Calculus

Problem 16901

Die Funktion $f(x)=|x|$ hat bei $x=0$ ein Minimum. Erkläre, warum man dieses Minimum nicht mit Extremabedingungen findet.

See Solution

Problem 16902

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

See Solution

Problem 16903

How much water is left in a tank after 3 minutes if it starts with 70 gallons and leaks at $r(t)=10-\frac{t^{2}}{3}$ ?

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Problem 16904

Bestimme die Füllhöhe des Pools nach 10 Minuten mit $v(t)=0,1 t^{2}-\frac{1}{t^{3}+1}+40$ und Grundfläche 3,6 $\mathrm{m}^{2}$ .

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Problem 16905

1.) Initial amount of Carbon-14? 2.) Amount after 16294 years? 3.) Half-life of Carbon-14?
Given: $A(t)=934 e^{-0.00012 t}$

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Problem 16906

Find the equation of $F(x)$ if its slope is $6 e^{3 x}+5 e^{-x}$ and it has a $y$ -intercept of 5.

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Problem 16907

A rocket launches 300 feet away from Isaac. At 400 feet high, the distance to Isaac increases at 40 ft/s. Find the rocket's speed.

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Problem 16908

Calculate the integral: $\int\left(\cot^{2} x + \sin^{3} x \sqrt{\cos x}\right) dx$

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Problem 16909

Find the rate of area increase of a circle when the radius is $6 \text{ ft}$ .

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Problem 16910

A circle's area increases at $3 \frac{f t^{2}}{s e c}$ . Find the radius's increase rate when the radius is $6 f t$ .

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Problem 16911

Find the limit as $x$ approaches infinity of $(2+x)^{1/x}$ .

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Problem 16912

Evaluate the integral: $\int\left(5 x \sec (x) \tan (x)-7 x 4^{x}+\frac{12}{\sqrt{x^{2}-9}}\right) \frac{\mathrm{d} x}{x}$ .

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Problem 16913

Find the quantity $Q$ that maximizes profit given $P=102-2Q$ and $C=2Q+0.5Q^2$ . Calculate the max profit.

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Problem 16914

Given $f(a)=g(a)=0$ , $f'(a)=0.5$ , and $g'(a)=1.25$ , find $\lim_{x \rightarrow a} \frac{f(x)}{g(x)}$ rounded to the nearest thousandth.

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Problem 16915

Find the area between the graph of $y=f^{-1}(x)$ and the $y$ -axis for $0 \leq y \leq 2$ where $f(x)=x^{5}+3 x+1$ .

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Problem 16916

Find the half-life of a radioactive substance with a decay rate of $9.1\%$ per day using the continuous decay model.

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Problem 16917

Bestimme die Änderungsrate der unzerfallenen Radon 222 Atome mit $N(t) \approx N_{0} \cdot 0,834058161^{t}$ für a) $t=2$ und b) $t=4$ .

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Problem 16918

Evaluate the integral $\int \ln \left(x^{7}\right) d x$ .

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Problem 16919

Find the integral of the function: $\int 4 x e^{5 x} d x$ .

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Problem 16920

Calculate the limit: $\lim _{n \rightarrow \infty}\left(1+\frac{\pi}{2^{n+1}}\right)^{2^{n}}$ .

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Problem 16921

Find intercepts, relative min/max, inflection points, and the asymptote for $y=\frac{x^{2}}{x^{2}+27}$ .

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Problem 16922

Calcular el límite: $\lim _{x \rightarrow 7} \frac{x^{2}-4 x-21}{x-7}$

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Problem 16923

Bestimmen Sie die Ableitung und die Steigung von $f_{a}(x)$ an $x=0$ . Für welchen Wert von $a$ ist die Steigung 1?

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Problem 16924

Calculate the integral: $\int \frac{d x}{\sqrt[4]{(x+1)^{3}}-\sqrt{x+1}}$

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Problem 16925

Find the derivative of $f(x)=\frac{x}{\sec(3x)+1}$ .

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Problem 16926

Gegeben ist die Funktion $f(x)=x \cdot e^{x}$ . Finde den Tiefpunkt, die Normale im Ursprung und den Wendepunkt.

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Problem 16927

Evaluate limits of rational functions as $x \rightarrow \infty$ or $x \rightarrow -\infty$ . Find:
b) $\lim _{x \rightarrow \infty} \frac{4 x^{3}-2 x^{2}}{5 x^{3}+30 x+2}$ , c) $\lim _{x \rightarrow -\infty} \frac{6 x^{2}-4 x+3}{7 x^{2}+5 x-10}$ , d) $\lim _{x \rightarrow -\infty} \frac{6 x^{2}-4 x+3}{7 x^{3}+5 x-10}$ , $\lim _{x \rightarrow \infty} \frac{2 x^{5}-x^{2}}{7 x^{3}+3 x^{2}+5}$ .

See Solution

Problem 16928

Find the derivative $\frac{d}{d m}\left(x^{3} m^{3}+y^{5} m+z^{6}\right)$ .

