Calculus

Problem 26601

Find the derivative of $f(x) = \ln \left( \frac{1}{x^{2}} \right)$ and explain each step.

See Solution

Problem 26602

Find the limit: $\lim _{x \rightarrow \pm \infty} \frac{3-11 x^{3}}{10 x^{3}}$

See Solution

Problem 26603

Logistic growth function $f(t)=\frac{103,000}{1+5700 e^{-t}}$ describes flu cases over time.
a. Initial cases? b. Cases after 4 weeks? c. Maximum possible cases?

See Solution

Problem 26604

Find the steepest tangent slope of $f(x)=\sqrt{x+2}$ on $[0,1]$ , the secant line equation, and compare slopes.

See Solution

Problem 26605

Find the derivative of $f(x) = \ln \sqrt{2 + \cos^2 x}$ and explain each step.

See Solution

Problem 26606

Find the derivatives of these functions: $y=\frac{6}{\sqrt[4]{x}}$ , $y=\frac{-2}{\sqrt[3]{x}}$ , $f(x)=\frac{x^{3}+5}{x}$ , $g(x)=\frac{x^{3}-4 x}{\sqrt{x}}$ , $g(x)=(8 x^{2}-4 x)^{2}$ .

See Solution

Problem 26607

Find the tangent line equation to $y=(3 x^{-2}-2 x^{3})^{5}$ at the point $(1,1)$ .

See Solution

Problem 26608

Differentiate $h(x)=(x^{2}+3)^{4}(4x-5)^{3}$ .

See Solution

Problem 26609

Find the arc length of $y=\frac{x^{3}}{6}+\frac{1}{2 x}$ from $x=\frac{1}{2}$ to $x=2$ .

See Solution

Problem 26610

Differentiate the function $\sqrt{8-\frac{1}{x^{3}}}$ with respect to $x$ .

See Solution

Problem 26611

Find the derivative of $f(x) = e^x(\sin x + \cos x)$ and explain each step.

See Solution

Problem 26612

Bestimme die Punkte, an denen die Tangente von $\mathrm{f}$ parallel zur Linie $y=3x+4$ ist für: a) $f(x)=x^{3}$ , b) $f(x)=x^{2}$ , c) $f(x)=x^{\frac{3}{2}}$ , d) $f(x)=x^{6}$ .

See Solution

Problem 26613

Find $c$ where $f_{a v e}=f(c)$ and calculate the arc length of $y=\frac{x^{3}}{6}+\frac{1}{2 x}$ from $x=\frac{1}{2}$ to $x=2$ .

See Solution

Problem 26614

Find $k$ so that the line $y = -\frac{3}{4}x + 3$ is tangent to $f(x) = \frac{k}{x}$ .

See Solution

Problem 26615

Differenzieren Sie die folgenden Funktionen mit der Quotientenregel: a) $f(x)=\frac{x}{x-1}$ , b) $f(x)=\frac{1+2 x}{1+x}$ , c) $f(x)=\frac{1-5 x}{1+10 x}$ , d) $f(x)=\frac{x+1}{x^{2}}$ , e) $f(x)=\frac{a \sqrt{x}}{x}$ , f) $f(x)=\frac{a+b x}{a-b x}$ .

See Solution

Problem 26616

Soit $u_{0}=0$ et $u_{n+1}=\frac{2 u_{n}+3}{u_{n}+4}$ . Étudiez $f(x)=\frac{2x+3}{x+4}$ sur $I=[0;1]$ , montrez que $u_n \in [0;1]$ et que $\left(u_n\right)$ est croissante, puis calculez sa limite.

See Solution

Problem 26617

Differenzieren Sie die folgenden Funktionen mit der Produktregel und überprüfen Sie die Ergebnisse:
a) $f(x)=x(1+x^{2})$ b) $f(x)=\sqrt{x} \cdot \sqrt{x}$ c) $f(x)=(x^{2}-1)(2 x^{2}+5)$ d) $f(x)=a x(a x^{2}+b)$ e) $f(x)=(x^{2}+1) \cdot \frac{1}{x}$ f) $f(x)=\sqrt{x} \cdot x$

See Solution

Problem 26618

Ein Handwerker trinkt giftige Flüssigkeit. Konzentration im Blut: $2 \mu \mathrm{g} / \mathrm{dl}$ und $3 \mu \mathrm{g} / \mathrm{dl}$ . Bestimmen Sie $a$ und $b$ aus $h(t)=(a t+b) \cdot e^{-0,1 t}$ . Berechnen Sie die Maximalkonzentration und prüfen Sie die Gefahrenzone ab $6 \mu \mathrm{g} / \mathrm{dl}$ .

