Calculus

Problem 23001

Find the dimensions $x$ and $y$ of a Norman window with a perimeter of 20 ft for maximum area.

See Solution

Problem 23002

Find the dimensions of a Norman window with a semicircle on top that has a maximum area and a total perimeter of 20 feet.

See Solution

Problem 23003

Use Newton's Method to find $x$ where $f(x)=x^4$ and $g(x)=\cos(x)$ , stopping when values differ by less than 0.001.

See Solution

Problem 23004

Analyze the function $f(x)=15 x^{2 / 3}-10 x$ . Find intercepts, extrema, inflection points, and asymptotes.

See Solution

Problem 23005

Find the average value of $f(x)=\frac{1}{x}$ on $[7,7e]$ and graph it. The average value is $f=\square$ . Choose the correct graph.

See Solution

Problem 23006

Evaluate the integral $\int_{0}^{1}(x^{2}-2 x+5) dx$ and check if it matches the figure. Is it A, B, C, or D?

See Solution

Problem 23007

Use Newton's Method with $f(x)=x^{9}-9$ and initial guess $x_{1}=1.4$ to find two approximations. Round to three decimals.

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Problem 23008

Find the area function $A(x)=\int_{-9}^{x} (t+9) dt$ and verify $A'(x)=f(x)$ . Graph $A(x)$ .

See Solution

Problem 23009

Find and graph the area function $A(x)=\int_{-6}^{x} (t+6) dt$ and verify that $A'(x)=f(x)$ .

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Problem 23010

Find the tangent line $T$ to $f(x)=x^{2}$ at (9,81) and complete the table. Round answers to four decimal places. $T(x)=\square$

See Solution

Problem 23011

Find the average value of $f(x)=\frac{3}{x}$ on $[2,2e]$ and graph the function with the average value indicated.

See Solution

Problem 23012

Evaluate the integrals: (a) $\int_{1}^{2}\left(x^{5 / 4}-x^{5 / 6}\right) d x$ (b) $\int_{-\pi}^{\pi} \sin (x) \cos (x) d x$

See Solution

Problem 23013

Find the initial velocity needed to throw an object to 580 feet with acceleration $a(t)=-32$ ft/s². Round to three decimal places.

See Solution

Problem 23014

A particle moves along the $x$ -axis with $x(t)=t^{3}-12t^{2}+21t-7$ for $0 \leq t \leq 10$ . Find: (a) $x^{\prime}(t)$ , $x^{\prime \prime}(t)$ ; (b) intervals moving right; (c) velocity when $x^{\prime \prime}(t)=0$ .

See Solution

Problem 23015

Evaluate the integral $\int_{0}^{1}(x^{2}-3x+4) \, dx$ and check if your result matches the graph.

See Solution

Problem 23016

A rock is dropped from a 1400 m canyon. How long will it take to hit the floor? Use $a(t)=-9.8$ m/s². Round to 1 decimal place.

See Solution

Problem 23017

Find the inflection points of $f(x) = \int_{0}^{x} e^{-t^{2}} \, dt$ and evaluate $\int_{0}^{\frac{\pi}{2}} \frac{d}{dx} \left(\sin(\cos(x)) \cos(\sin(x)) x^{10000000000}\right) \, dx$ .

See Solution

Problem 23018

Determine if the series $-\frac{15}{8}+\frac{45}{64}-\frac{135}{512}+\frac{405}{4096} \cdots$ converges or diverges. If it converges, find the sum.

See Solution

Problem 23019

Calculate the future value of a 16-year continuous income stream of \$230,000 at a continuous compound rate of 4.4%.

See Solution

Problem 23020

Approximate the area under $g(x)=9 \sin x$ from $[0, \pi]$ using 6 rectangles (left and right endpoints). Round to four decimals.

See Solution

Problem 23021

Approximate the area under $g(x)=9 \sin x$ over $[0, \pi]$ using 6 rectangles with left and right endpoints. Round to four decimals.

See Solution

Problem 23022

Approximate the area under $g(x) = 9 \sin x$ on $[0, \pi]$ using 6 rectangles (left and right endpoints).

See Solution

Problem 23023

Approximate the area under $g(x)=9 \sin x$ on $[0, \pi]$ using 6 rectangles with left and right endpoints. Round to four decimals.

See Solution

Problem 23024

Approximate the area under $g(x)=9 \sin x$ over $[0, \pi]$ using 6 rectangles (left and right endpoints). Round to 4 decimal places.

See Solution

Problem 23025

Evaluate the integral from 0 to 8 of $|x^{2}-6x+5|$ . Use a graphing tool to check your answer.

See Solution

Problem 23026

Evaluate the integral from $-\frac{\pi}{6}$ to $\frac{\pi}{6}$ of $5 \sec^2 x \, dx$ and verify with a graphing tool.

See Solution

Problem 23027

Calculate the right Riemann sum $R_{4}$ for $g(x)=\cos (\pi x)$ on the interval $[0,1]$ .