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Problem 16929

Find the first and second derivatives of $y=3x+9$ : $\frac{dy}{dx}=$ and $\frac{d^{2}y}{dx^{2}}=$ .

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Problem 16930

Gegeben ist die Funktion $f(x)=x \cdot e^{x}$ . Finde: a) den Tiefpunkt, b) die Normale im Ursprung, c) den Wendepunkt.

See Solution

Problem 16931

Find the first derivative of the function $f(t)=(t^{3}+2)^{8}$ .

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Problem 16932

Find the average velocity of an object with $v=4t-3t^{2}$ from $t=0$ to $t=2$ seconds. Options: a. 0 b. $-2 \mathrm{~m/s}$ c. $2 \mathrm{~m/s}$ d. -4ri's e. can't calculate without initial position.

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Problem 16933

Estimate the cost of making 41 bicycles if $C(40)=7000$ and $C^{\prime}(40)=55$ .

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Problem 16934

At what point are the velocity and acceleration vectors of a projectile thrown upward perpendicular? (a) Nowhere (b) highest point (c) launch point

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Problem 16935

1) Optimize $f(x)=3 x^{2/3}+2 x$ on $[-1,8]$ . 2) a. State Rolle's theorem. b. Verify Rolle's theorem for $f(x)=(x-2)^2(x+3)$ on $[-3,2]$ . c. Find $c$ values from Rolle's conclusions. 3) For $f(x)=\frac{x^5}{5}-16x$ : a. Find critical points. b. Solve $f'(x)>0$ and $f'(x)<0$ for increasing/decreasing intervals. c. Classify critical points using the First Derivative Test. 4) For $f(x)=(\sin x)(\cos x)+2$ : a. Find critical points. b. Solve $f'(x)>0$ and $f'(x)<0$ for increasing/decreasing intervals. c. Classify critical points using the First Derivative Test. BONUS: Minimize distance from $P(10,0)$ to $y=\sqrt{2x-9}$ : a. Set $g(x,y)=\text{dist}^2[(x,y),(10,0)]$ . b. Substitute $y=\sqrt{2x-9}$ to minimize $G(x)$ .

See Solution

Problem 16936

State Rolle's theorem. For $f(x)=(x-2)^{\wedge} 2(x+3)$ , verify it on $[-3,2]$ and find values of $c$ from the theorem.

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Problem 16937

Bestimme die Ableitung von $f$ mit der Produktregel für die gegebenen Funktionen: a) $f(x)=(2x^{3}+5)x^{2}$ , b) $f(x)=(x^{2}-5)(2x+1)$ , c) $f(x)=(x^{2}-1)(x^{2}+1)$ , d) $f(x)=(2x^{3}-3x^{2}+x)(3x+5)$ , e) $f(x)=(x^{2}+3)\sqrt{x}$ , f) $f(x)=3x^{5}(\cos(x)+x)$ , g) $f(x)=\frac{1}{x}\sin(x)$ , h) $f(x)=\sin(x)\cos(x)$ , i) $f(x)=\frac{1}{x^{2}}\sqrt{x}$ , j) $f(x)=(\sin(x)+1)(\cos(x)+1)$ , k) $f(t)=(3t^{3}+t^{2})t^{-3}$ , l) $f(t)=\frac{t^{3}}{2}(2t+1)+5t$ .

See Solution

Problem 16938

Bestimme die Ableitung der Funktionen mit der Produktregel: a) $f(x)=(2 x^{3}+5) \cdot x^{2}$ , b) $f(x)=(x^{2}-5) \cdot(2 x+1)$ , c) $f(x)=(x^{2}-1)(x^{2}+1)$ , d) $f(x)=(2 x^{3}-3 x^{2}+x)(3 x+5)$ , e) $f(x)=(x^{2}+3)\sqrt{x}$ , f) $f(x)=3 x^{5}(\cos(x)+x)$ , g) $f(x)=\frac{1}{x} \sin(x)$ , h) $f(x)=\sin(x) \cos(x)$ , i) $f(x)=\frac{1}{x^{2}} \sqrt{x}$ , j) $f(x)=(\sin(x)+1)(\cos(x)+1)$ , k) $f(t)=(3 t^{3}+t^{2}) t^{-3}$ , l) $f(t)=\frac{t^{3}}{2}(2 t+1)+5 t$ .

See Solution

Problem 16939

Bestimmen Sie die Ableitung von $f$ mit der Produktregel für die Funktionen a) bis l).

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Problem 16940

Check convergence or divergence of the series and identify if it's conditional or absolute: $\sum_{n=1}^{\infty}(-1)^{n} \ln \left(1+\frac{1}{n}\right)$

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Problem 16941

Find critical points of $f(x)=\frac{x^{5}}{5}-16x$ , determine intervals of increase/decrease, and classify critical points.

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Problem 16942

Calculate the integral $\int 2 x e^{-5 t} d x$ . Choose the correct answer from the options provided.

