Calculus

Problem 11001

Find the derivative of the function $P(x)=\frac{1}{3} x^{3} \cos 5 x$ . What is $P^{\prime}(x)$ ?

See Solution

Problem 11002

Match functions with their derivatives:
1. $y=\sin (x) \tan (x)$
2. $y=\cos ^{3}(x)$
3. $y=\tan (x)$
4. $y=\cos (\tan (x))$ A. $y^{\prime}=-3 \cos ^{3}(x) \tan (x)$ B. $y^{\prime}=1+\tan ^{2}(x)$ C. $y^{\prime}=-\sin (\tan (x)) / \cos ^{2}(x)$ D. $y^{\prime}=\sin (x)+\tan (x) \sec (x)$

See Solution

Problem 11003

Find the slope of the tangent line to the curve $2\left(x^{2}+y^{2}\right)^{2}=25\left(x^{2}-y^{2}\right)$ at $(3,-1)$ .

See Solution

Problem 11004

Find the derivative $y'$ , where $y=\frac{5+\sin x}{\cos x}$ .

See Solution

Problem 11005

Find the intervals where $f(x)=x^{4} e^{-x}$ is increasing, decreasing, concave up, and concave down.

See Solution

Problem 11006

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ for the equation $a x^{4}-b y^{4}=c^{4}$ , where $a, b, c$ are constants.

See Solution

Problem 11007

Find points $(x, y)$ on $f(x)=3 x^{2}-4 x$ where the tangent is parallel to $y=8 x+5$ . Also, find the tangent line equations.

See Solution

Problem 11008

Tentukan titik stasioner dan nilai stasioner dari $F(x)=\cos x-\sin x$ dalam interval $0 \leq x \leq 2\pi$ .

See Solution

Problem 11009

Find the derivative of the inverse function at 2 for $f(x)=x+\sqrt{x}$ : compute $\left(f^{-1}\right)^{\prime}(2)$ .

See Solution

Problem 11010

Find the monthly cost $C(x)$ and profit $P(x)$ functions for a gaming website. Determine the optimal log-on fee $x$ for max profit.

See Solution

Problem 11011

Given $g(x)$ with $g(\pi)=-2$ , $g^{\prime}(\pi)=0$ , $g^{\prime \prime}(\pi)=-3$ , determine the nature of $f(\pi)=2 \cos (x)+g(x)$ .

See Solution

Problem 11012

Find points $(x, y)$ on $g(x)=\frac{1}{3} x^{3}-\frac{3}{2} x^{2}+1$ where the tangent is parallel to $8 x-2 y=1$ and find the tangent line equations.

See Solution

Problem 11013

A book drops from a height of $75.4 \mathrm{~m}$ . Find the time it takes to reach the ground.

See Solution

Problem 11014

Compute the derivatives of these functions without using quotient or chain rules: (a) $f(x)=12 x^{4}-14 x^{2}+5 x+2$ (b) $g(t)=\sqrt{t^{3}}-\sqrt{t^{5}}$ (c) $h(s)=\frac{5 s+5}{2 \sqrt{s}}$ (d) $r(x)=\frac{1+x-4 \sqrt{x}}{x}$ (e) $p(t)=(t^{2}-2 t+5)(t^{2}-1)$

See Solution

Problem 11015

Determine the local extrema of the function $f(x)=e^{c|x|+3}$ , treating it as a piecewise function.

See Solution

Problem 11016

Given a twice differentiable function $g(x)$ , which statement about $f(x)=g(x)-x^{3}$ is TRUE?

See Solution

Problem 11017

Anthony kicks a ball upwards at $24.3 \frac{\mathrm{m}}{\mathrm{s}}$ . Find: time to max height, total time, max height, and impact velocity.

See Solution

Problem 11018

Calculate the average rate of change of $f(x)=3 x^{2}+2$ for these intervals: (-1,1), (-2,0), (1,4).

See Solution

Problem 11019

Given $u(0)=5$ , $u'(0)=-3$ , $v(0)=-1$ , $v'(0)=2$ , find derivatives at $x=0$ for: (a) $\frac{d}{dx}(u-v)$ (b) $\frac{d}{dx}(uv)$ (c) $\frac{d}{dx}(\sin(x)+3u+5v)$ (d) $\frac{d}{dx}(uv-u-\cos(x))$

See Solution

Problem 11020

A stream with insecticide concentration $9 \mathrm{~g} / \mathrm{m}^{3}$ flows into a pond at $25 \mathrm{~m}^{3}$ /day. The pond has volume $2000 \mathrm{~m}^{3}$ and initial concentration $2.5 \mathrm{~g} / \mathrm{m}^{3}$ .
(a) Find and solve the differential equation for $y(t)$ , the insecticide amount in grams. (b) What happens to the insecticide concentration over time? (c) How many days until concentration reaches $7 \mathrm{~g} / \mathrm{m}^{3}$ ?

See Solution

Problem 11021

Given the ideal gas law $p V=n R T$ , find $\frac{d p}{d V}$ in terms of $p, V, n, R, T$ and simplify it. What happens to pressure if volume increases?

