$ : $\int(\sqrt[6]{x})^{3} d x=\square +C$ .

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Problem 26909

Calculate the integral of $(\sqrt[6]{x})^{3}$ with respect to $x$ : $\int(\sqrt[6]{x})^{3} d x$ .

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Problem 26910

Find the average rate of change of $f(x)=\tan(2x)$ from 0 to $\frac{\pi}{6}$ .

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Problem 26911

Analyze the function $f(x)$ and its derivative $f'(x)$ given their behavior on specified intervals.

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Problem 26912

Discuss these questions: 1. What does a derivative represent? 2. Using product rule vs. quotient rule? 3. Using product rule vs. chain rule? 4. Types of discontinuities? 5. Requirements for function continuity?

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Problem 26913

If a 1000 mg drug dose loses 25% every 4 hours, will it ever be fully eliminated from your system?

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Problem 26914

Find the height $\mathrm{h}$ (1000 < $\mathrm{h}$ < 20000) that minimizes the operating cost $C=4000+\mathrm{h}/15+15000000/\mathrm{h}$ and calculate $C$ at this height.

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Problem 26915

Find when the velocity of the particle given by $s(t)=3 t^{3}-40.5 t^{2}+162 t$ is increasing, decreasing, or constant, and find acceleration at $t=6$ s.

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Problem 26916

Find the derivative of the function $y=x^{2}-x \sin x$ , expressed as $\frac{d y}{d x}$ .

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Problem 26917

How many years will it take for Brianna's investment to double with a continuous return of $34\%$ per year? $t=$

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Problem 26918

A 70m building, a l-kg ball thrown down at 10m/s, hits ground at 30m/s. Find energy lost to friction: a) 150 J b) 300 J c) 700 J d) 400 J e) 0 J.

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Problem 26919

Find the derivative of the function $y=x^{2}-2 x \cos x$ , i.e., compute $\frac{d y}{d x}$ .

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Problem 26920

Find the tangent line equation for the curve defined by $x = \sec^2(t) - 1$ , $y = \tan(t)$ at $t = -\frac{\pi}{4}$ and compute $\frac{d^2y}{dx^2}$ .

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Problem 26921

Calcula $\left(\frac{\partial U}{\partial V}\right)_{T}$ y el coeficiente de Joule $\mu_{J}$ para un gas de Van der Waals.

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Problem 26922

Determine if the function $f(x)=x^{5} e^{\sinh (x)} \cos (x)$ has a local min, max, or neither at $x=0$ .

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Problem 26923

Solve for $d$ in the equation $\frac{d^{x}}{d^{9}} = \frac{1}{d^{3}}$ .

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Problem 26924

Find the derivative of the function $-\sin(-x + 8)$ .

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Problem 26925

Find the tangent line equation for $y=4 \sin ^{2}(x)$ at $x=\frac{\pi}{6}$ .

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Problem 26926

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ given the equation $3 x y^{2}-3 y+2 x^{3}=0$ .

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Problem 26927

What is true about the function $f$ on the interval $3<x<4$ if its rate of change is positive and decreasing?

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Problem 26928

Determine if the particle is speeding up or slowing down at $t=2$ given its velocity $v(t)=-2t-2$ .

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Problem 26929

Find the equations of all tangent lines to the curve defined by $-x y + y^{4} = -2$ .

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Problem 26930

Find $\frac{dy}{dx}$ for $y = \cos(f(x^3))$ at $x = 2$ , where $f$ is a differentiable function.

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Problem 26931

Boats A and B leave together; A goes north at 12 km/hr, B east at 18 km/hr. After 2.5 hrs, find distance increase rate in km/hr.

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Problem 26932

Solve the IVP $\frac{d x}{d t}=x(1-x)$ with $x(0)=x_{0}$ . Find $x(t)$ for $x_{0}>0$ , $x_{0} \neq 1$ . Also, find solutions for $x_{0}=0$ or $x_{0}=1$ . Compute $\lim_{t \to \infty} x(t)$ .

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Problem 26933

Find the line perpendicular to $g'(x)$ at $x=-2$ for $g(x)=\int(5x^{3}+4x)dx$ .

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Problem 26934

Gegeben ist die Funktion $f(x)=(-x-2) \cdot e^{-x}$ . Finde den Tiefpunkt, Wendepunkt und die Tangentengleichung an der x-Achse.