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Problem 16943

Calculate the tumour mass change from $t=0$ to $t=2$ days given $G(t)=\frac{8}{9} t^{\frac{1}{3}}$ . Options: a) $\sqrt[3]{16}$ , b) $\frac{2}{3} \sqrt[3]{16}$ , c) $\frac{1}{4} \sqrt[3]{16}$ , d) $\frac{3}{4} \sqrt[3]{16}$ , e) None.

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Problem 16944

Berechnen Sie die Ableitung von $f$ an den angegebenen Stellen: a) $f(x)=x^{2}, x_{0}=4$ ; b) $f(x)=-2 x^{2}, x_{0}=3$ ; c) $f(x)=2 x^{2}, x_{0}=3$ ; d) $f(x)=2 x^{2}, x_{0}=4$ ; e) $f(x)=\frac{1}{x}, x_{0}=-1$ ; f) $f(x)=2 x^{2}, x_{0}=-2$ ; g) $f(x)=0,5 x^{2}, x_{0}=2$ ; h) $f(x)=-x+2, x_{0}=3$ ; i) $f(x)=4, x_{0}=7$ .

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Problem 16945

Find the integral of $\sin \left(\frac{1}{4} \pi\right) dx$ . What is the result?

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Problem 16946

Approximate $\int_{1}^{9}(x-6)^{2} d x$ using Riemann sum with left endpoints $u_{i}$ for intervals $[1,3],[3,5],[5,7],[7,9]$ . Options: a) 112 b) 308 c) 72 d) 36 e) None.

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Problem 16947

Find the integral: $\int \frac{5}{3} t^{\frac{1}{3}} d t=$ a) $\frac{5}{4} t^{\frac{4}{3}}+C$ b) $\frac{15}{8} t^{\frac{4}{3}}+C$ c) $\frac{1}{2} t^{\frac{4}{3}}+C$ d) $t^{\frac{5}{3}}+C$ e) None of the above

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Problem 16948

Calculate the integral $\int_{-5}^{5}(3 x)^{2} d x$ .

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Problem 16949

Estimate the integral $\int_{1}^{9}(x-5)^{2} dx$ using Riemann sums with right endpoints for intervals $[1,3],[3,5],[5,7],[7,9]$ . Choose from a) 40 b) 48 c) 24 d) 34 e) None.

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Problem 16950

Bestimme die Ableitung und die Steigung von $f_{a}(x)$ an $x=0$ für die Funktionen a) bis c) und finde $a$ für Steigung 1.

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Problem 16951

Find critical points of $f(x)=(\sin x)(\cos x)+2$ , determine intervals of increase/decrease, and classify critical points.

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Problem 16952

Find critical points of $f(x)=(\sin x)(\cos x)+2$ , analyze $f'(x)$ for increasing/decreasing intervals, and classify critical points.
BONUS: Minimize distance from point $P(10,0)$ to graph $y=\sqrt{(2x-9)}$ using $g(x,y)=\operatorname{dist}^{2}[(x,y),(10,0)]$ . Substitute to find $G(x)$ and its minimum.

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Problem 16953

Evaluate the integral: $\int \frac{r^{2}}{r+6} d r$ using absolute values.

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Problem 16954

Find $\frac{d y}{d x}$ for $\cos (x y)=y-1$ at $x=\frac{\pi}{2}$ , $y=1$ . Options: (A) $\frac{-2}{2-\pi}$ , (B) $\frac{-2}{2+\pi}$ , (C) 0, (D) $\frac{2}{2-\pi}$ , (E) $\frac{2}{2+\pi}$ .

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Problem 16955

Bestimme die Ableitung von $f(x)=\frac{(x^{3}+x) \cdot \sin (x)}{\ln (x)}$ .

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Problem 16956

Determine convergence/divergence of the series: $\sum_{n=1}^{\infty}(-1)^{n+1} \frac{3 \sqrt{n+1}}{\sqrt{n}+1}$ .

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Problem 16957

Die Funktion $f(t)=\frac{1}{4} t^{3}-\frac{11}{4} t^{2}-4 t+44$ beschreibt die Wassermenge im Stausee.
a) Bestimme Zeiträume mit Zunahme der Wassermenge. b) Finde den Zeitpunkt mit der geringsten Änderungsrate. c) Erläutere die Bedeutung dieses Zeitpunkts für die Funktion $g(t)$ .

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Problem 16958

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

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Problem 16959

Find the derivative of $s(t)=\left(\frac{t}{2 t+1}\right)^{3/2}$ .

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Problem 16960

Find $f'(x)$ for the function $f(x)=\frac{x^{2}}{(x^{2}-1)^{4}}$ .

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Problem 16961

Find the derivative of $h(x)=\frac{(3 x^{2}+1)^{3}}{(x^{2}-1)^{4}}$ .

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Problem 16962

Ein leerer Pool wird mit $v(t)=0,1 t^{2}-\frac{1}{t^{3}+1}+40$ befüllt. Bestimmen Sie die Füllhöhe und das Volumen nach 10 Minuten.

