Calculus

Problem 25001

Estimate $e^{-0.01}$ using linear approximation or differentials.

See Solution

Problem 25002

Find the linearization $L(x)$ of $f(x)=x^{4}+4x^{2}$ at $a=1$ . What is $L(x)$ ?

See Solution

Problem 25003

Find the linear approximation of $g(x)=\sqrt[3]{1+x}$ at $a=0$ . Use it to estimate $\sqrt[3]{0.95}$ and $\sqrt[3]{1.1}$ (3 decimal places).

See Solution

Problem 25004

If nicotine constricts artery diameter by 11%, what pressure difference (in mm Hg) is needed for blood flow if normal is 9.1 mm Hg?

See Solution

Problem 25005

Find the linear approximation of $f(x)=\sqrt{4-x}$ at $a=0$ . Use $L(x)$ to estimate $\sqrt{3.9}$ and $\sqrt{3.99}$ .

See Solution

Problem 25006

Find the differential of each function: (a) $y=x^{2} \sin 6 x$ , (b) $y=\ln \sqrt{5+t^{2}}$ .

See Solution

Problem 25007

Find the differential of the functions: (a) $y=e^{\tan \pi t}$ ; (b) $y=\sqrt{3+\ln z}$ .

See Solution

Problem 25008

Find the linear approximation of $f(x)=\sqrt{4-x}$ at $a=0$ : $L(x)=2-\frac{1}{4} x$ . Use $L(x)$ to estimate $\sqrt{3.9}$ and $\sqrt{3.99}$ .

See Solution

Problem 25009

Find the differential $d y$ for $y=\tan x$ . Then evaluate $d y$ at $x=\pi/3$ and $d x=-0.05$ .

See Solution

Problem 25010

Find the value of $c$ guaranteed by the Mean Value Theorem for $q(x) = 4x^2 + 5x - 10$ on $[-4,1]$ .

See Solution

Problem 25011

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{1}{\sqrt[3]{x^{2}+x+3}}$ .

See Solution

Problem 25012

Find the first and second derivatives of $y=\sin(x^2)$ . What are $y'$ and $y''$ ?

See Solution

Problem 25013

Find the derivative of the function $F(x)=(x^{4}+9 x^{2}-7)^{3}$ . What is $F^{\prime}(x)$ ?

See Solution

Problem 25014

Find the derivative of the function $y=\left(\frac{x^{2}+3}{x^{2}-3}\right)^{3}$ . What is $y^{\prime}$ ?

See Solution

Problem 25015

Find the derivative of $f(x)=(2x-2)^{4}(x^{2}+x+1)^{5}$ . What is $f'(x)$ ?

See Solution

Problem 25016

Find $F^{\prime}(2)$ for $F(x)=f(g(x))$ with given values: $f(3)=8$ , $f^{\prime}(3)=9$ , $g(2)=3$ , $g^{\prime}(2)=5$ .

See Solution

Problem 25017

Plot $f(x) = e^{-x^2}$ on $[0,1]$ and solve $f(x) = \text{(average value)}$ using $n=500$ partitions.

See Solution

Problem 25018

Find the derivative of $f(x)=(2x-2)^{4}(x^{2}+x+1)^{5}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 25019

Find the derivative of the function $y=\cos(a^{3}+x^{3})$ . What is $y'(x)$ ?

See Solution

Problem 25020

Find the derivative of $y=\cot^{2}(\cos \theta)$ . What is $y'$ ?

See Solution

Problem 25021

Find the derivative of the function $y=x \sin \frac{2}{x}$ . What is $y'(x)$ ?

See Solution

Problem 25022

Find the second partial derivatives of $f(x, y) = \ln(1-x-y) + 2\ln(x) + \ln(y)$ for $x, y \in (0,1]$ . Specifically, compute $\frac{\partial^2 f}{\partial x^2}$ , $\frac{\partial^2 f}{\partial y^2}$ , and $\frac{\partial^2 f}{\partial x \partial y}$ .

See Solution

Problem 25023

Differentiate $y=\frac{2x}{5-\cot x}$ and find $y'=\square$ .

See Solution

Problem 25024

Find the limit: $\lim _{x \rightarrow 0} \frac{\cos (7 x)-\cos (8 x)}{x^{2}}$ using I'Hospital's Rule.

See Solution

Problem 25025

Differentiate $F(y)=\left(\frac{1}{y^{2}}-\frac{7}{y^{4}}\right)\left(y+3 y^{3}\right)$ .

See Solution

Problem 25026

Find the second derivative $f^{\prime \prime}(3)$ for the function $f(x)=\frac{x^{2}}{3+x}$ .

See Solution

Problem 25027

Differentiate $y=7 x^{2} \sin x \tan x$ using the Product Rule. What is $y'$ ?

See Solution

Problem 25028

Differentiate $y = \frac{2x}{5 - \cot x}$ and find $y'$ .

See Solution

Problem 25029

Find the tangent line equation for the curve $y=\frac{x^{2}-1}{x^{2}+x+1}$ at the point (1,0).