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Problem 26935

Approximate the volume change of a sphere when the radius goes from 10 cm to 10.02 cm. Choices: A) 4213973 B) 1261.669 C) -1256.637 D) 25.233 E) 25.1

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Problem 26936

Find $p$ values for which the integrals converge: (a) $\int_{1}^{2} \frac{dx}{x(\ln x)^{p}}$ , (b) $\int_{2}^{\infty} \frac{dx}{x(\ln x)^{p}}$ .

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Problem 26937

Bestimme die zweite Ableitung der Funktion $f(x) = 2tx^3 - \frac{3}{2}t^2x^2$ und erkläre den Prozess.

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Problem 26938

Find the relative max and min of $f(x)=x^{3}+12x^{2}+2$ using the first-derivative test. What is $f^{\prime}(x)$ ?

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Problem 26939

Find the relative maximum and minimum points of the function $f(x)=x^{3}+12 x^{2}+45 x+1$ using the first derivative test.

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Problem 26940

Find the relative max and min of the function $f(x)=x^{3}+12x^{2}+2$ using the first-derivative test. First, find $f^{\prime}(x)$ .

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Problem 26941

Find where $f(x)$ is increasing if $f'(x)=x^3-2x^2+x$ . Options: A. $(0,1)$ B. $(0, \infty)$ C. $(-\infty, 0)$ D. $\left(\frac{1}{3}, 1\right)$ E. $\left(-\infty, \frac{1}{3}\right),(1, \infty)$ .

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Problem 26942

Bestimme die Ableitung $g$ von $f(x)=(2x-1)\cdot\cos(x)$ .

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Problem 26943

Find the relative maximum and minimum points of the function $f(x)=x^{3}+6x^{2}+4$ using the first-derivative test.

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Problem 26944

Solve the differential equation: $\frac{d y}{d x}=\frac{4 y+2 x+3}{2 y+x+5}$ .

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Problem 26945

Which statements about limits of $\ln x$ are true? I. $\lim _{x \rightarrow \infty} \ln x=\infty$ , II. $\lim _{x \rightarrow 1} \ln x=0$ , III. $\lim _{x \rightarrow 0^{+}} \ln x=0$ . Choose A-E.

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Problem 26946

A balloon rises at $8 \mathrm{ft/sec}$ from $150 \mathrm{ft}$ away. How fast is the distance to the balloon increasing at $50 \mathrm{ft}$ ?

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Problem 26947

Find the limit: $\lim _{x \rightarrow \infty} \frac{x^{2}}{e^{x}}=$ (A) 0 (B) 1 (C) -e (D) e (E) Does not exist.

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Problem 26948

A quantity starts at 180 and decays at $70\%$ per minute. Find its value after 333 seconds, rounded to the nearest hundredth.

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Problem 26949

A stone creates ripples in a pond. If the radius grows at $1.5 \mathrm{ft} / \mathrm{sec}$ , find the area increase rate when radius is 3 ft.

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Problem 26950

A 6 ft tall man is 10 ft from an 18 ft light pole, walking at 1 ft/sec. Find the rates of his shadow's length and tip movement.

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Problem 26951

A quantity starts at 2300 and grows at $45\%$ per minute. Find its value after 0.2 hours, rounded to two decimal places.

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Problem 26952

Untersuche die Stetigkeit der Funktion $f(x)=\left\{\begin{array}{cc}\cos \frac{1}{x}, & x<0 \\ x^{2}+1, & x \geq 0.\end{array}\right.$ an $x_{0}=0$ . Was ist korrekt an der Argumentation? Was muss allgemein gezeigt werden? Ist das Ergebnis korrekt?

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Problem 26953

Ben launches a potato to 17 m. What is its velocity when it hits the ground? Use $g = 9.8 \frac{\mathrm{m}}{\mathrm{s}^{2}}$ .

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Problem 26954

Gegeben die Funktionenschar $f_{a}(a>0)$ , finde die Ableitung und die Steigung bei $x=0$ . Wann ist die Steigung $1$ ?

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Problem 26955

Bestimme die Asymptotengleichung für den Graphen von $f(x)=x^{2} \cdot e^{-3 x}+5$ .

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Problem 26956

Find the derivative $P^{\prime}(t)$ of the function $P(t) = 50000(0.98)^{t}$ .

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Problem 26957

Find the derivative of the function $e^{x / 2}$ with respect to $x$ .

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Problem 26958

Graph the car's velocity $v(t)=120(1-0.85^{t})$ and describe its acceleration.