See Solution

Problem 11022

Identify which limits cannot be solved using I'Hopital's Rule:
1. $\lim _{x \rightarrow 0} \frac{x^{2}+2}{\sin x}$
2. $\lim _{x \rightarrow \infty} \frac{e^{x}}{\sin x}$
3. $\lim _{x \rightarrow \infty} \frac{e^{x}}{5 x}$
4. $\lim _{x \rightarrow 0} \frac{1-\cos x}{x}$
5. $\lim _{x \rightarrow 0} \frac{\sin x}{x}$
6. $\lim _{x \rightarrow 1} \frac{x^{2}-1}{x^{2}+2}$

See Solution

Problem 11023

Evaluate the limit: $\lim _{x \rightarrow 0}(\csc x-\cot x)$ . Which option matches its value? A: $\lim _{x \rightarrow 0} \frac{\cos x}{\sin x}$ B: $\lim _{x \rightarrow 0} \frac{1-\cos x}{\sin x}$ C: $\lim _{x \rightarrow 0} \frac{1-\sin x}{\cos x}$

See Solution

Problem 11024

True or false: L'Hopital's Rule statements about limits and derivatives. Evaluate I-V.

See Solution

Problem 11025

Find the derivative of $y=\tan^{2}x$ with respect to $x$ .

See Solution

Problem 11026

Find the derivative of $y=\frac{x+3}{\sqrt{x}+2}$ using the chain rule.

See Solution

Problem 11027

Find the derivative of $y=\sin x \cos x$ with respect to $x$ .

See Solution

Problem 11028

Find intervals where $f(x)=\cos^{2}(x)-2\sin(x)$ is increasing, decreasing, concave up, and concave down on $[0,2\pi]$ .

See Solution

Problem 11029

Calculate the average rate of change of $f(x)=-2 x^{2}+5$ for these intervals: (a) 4 to 6, (b) -2 to 0, (c) 0 to 3.

See Solution

Problem 11030

Find the derivative $f^{\prime}\left(\frac{\pi}{2}\right)$ for $f(x)=\frac{x^{3}+2 \cos (x)}{3 \sin (x)}$ .

See Solution

Problem 11031

Find the derivative of $f(x)=6 x^{3}-9 x+4$ and state the derivative rule(s) used.

See Solution

Problem 11032

Find the derivative of the function $f(x)=3 \arcsin (x) \arccos (x)-3 \arctan (x)$ at $x=0$ .

See Solution

Problem 11033

Find the derivatives of these functions: (a) $f(x)=(2 \cos ^{2} x+3)^{5/2}$ , (b) $g(t)=\tan(t^{2}-2 e^{-4t})$ , (c) $x^{2} \cos y+\sin(3x-4y)=3$ , (d) $h(x)=\cos^{-1}(x^{5/3})$ .

See Solution

Problem 11034

A right triangle has base $x$ m and height $h$ m. How does $\frac{d \theta}{d t}$ relate to $\frac{d x}{d t}$ ?

See Solution

Problem 11035

Find the derivative of $f(x)=(6 x+1)^{\ln (2 x+6)}$ and evaluate it at $x=0$ .

See Solution

Problem 11036

Find the derivative of $y$ with respect to $x$ from the equation $y^{5} \ln (y)-x^{4} \ln (x)=5$ .

See Solution

Problem 11037

Find the exact integrals: (a) $\int_{0}^{\pi / 4} \cos (2 x) e^{\sin (2 x)} d x$ and (b) $\int 3 x(1+x^{4})^{-1} d x$ .

See Solution

Problem 11038

Solve the initial value problem: $\frac{d y}{d x}=x^{2}+1+\frac{1}{x^{2}+1}$ , with $y=9$ when $x=0$ .

See Solution

Problem 11039

Hourly internet usage $f(t)$ is sinusoidal with max $5 \mathrm{~TB}$ & min $1 \mathrm{~TB}$ . Find period, formula, min times, & $f^{\prime}(3)$ .

See Solution

Problem 11040

Find $\int_{4}^{1} f(x) \mathrm{d} x$ given areas: $[-4,-3] = \frac{7}{3}$ , $[-3,1] = \frac{32}{3}$ , $[1,2] = \frac{7}{3}$ .

See Solution

Problem 11041

Find the integral $\int_{4}^{1} f(x) dx$ given areas on intervals $[-4,-3]$ , $[-3,1]$ , and $[1,2]$ .

See Solution

Problem 11042

Given the revenue function $R(x)=1.33 x$ , find the slope, marginal revenue, and interpret it. Choices for interpretation: A, B, C, D.

See Solution

Problem 11043

Find the derivative of $y=\sin x \cos x$ using the product rule.

See Solution

Problem 11044

Given $R(x)=1.79 x$ , find the slope, marginal revenue, and interpret it. Options for interpretation: A, B, C, D.

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Problem 11045

Sketch the area between $y=6 x^{3}$ , $y=6$ , and $x=0$ . Find the volume when this area is rotated around the $x$ -axis.