See Solution

Problem 25030

Evaluate the Riemann sum for $f(x)=3x^{2}-2x$ from $0$ to $3$ with $n=6$ using right endpoints.

See Solution

Problem 25031

Given $f(5)=1, f^{\prime}(5)=8, g(5)=-6$ , and $g^{\prime}(5)=7$ , find: (a) $(f g)^{\prime}(5)$ , (b) $(f / g)^{\prime}(5)$ , (c) $(g / f)^{\prime}(5)$ .

See Solution

Problem 25032

Find the limits of the sequences $a_n = (q^n + 5^n + 11^n)^{1/n}$ , $b_n = \sqrt[n]{q^n + n^2}$ , and $c_n = \frac{\sin(q^n \pi)}{n}$ .

See Solution

Problem 25033

Funktion f beschreibt tägliche Übernachtungen (in 1000) ab 2020. Analysiere die Aussagen (1)-(6) für $f(t)=0,15 \cdot(t-15) \cdot e^{-0,1 t}+8$ .

See Solution

Problem 25034

Calculate the integral from 1 to 4 of the function $x^{2}-4 x+2$ .

See Solution

Problem 25035

Find the general antiderivative of $f(x)=\frac{5-5x-x^{3}}{\sqrt{x}}$ . Use $x^{n} = \frac{x^{n+1}}{n+1}$ .

See Solution

Problem 25036

Calculate the integral $\int\left(\cos (x) \sin ^{3}(x)\right) d x$ .

See Solution

Problem 25037

Find the integral of the function: $\int\left(8 x^{3}+\frac{1}{x^{2}}+\sqrt{x}\right) d x$ .

See Solution

Problem 25038

Bestimme das Maximum der Erkrankten $e(t)=-\frac{1}{400} t^{2}(t-48)$ und die maximale Wachstumsrate. Wann endet die Epidemie?

See Solution

Problem 25039

Evaluate the integral: $\int_{1}^{4} \frac{(1+\sqrt{x})^{1 / 2}}{\sqrt{x}} d x$

See Solution

Problem 25040

Find the limit: $\lim _{x \rightarrow \infty}\left(x \sin \left(\frac{\pi}{x}\right)\right)$ .

See Solution

Problem 25041

Find the area between the curves $y = x^2 - 4$ and $y = -2x^2 + 3x + 2$ .

See Solution

Problem 25042

Evaluate the integral $I=\int_{0}^{12} f(t) dt=-\int_{0}^{12} t(t-21)(t+1) dt$ .

See Solution

Problem 25043

A projectile is fired at $v_0 = 150\ \text{m/s}$ and $\theta = 47^\circ$ . Find the max height $H$ with $g = 9.81\ \text{m/s}^2$ .

See Solution

Problem 25044

Find the volume of the solid formed by $y=\sqrt{x}$ , $x=4$ , $x=9$ , and the $x$ axis, when rotated about the $x$ axis.

See Solution

Problem 25045

Find when the velocity of the object given by $g(t) = \frac{2}{3} t^3 - \frac{5}{2} t^2 + 3t + 7$ is decreasing.

See Solution

Problem 25046

Calculate the integral from 1 to e: $\int_{1}^{e} \ln x \, dx$ .

See Solution

Problem 25047

Find the integral of the function: $\int \left(8x^3 + \frac{1}{x^2} + \sqrt{x}\right) dx$ .

See Solution

Problem 25048

Find where $f(x)$ is concave up and down, and determine the points of inflection for $f(x)=x^{4}-6 x^{3}$ .

See Solution

Problem 25049

Bestimmen Sie die Ableitungen der folgenden Funktionen mit der linearen Kettenregel: a) $f(x)=\sin (3 x-2)$ , b) $f(x)=\cos (0,5 x+3)$ , c) $f(x)=\sin (7 x-1)$ , d) $f(x)=\sqrt{7 x-3}$ , e) $h(z)=(3 z-4)^{4 z}$ , f) $f(a)=\frac{1}{3 a-5}$ .

See Solution

Problem 25050

Find the limit: $\lim _{x \rightarrow \infty}\left(x \sin \left(\frac{\pi}{x}\right)\right)$ .

See Solution

Problem 25051

Find the derivative of $f(x) = (x^{2} - 3x)^{5}$ .

See Solution

Problem 25052

Find $f^{\prime}(x)$ for: a) $f(x)=(x^{2}-3x)^{5}$ b) $f(x)=\sqrt{x} \tan^{2}(x)$ .

See Solution

Problem 25053

Find the derivative of the function $f(x) = \sqrt{x} \tan^2(x)$ .

See Solution

Problem 25054

Find the integral: $\int 3 \csc (7 x) \cot (7 x) \, dx$

See Solution

Problem 25055

Evaluate the integral using substitution: $\int_{0}^{\sqrt{\pi}} 2 x \sin(5 x^{2}) \, dx$ .

See Solution

Problem 25056

Find the average rate of change of power output from $x=4$ hours (20 MW) to $x=12$ hours (80 MW).