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Problem 16963

Given that $h(x) = h(2-x)$ for all $x$ , which statements are true? I. $\int_{0}^{2} h(x) d x > 0$ II. $h^{\prime}(1) = 0$ III. $h^{\prime}(0) = h^{\prime}(2) = 1$ Choose from: (A) I only, (B) II only, (C) III only, (D) II and III only, (E) I, II, and III.

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Problem 16964

Find the derivative of $f(x)=(2x+1)^{-2}$ .

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Problem 16965

Solve the differential equation $dy/dx = \frac{y - y^2}{x}$ for $x \neq 0$ . Find the general solution $y = f(x)$ .

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Problem 16966

Find the derivative of $g(t)=\frac{\sqrt{t+1}}{\sqrt{t^{2}+1}}$ .

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Problem 16967

Given $f$ is twice differentiable with $f(4)=6$ , $f'(4)=5$ , and $f''(4)=8$ . Find $h'(4)$ for $h(x)=\sqrt{1+f'(x)^{2}}$ . Round to three decimal places.

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Problem 16968

Find the maximum number of extreme values for the function $f(x)=x^{3}-7x-6$ . A. 4 B. 3 C. 1 D. 2

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Problem 16969

Find $\frac{d y}{d x}$ at $(6,2)$ given $f(6)=1$ , $f^{\prime}(6)=1.6$ , and $f(x)^{2}+y^{2}=6y-7f(x)$ . Round to three decimal places.

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Problem 16970

Berechnen Sie die Fläche zwischen dem Graphen von $f$ und der $x$ -Achse für die Funktionen a) bis f).

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Problem 16971

Evaluate the integral: $\int \frac{9}{(x+a)(x+b)} d x$ (Assume $a \neq b$ ).

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Problem 16972

James drops a penny from a skyscraper; it lands in 6.7 seconds. Find the height in feet. Choices: a. 113.9 b. 214.4 c. 107.2 d. 718.24

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Problem 16973

Find the value of the nested radical $\sqrt{72+\sqrt{72+\sqrt{72+\ldots}}}$ and its recursive sequence $a_{n}$ . What is $a_{0}$ ? Also, find $\lim _{n \rightarrow \infty} a_{n}$ and explain your reasoning for the limit.

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Problem 16974

Trouvez les abscisses des points critiques de la fonction $f(x)=x^{4}(4 x+9)$ .

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Problem 16975

Une échelle de $10 \mathrm{~m}$ glisse contre un mur. Si le bas s'éloigne à $1 \mathrm{~m/s}$ , quelle est la vitesse du sommet quand le bas est à $8 \mathrm{~m}$ du mur ?

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Problem 16976

Construire une boîte cylindrique sans couvercle de volume $30 \mathrm{~cm}^{3}$ . Trouver $r$ et $h$ pour minimiser la surface $2 \pi r h$ .

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Problem 16977

A mass $m$ on a table connected by springs to points $A$ and $B$ (6 $l_0$ apart). Find motion equation, equilibrium, frequency, and displacement $x(t)$ .

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Problem 16978

A substance grows at $19\%$ daily. If it starts at 75 grams, find its mass after 2 days, rounded to the nearest tenth.

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Problem 16979

Analyze the function $f(x) = \frac{x^2+4}{x+3}$ for asymptotes and extrema. Which option fits its behavior?

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Problem 16980

Find the intervals where $k(x)=\frac{-3}{2 x+4}$ is increasing or decreasing.

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Problem 16981

Identify the correct intervals of increase and decrease for $h(x)=\frac{-1}{x^{2}+10 x+25}$ .

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Problem 16982

Find the max and min of the function $f(x)=\frac{|2 x-3|}{x^{2}+4}$ on the interval $[-10,10]$ .

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Problem 16983

Find the derivative of $f(x)=\frac{\sin (\theta)}{\cos (\theta)}+\frac{\cos (\theta)}{\sin (\theta)}$ .

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Problem 16984

Find the area between the curve $4-x^{2}$ and the $x$ -axis from $x=-2$ to $x=2$ using a Riemann sum with $n$ subintervals.

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Problem 16985

Gegeben ist die Funktion $g$ mit $\int_{-2}^{2} g(x) d x=1$ .
1. Zeichnen Sie den Graphen von $v$ .
2. Bestimmen Sie die Position des Aufzugs nach 20 s.
3. Schreiben Sie die Positionsänderung aus Teil 2 in Integralschreibweise.

See Solution

Problem 16986

Find the anti-derivative of $f(x) = \frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)}$ .

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Problem 16987

Find the anti-derivative of $\frac{\sin (\theta)}{\cos (\theta)+1}+\frac{\cos (\theta)}{\sin (\theta)}$ .

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Problem 16988

Analyze the behavior of the functions $f(x)=\frac{1}{x-2}$ and $f(x)=\frac{-1}{x^{2}-4}$ as $x \rightarrow 2^{+}$ and $x \rightarrow+\infty$ .