See Solution

Problem 23028

Calculate the area using the integral $\int_{\pi / 12}^{\pi / 4} \csc 2 x \cot 2 x \, dx$ and verify with a graphing tool.

See Solution

Problem 23029

Find the area between the curve $y = x + \sin x$ and the x-axis from $0$ to $\pi$ .

See Solution

Problem 23030

Evaluate the integral: $\int_{-6}^{6} x\left(7 x^{2}+4\right)^{3} d x$ using even and odd function properties.

See Solution

Problem 23031

Find the average of $f(x)=16-x^{2}$ over $[-4,4]$ . Then, find $x$ where $f(x)$ equals this average value.

See Solution

Problem 23032

Find $F(x)=\int_{\pi / 3}^{x} \sec ^{2} t \, dt$ . (a) Integrate to get $F(x)$ . (b) Differentiate $F(x)$ to show the Second Fundamental Theorem.

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Problem 23033

Find the limit as x approaches 6 from the right: $\lim _{x \rightarrow 6^{+}} \ln (x-6)$ . Round to four decimal places.

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Problem 23034

Find the derivative of $f(x)=\ln \left(\frac{6 x}{x+1}\right)$ . What is $f^{\prime}(x)$ ?

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Problem 23035

Find the derivative of $f(x)=\ln(x+\sqrt{8+x^{2}})$ . What is $f'(x)$ ?

See Solution

Problem 23036

Find $d y / d x$ using logarithmic differentiation for $y=\frac{(x+1)(x-4)}{(x-1)(x+4)}$ , $x>4$ .

See Solution

Problem 23037

Find the derivative of $y=\ln (|\csc (5 x)|)$ . What is $y^{\prime}(x)$ ?

See Solution

Problem 23038

Calculate the area between $y=2 \sec \left(\frac{\pi x}{6}\right)$ , $x=0$ , $x=2$ , and $y=0$ . Round to three decimal places.

See Solution

Problem 23039

Differentiate the integral $\int_{0}^{x^{3}} e^{-3 t} d t$ with respect to $x$ .

See Solution

Problem 23040

Find the derivative of $\int_{0}^{x^{3}} e^{-3t} dt$ using two methods: a. evaluate then differentiate, b. differentiate directly.

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Problem 23041

Calculate the cooling time from $300^{\circ} \mathrm{F}$ to $220^{\circ} \mathrm{F}$ using the formula: $t=\frac{10}{\ln 2} \int_{220}^{300} \frac{1}{T-100} d T$ , rounding to four decimal places.

See Solution

Problem 23042

Evaluate the integral: $\int_{1}^{8} \frac{2 \ln x}{x} dx = \square$

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Problem 23043

Find the area of the shaded region between the curves $x=6y^2-6y^3$ and $x=4y^2-4y$ .

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Problem 23044

Find the derivative of the integral: $\frac{\mathrm{d}}{\mathrm{dx}} \int_{0}^{\mathrm{x}^{3}} e^{-3 \mathrm{t}} \mathrm{dt}.$

See Solution

Problem 23045

Rewrite the integral using $u$ : $\int_{1}^{8} \frac{2 \ln x}{x} dx=\int_{0}^{\square} \square du$

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Problem 23046

Find $\left(f^{-1}\right)^{\prime}(1)$ for $f(x)=\cos(4x)$ , $0 \leq x \leq \frac{\pi}{4}$ . Answer DNE if not exist.

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Problem 23047

Graph $V=11000 e^{-0.6250 t}$ for $0 \leq t \leq 10$ . Find $V^{\prime}(4)$ and $V^{\prime}(6)$ , rounded to two decimals.

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Problem 23048

Find the area under the curve $y=2(\sin x) \sqrt{1+\cos x}$ from $x=-\pi$ to $x=\pi$ .

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Problem 23049

Given functions $f(x)=\sqrt{x-1}$ and $f^{-1}(x)=x^{2}+1$ (for $x \geq 0$ ), find their domains, ranges, graph them, and show slopes at points $(5,2)$ and $(2,5)$ .

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Problem 23050

Find the extrema and inflection points of $g(x)=\frac{1}{\sqrt{2 \pi}} e^{-(x-6)^{2} / 2}$ . Round to three decimals.

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Problem 23051

Find the derivative of $y=e^{e^{\sqrt{x}}}+\cosh \sqrt{x}$ . Select the correct expression for $y'$ .

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Problem 23052

Is it true that $1 \leq \int_{0}^{1} e^{x^{2}} dx \leq e$ ? Select True or False.

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Problem 23053

The integral of $\cot(x)$ is $\ln |\csc x| + C$ . Is this statement true or false?

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Problem 23054

Calculate the integral from 1 to 3 of $\frac{x+3}{x^2+6x} \, dx$ . Choose the correct answer: a, b, c, d, or e.

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Problem 23055

Calculate the integral from 0 to 1 of $(x^{2}-1)^{3}$ .

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Problem 23056

Berechnen Sie die Ableitung von $f(x) = (5-x)(3+x)$ mit der Produktregel.