See Solution

Problem 25057

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

See Solution

Problem 25058

Zeichne ein Flächenstück unter $f(x)=2 e^{-\frac{1}{2} x}$ , das Fläche 1,5 hat, und gib den Berechnungsterm an.

See Solution

Problem 25059

Evaluate these integrals: a. $\int(8 x^{3}+\frac{1}{x^{2}}+\sqrt{x}) dx$ b. $\int(\cos(x) \sin^{3}(x)) dx$ c. $\int_{1}^{4} \frac{(1+\sqrt{x})^{1/2}}{\sqrt{x}} dx$

See Solution

Problem 25060

Calculate the integral of $(5 x-3)^{2}$ with respect to $x$ .

See Solution

Problem 25061

Find the integral of $(2 x+6)^{8}$ with respect to $x$ .

See Solution

Problem 25062

Find the intervals where the function $f$ is decreasing given that $f^{\prime}$ has two zeros and selected values of $f^{\prime}(x)$ .

See Solution

Problem 25063

Find the integral of $\cos (\pi \theta)$ with respect to $\theta$ .

See Solution

Problem 25064

Calculate the integral: $\int_{-2}^{1} \sqrt{6-5 x} \, dx$

See Solution

Problem 25065

Find the fluid force on a semicircular plate (radius $14 \mathrm{ft}$ ) submerged in water (depth $15 \mathrm{ft}$ ).

See Solution

Problem 25066

Find the derivative of the function $f(x) = 2 + \left(\frac{-1}{\sqrt{1-x^{2}}}\right)$ . What is $f'(x)$ ?

See Solution

Problem 25067

d. Find $\int 3 \csc (7 x) \cot (7 x) d x$ . e. Calculate $\int(2 x+6)^{8} d x$ . f. Evaluate $\int \cos (\pi \theta) d \theta$ . g. Solve $\int_{-2}^{1} \sqrt{6-5 x} d x$ .

See Solution

Problem 25068

Berechne die Elastizität von $y=3 x \exp (4-2 x)$ für $x=0.01$ und $x=4.39$ . Ergebnisse als Dezimalzahlen, gerundet auf zwei Nachkommastellen.

See Solution

Problem 25069

Consider the vector field $\mathbf{F}=4 y \tan x \sec ^{2} x \mathbf{i}+\left(2 \tan ^{2} x+4 y\right) \mathbf{j}$ .
(a) Prove $\mathbf{F}$ is conservative. (b) Determine the potential function $f$ .

See Solution

Problem 25070

Bestimme die jährliche Fahrstrecke $x$ (in $1000 \mathrm{~km}$ ), um die Durchschnittskosten minimal zu halten. Kostenformel: $K = 1302 + 120x + 32x \ln(x) + 12x^2$ .

See Solution

Problem 25071

Find the gradient $\mathbf{F}=\nabla f$ for the function $f(x, y, z)=z e^{x+y}+z e^{x-z}$ .

See Solution

Problem 25072

Finde das Minimum der Funktion $f(x)=6 x^{2}+12 x+3$ . Ergebnisse als Dezimalzahl (4 Nachkommastellen) angeben: $x_{\text{min}}=$ , $y_{\text{min}}=$ . Bestimme $f^{\prime \prime}(x_{\min })=$ .

See Solution

Problem 25073

Bearbeite eine der drei Aufgaben: Zeige, dass das Volumen der Flüssigkeit in den ersten 4 Stunden zunimmt und finde $t$ , wenn $f(t)=7$ .

See Solution

Problem 25074

Ein Behälter hat anfangs 2 Liter. Berechne, wann er 7 Liter erreicht, wenn $f(t)=-t(t-4)$ die Zuflussrate ist.

See Solution

Problem 25075

Given the vector function $\mathbf{F}=3xy\mathbf{i}+2yz\mathbf{j}+xz\mathbf{k}$ , find:
(a) curl $\mathbf{F}$ .
(b) Is $\mathbf{F}$ conservative? Explain.
(c) Calculate work done by $\mathbf{F}$ along curve $C$ from $(1,0,0)$ to $(0,2,0)$ to $(0,0,3)$ to $(-4,0,0)$ .

See Solution

Problem 25076

Sketch the area between $y = \sqrt{x}$ , $x = 4$ , and $x = 9$ . Find the volume when this area is revolved around the x-axis.

See Solution

Problem 25077

Berechnen Sie die Fläche zwischen $g(x) = \frac{1}{2} x + 2$ und der x-Achse von $x=0$ bis $x=4$ .

See Solution

Problem 25078

Find the average rate of change of $f(x)=2 x^{2}+8$ from $x=2$ to $x=6$ . A. 2 B. 4 C. 12 D. 16

See Solution

Problem 25079

Calculez $f^{\prime}(x)$ pour $f(x)=e^{x+\ln x}+\ln \left(x+e^{x}\right)+\pi^{-2 x^{2}}$ .

See Solution

Problem 25080

Find values of $a$ and $b$ for the function $f(x)=\left\{\begin{array}{ll}a x & \text { if } x \leq 1 \\ b x^{2}+x+1 & \text { if } x>1\end{array}\right.$ to ensure continuity at $x=1$ . Express $a$ in terms of $b$ .