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Problem 26959

Find the max effectiveness $E(t)=0.5\left[10+t e^{-\frac{1}{5}}\right]$ for studying up to 30 hours.

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Problem 26960

Find the horizontal asymptote of $f(x)=\frac{5 x^{2}+3}{4 x^{2}-5}$ . Choose A, B, or C and fill in the boxes if needed.

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Problem 26961

Find the value of $b$ for which $\lim _{x \rightarrow-2} \frac{b x^{2}+15 x+15+b}{x^{2}+x-2}$ exists, and determine the limit.

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Problem 26962

Evaluate the integral: $\int_{0}^{2} \frac{2}{y^{2}+y-20} dy = \int_{0}^{2} \square dy$ .

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Problem 26963

Berechne die Ableitung von $f(x)=\frac{1}{x^{2}+1}$ und vereinfache das Ergebnis.

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Problem 26964

Evaluate the integral using trigonometric substitution: $\int \frac{d x}{\sqrt{x^{2}-256}}, x>16$ .

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Problem 26965

Find the value of $t$ for the function $y=3 \cos (t-\pi)+2$ where the rate of change is 0. Options: a. 0, b. $\frac{\pi}{2}$ , c. $\frac{\pi}{3}$ , d. $\frac{\pi}{4}$ .

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Problem 26966

Find how many terms of the series $\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k^{6}}$ are needed for the remainder to be < $10^{-3}$ .

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Problem 26967

Untersuchen Sie das Verhalten der Funktionen für $x \rightarrow+\infty$ und $x \rightarrow-\infty$ für die folgenden Fälle: a) $f(x)=x \cdot e^{x}$ b) $f(x)=x^{3} \cdot e^{-x}$ c) $f(x)=e^{x}+4$ d) $f(x)=3-x \cdot e^{x}$ e) $f(x)=1000 \cdot e^{-0,1 x}-4$ f) $f(x)=x \cdot e^{2 x+1}$

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Problem 26968

Finde die erste Ableitung von $f(x)=\sqrt{2x-1}$ und vereinfache das Ergebnis.

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Problem 26969

Ein Patient nimmt $6 \mathrm{mg}$ eines Medikaments. a. Formuliere den exponentiellen Abbau. b. Nach wie vielen Tagen sind noch $3 \mathrm{mg}$ ?

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Problem 26970

Berechne die erste Ableitung von $f(x) = 2 \cos(x^{2})$ und vereinfache das Ergebnis.

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Problem 26971

Bestimme die Länge eines Tigerpythons mit $l(t)=3,5-3,1 \cdot e^{-0,02 t}$ . Beantworte Fragen a) bis e).

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Problem 26972

Bestimmen Sie die Ableitungen der Funktionen: a) $f(x)=(5x+0,5x^{3})^{4}$ , b) $f(x)=\frac{e^{3x}}{x+1}$ , c) $f(x)=e^{x^{2}} \cdot \sqrt{x}$ .

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Problem 26973

Graph the car's velocity $v(t)=120(1-0.85^{t})$ and describe its acceleration.

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Problem 26974

Find the limits:
1. $\lim_{x \to \infty} \left(-\frac{8}{x}\right)$
2. $\lim_{x \to -\infty} \left(-\frac{8}{x}\right)$

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Problem 26975

What is the effect on $f(x)=(x-1)^{2}+1$ when it becomes $f^{\prime}(x)=\frac{1}{2}\left((x-1)^{2}+1\right)$ ?

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Problem 26976

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{x}-1}{x}$ .

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Problem 26977

A cylindrical tank with radius A meters fills with water at 3 m³/min. Find the height increase rate and leak amount in 5 min.

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Problem 26978

Evaluate the limit: $\lim _{x \rightarrow 0}(1+3 \sin x)^{1 / x}$ .

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Problem 26979

Evaluate the limit: $\lim _{x \rightarrow 0}(1+3 \sin x)^{1 / x}$ .

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Problem 26980

Calculate how many mL of ice-water you need to consume to expend 150 calories from a 1 oz bag of chips. Is 700 mL enough?

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Problem 26981

Find the interval of convergence for the series $\sum_{n=1}^{\infty} \cos (\pi n) \frac{n}{n^{2}+1}(x+2)^{n}$ .

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Problem 26982

Bestimme die Steigung der Funktion $f(x)=\frac{1}{2} x^{2}-3 x$ an der Stelle $x=4$ .

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Problem 26983

Find the derivative of $f(x)=\frac{x}{3}-2+e \cdot e^{x}$ .