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Problem 23057

Berechnen Sie die Ableitung von $g(x) = 3x \cdot (0.5x + 1)^2$ .

See Solution

Problem 23058

Berechnen Sie die Ableitung von $g(x) = 3x \cdot (0.5x + 1)^2$ mit Produkt- und Kettenregel, dann mit Potenz- und Summenregel.

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Problem 23059

Find the derivative of the function $f(x)=\frac{4}{x \sqrt{x}}$ .

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Problem 23060

Zeichne die Tangente an $f(x)=-x^{3}+3x^{2}$ im Punkt $P(1, 2)$ und finde die Steigung $m$ durch $f'(1)$ . Bestimme die Tangentengleichung.

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Problem 23061

Calcula la población a largo plazo de mosquitos Aedes aegypti dada por $P_{(t)}=\frac{5+27 t^{2}}{(3 t-2)^{2}}$ .

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Problem 23062

Halla la constante de integración de $G(x)$ si $\int(12 x^{5}-12 x^{3}-x) dx=G(x)$ y $G(2)=90$ .

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Problem 23063

Ein Heißluftballon sinkt mit $v(t)=0,0015 t^{2}-0,3 t$ . Finde $h(t)$ , Höhe nach 2 Minuten und Landungszeit/Höhe.

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Problem 23064

Gegeben ist die Funktion $f_{a}(x) = \frac{1}{a} \cdot x^{3} - x$ . Bestimme (a) den Graphen von $f_{a}$ und (b) den Wert von $a$ , sodass $f_{a}$ bei $x=3$ einen Extrempunkt hat.

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Problem 23065

Un móvil tiene velocidad $V(t)=3 t^{2}-2 t-3$ m/s. ¿Cuál es su desplazamiento en el cuarto segundo? 28, 30, 27, 25 o 45 m?

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Problem 23066

Find the sum of the infinite geometric series: $\sum_{n=0}^{\infty} 5\left(-\frac{1}{4}\right)^{n}$ .

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Problem 23067

Gegeben ist die Funktion $f(x)=\frac{2}{3} x^{2}-4 x+1$ . Finde die Tangentengleichung in $B(6, f(6))$ und einen Punkt $Q$ mit paralleler Tangente zur Linie $y=-8 x+5$ .

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Problem 23068

Given a function $f(x)$ with a critical point at $x=5$ , analyze $g(x)=-30 f(20 x+10)$ on $[-1,0]$ . Which statements are true? A, B, C, or D?

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Problem 23069

A spherical cell's volume $V$ grows at a rate proportional to surface area $S$ . Which rate $\frac{d r}{d t}$ is true? (A) 1, (B) 1/2, (C) 1/3, (D) 1/4.

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Problem 23070

Which statement is always true? (A) Continuous at $a$ implies differentiable at $a$ (B) Continuous function has a global maximum (C) If $f$ max at $p$ , then $f'(p)=0$ (D) Differentiable at $a$ implies continuous at $a$ (E) $f''(p)=0$ means $f$ has an inflection point at $p$ (F) Critical point $p$ means local extremum at $p$

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Problem 23071

Formuliere eine Leitfrage zur Funktion $h(t)=0,2 \cdot e^{0,1 \cdot t-0,9}$ . Berechne $h(0)$ und den Zeitpunkt für $h=50 \mathrm{~cm}$ . Bestimme das Wachstumshöchstmaß.

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Problem 23072

A house plan has a passageway width of $\frac{3}{4} \mathrm{~m}$ and a room width of $6 \mathrm{~m}$ .
(a) Prove that $L=\frac{3}{4} \sec \alpha+6 \operatorname{cosec} \alpha$ . (b.i) Calculate $\frac{\mathrm{d} L}{\mathrm{~d} \alpha}$ . (b.ii) Show that when $\frac{\mathrm{d} L}{\mathrm{~d} \alpha}=0$ , then $\alpha=\arctan 2$ .

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Problem 23073

Find the quantity $q$ that maximizes the profit function $P(q)=16q-2q^{2}-24$ and the corresponding profit.

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Problem 23074

Find local and absolute extrema of $f(x)=-3 x^{4}-8 x^{3}+11 x^{2}+26 x$ . List local minima or state none exist.

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Problem 23075

Calculate the integral $\int \sin^{2} \frac{x}{2} \cos \frac{x}{2} \, dx$ .

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Problem 23076

Calculate the integral from 20 to 30 of the function $0.02 e^{-0.1 t + 3.1}$ .

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Problem 23077

Find where the function $f(x)=xe^{-2x^2+3x}$ has relative extrema. Steps: 1) Derivative $f'(x)$ , 2) Critical points, 3) Maxima or minima.