See Solution

Problem 25081

Bestimme den Betrag der Elastizität von $f(x)=(4 x^{4}+7)^{2}$ bei $x^{*}=1.8$ und runde auf zwei Dezimalstellen!

See Solution

Problem 25082

Find the derivative of $f(x) = 3\left(x^{2} + 5\right)^{4}$ .

See Solution

Problem 25083

Find the tangent line to $f(x)=x^{2}-2x+3$ that is parallel to $4x-y+1=0$ .

See Solution

Problem 25084

Find $a$ and $b$ for the function $f(x)=\begin{cases}a x & x \leq 1 \\ b x^{2}+x+1 & x>1\end{cases}$ to be continuous and differentiable at $x=1$ .
Continuity: $a=b+2$ ; Differentiability: $a=\text{Number}$ .

See Solution

Problem 25085

Die Elastizität von $P(x)$ bei $x_0 = 63$ beträgt 0.27. Welche Aussage ist korrekt?
1. Bei Verdopplung von $x$ steigt $P$ auf $1.2058$ .
2. Bei $2\%$ Rückgang von $P$ sinkt $x$ um $0.54\%$ .
3. Steigt $x$ auf $65.52$ , steigt $P$ um $1.08\%$ .
4. Sinkt $x$ auf $62$ , sinkt $P$ um $27\%$ .

See Solution

Problem 25086

Ableitungsregeln und Differentiation: 1. Nennen Sie Ableitungsregeln. 2. Leiten Sie folgende Funktionen ab. 3. Finden Sie Fehler in den Ableitungen. 4. Für $f(x)=(x-6) \cdot \sqrt{x}$ : a) Nullstellen, b) Ableitung, c) Extremum, d) Steigungswinkel bei $x=6$ , e) Graph skizzieren. 5. Bestimmen Sie die Anleitung von $f$ .

See Solution

Problem 25087

Find the average rate of change of $f(x)=5 x^{2}-2 x+6$ between $x=2$ and $x=4$ .

See Solution

Problem 25088

Given the position function $s(t)=2 t^{4}-16 t^{3}-64 t^{2}+64 t$ , find: a) $v(t)$ , b) when $v(t)=64$ , c) when $a(t)=0$ .

See Solution

Problem 25089

Find the derivative of the Phillips curve $y=9.638 x^{-1.394}-0.900$ at $x=2\%$ and $x=7\%$ . Round to two decimal places. Interpret results.

See Solution

Problem 25090

Soit $f(x)=\frac{9 x^{2}}{(x-4)^{2}}$ . Trouvez le domaine, les asymptotes, et les nombres critiques de $f$ .

See Solution

Problem 25091

Find the integral of $\cos^3\left(\frac{1}{4} x\right)$ with respect to $x$ .

See Solution

Problem 25092

Differentiate the function $y=\frac{x^{\frac{3}{4}} \sqrt{x^{2}-9}}{(3 x+5)^{5}}$ using logarithmic differentiation.

See Solution

Problem 25093

A company’s cost function is $C(x)=7x+80$ . Find the average cost $AC(x)$ , marginal average cost $MAC(x)$ , and $MAC(22)$ .

See Solution

Problem 25094

Find the derivative of $y=\left(x^{5}+2\right)^{2}\left(x^{3}+4\right)^{5}$ using logarithmic differentiation.

See Solution

Problem 25095

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{1}{3} x^{3}-x^{4}$ and evaluate it at $x=-2$ .

See Solution

Problem 25096

Gegeben ist $f(x)=(x-6) \cdot \sqrt{x}, x \geq 0$ . Finde Nullstellen, Ableitung, Extremum, Steigungswinkel bei $x=6$ und skizziere den Graphen.

See Solution

Problem 25097

Vereinfachen Sie den Integranden und berechnen Sie das Integral:
$\int_{1}^{3}\left(4 x^{3} \cdot \frac{0,25}{x^{2}}\right) d x, \quad \int_{8}^{64}\left(\frac{1}{x^{\frac{2}{3}}}\right) d x, \quad \int_{1}^{2}\left(\frac{5}{4 x^{4}}-0,25 x^{-4}\right) d x$

See Solution

Problem 25098

Find the curvature $K$ of the curve $y=5x^{2}+7$ at $x=-1$ .

See Solution

Problem 25099

Soit $f(x)=\frac{9 x^{2}}{(x-4)^{2}}$ .
(a) Déterminez le domaine de $f$ . (b) Trouvez les asymptotes (verticales et horizontales). (c) Montrez que $f^{\prime}(x)=\frac{-72 x}{(x-4)^{3}}$ et trouvez les nombres critiques. (d) Montrez que $f^{\prime \prime}(x)=\frac{144(x+2)}{(x-4)^{4}}$ et trouvez les nombres critiques de $f^{\prime}$ .