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Problem 26984

Approximate $4.6^4$ using the tangent line of $f(x) = x^6$ at $x=5$ with $m=3125$ . Find $b$ .

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Problem 26985

Rotate the area between $y=x$ , $x^{2}+y^{2}=1$ , and the $x$ -axis in the first quadrant about the $y$ -axis.
(a) Set up the volume integral with respect to $y$ . (b) Find the probability that a point's $y$ -coordinate is greater than $\frac{1}{2}$ . (c) What is the probability that the $y$ -coordinate is exactly $\frac{1}{2}$ ?

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Problem 26986

Find the horizontal asymptote of the function $\sqrt{x^{2}+10x+20}-x$ . What is $y=$ ?

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Problem 26987

Bestimme $k$ für 5900 Füchse, finde den Zeitpunkt für 5000 Füchse, langfristige Anzahl und interpretiere $p(20)-p(25)$ .

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Problem 26988

Bewerten Sie 10 Aussagen als "wahr" oder "unwahr" und notieren Sie Punkte: 2 für richtig, -2 für falsch, 0 für keine Antwort.

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Problem 26989

Bestimme die Ableitungen der Funktionen: a) $f(x)=5 e^{7 x^{2}+4 x}$ , b) $g(x)=(2 x^{4}-7 x)(x^{3}+x)$ , c) $h(x)=3(2 x^{4}+7)^{13}$ , d) $j(x)=e^{5 x}(2 x^{3}+2)^{4}$ .

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Problem 26990

Find the average slope of $f(x)=2 x^{3}-9 x^{2}-60 x+1$ on $(-5,7)$ and list values $c$ where $f'(c)=0$ .

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Problem 26991

Find the horizontal asymptote of the function $\sqrt{x^{2}+9 x+12}-x$ . What is $y=$ ?

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Problem 26992

Find the average slope of $f(x)=2 x^{3}-9 x^{2}-60 x+1$ on $(-5,7)$ and list all $c$ where $f^{\prime}(c)=0$ .

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Problem 26993

Entscheide, ob die Aussagen wahr oder unwahr sind. Punkte: richtig: 2, falsch: -2, keine: 0. Aussagen: a) Wendestellen und Ableitungen, b) Wendestellen zwischen Extremstellen, c) Funktion 8. Grades max. 6 Wendestellen, d) n-te Ableitung ist 0, e) Ableitung gibt Geschwindigkeit, f) Sekante durch 3 Punkte, g) Ableitung $f(x)=a \cdot x^{n}$ ist $f^{\prime}(x)=\frac{a}{n} \cdot x^{n-1}$ , h) Tangente schneidet Graph, i) zweite Ableitung gibt Steigung der ersten an, j) Sattelpunkte haben Steigung 0, Vorzeichen ändert sich nicht.

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Problem 26994

Find the derivative of the function $f(x) = 2 \cdot x^{-3}$ .

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Problem 26995

i) Die zweite Ableitung zeigt die Steigung der ersten Ableitung. $\square$ wahr $\square$ unwahr
j) An Sattelpunkten ist die Steigung Null, ändert sich aber nicht. $\square$ wahr $\square$ unwahr

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Problem 26996

Find the height $h$ (1000 < h < 20000) that minimizes the operating cost $C = 4000 + \frac{h}{15} + \frac{15000000}{h}$ . What is the minimum cost?

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Problem 26997

Zeichnen Sie die Ableitungen der Funktionen $f(x) = 3^{x}$ , $f(x) = 2^{x}$ und $f(x) = e^{x}$ in einem Koordinatensystem.

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Problem 26998

Find the electric potential at point $P$ above a charged ring with $r=1.0 \, \mathrm{m}$ and $d=75 \, \mathrm{cm}$ . Choices: a) $-25 \, \mathrm{V}$ , b) $9.0 \, \mathrm{V}$ , c) $11 \, \mathrm{V}$ , d) $15 \, \mathrm{V}$ , e) $34 \, \mathrm{V}$ .

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Problem 26999

Is $g(x)=2 x^{3}-10 x$ decreasing at $x=1$ ? Use calculus to explain your answer.

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Problem 27000

Finde die Ableitung von $f(t) = t^{2} \cdot e^{\frac{1}{2} t} + 5$ .