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Problem 23078

Öltank:
1. Bestimme die Funktion für Ölmenge nach $x$ Stunden: $f(x)=0,12 x^{3}-2,64 x^{2}+12,6 x$ .
2. Zuflussrate nach 6 und 9 Stunden.
3. Finde Zeitpunkte für max. Zufluss und wann $12 \mathrm{~m}^{3} / \mathrm{h}$ erreicht wird.
4. Bestimme max. und min. Füllmenge.
5. Ölmenge nach 5 und 12 Stunden; Ölfluss zwischen 1. und 4. Stunde.
6. Berechne $\int_{8}^{10} f(x) dx$ und erläutere.

See Solution

Problem 23079

Find the horizontal asymptote of $f(x)=e^{-x}-2$ . Provide your answer as an equation.

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Problem 23080

Pflanzenwachstum:
1. Funktionsgleichung der Höhe nach $x$ Tagen.
2. Wachstum am 6. Tag: $f(6)=$
3. Höhe nach 6 Tagen: $F(6)=$
4. Wann ist die Pflanze ausgewachsen?
5. (1) Durchschnittliche Höhenzunahme von Tag 3 bis 6. (2) Berechne das Integral.

See Solution

Problem 23081

Determine the horizontal asymptote of the function $f(x)=\frac{2}{x-3}$ .

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Problem 23082

Wachstum einer Pflanze:
a) Funktionsgleichung für Höhe nach $x$ Tagen. b) Wachstum am 6. Tag: $f(6)=$ c) Höhe nach 6 Tagen: $F(6)=$ d) Wann ist die Pflanze ausgewachsen und wie groß? e) (1) Durchschnittliche Höhenzunahme zwischen Tag 3 und 6. (2) Berechne $\int^{9} f(x) d x$ .

See Solution

Problem 23083

Find $x$ that minimizes the average cost given $C(x)=10,000+250x+10x^2$ . Round to two decimals.

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Problem 23084

A particle moves along a line with position $s(t)=5t-2t^2$ . When does it revisit $s(0)$ , and how far right does it travel?

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Problem 23085

A train's location is $s(t)=\frac{140}{t}$ for $2 \leq t \leq 8$ . Graph it and find the average velocity from $t=2$ to $t=8$ .

See Solution

Problem 23086

A train's position is given by $s(t)=\frac{140}{t}$ for $2 \leq t \leq 8$ . Find the average velocity between $t=2$ and $t=8$ . Where is it on the graph?

See Solution

Problem 23087

A balloon inflates at $27 \pi \mathrm{in}^{3}/\mathrm{min}$ . Find the radius increase rate at $r=3$ in using $V=\frac{4}{3} \pi r^{3}$ . Options: a. $1.24 \mathrm{in}/\mathrm{sec}$ b. $0.75 \mathrm{in}/\mathrm{sec}$ c. $0.54 \mathrm{in}/\mathrm{sec}$ d. $0.24 \mathrm{in}/\mathrm{sec}$ .

See Solution

Problem 23088

For the function $f(x)=\frac{1}{4} x^{3}-3 x$ , complete (a) graph $y=f(x)$ , (b) find turning points, (c) estimate local extrema.

See Solution

Problem 23089

Stromverbrauch:
Gegeben ist die Funktion $p(t)=-\frac{3}{51200} t^{4}+\frac{11}{3200} t^{3}-\frac{53}{800} t^{2}+\frac{21}{50} t+\frac{1}{10}$ für die Leistung.
a) Finde die Funktion für die verbrauchte Energie. b) (1) Bestimme die Leistung um 6 Uhr. (2) Wann war die Leistung 0,5265625 kW? (3) Wann war die maximale/minimale Leistung? c) (1) Wie viel Energie wurde an einem Tag verbraucht? (2) Bis wann waren 8,184 kWh verbraucht? d) (1) Energieverbrauch zwischen 6 und 9 Uhr? (2) Berechne $\int_{10}^{12} p(t) d t$ und erkläre die Bedeutung.

See Solution

Problem 23090

Find where the function $g(x) = 2 + \frac{x-1}{4-2x}$ is increasing or decreasing and identify any extremum points.

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Problem 23091

What is the derivative of $\epsilon^{1/2}$ with respect to $\epsilon$ ?

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Problem 23092

Find the velocity $v(1)$ and acceleration $a(1)$ for $s(t)=t^{2}-3t$ at $t=1 \mathrm{~s}$ . $v(1)=\square \square$

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Problem 23093

Calculate the growth rate $\frac{\mathrm{dH}}{\mathrm{dt}}$ for head circumference $H=-30.22+1.811 t^{2}-0.0214 t^{2} \log t$ . Compare at $t=8$ and $t=36$ weeks. Also find the fractional rate $\frac{1}{H} \frac{\mathrm{dH}}{\mathrm{dt}}$ .

See Solution

Problem 23094

The function $f(t)=\frac{104,000}{1+4800 e^{-t}}$ models flu cases. Find: a) initial cases, b) cases after 4 weeks, c) max cases.

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Problem 23095

Determine if the piecewise function $f(x)=\begin{cases} x^{2}-1 & \text{if } -2 \leq x \leq 0 \\ 1-x^{2} & \text{if } 0<x \leq 2 \end{cases}$ is continuous on its domain.