See Solution

Problem 25100

Find values of $a$ and $b$ so that the function $f$ is differentiable at $x=2$ where:
$f(x) = \begin{cases} b x^{2} + a x - 3a & \text{if } x < 2 \\ x^{3} + a x^{2} + 8 & \text{if } x \geq 2 \end{cases}$

See Solution

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Calculus

Problem 25001

Estimate e−0.01e^{-0.01}e−0.01 using linear approximation or differentials.

Problem 25002

Find the linearization L(x)L(x)L(x) of f(x)=x4+4x2f(x)=x^{4}+4x^{2}f(x)=x4+4x2 at a=1a=1a=1. What is L(x)L(x)L(x)?

Problem 25003

Find the linear approximation of g(x)=1+x3g(x)=\sqrt[3]{1+x}g(x)=31+x​ at a=0a=0a=0. Use it to estimate 0.953\sqrt[3]{0.95}30.95​ and 1.13\sqrt[3]{1.1}31.1​ (3 decimal places).

Problem 25004

If nicotine constricts artery diameter by 11%, what pressure difference (in mm Hg) is needed for blood flow if normal is 9.1 mm Hg?

Problem 25005

Find the linear approximation of f(x)=4−xf(x)=\sqrt{4-x}f(x)=4−x​ at a=0a=0a=0. Use L(x)L(x)L(x) to estimate 3.9\sqrt{3.9}3.9​ and 3.99\sqrt{3.99}3.99​.

Problem 25006

Find the differential of each function: (a) y=x2sin⁡6xy=x^{2} \sin 6 xy=x2sin6x, (b) y=ln⁡5+t2y=\ln \sqrt{5+t^{2}}y=ln5+t2​.

Problem 25007

Find the differential of the functions: (a) y=etan⁡πty=e^{\tan \pi t}y=etanπt; (b) y=3+ln⁡zy=\sqrt{3+\ln z}y=3+lnz​.

Problem 25008

Find the linear approximation of f(x)=4−xf(x)=\sqrt{4-x}f(x)=4−x​ at a=0a=0a=0: L(x)=2−14xL(x)=2-\frac{1}{4} xL(x)=2−41​x. Use L(x)L(x)L(x) to estimate 3.9\sqrt{3.9}3.9​ and 3.99\sqrt{3.99}3.99​.

Problem 25009

Find the differential dyd ydy for y=tan⁡xy=\tan xy=tanx. Then evaluate dyd ydy at x=π/3x=\pi/3x=π/3 and dx=−0.05d x=-0.05dx=−0.05.

Problem 25010

Find the value of ccc guaranteed by the Mean Value Theorem for q(x)=4x2+5x−10q(x) = 4x^2 + 5x - 10q(x)=4x2+5x−10 on [−4,1][-4,1][−4,1].

Problem 25011

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=1x2+x+33f(x)=\frac{1}{\sqrt[3]{x^{2}+x+3}}f(x)=3x2+x+3​1​.

Problem 25012

Find the first and second derivatives of y=sin⁡(x2)y=\sin(x^2)y=sin(x2). What are y′y'y′ and y′′y''y′′?

Problem 25013

Find the derivative of the function F(x)=(x4+9x2−7)3F(x)=(x^{4}+9 x^{2}-7)^{3}F(x)=(x4+9x2−7)3. What is F′(x)F^{\prime}(x)F′(x)?

Problem 25014

Find the derivative of the function y=(x2+3x2−3)3y=\left(\frac{x^{2}+3}{x^{2}-3}\right)^{3}y=(x2−3x2+3​)3. What is y′y^{\prime}y′?

Problem 25015

Find the derivative of f(x)=(2x−2)4(x2+x+1)5f(x)=(2x-2)^{4}(x^{2}+x+1)^{5}f(x)=(2x−2)4(x2+x+1)5. What is f′(x)f'(x)f′(x)?

Problem 25016

Find F′(2)F^{\prime}(2)F′(2) for F(x)=f(g(x))F(x)=f(g(x))F(x)=f(g(x)) with given values: f(3)=8f(3)=8f(3)=8, f′(3)=9f^{\prime}(3)=9f′(3)=9, g(2)=3g(2)=3g(2)=3, g′(2)=5g^{\prime}(2)=5g′(2)=5.

Problem 25017

Plot f(x)=e−x2f(x) = e^{-x^2}f(x)=e−x2 on [0,1][0,1][0,1] and solve f(x)=(average value)f(x) = \text{(average value)}f(x)=(average value) using n=500n=500n=500 partitions.

Problem 25018

Find the derivative of f(x)=(2x−2)4(x2+x+1)5f(x)=(2x-2)^{4}(x^{2}+x+1)^{5}f(x)=(2x−2)4(x2+x+1)5. What is f′(x)f^{\prime}(x)f′(x)?

Problem 25019

Find the derivative of the function y=cos⁡(a3+x3)y=\cos(a^{3}+x^{3})y=cos(a3+x3). What is y′(x)y'(x)y′(x)?

Problem 25020

Find the derivative of y=cot⁡2(cos⁡θ)y=\cot^{2}(\cos \theta)y=cot2(cosθ). What is y′y'y′?