See Solution

Problem 16989

Analyze the behavior of the functions $f(x)=\frac{1}{x-2}$ and $f(x)=\frac{-1}{x^{2}-4}$ as $x \to 2^{+}$ and $x \to +\infty$ .

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Problem 16990

Zeigen Sie, dass die folgenden Integrale den Wert 0 haben: a) $\int_{-4}^{4} 0,25 x d x$ , b) $\int_{-\pi}^{\pi} \sin (x) d x$ , c) $\int_{0}^{\pi} \cos (x) d x$ , d) $\int_{1}^{3}(x-2)^{3} d x$ .

See Solution

Problem 16991

Find dimensions of a rectangular area of $2400 \mathrm{~m}^{2}$ to minimize fencing cost: outside \$30/m, inside \$10/m.

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Problem 16992

Find the absolute max and min of $f(x)=2x^3-3x^2-36x$ on the interval $-5 \leq x \leq 5$ .

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Problem 16993

Beweisen Sie die folgenden Gleichungen: a) $\int_{-5}^{-2} x^{2} d x=\int_{2}^{5} x^{2} d x$ b) $\int_{-4}^{4} 3 x^{2} d x=2 \cdot \int_{0}^{4} 3 x^{2} d x$ c) $\int_{-2}^{0}(x-1)^{2} d x=\int_{2}^{4}(x-1)^{2} d x$

See Solution

Problem 16994

Find the time $t$ (0 ≤ $t$ ≤ 1) when the pollutant level $P(t)=\frac{2t+1}{162t+1}$ is lowest.

See Solution

Problem 16995

Find $y^{\prime}$ for $y=(x^{2}+3)^{2}(x^{3}-2x)^{3}$ .

See Solution

Problem 16996

Find the value of $a$ for the function $f(x)=3 x^{3}+a x^{2}+4 a^{2} x+9 a$ with an inflection point at $x=-1$ .

See Solution

Problem 16997

Find the critical point(s) of the function $f(x)=-\frac{3}{x^{2}-9}$ .

See Solution

Problem 16998

Find $\int_{3}^{4} f(z) d z$ given $\int_{1}^{5} f(z) dz=6$ and $\int_{1}^{4} f(z) dz=9$ .

See Solution

Problem 16999

Sketch possible graphs of $f'(x)$ and $f(x)$ given $f''(x)$ is decreasing and has a point at (2,0). Explain key features.

See Solution

Problem 17000

Find the derivatives of $f(x)=(x+4)^{3}(x-3)^{6}$ and $y=\frac{3x^{2}+2x}{x^{2}+1}$ .

See Solution

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Calculus

Problem 16901

Die Funktion f(x)=∣x∣f(x)=|x|f(x)=∣x∣ hat bei x=0x=0x=0 ein Minimum. Erkläre, warum man dieses Minimum nicht mit Extremabedingungen findet.

Problem 16902

Evaluate the integral: ∫e1/x2x3dx\int \frac{e^{1 / x^{2}}}{x^{3}} d x∫x3e1/x2​dx.

Problem 16903

How much water is left in a tank after 3 minutes if it starts with 70 gallons and leaks at r(t)=10−t23r(t)=10-\frac{t^{2}}{3}r(t)=10−3t2​?

Problem 16904

Bestimme die Füllhöhe des Pools nach 10 Minuten mit v(t)=0,1t2−1t3+1+40v(t)=0,1 t^{2}-\frac{1}{t^{3}+1}+40v(t)=0,1t2−t3+11​+40 und Grundfläche 3,6 m2\mathrm{m}^{2}m2.

Problem 16905

1.) Initial amount of Carbon-14? 2.) Amount after 16294 years? 3.) Half-life of Carbon-14? Given: A(t)=934e−0.00012tA(t)=934 e^{-0.00012 t}A(t)=934e−0.00012t

Problem 16906

Find the equation of F(x)F(x)F(x) if its slope is 6e3x+5e−x6 e^{3 x}+5 e^{-x}6e3x+5e−x and it has a yyy-intercept of 5.

Problem 16907

A rocket launches 300 feet away from Isaac. At 400 feet high, the distance to Isaac increases at 40 ft/s. Find the rocket's speed.

Problem 16908

Calculate the integral: ∫(cot⁡2x+sin⁡3xcos⁡x)dx\int\left(\cot^{2} x + \sin^{3} x \sqrt{\cos x}\right) dx∫(cot2x+sin3xcosx​)dx

Problem 16909

Find the rate of area increase of a circle when the radius is 6 ft6 \text{ ft}6 ft.

Problem 16910

A circle's area increases at 3ft2sec3 \frac{f t^{2}}{s e c}3secft2​. Find the radius's increase rate when the radius is 6ft6 f t6ft.

Problem 16911

Find the limit as xxx approaches infinity of (2+x)1/x(2+x)^{1/x}(2+x)1/x.