See Solution

Problem 23096

Determine the continuity of the function $f$ at $x=1$ based on its definition. What can you conclude?

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Problem 23097

A bacteria population of 2,000 grows at $50\%$ daily. Find the number of bacteria after one week.

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Problem 23098

Differentiate $f(x^3)$ and show it equals $f'(3x^2)$ .

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Problem 23099

True or false: If $f$ is differentiable, is $\frac{d}{dx}[f(x^{3})]=f'(3x^{2})$ ?

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Problem 23100

True or False: If $f$ is differentiable, is $\frac{d}{d x}[3+\sqrt{f(\sin (x))}] = \frac{\cos (x) f^{\prime}(\sin (x))}{2 \sqrt{f(\sin (x))}}$ ?

See Solution

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Calculus

Problem 23001

Find the dimensions xxx and yyy of a Norman window with a perimeter of 20 ft for maximum area.

Problem 23002

Find the dimensions of a Norman window with a semicircle on top that has a maximum area and a total perimeter of 20 feet.

Problem 23003

Use Newton's Method to find xxx where f(x)=x4f(x)=x^4f(x)=x4 and g(x)=cos⁡(x)g(x)=\cos(x)g(x)=cos(x), stopping when values differ by less than 0.001.

Problem 23004

Analyze the function f(x)=15x2/3−10xf(x)=15 x^{2 / 3}-10 xf(x)=15x2/3−10x. Find intercepts, extrema, inflection points, and asymptotes.

Problem 23005

Find the average value of f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ on [7,7e][7,7e][7,7e] and graph it. The average value is f=□f=\squaref=□. Choose the correct graph.

Problem 23006

Evaluate the integral ∫01(x2−2x+5)dx\int_{0}^{1}(x^{2}-2 x+5) dx∫01​(x2−2x+5)dx and check if it matches the figure. Is it A, B, C, or D?

Problem 23007

Use Newton's Method with f(x)=x9−9f(x)=x^{9}-9f(x)=x9−9 and initial guess x1=1.4x_{1}=1.4x1​=1.4 to find two approximations. Round to three decimals.

Problem 23008

Find the area function A(x)=∫−9x(t+9)dtA(x)=\int_{-9}^{x} (t+9) dtA(x)=∫−9x​(t+9)dt and verify A′(x)=f(x)A'(x)=f(x)A′(x)=f(x). Graph A(x)A(x)A(x).

Problem 23009

Find and graph the area function A(x)=∫−6x(t+6)dtA(x)=\int_{-6}^{x} (t+6) dtA(x)=∫−6x​(t+6)dt and verify that A′(x)=f(x)A'(x)=f(x)A′(x)=f(x).

Problem 23010

Find the tangent line TTT to f(x)=x2f(x)=x^{2}f(x)=x2 at (9,81) and complete the table. Round answers to four decimal places. T(x)=□T(x)=\squareT(x)=□

Problem 23011

Find the average value of f(x)=3xf(x)=\frac{3}{x}f(x)=x3​ on [2,2e][2,2e][2,2e] and graph the function with the average value indicated.

Problem 23012

Evaluate the integrals: (a) ∫12(x5/4−x5/6)dx\int_{1}^{2}\left(x^{5 / 4}-x^{5 / 6}\right) d x∫12​(x5/4−x5/6)dx (b) ∫−ππsin⁡(x)cos⁡(x)dx\int_{-\pi}^{\pi} \sin (x) \cos (x) d x∫−ππ​sin(x)cos(x)dx

Problem 23013

Find the initial velocity needed to throw an object to 580 feet with acceleration a(t)=−32a(t)=-32a(t)=−32 ft/s². Round to three decimal places.

Problem 23014

Problem 23015

Evaluate the integral ∫01(x2−3x+4) dx\int_{0}^{1}(x^{2}-3x+4) \, dx∫01​(x2−3x+4)dx and check if your result matches the graph.

Problem 23016

A rock is dropped from a 1400 m canyon. How long will it take to hit the floor? Use a(t)=−9.8a(t)=-9.8a(t)=−9.8 m/s². Round to 1 decimal place.

Problem 23017

Problem 23018

Determine if the series −158+4564−135512+4054096⋯-\frac{15}{8}+\frac{45}{64}-\frac{135}{512}+\frac{405}{4096} \cdots−815​+6445​−512135​+4096405​⋯ converges or diverges. If it converges, find the sum.

Problem 23019

Calculate the future value of a 16-year continuous income stream of \$230,000 at a continuous compound rate of 4.4%.

Problem 23020

Approximate the area under g(x)=9sin⁡xg(x)=9 \sin xg(x)=9sinx from [0,π][0, \pi][0,π] using 6 rectangles (left and right endpoints). Round to four decimals.

Problem 23021

Approximate the area under g(x)=9sin⁡xg(x)=9 \sin xg(x)=9sinx over [0,π][0, \pi][0,π] using 6 rectangles with left and right endpoints. Round to four decimals.