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Calculus

Problem 26901

Given the biomass change B(t)B(t)B(t) with ddtB(t)=2t+1\frac{d}{dt} B(t)=2t+1dtd​B(t)=2t+1 for t∈[0,6]t \in [0,6]t∈[0,6] and B(0)=10B(0)=10B(0)=10, find: a. cumulative change, b. solution, c. average biomass.

Problem 26902

Compute the average value of f(x)=cos⁡(πx)+3xf(x)=\cos (\pi x)+3^{x}f(x)=cos(πx)+3x over the interval [2,4][2,4][2,4].

Problem 26903

Find the limit: lim⁡x→0x+4−2x\lim _{x \rightarrow 0} \frac{\sqrt{x+4}-2}{x}limx→0​xx+4​−2​.

Problem 26904

Find the limit: lim⁡x→04x+10−25x\lim _{x \rightarrow 0} \frac{\frac{4}{x+10}-\frac{2}{5}}{x}limx→0​xx+104​−52​​.

Problem 26905

Find the limit as xxx approaches 2 for the expression 5x2+15x5x2+12x+4\frac{5 x^{2}+15 x}{5 x^{2}+12 x+4}5x2+12x+45x2+15x​.

Problem 26906

Die Funktion p(t)=0,2te−0,0625t+5p(t)=0,2 t e^{-0,0625 t}+5p(t)=0,2te−0,0625t+5 beschreibt Plastikmüll in Mio. Tonnen. Analysieren Sie p(0)p(0)p(0), p(10)−p(5)p(10)-p(5)p(10)−p(5) und p′(16)p^{\prime}(16)p′(16).

Problem 26907

Aufgabe: a) Bestimme den Grenzwert von h(x)h(x)h(x) für große xxx. b) Ab welcher Stückzahl ist das Kunststoffleitwerk günstiger als das Metallleitwerk?

Problem 26908

Find the integral of (x6)3(\sqrt[6]{x})^{3}(6x​)3 with respect to xxx: ∫(x6)3dx=□+C\int(\sqrt[6]{x})^{3} d x=\square +C∫(6x​)3dx=□+C.

Problem 26909

Calculate the integral of (x6)3(\sqrt[6]{x})^{3}(6x​)3 with respect to xxx: ∫(x6)3dx\int(\sqrt[6]{x})^{3} d x∫(6x​)3dx.

Problem 26910

Find the average rate of change of f(x)=tan⁡(2x)f(x)=\tan(2x)f(x)=tan(2x) from 0 to π6\frac{\pi}{6}6π​.

Problem 26911

Analyze the function f(x)f(x)f(x) and its derivative f′(x)f'(x)f′(x) given their behavior on specified intervals.

Problem 26912

Discuss these questions: 1. What does a derivative represent? 2. Using product rule vs. quotient rule? 3. Using product rule vs. chain rule? 4. Types of discontinuities? 5. Requirements for function continuity?

Problem 26913

If a 1000 mg drug dose loses 25% every 4 hours, will it ever be fully eliminated from your system?

Problem 26914

Find the height h\mathrm{h}h (1000 < h\mathrm{h}h < 20000) that minimizes the operating cost C=4000+h/15+15000000/hC=4000+\mathrm{h}/15+15000000/\mathrm{h}C=4000+h/15+15000000/h and calculate CCC at this height.

Problem 26915

Find when the velocity of the particle given by s(t)=3t3−40.5t2+162ts(t)=3 t^{3}-40.5 t^{2}+162 ts(t)=3t3−40.5t2+162t is increasing, decreasing, or constant, and find acceleration at t=6t=6t=6 s.

Problem 26916

Find the derivative of the function y=x2−xsin⁡xy=x^{2}-x \sin xy=x2−xsinx, expressed as dydx\frac{d y}{d x}dxdy​.

Problem 26917

How many years will it take for Brianna's investment to double with a continuous return of 34%34\%34% per year? t=t= t=

Problem 26918

A 70m building, a l-kg ball thrown down at 10m/s, hits ground at 30m/s. Find energy lost to friction: a) 150 J b) 300 J c) 700 J d) 400 J e) 0 J.

Problem 26919

Find the derivative of the function y=x2−2xcos⁡xy=x^{2}-2 x \cos xy=x2−2xcosx, i.e., compute dydx\frac{d y}{d x}dxdy​.

Problem 26920

Find the tangent line equation for the curve defined by x=sec⁡2(t)−1x = \sec^2(t) - 1x=sec2(t)−1, y=tan⁡(t)y = \tan(t)y=tan(t) at t=−π4t = -\frac{\pi}{4}t=−4π​ and compute d2ydx2\frac{d^2y}{dx^2}dx2d2y​.