Problem 16912

Evaluate the integral: ∫(5xsec⁡(x)tan⁡(x)−7x4x+12x2−9)dxx\int\left(5 x \sec (x) \tan (x)-7 x 4^{x}+\frac{12}{\sqrt{x^{2}-9}}\right) \frac{\mathrm{d} x}{x}∫(5xsec(x)tan(x)−7x4x+x2−9​12​)xdx​.

Problem 16913

Find the quantity QQQ that maximizes profit given P=102−2QP=102-2QP=102−2Q and C=2Q+0.5Q2C=2Q+0.5Q^2C=2Q+0.5Q2. Calculate the max profit.

Problem 16914

Given f(a)=g(a)=0f(a)=g(a)=0f(a)=g(a)=0, f′(a)=0.5f'(a)=0.5f′(a)=0.5, and g′(a)=1.25g'(a)=1.25g′(a)=1.25, find lim⁡x→af(x)g(x)\lim_{x \rightarrow a} \frac{f(x)}{g(x)}limx→a​g(x)f(x)​ rounded to the nearest thousandth.

Problem 16915

Find the area between the graph of y=f−1(x)y=f^{-1}(x)y=f−1(x) and the yyy-axis for 0≤y≤20 \leq y \leq 20≤y≤2 where f(x)=x5+3x+1f(x)=x^{5}+3 x+1f(x)=x5+3x+1.

Problem 16916

Find the half-life of a radioactive substance with a decay rate of 9.1%9.1\%9.1% per day using the continuous decay model.

Problem 16917

Bestimme die Änderungsrate der unzerfallenen Radon 222 Atome mit N(t)≈N0⋅0,834058161tN(t) \approx N_{0} \cdot 0,834058161^{t}N(t)≈N0​⋅0,834058161t für a) t=2t=2t=2 und b) t=4t=4t=4.

Problem 16918

Evaluate the integral ∫ln⁡(x7)dx\int \ln \left(x^{7}\right) d x∫ln(x7)dx.

Problem 16919

Find the integral of the function: ∫4xe5xdx\int 4 x e^{5 x} d x∫4xe5xdx.

Problem 16920

Calculate the limit: lim⁡n→∞(1+π2n+1)2n\lim _{n \rightarrow \infty}\left(1+\frac{\pi}{2^{n+1}}\right)^{2^{n}}limn→∞​(1+2n+1π​)2n.

Problem 16921

Find intercepts, relative min/max, inflection points, and the asymptote for y=x2x2+27y=\frac{x^{2}}{x^{2}+27}y=x2+27x2​.

Problem 16922

Calcular el límite: lim⁡x→7x2−4x−21x−7\lim _{x \rightarrow 7} \frac{x^{2}-4 x-21}{x-7}x→7lim​x−7x2−4x−21​

Problem 16923

Bestimmen Sie die Ableitung und die Steigung von fa(x)f_{a}(x)fa​(x) an x=0x=0x=0. Für welchen Wert von aaa ist die Steigung 1?

Problem 16924

Calculate the integral: ∫dx(x+1)34−x+1\int \frac{d x}{\sqrt[4]{(x+1)^{3}}-\sqrt{x+1}}∫4(x+1)3​−x+1​dx​

Problem 16925

Find the derivative of f(x)=xsec⁡(3x)+1f(x)=\frac{x}{\sec(3x)+1}f(x)=sec(3x)+1x​.

Problem 16926

Gegeben ist die Funktion f(x)=x⋅exf(x)=x \cdot e^{x}f(x)=x⋅ex. Finde den Tiefpunkt, die Normale im Ursprung und den Wendepunkt.

Problem 16927

Problem 16928

Find the derivative ddm(x3m3+y5m+z6)\frac{d}{d m}\left(x^{3} m^{3}+y^{5} m+z^{6}\right)dmd​(x3m3+y5m+z6).

Problem 16929

Find the first and second derivatives of y=3x+9y=3x+9y=3x+9: dydx=\frac{dy}{dx}=dxdy​= and d2ydx2=\frac{d^{2}y}{dx^{2}}=dx2d2y​=.

Problem 16930

Gegeben ist die Funktion f(x)=x⋅exf(x)=x \cdot e^{x}f(x)=x⋅ex. Finde: a) den Tiefpunkt, b) die Normale im Ursprung, c) den Wendepunkt.

Problem 16931

Find the first derivative of the function f(t)=(t3+2)8f(t)=(t^{3}+2)^{8}f(t)=(t3+2)8.

Problem 16932

Find the average velocity of an object with v=4t−3t2v=4t-3t^{2}v=4t−3t2 from t=0t=0t=0 to t=2t=2t=2 seconds. Options: a. 0 b. −2 m/s-2 \mathrm{~m/s}−2 m/s c. 2 m/s2 \mathrm{~m/s}2 m/s d. -4ri's e. can't calculate without initial position.

Problem 16933

Estimate the cost of making 41 bicycles if C(40)=7000C(40)=7000C(40)=7000 and C′(40)=55C^{\prime}(40)=55C′(40)=55.