Problem 23022

Approximate the area under g(x)=9sin⁡xg(x) = 9 \sin xg(x)=9sinx on [0,π][0, \pi][0,π] using 6 rectangles (left and right endpoints).

Problem 23023

Approximate the area under g(x)=9sin⁡xg(x)=9 \sin xg(x)=9sinx on [0,π][0, \pi][0,π] using 6 rectangles with left and right endpoints. Round to four decimals.

Problem 23024

Approximate the area under g(x)=9sin⁡xg(x)=9 \sin xg(x)=9sinx over [0,π][0, \pi][0,π] using 6 rectangles (left and right endpoints). Round to 4 decimal places.

Problem 23025

Evaluate the integral from 0 to 8 of ∣x2−6x+5∣|x^{2}-6x+5|∣x2−6x+5∣. Use a graphing tool to check your answer.

Problem 23026

Evaluate the integral from −π6-\frac{\pi}{6}−6π​ to π6\frac{\pi}{6}6π​ of 5sec⁡2x dx5 \sec^2 x \, dx5sec2xdx and verify with a graphing tool.

Problem 23027

Calculate the right Riemann sum R4R_{4}R4​ for g(x)=cos⁡(πx)g(x)=\cos (\pi x)g(x)=cos(πx) on the interval [0,1][0,1][0,1].

Problem 23028

Calculate the area using the integral ∫π/12π/4csc⁡2xcot⁡2x dx\int_{\pi / 12}^{\pi / 4} \csc 2 x \cot 2 x \, dx∫π/12π/4​csc2xcot2xdx and verify with a graphing tool.

Problem 23029

Find the area between the curve y=x+sin⁡xy = x + \sin xy=x+sinx and the x-axis from 000 to π\piπ.

Problem 23030

Evaluate the integral: ∫−66x(7x2+4)3dx\int_{-6}^{6} x\left(7 x^{2}+4\right)^{3} d x∫−66​x(7x2+4)3dx using even and odd function properties.

Problem 23031

Find the average of f(x)=16−x2f(x)=16-x^{2}f(x)=16−x2 over [−4,4][-4,4][−4,4]. Then, find xxx where f(x)f(x)f(x) equals this average value.

Problem 23032

Find F(x)=∫π/3xsec⁡2t dtF(x)=\int_{\pi / 3}^{x} \sec ^{2} t \, dtF(x)=∫π/3x​sec2tdt. (a) Integrate to get F(x)F(x)F(x). (b) Differentiate F(x)F(x)F(x) to show the Second Fundamental Theorem.

Problem 23033

Find the limit as x approaches 6 from the right: lim⁡x→6+ln⁡(x−6)\lim _{x \rightarrow 6^{+}} \ln (x-6)limx→6+​ln(x−6). Round to four decimal places.

Problem 23034

Find the derivative of f(x)=ln⁡(6xx+1)f(x)=\ln \left(\frac{6 x}{x+1}\right)f(x)=ln(x+16x​). What is f′(x)f^{\prime}(x)f′(x)?

Problem 23035

Find the derivative of f(x)=ln⁡(x+8+x2)f(x)=\ln(x+\sqrt{8+x^{2}})f(x)=ln(x+8+x2​). What is f′(x)f'(x)f′(x)?

Problem 23036

Find dy/dxd y / d xdy/dx using logarithmic differentiation for y=(x+1)(x−4)(x−1)(x+4)y=\frac{(x+1)(x-4)}{(x-1)(x+4)}y=(x−1)(x+4)(x+1)(x−4)​, x>4x>4x>4.

Problem 23037

Find the derivative of y=ln⁡(∣csc⁡(5x)∣)y=\ln (|\csc (5 x)|)y=ln(∣csc(5x)∣). What is y′(x)y^{\prime}(x)y′(x)?

Problem 23038

Calculate the area between y=2sec⁡(πx6)y=2 \sec \left(\frac{\pi x}{6}\right)y=2sec(6πx​), x=0x=0x=0, x=2x=2x=2, and y=0y=0y=0. Round to three decimal places.

Problem 23039

Differentiate the integral ∫0x3e−3tdt\int_{0}^{x^{3}} e^{-3 t} d t∫0x3​e−3tdt with respect to xxx.

Problem 23040

Find the derivative of ∫0x3e−3tdt\int_{0}^{x^{3}} e^{-3t} dt∫0x3​e−3tdt using two methods: a. evaluate then differentiate, b. differentiate directly.

Problem 23041

Find the dimensions $x$ and $y$ of a Norman window with a perimeter of 20 ft for maximum area.

Use Newton's Method to find $x$ where $f(x)=x^4$ and $g(x)=\cos(x)$ , stopping when values differ by less than 0.001.

Analyze the function $f(x)=15 x^{2 / 3}-10 x$ . Find intercepts, extrema, inflection points, and asymptotes.

Find the average value of $f(x)=\frac{1}{x}$ on $[7,7e]$ and graph it. The average value is $f=\square$ . Choose the correct graph.

Evaluate the integral $\int_{0}^{1}(x^{2}-2 x+5) dx$ and check if it matches the figure. Is it A, B, C, or D?