Problem 25021

Find the derivative of the function y=xsin⁡2xy=x \sin \frac{2}{x}y=xsinx2​. What is y′(x)y'(x)y′(x)?

Problem 25022

Problem 25023

Differentiate y=2x5−cot⁡xy=\frac{2x}{5-\cot x}y=5−cotx2x​ and find y′=□y'=\squarey′=□.

Problem 25024

Find the limit: lim⁡x→0cos⁡(7x)−cos⁡(8x)x2\lim _{x \rightarrow 0} \frac{\cos (7 x)-\cos (8 x)}{x^{2}}x→0lim​x2cos(7x)−cos(8x)​ using I'Hospital's Rule.

Problem 25025

Differentiate F(y)=(1y2−7y4)(y+3y3)F(y)=\left(\frac{1}{y^{2}}-\frac{7}{y^{4}}\right)\left(y+3 y^{3}\right)F(y)=(y21​−y47​)(y+3y3).

Problem 25026

Find the second derivative f′′(3)f^{\prime \prime}(3)f′′(3) for the function f(x)=x23+xf(x)=\frac{x^{2}}{3+x}f(x)=3+xx2​.

Problem 25027

Differentiate y=7x2sin⁡xtan⁡x y=7 x^{2} \sin x \tan x y=7x2sinxtanx using the Product Rule. What is y′ y' y′?

Problem 25028

Differentiate y=2x5−cot⁡x y = \frac{2x}{5 - \cot x} y=5−cotx2x​ and find y′ y' y′.

Problem 25029

Find the tangent line equation for the curve y=x2−1x2+x+1y=\frac{x^{2}-1}{x^{2}+x+1}y=x2+x+1x2−1​ at the point (1,0).

Problem 25030

Evaluate the Riemann sum for f(x)=3x2−2xf(x)=3x^{2}-2xf(x)=3x2−2x from 000 to 333 with n=6n=6n=6 using right endpoints.

Problem 25031

Given f(5)=1,f′(5)=8,g(5)=−6f(5)=1, f^{\prime}(5)=8, g(5)=-6f(5)=1,f′(5)=8,g(5)=−6, and g′(5)=7g^{\prime}(5)=7g′(5)=7, find: (a) (fg)′(5)(f g)^{\prime}(5)(fg)′(5), (b) (f/g)′(5)(f / g)^{\prime}(5)(f/g)′(5), (c) (g/f)′(5)(g / f)^{\prime}(5)(g/f)′(5).

Problem 25032

Find the limits of the sequences an=(qn+5n+11n)1/na_n = (q^n + 5^n + 11^n)^{1/n}an​=(qn+5n+11n)1/n, bn=qn+n2nb_n = \sqrt[n]{q^n + n^2}bn​=nqn+n2​, and cn=sin⁡(qnπ)nc_n = \frac{\sin(q^n \pi)}{n}cn​=nsin(qnπ)​.

Problem 25033

Funktion f beschreibt tägliche Übernachtungen (in 1000) ab 2020. Analysiere die Aussagen (1)-(6) für f(t)=0,15⋅(t−15)⋅e−0,1t+8f(t)=0,15 \cdot(t-15) \cdot e^{-0,1 t}+8f(t)=0,15⋅(t−15)⋅e−0,1t+8.

Problem 25034

Calculate the integral from 1 to 4 of the function x2−4x+2x^{2}-4 x+2x2−4x+2.

Problem 25035

Find the general antiderivative of f(x)=5−5x−x3xf(x)=\frac{5-5x-x^{3}}{\sqrt{x}}f(x)=x​5−5x−x3​. Use xn=xn+1n+1x^{n} = \frac{x^{n+1}}{n+1}xn=n+1xn+1​.

Problem 25036

Calculate the integral ∫(cos⁡(x)sin⁡3(x))dx\int\left(\cos (x) \sin ^{3}(x)\right) d x∫(cos(x)sin3(x))dx.

Problem 25037

Find the integral of the function: ∫(8x3+1x2+x)dx\int\left(8 x^{3}+\frac{1}{x^{2}}+\sqrt{x}\right) d x∫(8x3+x21​+x​)dx.

Problem 25038

Bestimme das Maximum der Erkrankten e(t)=−1400t2(t−48)e(t)=-\frac{1}{400} t^{2}(t-48)e(t)=−4001​t2(t−48) und die maximale Wachstumsrate. Wann endet die Epidemie?

Problem 25039

Evaluate the integral: ∫14(1+x)1/2xdx\int_{1}^{4} \frac{(1+\sqrt{x})^{1 / 2}}{\sqrt{x}} d x∫14​x​(1+x​)1/2​dx

Problem 25040

Find the limit: lim⁡x→∞(xsin⁡(πx))\lim _{x \rightarrow \infty}\left(x \sin \left(\frac{\pi}{x}\right)\right)limx→∞​(xsin(xπ​)).

Estimate $e^{-0.01}$ using linear approximation or differentials.

Find the linearization $L(x)$ of $f(x)=x^{4}+4x^{2}$ at $a=1$ . What is $L(x)$ ?