Problem 16934

At what point are the velocity and acceleration vectors of a projectile thrown upward perpendicular? (a) Nowhere (b) highest point (c) launch point

Problem 16935

Problem 16936

State Rolle's theorem. For f(x)=(x−2)∧2(x+3)f(x)=(x-2)^{\wedge} 2(x+3)f(x)=(x−2)∧2(x+3), verify it on [−3,2][-3,2][−3,2] and find values of ccc from the theorem.

Problem 16937

Problem 16938

Problem 16939

Bestimmen Sie die Ableitung von fff mit der Produktregel für die Funktionen a) bis l).

Problem 16940

Check convergence or divergence of the series and identify if it's conditional or absolute: ∑n=1∞(−1)nln⁡(1+1n)\sum_{n=1}^{\infty}(-1)^{n} \ln \left(1+\frac{1}{n}\right)n=1∑∞​(−1)nln(1+n1​)

Problem 16941

Find critical points of f(x)=x55−16xf(x)=\frac{x^{5}}{5}-16xf(x)=5x5​−16x, determine intervals of increase/decrease, and classify critical points.

Problem 16942

Die Funktion $f(x)=|x|$ hat bei $x=0$ ein Minimum. Erkläre, warum man dieses Minimum nicht mit Extremabedingungen findet.

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

How much water is left in a tank after 3 minutes if it starts with 70 gallons and leaks at $r(t)=10-\frac{t^{2}}{3}$ ?

Bestimme die Füllhöhe des Pools nach 10 Minuten mit $v(t)=0,1 t^{2}-\frac{1}{t^{3}+1}+40$ und Grundfläche 3,6 $\mathrm{m}^{2}$ .

1.) Initial amount of Carbon-14? 2.) Amount after 16294 years? 3.) Half-life of Carbon-14?
Given: $A(t)=934 e^{-0.00012 t}$

Find the equation of $F(x)$ if its slope is $6 e^{3 x}+5 e^{-x}$ and it has a $y$ -intercept of 5.

Calculate the integral: $\int\left(\cot^{2} x + \sin^{3} x \sqrt{\cos x}\right) dx$

Find the rate of area increase of a circle when the radius is $6 \text{ ft}$ .

A circle's area increases at $3 \frac{f t^{2}}{s e c}$ . Find the radius's increase rate when the radius is $6 f t$ .

Find the limit as $x$ approaches infinity of $(2+x)^{1/x}$ .

Evaluate the integral: $\int\left(5 x \sec (x) \tan (x)-7 x 4^{x}+\frac{12}{\sqrt{x^{2}-9}}\right) \frac{\mathrm{d} x}{x}$ .

Find the quantity $Q$ that maximizes profit given $P=102-2Q$ and $C=2Q+0.5Q^2$ . Calculate the max profit.

Given $f(a)=g(a)=0$ , $f'(a)=0.5$ , and $g'(a)=1.25$ , find $\lim_{x \rightarrow a} \frac{f(x)}{g(x)}$ rounded to the nearest thousandth.

Find the area between the graph of $y=f^{-1}(x)$ and the $y$ -axis for $0 \leq y \leq 2$ where $f(x)=x^{5}+3 x+1$ .

Find the half-life of a radioactive substance with a decay rate of $9.1\%$ per day using the continuous decay model.

Bestimme die Änderungsrate der unzerfallenen Radon 222 Atome mit $N(t) \approx N_{0} \cdot 0,834058161^{t}$ für a) $t=2$ und b) $t=4$ .

Evaluate the integral $\int \ln \left(x^{7}\right) d x$ .

Find the integral of the function: $\int 4 x e^{5 x} d x$ .

Calculate the limit: $\lim _{n \rightarrow \infty}\left(1+\frac{\pi}{2^{n+1}}\right)^{2^{n}}$ .

Find intercepts, relative min/max, inflection points, and the asymptote for $y=\frac{x^{2}}{x^{2}+27}$ .

Calcular el límite: $\lim _{x \rightarrow 7} \frac{x^{2}-4 x-21}{x-7}$

Bestimmen Sie die Ableitung und die Steigung von $f_{a}(x)$ an $x=0$ . Für welchen Wert von $a$ ist die Steigung 1?

Calculate the integral: $\int \frac{d x}{\sqrt[4]{(x+1)^{3}}-\sqrt{x+1}}$

Find the derivative of $f(x)=\frac{x}{\sec(3x)+1}$ .

Gegeben ist die Funktion $f(x)=x \cdot e^{x}$ . Finde den Tiefpunkt, die Normale im Ursprung und den Wendepunkt.

Find the derivative $\frac{d}{d m}\left(x^{3} m^{3}+y^{5} m+z^{6}\right)$ .

Find the first and second derivatives of $y=3x+9$ : $\frac{dy}{dx}=$ and $\frac{d^{2}y}{dx^{2}}=$ .

Gegeben ist die Funktion $f(x)=x \cdot e^{x}$ . Finde: a) den Tiefpunkt, b) die Normale im Ursprung, c) den Wendepunkt.

Find the first derivative of the function $f(t)=(t^{3}+2)^{8}$ .