Use Newton's Method with $f(x)=x^{9}-9$ and initial guess $x_{1}=1.4$ to find two approximations. Round to three decimals.

Find the area function $A(x)=\int_{-9}^{x} (t+9) dt$ and verify $A'(x)=f(x)$ . Graph $A(x)$ .

Find and graph the area function $A(x)=\int_{-6}^{x} (t+6) dt$ and verify that $A'(x)=f(x)$ .

Find the tangent line $T$ to $f(x)=x^{2}$ at (9,81) and complete the table. Round answers to four decimal places. $T(x)=\square$

Find the average value of $f(x)=\frac{3}{x}$ on $[2,2e]$ and graph the function with the average value indicated.

Evaluate the integrals: (a) $\int_{1}^{2}\left(x^{5 / 4}-x^{5 / 6}\right) d x$ (b) $\int_{-\pi}^{\pi} \sin (x) \cos (x) d x$

Find the initial velocity needed to throw an object to 580 feet with acceleration $a(t)=-32$ ft/s². Round to three decimal places.

Evaluate the integral $\int_{0}^{1}(x^{2}-3x+4) \, dx$ and check if your result matches the graph.

A rock is dropped from a 1400 m canyon. How long will it take to hit the floor? Use $a(t)=-9.8$ m/s². Round to 1 decimal place.

Determine if the series $-\frac{15}{8}+\frac{45}{64}-\frac{135}{512}+\frac{405}{4096} \cdots$ converges or diverges. If it converges, find the sum.

Approximate the area under $g(x)=9 \sin x$ from $[0, \pi]$ using 6 rectangles (left and right endpoints). Round to four decimals.

Approximate the area under $g(x)=9 \sin x$ over $[0, \pi]$ using 6 rectangles with left and right endpoints. Round to four decimals.

Approximate the area under $g(x) = 9 \sin x$ on $[0, \pi]$ using 6 rectangles (left and right endpoints).

Approximate the area under $g(x)=9 \sin x$ on $[0, \pi]$ using 6 rectangles with left and right endpoints. Round to four decimals.

Approximate the area under $g(x)=9 \sin x$ over $[0, \pi]$ using 6 rectangles (left and right endpoints). Round to 4 decimal places.

Evaluate the integral from 0 to 8 of $|x^{2}-6x+5|$ . Use a graphing tool to check your answer.

Evaluate the integral from $-\frac{\pi}{6}$ to $\frac{\pi}{6}$ of $5 \sec^2 x \, dx$ and verify with a graphing tool.

Calculate the right Riemann sum $R_{4}$ for $g(x)=\cos (\pi x)$ on the interval $[0,1]$ .

Calculate the area using the integral $\int_{\pi / 12}^{\pi / 4} \csc 2 x \cot 2 x \, dx$ and verify with a graphing tool.

Find the area between the curve $y = x + \sin x$ and the x-axis from $0$ to $\pi$ .

Evaluate the integral: $\int_{-6}^{6} x\left(7 x^{2}+4\right)^{3} d x$ using even and odd function properties.

Find the average of $f(x)=16-x^{2}$ over $[-4,4]$ . Then, find $x$ where $f(x)$ equals this average value.

Find $F(x)=\int_{\pi / 3}^{x} \sec ^{2} t \, dt$ . (a) Integrate to get $F(x)$ . (b) Differentiate $F(x)$ to show the Second Fundamental Theorem.

Find the limit as x approaches 6 from the right: $\lim _{x \rightarrow 6^{+}} \ln (x-6)$ . Round to four decimal places.

Find the derivative of $f(x)=\ln \left(\frac{6 x}{x+1}\right)$ . What is $f^{\prime}(x)$ ?

Find the derivative of $f(x)=\ln(x+\sqrt{8+x^{2}})$ . What is $f'(x)$ ?

Find $d y / d x$ using logarithmic differentiation for $y=\frac{(x+1)(x-4)}{(x-1)(x+4)}$ , $x>4$ .

Find the derivative of $y=\ln (|\csc (5 x)|)$ . What is $y^{\prime}(x)$ ?

Calculate the area between $y=2 \sec \left(\frac{\pi x}{6}\right)$ , $x=0$ , $x=2$ , and $y=0$ . Round to three decimal places.

Differentiate the integral $\int_{0}^{x^{3}} e^{-3 t} d t$ with respect to $x$ .

Find the derivative of $\int_{0}^{x^{3}} e^{-3t} dt$ using two methods: a. evaluate then differentiate, b. differentiate directly.

Calculate the cooling time from $300^{\circ} \mathrm{F}$ to $220^{\circ} \mathrm{F}$ using the formula: $t=\frac{10}{\ln 2} \int_{220}^{300} \frac{1}{T-100} d T$ , rounding to four decimal places.

Evaluate the integral: $\int_{1}^{8} \frac{2 \ln x}{x} dx = \square$

Find the area of the shaded region between the curves $x=6y^2-6y^3$ and $x=4y^2-4y$ .