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Problem 11046

Find the intervals where the function $f(x)=x^{5}-5 x^{4}$ is decreasing and concave up. Provide your answer as intervals.

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Problem 11047

Calculate the area between the $x$ -axis and $y=7 \sin (x)$ , $y=7 \cos (x)$ for $x \in [0, \frac{\pi}{3}]$ .

See Solution

Problem 11048

Find the intervals where the function $f(x)=x^{5}-5 x^{4}$ is decreasing and concave up. Provide your intervals. $\text { interval }(\mathrm{s})=$

See Solution

Problem 11049

Find the absolute maximum of $f(x)=3 \sin (2 x)+6 \cos (x)$ on $[0, \frac{\pi}{2}]$ .

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Problem 11050

Given the function $A(x)=x \sqrt{x+5}$ , determine the intervals where $A$ is increasing and decreasing, and find local maxima and minima.

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Problem 11051

Find values of $c$ for $f(x)=x^{2}+2x+3$ in $[-3,1]$ that meet Rolle's Theorem: a. $-2,0$ b. -2 c. $-1,0$ d. -1

See Solution

Problem 11052

Find if the limit $\lim _{x \rightarrow-\infty} x^{2} e^{3 x}$ exists and calculate its value if it does.

See Solution

Problem 11053

Find where the function $f(x)=4+7 x^{1/3}$ is concave up/down and identify its inflection points.

See Solution

Problem 11054

Given $f(x)=2 x^{3}-3 x^{2}-36 x-9$ , find where $f$ is increasing/decreasing and its relative maxima/minima.

See Solution

Problem 11055

Find the relative extrema of $f(x)=x+\frac{36}{x}$ using the Second Derivative Test.

See Solution

Problem 11056

Find the population of Rhodobacter sphaeroides after 7 hours, starting with 23 and increasing at $3.4657 e^{0.1386 t}$ .

See Solution

Problem 11057

Find the derivative of the function $f(x)=x^{2}+x$ using the limit definition.

See Solution

Problem 11058

Find where the function $f(u)=\frac{2 u}{u^{2}-49}$ is concave up/down and identify its inflection points.

See Solution

Problem 11059

Find the population of Rhodobacter sphaeroides after 7 hours, starting with 23 and increasing at $3.4657 e^{0.1386 t}$ .

See Solution

Problem 11060

Find points on the graph of $y^{2}=x^{3}-3 x+1$ where the tangent line is horizontal.

See Solution

Problem 11061

Find the function $f$ given $f^{\prime \prime}(\theta)=\sin \theta+\cos \theta$ , $f(0)=4$ , $f^{\prime}(0)=3$ .

See Solution

Problem 11062

Find points on the graph of $3 x^{2}+4 y^{2}+3 x y=24$ where the tangent line is horizontal.

See Solution

Problem 11063

Find points on the graph of $3 x^{2}+4 y^{2}+3 x y=24$ where the tangent line is horizontal.

See Solution

Problem 11064

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{(\ln x)^{5}}{6 x^{4}}$ using l'Hôpital's Rule.

See Solution

Problem 11065

Find if the limit exists: $\lim _{x \rightarrow 0+}(1+5 x)^{\frac{1}{3 x}}$ and determine its value if it does.

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Problem 11066

Find $g'(2)$ for $g(x)=3f(x)+x^5f(x)$ and $h'(2)$ for $h(x)=(9x+8)f(x)$ , given $f(2)=5$ and $f'(2)=-2$ .

See Solution

Problem 11067

Find the value of $\frac{d}{d x}\left(f(x)+5 x^{4}-5 x^{2}+4\right)^{5}$ at $x=3$ given $f(3)=5$ and $f^{\prime}(3)=-9$ .

See Solution

Problem 11068

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{(\sin x)^{2}}{1-\sec x}$ using l'Hôpital's Rule.

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Problem 11069

Gegeben ist die Funktion $f(x)=\frac{3}{5} x^{4}-\frac{5}{2} x^{3}+2 x$ . Finde Nullstellen, Extrema, Wendepunkte, Krümmung, Steigung -1 und $f(x)=0,5$ .

See Solution

Problem 11070

Finde den höchsten Punkt der Funktion $f(x)=5x^{3}-26x^{2}+21x+36$ im Bereich $[-0,8; 3]$ und bestimme die Höhe über Normalnull.

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Problem 11071

Approximate $\sqrt{9.04}$ using the linear approximation of $f(x)=\sqrt{9-x}$ at $a=0$ .

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Problem 11072

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{4}+x^{3}}$

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Problem 11073

Estimate the paint needed for a $0.0015 \mathrm{~m}$ thick coat on a hemispherical dome with a diameter of $50 \mathrm{~m}$ .

See Solution

Problem 11074

Bestimme die Stammfunktion von $f$ : a) $f(x)=4 x^{3}+2 x^{2}-1$ ; b) $f(x)=x^{5}-4 x^{3}$ .