Find the average velocity of an object with $v=4t-3t^{2}$ from $t=0$ to $t=2$ seconds. Options: a. 0 b. $-2 \mathrm{~m/s}$ c. $2 \mathrm{~m/s}$ d. -4ri's e. can't calculate without initial position.

Estimate the cost of making 41 bicycles if $C(40)=7000$ and $C^{\prime}(40)=55$ .

State Rolle's theorem. For $f(x)=(x-2)^{\wedge} 2(x+3)$ , verify it on $[-3,2]$ and find values of $c$ from the theorem.

Bestimmen Sie die Ableitung von $f$ mit der Produktregel für die Funktionen a) bis l).

Check convergence or divergence of the series and identify if it's conditional or absolute: $\sum_{n=1}^{\infty}(-1)^{n} \ln \left(1+\frac{1}{n}\right)$

Find critical points of $f(x)=\frac{x^{5}}{5}-16x$ , determine intervals of increase/decrease, and classify critical points.

Calculate the integral $\int 2 x e^{-5 t} d x$ . Choose the correct answer from the options provided.

Find the integral of $\sin \left(\frac{1}{4} \pi\right) dx$ . What is the result?

Approximate $\int_{1}^{9}(x-6)^{2} d x$ using Riemann sum with left endpoints $u_{i}$ for intervals $[1,3],[3,5],[5,7],[7,9]$ . Options: a) 112 b) 308 c) 72 d) 36 e) None.

Calculate the integral $\int_{-5}^{5}(3 x)^{2} d x$ .

Estimate the integral $\int_{1}^{9}(x-5)^{2} dx$ using Riemann sums with right endpoints for intervals $[1,3],[3,5],[5,7],[7,9]$ . Choose from a) 40 b) 48 c) 24 d) 34 e) None.

Bestimme die Ableitung und die Steigung von $f_{a}(x)$ an $x=0$ für die Funktionen a) bis c) und finde $a$ für Steigung 1.

Find critical points of $f(x)=(\sin x)(\cos x)+2$ , determine intervals of increase/decrease, and classify critical points.

Evaluate the integral: $\int \frac{r^{2}}{r+6} d r$ using absolute values.

Find $\frac{d y}{d x}$ for $\cos (x y)=y-1$ at $x=\frac{\pi}{2}$ , $y=1$ . Options: (A) $\frac{-2}{2-\pi}$ , (B) $\frac{-2}{2+\pi}$ , (C) 0, (D) $\frac{2}{2-\pi}$ , (E) $\frac{2}{2+\pi}$ .

Bestimme die Ableitung von $f(x)=\frac{(x^{3}+x) \cdot \sin (x)}{\ln (x)}$ .

Determine convergence/divergence of the series: $\sum_{n=1}^{\infty}(-1)^{n+1} \frac{3 \sqrt{n+1}}{\sqrt{n}+1}$ .

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

Find the derivative of $s(t)=\left(\frac{t}{2 t+1}\right)^{3/2}$ .

Find $f'(x)$ for the function $f(x)=\frac{x^{2}}{(x^{2}-1)^{4}}$ .

Find the derivative of $h(x)=\frac{(3 x^{2}+1)^{3}}{(x^{2}-1)^{4}}$ .

Ein leerer Pool wird mit $v(t)=0,1 t^{2}-\frac{1}{t^{3}+1}+40$ befüllt. Bestimmen Sie die Füllhöhe und das Volumen nach 10 Minuten.

Find the derivative of $f(x)=(2x+1)^{-2}$ .

Solve the differential equation $dy/dx = \frac{y - y^2}{x}$ for $x \neq 0$ . Find the general solution $y = f(x)$ .

Find the derivative of $g(t)=\frac{\sqrt{t+1}}{\sqrt{t^{2}+1}}$ .

Given $f$ is twice differentiable with $f(4)=6$ , $f'(4)=5$ , and $f''(4)=8$ . Find $h'(4)$ for $h(x)=\sqrt{1+f'(x)^{2}}$ . Round to three decimal places.

Find the maximum number of extreme values for the function $f(x)=x^{3}-7x-6$ . A. 4 B. 3 C. 1 D. 2

Find $\frac{d y}{d x}$ at $(6,2)$ given $f(6)=1$ , $f^{\prime}(6)=1.6$ , and $f(x)^{2}+y^{2}=6y-7f(x)$ . Round to three decimal places.

Berechnen Sie die Fläche zwischen dem Graphen von $f$ und der $x$ -Achse für die Funktionen a) bis f).

Evaluate the integral: $\int \frac{9}{(x+a)(x+b)} d x$ (Assume $a \neq b$ ).

Find the value of the nested radical $\sqrt{72+\sqrt{72+\sqrt{72+\ldots}}}$ and its recursive sequence $a_{n}$ . What is $a_{0}$ ? Also, find $\lim _{n \rightarrow \infty} a_{n}$ and explain your reasoning for the limit.

Trouvez les abscisses des points critiques de la fonction $f(x)=x^{4}(4 x+9)$ .