Find the linear approximation of $g(x)=\sqrt[3]{1+x}$ at $a=0$ . Use it to estimate $\sqrt[3]{0.95}$ and $\sqrt[3]{1.1}$ (3 decimal places).

Find the linear approximation of $f(x)=\sqrt{4-x}$ at $a=0$ . Use $L(x)$ to estimate $\sqrt{3.9}$ and $\sqrt{3.99}$ .

Find the differential of each function: (a) $y=x^{2} \sin 6 x$ , (b) $y=\ln \sqrt{5+t^{2}}$ .

Find the differential of the functions: (a) $y=e^{\tan \pi t}$ ; (b) $y=\sqrt{3+\ln z}$ .

Find the linear approximation of $f(x)=\sqrt{4-x}$ at $a=0$ : $L(x)=2-\frac{1}{4} x$ . Use $L(x)$ to estimate $\sqrt{3.9}$ and $\sqrt{3.99}$ .

Find the differential $d y$ for $y=\tan x$ . Then evaluate $d y$ at $x=\pi/3$ and $d x=-0.05$ .

Find the value of $c$ guaranteed by the Mean Value Theorem for $q(x) = 4x^2 + 5x - 10$ on $[-4,1]$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{1}{\sqrt[3]{x^{2}+x+3}}$ .

Find the first and second derivatives of $y=\sin(x^2)$ . What are $y'$ and $y''$ ?

Find the derivative of the function $F(x)=(x^{4}+9 x^{2}-7)^{3}$ . What is $F^{\prime}(x)$ ?

Find the derivative of the function $y=\left(\frac{x^{2}+3}{x^{2}-3}\right)^{3}$ . What is $y^{\prime}$ ?

Find the derivative of $f(x)=(2x-2)^{4}(x^{2}+x+1)^{5}$ . What is $f'(x)$ ?

Find $F^{\prime}(2)$ for $F(x)=f(g(x))$ with given values: $f(3)=8$ , $f^{\prime}(3)=9$ , $g(2)=3$ , $g^{\prime}(2)=5$ .

Plot $f(x) = e^{-x^2}$ on $[0,1]$ and solve $f(x) = \text{(average value)}$ using $n=500$ partitions.

Find the derivative of $f(x)=(2x-2)^{4}(x^{2}+x+1)^{5}$ . What is $f^{\prime}(x)$ ?

Find the derivative of the function $y=\cos(a^{3}+x^{3})$ . What is $y'(x)$ ?

Find the derivative of $y=\cot^{2}(\cos \theta)$ . What is $y'$ ?

Find the derivative of the function $y=x \sin \frac{2}{x}$ . What is $y'(x)$ ?

Differentiate $y=\frac{2x}{5-\cot x}$ and find $y'=\square$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\cos (7 x)-\cos (8 x)}{x^{2}}$ using I'Hospital's Rule.

Differentiate $F(y)=\left(\frac{1}{y^{2}}-\frac{7}{y^{4}}\right)\left(y+3 y^{3}\right)$ .

Find the second derivative $f^{\prime \prime}(3)$ for the function $f(x)=\frac{x^{2}}{3+x}$ .

Differentiate $y=7 x^{2} \sin x \tan x$ using the Product Rule. What is $y'$ ?

Differentiate $y = \frac{2x}{5 - \cot x}$ and find $y'$ .

Find the tangent line equation for the curve $y=\frac{x^{2}-1}{x^{2}+x+1}$ at the point (1,0).

Evaluate the Riemann sum for $f(x)=3x^{2}-2x$ from $0$ to $3$ with $n=6$ using right endpoints.

Given $f(5)=1, f^{\prime}(5)=8, g(5)=-6$ , and $g^{\prime}(5)=7$ , find: (a) $(f g)^{\prime}(5)$ , (b) $(f / g)^{\prime}(5)$ , (c) $(g / f)^{\prime}(5)$ .

Find the limits of the sequences $a_n = (q^n + 5^n + 11^n)^{1/n}$ , $b_n = \sqrt[n]{q^n + n^2}$ , and $c_n = \frac{\sin(q^n \pi)}{n}$ .

Funktion f beschreibt tägliche Übernachtungen (in 1000) ab 2020. Analysiere die Aussagen (1)-(6) für $f(t)=0,15 \cdot(t-15) \cdot e^{-0,1 t}+8$ .

Calculate the integral from 1 to 4 of the function $x^{2}-4 x+2$ .

Find the general antiderivative of $f(x)=\frac{5-5x-x^{3}}{\sqrt{x}}$ . Use $x^{n} = \frac{x^{n+1}}{n+1}$ .

Calculate the integral $\int\left(\cos (x) \sin ^{3}(x)\right) d x$ .

Find the integral of the function: $\int\left(8 x^{3}+\frac{1}{x^{2}}+\sqrt{x}\right) d x$ .

Bestimme das Maximum der Erkrankten $e(t)=-\frac{1}{400} t^{2}(t-48)$ und die maximale Wachstumsrate. Wann endet die Epidemie?