Problem 26921

Calcula (∂U∂V)T\left(\frac{\partial U}{\partial V}\right)_{T}(∂V∂U​)T​ y el coeficiente de Joule μJ\mu_{J}μJ​ para un gas de Van der Waals.

Problem 26922

Determine if the function f(x)=x5esinh⁡(x)cos⁡(x)f(x)=x^{5} e^{\sinh (x)} \cos (x)f(x)=x5esinh(x)cos(x) has a local min, max, or neither at x=0x=0x=0.

Problem 26923

Solve for ddd in the equation dxd9=1d3\frac{d^{x}}{d^{9}} = \frac{1}{d^{3}}d9dx​=d31​.

Problem 26924

Find the derivative of the function −sin⁡(−x+8)-\sin(-x + 8)−sin(−x+8).

Problem 26925

Find the tangent line equation for y=4sin⁡2(x)y=4 \sin ^{2}(x)y=4sin2(x) at x=π6x=\frac{\pi}{6}x=6π​.

Problem 26926

Find dydx\frac{d y}{d x}dxdy​ in terms of xxx and yyy given the equation 3xy2−3y+2x3=03 x y^{2}-3 y+2 x^{3}=03xy2−3y+2x3=0.

Problem 26927

What is true about the function fff on the interval 3<x<43<x<43<x<4 if its rate of change is positive and decreasing?

Problem 26928

Determine if the particle is speeding up or slowing down at t=2t=2t=2 given its velocity v(t)=−2t−2v(t)=-2t-2v(t)=−2t−2.

Problem 26929

Find the equations of all tangent lines to the curve defined by −xy+y4=−2-x y + y^{4} = -2−xy+y4=−2.

Problem 26930

Find dydx\frac{dy}{dx}dxdy​ for y=cos⁡(f(x3))y = \cos(f(x^3))y=cos(f(x3)) at x=2x = 2x=2, where fff is a differentiable function.

Problem 26931

Boats A and B leave together; A goes north at 12 km/hr, B east at 18 km/hr. After 2.5 hrs, find distance increase rate in km/hr.

Problem 26932

Problem 26933

Find the line perpendicular to g′(x)g'(x)g′(x) at x=−2x=-2x=−2 for g(x)=∫(5x3+4x)dxg(x)=\int(5x^{3}+4x)dxg(x)=∫(5x3+4x)dx.

Problem 26934

Gegeben ist die Funktion f(x)=(−x−2)⋅e−xf(x)=(-x-2) \cdot e^{-x}f(x)=(−x−2)⋅e−x. Finde den Tiefpunkt, Wendepunkt und die Tangentengleichung an der x-Achse.

Problem 26935

Approximate the volume change of a sphere when the radius goes from 10 cm to 10.02 cm. Choices: A) 4213973 B) 1261.669 C) -1256.637 D) 25.233 E) 25.1

Problem 26936

Find ppp values for which the integrals converge: (a) ∫12dxx(ln⁡x)p\int_{1}^{2} \frac{dx}{x(\ln x)^{p}}∫12​x(lnx)pdx​, (b) ∫2∞dxx(ln⁡x)p\int_{2}^{\infty} \frac{dx}{x(\ln x)^{p}}∫2∞​x(lnx)pdx​.

Problem 26937

Bestimme die zweite Ableitung der Funktion f(x)=2tx3−32t2x2f(x) = 2tx^3 - \frac{3}{2}t^2x^2f(x)=2tx3−23​t2x2 und erkläre den Prozess.

Problem 26938

Find the relative max and min of f(x)=x3+12x2+2f(x)=x^{3}+12x^{2}+2f(x)=x3+12x2+2 using the first-derivative test. What is f′(x)f^{\prime}(x)f′(x)?

Problem 26939

Find the relative maximum and minimum points of the function f(x)=x3+12x2+45x+1f(x)=x^{3}+12 x^{2}+45 x+1f(x)=x3+12x2+45x+1 using the first derivative test.

Problem 26940

Find the relative max and min of the function f(x)=x3+12x2+2f(x)=x^{3}+12x^{2}+2f(x)=x3+12x2+2 using the first-derivative test. First, find f′(x)f^{\prime}(x)f′(x).

Given the biomass change $B(t)$ with $\frac{d}{dt} B(t)=2t+1$ for $t \in [0,6]$ and $B(0)=10$ , find: a. cumulative change, b. solution, c. average biomass.