See Solution

Problem 11075

Finde eine Stammfunktion für die Funktion $f(x)=x^{5}-4 x^{3}$ .

See Solution

Problem 11076

Finde eine Stammfunktion von $f(x)=-x^{3}-2x+7$ .

See Solution

Problem 11077

Finde eine Stammfunktion von $f(x)=4x^3+2x^2-1$ .

See Solution

Problem 11078

Estimate the relative error in the volume of a cube with edge $20 \mathrm{~cm}$ and measurement error $0.2 \mathrm{~cm}$ .

See Solution

Problem 11079

Identify the interval where the function is increasing given its low point at -2. Choose A or B for your answer.

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Problem 11080

Untersuchen Sie die Integrale $\int_{2}^{\infty} \frac{1}{x^{3}} d x$ und $\int_{0}^{1} \frac{1}{x^{3}} d x$ auf Existenz und Wert.

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Problem 11081

Bewerten Sie die Rechnung $\int_{-1}^{1} \frac{1}{x^{2}} d x=\left[-\frac{1}{x}\right]_{-1}^{1}=-2$ und die Anmerkung dazu.

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Problem 11082

Find the interval where the function is increasing, given it has a lowest point of -2.
A. Increasing on interval(s): __________ B. No increasing interval.

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Problem 11083

Gegeben sind zwei Kurven $K_{1}$ und $K_{2}$ . Skizzieren Sie sie und berechnen Sie die Volumina $V_{1}$ , $V_{2}$ und $V$ der Schale.

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Problem 11084

Gjeni shpejtësitë e ndryshimit të funksioneve: a $f(x)=x^{2}+10x$ në $x=4$ b $g(x)=x^{3}+2x^{2}+x+1$ në $x=-1$ c $h(x)=5x+6$ në $x=1$ d $k(x)=x+\frac{1}{x}$ në $x=3$ .

See Solution

Problem 11085

Njehsoni shpejtësitë e ndryshimit të funksioneve në pikat e dhëna: a) $f(x)=x^{2}+10x$ në $x=4$ , b) $g(x)=x^{3}+2x^{2}+x+1$ në $x=-1$ , c) $h(x)=5x+6$ në $x=1$ , d) $k(x)=x+\frac{1}{x}$ në $x=3$ , e) $m(x)=9x^{2}+6x+\ln e$ në $x=1$ , f) $n(x)=x^{-1}+x^{-2}$ në $x=-2$ , g) $p(x)=\frac{1}{\sqrt{x}}$ në $x=\frac{1}{9}$ , h) $q(x)=x^{\frac{3}{2}}+\ln e$ në $x=36$ , i) $r(x)=x^{4}-8x^{2}$ në $x=-2$ , j) $s(x)=\sqrt{x}-x$ në $x=4$ .

See Solution

Problem 11086

Rotiert die Fläche zwischen $f(x)=\frac{1}{50}(x-2)^{2}+0,4$ und $g(x)=\frac{1}{4}\sqrt{x-1}$ von $[0;6]$ um die $x$ -Achse, a) zeichnen Sie die Graphen, b) wie viel Flüssigkeit passt in den Pokal? c) Masse bei Dichte $3,4 \mathrm{~g/cm^{3}}$ ? d) Neue Funktion $g^{*}$ nach Verschiebung um $5 \mathrm{~cm}$ und mögliche Probleme erläutern.

See Solution

Problem 11087

Gjeni shpejtësitë e ndryshimit të funksioneve: a $f(x)=x^{2}+10x$ në $x=4$ b $g(x)=x^{3}+2x^{2}+x+1$ në $x=-1$ c $h(x)=5x+6$ në $x=1$ d $k(x)=x+\frac{1}{x}$ në $x=3$ e $m(x)=9x^{2}+6x+1$ në $x=1$ f $n(x)=x^{-1}+x^{-2}$ në $x=-2$

See Solution

Problem 11088

Gjeni shpejtësinë e ndryshimit të funksioneve $m(x)=9x^2+6x+1$ në $x=1$ , $n(x)=x^{-1}+x^{-2}$ për $x<-2$ , $p(x)=\frac{1}{\sqrt{x}}$ në $x=\frac{1}{9}$ , dhe $g(x)=x^{3/2}+1$ në $x=36.

See Solution

Problem 11089

Bestimme den Wert von $x$ , der das Volumen einer offenen Box aus einem $16 \mathrm{~cm} \times 10 \mathrm{~cm}$ Rechteck maximiert.

See Solution

Problem 11090

Zeigen Sie, dass $F(x)=\frac{1}{2} x^{2} \cdot \ln x$ eine Stammfunktion von $f(x)=x \cdot\left(\frac{1}{2}+\ln x\right)$ ist.

See Solution

Problem 11091

Check if the Mean Value Theorem applies to $f(x)=x^{2}+x$ on $[-2,2]$ and find values of $c$ or state why it doesn't.

See Solution

Problem 11092

Bestimmen Sie die Stammfunktion $F$ von $f(x)=2 \cdot x^{3}-3 \cdot x^{2}$ , die durch $P(0/2)$ verläuft.