Evaluate the integral: $\int_{1}^{4} \frac{(1+\sqrt{x})^{1 / 2}}{\sqrt{x}} d x$

Find the limit: $\lim _{x \rightarrow \infty}\left(x \sin \left(\frac{\pi}{x}\right)\right)$ .

Find the area between the curves $y = x^2 - 4$ and $y = -2x^2 + 3x + 2$ .

Evaluate the integral $I=\int_{0}^{12} f(t) dt=-\int_{0}^{12} t(t-21)(t+1) dt$ .

A projectile is fired at $v_0 = 150\ \text{m/s}$ and $\theta = 47^\circ$ . Find the max height $H$ with $g = 9.81\ \text{m/s}^2$ .

Find the volume of the solid formed by $y=\sqrt{x}$ , $x=4$ , $x=9$ , and the $x$ axis, when rotated about the $x$ axis.

Find when the velocity of the object given by $g(t) = \frac{2}{3} t^3 - \frac{5}{2} t^2 + 3t + 7$ is decreasing.

Calculate the integral from 1 to e: $\int_{1}^{e} \ln x \, dx$ .

Find the integral of the function: $\int \left(8x^3 + \frac{1}{x^2} + \sqrt{x}\right) dx$ .

Find where $f(x)$ is concave up and down, and determine the points of inflection for $f(x)=x^{4}-6 x^{3}$ .

Find the limit: $\lim _{x \rightarrow \infty}\left(x \sin \left(\frac{\pi}{x}\right)\right)$ .

Find the derivative of $f(x) = (x^{2} - 3x)^{5}$ .

Find $f^{\prime}(x)$ for: a) $f(x)=(x^{2}-3x)^{5}$ b) $f(x)=\sqrt{x} \tan^{2}(x)$ .

Find the derivative of the function $f(x) = \sqrt{x} \tan^2(x)$ .

Find the integral: $\int 3 \csc (7 x) \cot (7 x) \, dx$

Evaluate the integral using substitution: $\int_{0}^{\sqrt{\pi}} 2 x \sin(5 x^{2}) \, dx$ .

Find the average rate of change of power output from $x=4$ hours (20 MW) to $x=12$ hours (80 MW).

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

Zeichne ein Flächenstück unter $f(x)=2 e^{-\frac{1}{2} x}$ , das Fläche 1,5 hat, und gib den Berechnungsterm an.

Evaluate these integrals: a. $\int(8 x^{3}+\frac{1}{x^{2}}+\sqrt{x}) dx$ b. $\int(\cos(x) \sin^{3}(x)) dx$ c. $\int_{1}^{4} \frac{(1+\sqrt{x})^{1/2}}{\sqrt{x}} dx$

Calculate the integral of $(5 x-3)^{2}$ with respect to $x$ .

Find the integral of $(2 x+6)^{8}$ with respect to $x$ .

Find the intervals where the function $f$ is decreasing given that $f^{\prime}$ has two zeros and selected values of $f^{\prime}(x)$ .

Find the integral of $\cos (\pi \theta)$ with respect to $\theta$ .

Calculate the integral: $\int_{-2}^{1} \sqrt{6-5 x} \, dx$

Find the fluid force on a semicircular plate (radius $14 \mathrm{ft}$ ) submerged in water (depth $15 \mathrm{ft}$ ).

Find the derivative of the function $f(x) = 2 + \left(\frac{-1}{\sqrt{1-x^{2}}}\right)$ . What is $f'(x)$ ?

Berechne die Elastizität von $y=3 x \exp (4-2 x)$ für $x=0.01$ und $x=4.39$ . Ergebnisse als Dezimalzahlen, gerundet auf zwei Nachkommastellen.

Consider the vector field $\mathbf{F}=4 y \tan x \sec ^{2} x \mathbf{i}+\left(2 \tan ^{2} x+4 y\right) \mathbf{j}$ .
(a) Prove $\mathbf{F}$ is conservative. (b) Determine the potential function $f$ .

Bestimme die jährliche Fahrstrecke $x$ (in $1000 \mathrm{~km}$ ), um die Durchschnittskosten minimal zu halten. Kostenformel: $K = 1302 + 120x + 32x \ln(x) + 12x^2$ .

Find the gradient $\mathbf{F}=\nabla f$ for the function $f(x, y, z)=z e^{x+y}+z e^{x-z}$ .

Finde das Minimum der Funktion $f(x)=6 x^{2}+12 x+3$ . Ergebnisse als Dezimalzahl (4 Nachkommastellen) angeben: $x_{\text{min}}=$ , $y_{\text{min}}=$ . Bestimme $f^{\prime \prime}(x_{\min })=$ .

Bearbeite eine der drei Aufgaben: Zeige, dass das Volumen der Flüssigkeit in den ersten 4 Stunden zunimmt und finde $t$ , wenn $f(t)=7$ .

Ein Behälter hat anfangs 2 Liter. Berechne, wann er 7 Liter erreicht, wenn $f(t)=-t(t-4)$ die Zuflussrate ist.