Find the derivative of the integral: $\frac{\mathrm{d}}{\mathrm{dx}} \int_{0}^{\mathrm{x}^{3}} e^{-3 \mathrm{t}} \mathrm{dt}.$

Rewrite the integral using $u$ : $\int_{1}^{8} \frac{2 \ln x}{x} dx=\int_{0}^{\square} \square du$

Find $\left(f^{-1}\right)^{\prime}(1)$ for $f(x)=\cos(4x)$ , $0 \leq x \leq \frac{\pi}{4}$ . Answer DNE if not exist.

Graph $V=11000 e^{-0.6250 t}$ for $0 \leq t \leq 10$ . Find $V^{\prime}(4)$ and $V^{\prime}(6)$ , rounded to two decimals.

Find the area under the curve $y=2(\sin x) \sqrt{1+\cos x}$ from $x=-\pi$ to $x=\pi$ .

Given functions $f(x)=\sqrt{x-1}$ and $f^{-1}(x)=x^{2}+1$ (for $x \geq 0$ ), find their domains, ranges, graph them, and show slopes at points $(5,2)$ and $(2,5)$ .

Find the extrema and inflection points of $g(x)=\frac{1}{\sqrt{2 \pi}} e^{-(x-6)^{2} / 2}$ . Round to three decimals.

Find the derivative of $y=e^{e^{\sqrt{x}}}+\cosh \sqrt{x}$ . Select the correct expression for $y'$ .

Is it true that $1 \leq \int_{0}^{1} e^{x^{2}} dx \leq e$ ? Select True or False.

The integral of $\cot(x)$ is $\ln |\csc x| + C$ . Is this statement true or false?

Calculate the integral from 1 to 3 of $\frac{x+3}{x^2+6x} \, dx$ . Choose the correct answer: a, b, c, d, or e.

Calculate the integral from 0 to 1 of $(x^{2}-1)^{3}$ .

Berechnen Sie die Ableitung von $f(x) = (5-x)(3+x)$ mit der Produktregel.

Berechnen Sie die Ableitung von $g(x) = 3x \cdot (0.5x + 1)^2$ .

Berechnen Sie die Ableitung von $g(x) = 3x \cdot (0.5x + 1)^2$ mit Produkt- und Kettenregel, dann mit Potenz- und Summenregel.

Find the derivative of the function $f(x)=\frac{4}{x \sqrt{x}}$ .

Zeichne die Tangente an $f(x)=-x^{3}+3x^{2}$ im Punkt $P(1, 2)$ und finde die Steigung $m$ durch $f'(1)$ . Bestimme die Tangentengleichung.

Calcula la población a largo plazo de mosquitos Aedes aegypti dada por $P_{(t)}=\frac{5+27 t^{2}}{(3 t-2)^{2}}$ .

Halla la constante de integración de $G(x)$ si $\int(12 x^{5}-12 x^{3}-x) dx=G(x)$ y $G(2)=90$ .

Ein Heißluftballon sinkt mit $v(t)=0,0015 t^{2}-0,3 t$ . Finde $h(t)$ , Höhe nach 2 Minuten und Landungszeit/Höhe.

Gegeben ist die Funktion $f_{a}(x) = \frac{1}{a} \cdot x^{3} - x$ . Bestimme (a) den Graphen von $f_{a}$ und (b) den Wert von $a$ , sodass $f_{a}$ bei $x=3$ einen Extrempunkt hat.

Un móvil tiene velocidad $V(t)=3 t^{2}-2 t-3$ m/s. ¿Cuál es su desplazamiento en el cuarto segundo? 28, 30, 27, 25 o 45 m?

Find the sum of the infinite geometric series: $\sum_{n=0}^{\infty} 5\left(-\frac{1}{4}\right)^{n}$ .

Gegeben ist die Funktion $f(x)=\frac{2}{3} x^{2}-4 x+1$ . Finde die Tangentengleichung in $B(6, f(6))$ und einen Punkt $Q$ mit paralleler Tangente zur Linie $y=-8 x+5$ .

Given a function $f(x)$ with a critical point at $x=5$ , analyze $g(x)=-30 f(20 x+10)$ on $[-1,0]$ . Which statements are true? A, B, C, or D?

A spherical cell's volume $V$ grows at a rate proportional to surface area $S$ . Which rate $\frac{d r}{d t}$ is true? (A) 1, (B) 1/2, (C) 1/3, (D) 1/4.

Formuliere eine Leitfrage zur Funktion $h(t)=0,2 \cdot e^{0,1 \cdot t-0,9}$ . Berechne $h(0)$ und den Zeitpunkt für $h=50 \mathrm{~cm}$ . Bestimme das Wachstumshöchstmaß.

Find the quantity $q$ that maximizes the profit function $P(q)=16q-2q^{2}-24$ and the corresponding profit.

Find local and absolute extrema of $f(x)=-3 x^{4}-8 x^{3}+11 x^{2}+26 x$ . List local minima or state none exist.