See Solution

Problem 11093

Determine true statements about a polynomial $g$ with an inflection point at $x=3$ .

See Solution

Problem 11094

Bestimme die Intervallgrenze $u$ für verschiedene Funktionen $f$ , sodass der Flächeninhalt $A$ über das Intervall $I$ erreicht wird.

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Problem 11095

Design a rectangular container with a square base and volume $3888 \mathrm{ft}^{3}$ . Minimize cost with top at \$2/ft², sides at \$9/ft², and bottom at \$10/ft². Find dimensions.

See Solution

Problem 11096

Given the function $r(x)=\frac{x^{2}-4}{x^{2}-x-2}$ , which statement about $\lim_{x \to 2}$ is true?

See Solution

Problem 11097

Skizzieren Sie den Graphen von $f$ und berechnen Sie die Fläche zwischen $f$ und der $x$ -Achse für die Intervalle I: a) $I=[-1 ; 6]$ , b) $I=[-2 ; 3]$ , c) $I=[-1 ; 2]$ , d) $I=[1 ; 5]$ , e) $I=[0,5 ; 2]$ , f) $I=[-1 ; 2,5]$ .

See Solution

Problem 11098

Bestimme die Tangentengleichung an $f(x)=2 \cdot e^{-\frac{1}{2} x}$ im Schnittpunkt mit der $y$ -Achse.

See Solution

Problem 11099

Find local maxima of $f(x)$ with critical points at $x=-0.64, 0.46, 1.54, 2.64$ and $f^{\prime \prime}(x)=20 x^{3}-60 x^{2}+30 x+10$ .

See Solution

Problem 11100

Bestimme die Tangentengleichung von $f(x) = 2e^{-\frac{1}{2}x}$ an der $y$ -Achse und finde den Flächeninhalt von 1,5.

See Solution

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Calculus

Problem 11001

Find the derivative of the function P(x)=13x3cos⁡5xP(x)=\frac{1}{3} x^{3} \cos 5 xP(x)=31​x3cos5x. What is P′(x)P^{\prime}(x)P′(x)?

Problem 11002

Problem 11003

Find the slope of the tangent line to the curve 2(x2+y2)2=25(x2−y2)2\left(x^{2}+y^{2}\right)^{2}=25\left(x^{2}-y^{2}\right)2(x2+y2)2=25(x2−y2) at (3,−1)(3,-1)(3,−1).

Problem 11004

Find the derivative y′y'y′, where y=5+sin⁡xcos⁡xy=\frac{5+\sin x}{\cos x}y=cosx5+sinx​.

Problem 11005

Find the intervals where f(x)=x4e−xf(x)=x^{4} e^{-x}f(x)=x4e−x is increasing, decreasing, concave up, and concave down.

Problem 11006

Find dydx\frac{d y}{d x}dxdy​ in terms of xxx and yyy for the equation ax4−by4=c4a x^{4}-b y^{4}=c^{4}ax4−by4=c4, where a,b,ca, b, ca,b,c are constants.

Problem 11007

Find points (x,y)(x, y)(x,y) on f(x)=3x2−4xf(x)=3 x^{2}-4 xf(x)=3x2−4x where the tangent is parallel to y=8x+5y=8 x+5y=8x+5. Also, find the tangent line equations.

Problem 11008

Tentukan titik stasioner dan nilai stasioner dari F(x)=cos⁡x−sin⁡xF(x)=\cos x-\sin xF(x)=cosx−sinx dalam interval 0≤x≤2π0 \leq x \leq 2\pi0≤x≤2π.

Problem 11009

Find the derivative of the inverse function at 2 for f(x)=x+xf(x)=x+\sqrt{x}f(x)=x+x​: compute (f−1)′(2)\left(f^{-1}\right)^{\prime}(2)(f−1)′(2).

Problem 11010

Find the monthly cost C(x)C(x)C(x) and profit P(x)P(x)P(x) functions for a gaming website. Determine the optimal log-on fee xxx for max profit.

Problem 11011

Given g(x)g(x)g(x) with g(π)=−2g(\pi)=-2g(π)=−2, g′(π)=0g^{\prime}(\pi)=0g′(π)=0, g′′(π)=−3g^{\prime \prime}(\pi)=-3g′′(π)=−3, determine the nature of f(π)=2cos⁡(x)+g(x)f(\pi)=2 \cos (x)+g(x)f(π)=2cos(x)+g(x).

Problem 11012

Find points (x,y)(x, y)(x,y) on g(x)=13x3−32x2+1g(x)=\frac{1}{3} x^{3}-\frac{3}{2} x^{2}+1g(x)=31​x3−23​x2+1 where the tangent is parallel to 8x−2y=18 x-2 y=18x−2y=1 and find the tangent line equations.

Problem 11013

A book drops from a height of 75.4 m75.4 \mathrm{~m}75.4 m. Find the time it takes to reach the ground.

Problem 11014

Problem 11015

Determine the local extrema of the function f(x)=ec∣x∣+3f(x)=e^{c|x|+3}f(x)=ec∣x∣+3, treating it as a piecewise function.