Compute the average value of $f(x)=\cos (\pi x)+3^{x}$ over the interval $[2,4]$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{x+4}-2}{x}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\frac{4}{x+10}-\frac{2}{5}}{x}$ .

Find the limit as $x$ approaches 2 for the expression $\frac{5 x^{2}+15 x}{5 x^{2}+12 x+4}$ .

Die Funktion $p(t)=0,2 t e^{-0,0625 t}+5$ beschreibt Plastikmüll in Mio. Tonnen. Analysieren Sie $p(0)$ , $p(10)-p(5)$ und $p^{\prime}(16)$ .

Aufgabe:
a) Bestimme den Grenzwert von $h(x)$ für große $x$ . b) Ab welcher Stückzahl ist das Kunststoffleitwerk günstiger als das Metallleitwerk?

Find the integral of $(\sqrt[6]{x})^{3}$ with respect to $x$ : $\int(\sqrt[6]{x})^{3} d x=\square +C$ .

Calculate the integral of $(\sqrt[6]{x})^{3}$ with respect to $x$ : $\int(\sqrt[6]{x})^{3} d x$ .

Find the average rate of change of $f(x)=\tan(2x)$ from 0 to $\frac{\pi}{6}$ .

Analyze the function $f(x)$ and its derivative $f'(x)$ given their behavior on specified intervals.

Find the height $\mathrm{h}$ (1000 < $\mathrm{h}$ < 20000) that minimizes the operating cost $C=4000+\mathrm{h}/15+15000000/\mathrm{h}$ and calculate $C$ at this height.

Find when the velocity of the particle given by $s(t)=3 t^{3}-40.5 t^{2}+162 t$ is increasing, decreasing, or constant, and find acceleration at $t=6$ s.

Find the derivative of the function $y=x^{2}-x \sin x$ , expressed as $\frac{d y}{d x}$ .

How many years will it take for Brianna's investment to double with a continuous return of $34\%$ per year? $t=$

Find the derivative of the function $y=x^{2}-2 x \cos x$ , i.e., compute $\frac{d y}{d x}$ .

Find the tangent line equation for the curve defined by $x = \sec^2(t) - 1$ , $y = \tan(t)$ at $t = -\frac{\pi}{4}$ and compute $\frac{d^2y}{dx^2}$ .

Calcula $\left(\frac{\partial U}{\partial V}\right)_{T}$ y el coeficiente de Joule $\mu_{J}$ para un gas de Van der Waals.

Determine if the function $f(x)=x^{5} e^{\sinh (x)} \cos (x)$ has a local min, max, or neither at $x=0$ .

Solve for $d$ in the equation $\frac{d^{x}}{d^{9}} = \frac{1}{d^{3}}$ .

Find the derivative of the function $-\sin(-x + 8)$ .

Find the tangent line equation for $y=4 \sin ^{2}(x)$ at $x=\frac{\pi}{6}$ .

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ given the equation $3 x y^{2}-3 y+2 x^{3}=0$ .

What is true about the function $f$ on the interval $3<x<4$ if its rate of change is positive and decreasing?

Determine if the particle is speeding up or slowing down at $t=2$ given its velocity $v(t)=-2t-2$ .

Find the equations of all tangent lines to the curve defined by $-x y + y^{4} = -2$ .

Find $\frac{dy}{dx}$ for $y = \cos(f(x^3))$ at $x = 2$ , where $f$ is a differentiable function.

Find the line perpendicular to $g'(x)$ at $x=-2$ for $g(x)=\int(5x^{3}+4x)dx$ .

Gegeben ist die Funktion $f(x)=(-x-2) \cdot e^{-x}$ . Finde den Tiefpunkt, Wendepunkt und die Tangentengleichung an der x-Achse.

Find $p$ values for which the integrals converge: (a) $\int_{1}^{2} \frac{dx}{x(\ln x)^{p}}$ , (b) $\int_{2}^{\infty} \frac{dx}{x(\ln x)^{p}}$ .

Bestimme die zweite Ableitung der Funktion $f(x) = 2tx^3 - \frac{3}{2}t^2x^2$ und erkläre den Prozess.

Find the relative max and min of $f(x)=x^{3}+12x^{2}+2$ using the first-derivative test. What is $f^{\prime}(x)$ ?

Find the relative maximum and minimum points of the function $f(x)=x^{3}+12 x^{2}+45 x+1$ using the first derivative test.

Find the relative max and min of the function $f(x)=x^{3}+12x^{2}+2$ using the first-derivative test. First, find $f^{\prime}(x)$ .

Bestimme die Ableitung $g$ von $f(x)=(2x-1)\cdot\cos(x)$ .

Find the relative maximum and minimum points of the function $f(x)=x^{3}+6x^{2}+4$ using the first-derivative test.

Solve the differential equation: $\frac{d y}{d x}=\frac{4 y+2 x+3}{2 y+x+5}$ .

A balloon rises at $8 \mathrm{ft/sec}$ from $150 \mathrm{ft}$ away. How fast is the distance to the balloon increasing at $50 \mathrm{ft}$ ?

Find the limit: $\lim _{x \rightarrow \infty} \frac{x^{2}}{e^{x}}=$ (A) 0 (B) 1 (C) -e (D) e (E) Does not exist.

A quantity starts at 180 and decays at $70\%$ per minute. Find its value after 333 seconds, rounded to the nearest hundredth.

A stone creates ripples in a pond. If the radius grows at $1.5 \mathrm{ft} / \mathrm{sec}$ , find the area increase rate when radius is 3 ft.

A quantity starts at 2300 and grows at $45\%$ per minute. Find its value after 0.2 hours, rounded to two decimal places.

Ben launches a potato to 17 m. What is its velocity when it hits the ground? Use $g = 9.8 \frac{\mathrm{m}}{\mathrm{s}^{2}}$ .

Gegeben die Funktionenschar $f_{a}(a>0)$ , finde die Ableitung und die Steigung bei $x=0$ . Wann ist die Steigung $1$ ?

Bestimme die Asymptotengleichung für den Graphen von $f(x)=x^{2} \cdot e^{-3 x}+5$ .

Find the derivative $P^{\prime}(t)$ of the function $P(t) = 50000(0.98)^{t}$ .

Find the derivative of the function $e^{x / 2}$ with respect to $x$ .

Graph the car's velocity $v(t)=120(1-0.85^{t})$ and describe its acceleration.

Find the max effectiveness $E(t)=0.5\left[10+t e^{-\frac{1}{5}}\right]$ for studying up to 30 hours.

Find the horizontal asymptote of $f(x)=\frac{5 x^{2}+3}{4 x^{2}-5}$ . Choose A, B, or C and fill in the boxes if needed.

Find the value of $b$ for which $\lim _{x \rightarrow-2} \frac{b x^{2}+15 x+15+b}{x^{2}+x-2}$ exists, and determine the limit.

Evaluate the integral: $\int_{0}^{2} \frac{2}{y^{2}+y-20} dy = \int_{0}^{2} \square dy$ .

Berechne die Ableitung von $f(x)=\frac{1}{x^{2}+1}$ und vereinfache das Ergebnis.

Evaluate the integral using trigonometric substitution: $\int \frac{d x}{\sqrt{x^{2}-256}}, x>16$ .

Find the value of $t$ for the function $y=3 \cos (t-\pi)+2$ where the rate of change is 0. Options: a. 0, b. $\frac{\pi}{2}$ , c. $\frac{\pi}{3}$ , d. $\frac{\pi}{4}$ .

Find how many terms of the series $\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k^{6}}$ are needed for the remainder to be < $10^{-3}$ .

Finde die erste Ableitung von $f(x)=\sqrt{2x-1}$ und vereinfache das Ergebnis.

Ein Patient nimmt $6 \mathrm{mg}$ eines Medikaments. a. Formuliere den exponentiellen Abbau. b. Nach wie vielen Tagen sind noch $3 \mathrm{mg}$ ?

Berechne die erste Ableitung von $f(x) = 2 \cos(x^{2})$ und vereinfache das Ergebnis.

Bestimme die Länge eines Tigerpythons mit $l(t)=3,5-3,1 \cdot e^{-0,02 t}$ . Beantworte Fragen a) bis e).

Bestimmen Sie die Ableitungen der Funktionen: a) $f(x)=(5x+0,5x^{3})^{4}$ , b) $f(x)=\frac{e^{3x}}{x+1}$ , c) $f(x)=e^{x^{2}} \cdot \sqrt{x}$ .

Graph the car's velocity $v(t)=120(1-0.85^{t})$ and describe its acceleration.

Find the limits:
1. $\lim_{x \to \infty} \left(-\frac{8}{x}\right)$
2. $\lim_{x \to -\infty} \left(-\frac{8}{x}\right)$