Problem 11016

Given a twice differentiable function g(x)g(x)g(x), which statement about f(x)=g(x)−x3f(x)=g(x)-x^{3}f(x)=g(x)−x3 is TRUE?

Problem 11017

Anthony kicks a ball upwards at 24.3ms24.3 \frac{\mathrm{m}}{\mathrm{s}}24.3sm​. Find: time to max height, total time, max height, and impact velocity.

Problem 11018

Calculate the average rate of change of f(x)=3x2+2f(x)=3 x^{2}+2f(x)=3x2+2 for these intervals: (-1,1), (-2,0), (1,4).

Problem 11019

Problem 11020

Problem 11021

Given the ideal gas law pV=nRTp V=n R TpV=nRT, find dpdV\frac{d p}{d V}dVdp​ in terms of p,V,n,R,Tp, V, n, R, Tp,V,n,R,T and simplify it. What happens to pressure if volume increases?

Problem 11022

Problem 11023

Problem 11024

True or false: L'Hopital's Rule statements about limits and derivatives. Evaluate I-V.

Problem 11025

Find the derivative of y=tan⁡2xy=\tan^{2}xy=tan2x with respect to xxx.

Problem 11026

Find the derivative of y=x+3x+2y=\frac{x+3}{\sqrt{x}+2}y=x​+2x+3​ using the chain rule.

Problem 11027

Find the derivative of y=sin⁡xcos⁡xy=\sin x \cos xy=sinxcosx with respect to xxx.

Problem 11028

Find intervals where f(x)=cos⁡2(x)−2sin⁡(x)f(x)=\cos^{2}(x)-2\sin(x)f(x)=cos2(x)−2sin(x) is increasing, decreasing, concave up, and concave down on [0,2π][0,2\pi][0,2π].

Problem 11029

Calculate the average rate of change of f(x)=−2x2+5f(x)=-2 x^{2}+5f(x)=−2x2+5 for these intervals: (a) 4 to 6, (b) -2 to 0, (c) 0 to 3.

Problem 11030

Find the derivative f′(π2)f^{\prime}\left(\frac{\pi}{2}\right)f′(2π​) for f(x)=x3+2cos⁡(x)3sin⁡(x)f(x)=\frac{x^{3}+2 \cos (x)}{3 \sin (x)}f(x)=3sin(x)x3+2cos(x)​.

Problem 11031

Find the derivative of f(x)=6x3−9x+4f(x)=6 x^{3}-9 x+4f(x)=6x3−9x+4 and state the derivative rule(s) used.

Problem 11032

Find the derivative of the function f(x)=3arcsin⁡(x)arccos⁡(x)−3arctan⁡(x)f(x)=3 \arcsin (x) \arccos (x)-3 \arctan (x)f(x)=3arcsin(x)arccos(x)−3arctan(x) at x=0x=0x=0.

Problem 11033

Problem 11034

A right triangle has base xxx m and height hhh m. How does dθdt\frac{d \theta}{d t}dtdθ​ relate to dxdt\frac{d x}{d t}dtdx​?

Problem 11035

Find the derivative of f(x)=(6x+1)ln⁡(2x+6)f(x)=(6 x+1)^{\ln (2 x+6)}f(x)=(6x+1)ln(2x+6) and evaluate it at x=0x=0x=0.

Problem 11036

Find the derivative of yyy with respect to xxx from the equation y5ln⁡(y)−x4ln⁡(x)=5y^{5} \ln (y)-x^{4} \ln (x)=5y5ln(y)−x4ln(x)=5.

Problem 11037

Find the exact integrals: (a) ∫0π/4cos⁡(2x)esin⁡(2x)dx\int_{0}^{\pi / 4} \cos (2 x) e^{\sin (2 x)} d x∫0π/4​cos(2x)esin(2x)dx and (b) ∫3x(1+x4)−1dx\int 3 x(1+x^{4})^{-1} d x∫3x(1+x4)−1dx.

Problem 11038

Solve the initial value problem: dydx=x2+1+1x2+1\frac{d y}{d x}=x^{2}+1+\frac{1}{x^{2}+1}dxdy​=x2+1+x2+11​, with y=9y=9y=9 when x=0x=0x=0.

Problem 11039

Hourly internet usage f(t)f(t)f(t) is sinusoidal with max 5 TB5 \mathrm{~TB}5 TB & min 1 TB1 \mathrm{~TB}1 TB. Find period, formula, min times, & f′(3)f^{\prime}(3)f′(3).

Problem 11040

Find ∫41f(x)dx\int_{4}^{1} f(x) \mathrm{d} x∫41​f(x)dx given areas: [−4,−3]=73[-4,-3] = \frac{7}{3}[−4,−3]=37​, [−3,1]=323[-3,1] = \frac{32}{3}[−3,1]=332​, [1,2]=73[1,2] = \frac{7}{3}[1,2]=37​.

Problem 11041

Find the integral ∫41f(x)dx\int_{4}^{1} f(x) dx∫41​f(x)dx given areas on intervals [−4,−3][-4,-3][−4,−3], [−3,1][-3,1][−3,1], and [1,2][1,2][1,2].

Problem 11042

Given the revenue function R(x)=1.33xR(x)=1.33 xR(x)=1.33x, find the slope, marginal revenue, and interpret it. Choices for interpretation: A, B, C, D.

Problem 11043

Find the derivative of y=sin⁡xcos⁡xy=\sin x \cos xy=sinxcosx using the product rule.

Find the derivative of the function $P(x)=\frac{1}{3} x^{3} \cos 5 x$ . What is $P^{\prime}(x)$ ?

Find the slope of the tangent line to the curve $2\left(x^{2}+y^{2}\right)^{2}=25\left(x^{2}-y^{2}\right)$ at $(3,-1)$ .

Find the derivative $y'$ , where $y=\frac{5+\sin x}{\cos x}$ .

Find the intervals where $f(x)=x^{4} e^{-x}$ is increasing, decreasing, concave up, and concave down.

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ for the equation $a x^{4}-b y^{4}=c^{4}$ , where $a, b, c$ are constants.

Find points $(x, y)$ on $f(x)=3 x^{2}-4 x$ where the tangent is parallel to $y=8 x+5$ . Also, find the tangent line equations.

Tentukan titik stasioner dan nilai stasioner dari $F(x)=\cos x-\sin x$ dalam interval $0 \leq x \leq 2\pi$ .

Find the derivative of the inverse function at 2 for $f(x)=x+\sqrt{x}$ : compute $\left(f^{-1}\right)^{\prime}(2)$ .

Find the monthly cost $C(x)$ and profit $P(x)$ functions for a gaming website. Determine the optimal log-on fee $x$ for max profit.

Given $g(x)$ with $g(\pi)=-2$ , $g^{\prime}(\pi)=0$ , $g^{\prime \prime}(\pi)=-3$ , determine the nature of $f(\pi)=2 \cos (x)+g(x)$ .

Find points $(x, y)$ on $g(x)=\frac{1}{3} x^{3}-\frac{3}{2} x^{2}+1$ where the tangent is parallel to $8 x-2 y=1$ and find the tangent line equations.

A book drops from a height of $75.4 \mathrm{~m}$ . Find the time it takes to reach the ground.

Determine the local extrema of the function $f(x)=e^{c|x|+3}$ , treating it as a piecewise function.

Given a twice differentiable function $g(x)$ , which statement about $f(x)=g(x)-x^{3}$ is TRUE?

Anthony kicks a ball upwards at $24.3 \frac{\mathrm{m}}{\mathrm{s}}$ . Find: time to max height, total time, max height, and impact velocity.

Calculate the average rate of change of $f(x)=3 x^{2}+2$ for these intervals: (-1,1), (-2,0), (1,4).

Given the ideal gas law $p V=n R T$ , find $\frac{d p}{d V}$ in terms of $p, V, n, R, T$ and simplify it. What happens to pressure if volume increases?

Find the derivative of $y=\tan^{2}x$ with respect to $x$ .

Find the derivative of $y=\frac{x+3}{\sqrt{x}+2}$ using the chain rule.

Find the derivative of $y=\sin x \cos x$ with respect to $x$ .

Find intervals where $f(x)=\cos^{2}(x)-2\sin(x)$ is increasing, decreasing, concave up, and concave down on $[0,2\pi]$ .

Calculate the average rate of change of $f(x)=-2 x^{2}+5$ for these intervals: (a) 4 to 6, (b) -2 to 0, (c) 0 to 3.

Find the derivative $f^{\prime}\left(\frac{\pi}{2}\right)$ for $f(x)=\frac{x^{3}+2 \cos (x)}{3 \sin (x)}$ .

Find the derivative of $f(x)=6 x^{3}-9 x+4$ and state the derivative rule(s) used.

Find the derivative of the function $f(x)=3 \arcsin (x) \arccos (x)-3 \arctan (x)$ at $x=0$ .

A right triangle has base $x$ m and height $h$ m. How does $\frac{d \theta}{d t}$ relate to $\frac{d x}{d t}$ ?

Find the derivative of $f(x)=(6 x+1)^{\ln (2 x+6)}$ and evaluate it at $x=0$ .

Find the derivative of $y$ with respect to $x$ from the equation $y^{5} \ln (y)-x^{4} \ln (x)=5$ .

Find the exact integrals: (a) $\int_{0}^{\pi / 4} \cos (2 x) e^{\sin (2 x)} d x$ and (b) $\int 3 x(1+x^{4})^{-1} d x$ .

Solve the initial value problem: $\frac{d y}{d x}=x^{2}+1+\frac{1}{x^{2}+1}$ , with $y=9$ when $x=0$ .

Hourly internet usage $f(t)$ is sinusoidal with max $5 \mathrm{~TB}$ & min $1 \mathrm{~TB}$ . Find period, formula, min times, & $f^{\prime}(3)$ .

Find $\int_{4}^{1} f(x) \mathrm{d} x$ given areas: $[-4,-3] = \frac{7}{3}$ , $[-3,1] = \frac{32}{3}$ , $[1,2] = \frac{7}{3}$ .

Find the integral $\int_{4}^{1} f(x) dx$ given areas on intervals $[-4,-3]$ , $[-3,1]$ , and $[1,2]$ .

Given the revenue function $R(x)=1.33 x$ , find the slope, marginal revenue, and interpret it. Choices for interpretation: A, B, C, D.

Find the derivative of $y=\sin x \cos x$ using the product rule.

Given $R(x)=1.79 x$ , find the slope, marginal revenue, and interpret it. Options for interpretation: A, B, C, D.

Sketch the area between $y=6 x^{3}$ , $y=6$ , and $x=0$ . Find the volume when this area is rotated around the $x$ -axis.

Find the intervals where the function $f(x)=x^{5}-5 x^{4}$ is decreasing and concave up. Provide your answer as intervals.

Calculate the area between the $x$ -axis and $y=7 \sin (x)$ , $y=7 \cos (x)$ for $x \in [0, \frac{\pi}{3}]$ .

Find the intervals where the function $f(x)=x^{5}-5 x^{4}$ is decreasing and concave up. Provide your intervals. $\text { interval }(\mathrm{s})=$

Find the absolute maximum of $f(x)=3 \sin (2 x)+6 \cos (x)$ on $[0, \frac{\pi}{2}]$ .

Given the function $A(x)=x \sqrt{x+5}$ , determine the intervals where $A$ is increasing and decreasing, and find local maxima and minima.

Find values of $c$ for $f(x)=x^{2}+2x+3$ in $[-3,1]$ that meet Rolle's Theorem: a. $-2,0$ b. -2 c. $-1,0$ d. -1

Find if the limit $\lim _{x \rightarrow-\infty} x^{2} e^{3 x}$ exists and calculate its value if it does.

Find where the function $f(x)=4+7 x^{1/3}$ is concave up/down and identify its inflection points.

Given $f(x)=2 x^{3}-3 x^{2}-36 x-9$ , find where $f$ is increasing/decreasing and its relative maxima/minima.

Find the relative extrema of $f(x)=x+\frac{36}{x}$ using the Second Derivative Test.

Find the population of Rhodobacter sphaeroides after 7 hours, starting with 23 and increasing at $3.4657 e^{0.1386 t}$ .

Find the derivative of the function $f(x)=x^{2}+x$ using the limit definition.

Find where the function $f(u)=\frac{2 u}{u^{2}-49}$ is concave up/down and identify its inflection points.

Find the population of Rhodobacter sphaeroides after 7 hours, starting with 23 and increasing at $3.4657 e^{0.1386 t}$ .

Find points on the graph of $y^{2}=x^{3}-3 x+1$ where the tangent line is horizontal.

Find the function $f$ given $f^{\prime \prime}(\theta)=\sin \theta+\cos \theta$ , $f(0)=4$ , $f^{\prime}(0)=3$ .

Find points on the graph of $3 x^{2}+4 y^{2}+3 x y=24$ where the tangent line is horizontal.

Find points on the graph of $3 x^{2}+4 y^{2}+3 x y=24$ where the tangent line is horizontal.

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{(\ln x)^{5}}{6 x^{4}}$ using l'Hôpital's Rule.

Find if the limit exists: $\lim _{x \rightarrow 0+}(1+5 x)^{\frac{1}{3 x}}$ and determine its value if it does.

Find $g'(2)$ for $g(x)=3f(x)+x^5f(x)$ and $h'(2)$ for $h(x)=(9x+8)f(x)$ , given $f(2)=5$ and $f'(2)=-2$ .

Find the value of $\frac{d}{d x}\left(f(x)+5 x^{4}-5 x^{2}+4\right)^{5}$ at $x=3$ given $f(3)=5$ and $f^{\prime}(3)=-9$ .

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{(\sin x)^{2}}{1-\sec x}$ using l'Hôpital's Rule.

Gegeben ist die Funktion $f(x)=\frac{3}{5} x^{4}-\frac{5}{2} x^{3}+2 x$ . Finde Nullstellen, Extrema, Wendepunkte, Krümmung, Steigung -1 und $f(x)=0,5$ .

Finde den höchsten Punkt der Funktion $f(x)=5x^{3}-26x^{2}+21x+36$ im Bereich $[-0,8; 3]$ und bestimme die Höhe über Normalnull.

Approximate $\sqrt{9.04}$ using the linear approximation of $f(x)=\sqrt{9-x}$ at $a=0$ .

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{4}+x^{3}}$

Estimate the paint needed for a $0.0015 \mathrm{~m}$ thick coat on a hemispherical dome with a diameter of $50 \mathrm{~m}$ .

Bestimme die Stammfunktion von $f$ : a) $f(x)=4 x^{3}+2 x^{2}-1$ ; b) $f(x)=x^{5}-4 x^{3}$ .