See Solution

Problem 26619

Find the derivative using the product rule for these functions: 1. $y=(3x^2+2)(2x-1)$ , 2. $y=(5x^2-1)(4x+3)$ , 3. $y=(2x-5)^2$ , 4. $y=(7x-6)^2$ , 5. $k(t)=(r^2-1)^2$ , 6. $g(t)=(3t^2+2)^2$ , 7. $y=(x+1)(\sqrt{x}+2)$ , 8. $y=(2x-3)(\sqrt{x}-1)$ , 9. $p(y)=(y^{-1}+y^{-2})(2y^{-3}-5y^{-4})$ , 10. $q(x)=(x^{-2}-x^{-3})(3x^{-1}+4x^{-4})$ .

See Solution

Problem 26620

Find the integrals: (a) $\int \tan ^{2} x \mathrm{~d} x$ , (b) $\int \frac{1}{x^{3}} \ln x \mathrm{~d} x$ , (c) show $\int \frac{\mathrm{e}^{3 x}}{1+\mathrm{e}^{x}} \mathrm{~d} x=\frac{1}{2} \mathrm{e}^{2 x}-\mathrm{e}^{x}+\ln(1+\mathrm{e}^{x})+k$ .

See Solution

Problem 26621

Für welche Funktionen $f$ gilt $f = f'$ ? a) $f(x)=-3 e^{x}$ b) $f(x)=2 e^{x+3}$ c) $f(x)=1,5 e^{x+3}-2 e^{x-2}$ d) $f(x)=2 e^{-x}$

See Solution

Problem 26622

Berechnen Sie $f^{\prime}(x)$ für $f(x)=\frac{6}{x}+3 \sqrt{x}-a x^{2}$ oder $f(x)=-\frac{3}{8} x^{-4}+a b^{2} x-\pi^{2}$ .

See Solution

Problem 26623

Berechne die Steigung der Funktion $f(x)=2 x^{2}+1$ bei $x_{0}=3$ mit der h-Methode oder der Ableitungsmethode.

See Solution

Problem 26624

Bestimme die Ableitungsfunktion von $f(x)=x^{15}$ . Fasse zuerst mit Potenzgesetzen zusammen.

See Solution

Problem 26625

Berechnen Sie den Flächeninhalt $A$ zwischen den Funktionen $f(x)=x^{2}+2$ und $g(x)=x+1$ im Intervall $[-1 ; 2]$ .

See Solution

Problem 26626

Find critical points of the function $f(x)=\frac{x^{5}-4 x^{4}+10 x}{x}$ .

See Solution

Problem 26627

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{|1+\Delta x-1|-|1-1|}{\Delta x}$ .

See Solution

Problem 26628

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{f(2+\Delta x)-f(2)}{\Delta x}$ for $f(x)=\frac{-4}{5-x}$ .

See Solution

Problem 26629

Gegeben ist die Funktion $f(x) = -\frac{1}{2} x^{3} + \frac{9}{2} x$ . Finde Nullstellen, Symmetrie, Grenzverhalten und skizziere den Graphen.

See Solution

Problem 26630

Find $\lim _{x \rightarrow c}(f(x) g(x))$ given $\lim _{x \rightarrow c}(f(x)+g(x))=3$ and $\lim _{x \rightarrow c}(f(x)-g(x))=-1$ .

See Solution

Problem 26631

Bestimme die Ableitung von $f(x)=(x^{-3})^{-5}$ und fasse sie zuerst zu einer Potenz zusammen.

See Solution

Problem 26632

If $f$ is odd and $f^{\prime}(c)=3$ , what is $f^{\prime}(-c)$ ? A) -3 B) $\frac{1}{3}$ C) 3 D) $\frac{-1}{3}$

See Solution

Problem 26633

Given the function $f(x) = \frac{-4}{5-x}$ , find its derivative and the limit as $x$ approaches 2.

See Solution

Problem 26634

Find the tangent gradient of $y=2 x^{3}+15 x^{2}-84 x$ at $x=-1$ and identify stationary points.

See Solution

Problem 26635

Bestimme die Tangentengleichung von $f(x)=x^{6}$ bei $x_{0}=2$ .

See Solution

Problem 26636

Bestimme die Ableitung der Funktionen: a) $f(x)=(3 x-2)^{3}$ b) $f(x)=\sqrt{24-x}$ c) $f(x)=(0,5 x^{2}-5 x)^{4}$ d) $f(x)=\sqrt{x^{2}-2 x}$ e) $f(x)=\frac{1}{x^{3}-2}$ f) $f(x)=\frac{36}{2 x^{2}-1}$

See Solution

Problem 26637

Find $h^{\prime}(-1)$ if $h(x)=f(x)+3g(x)$ , given $f^{\prime}(-1)=2g^{\prime}(-1)$ and $\lim_{x \to 1} \frac{f(x)-f(-1)}{x+1}=-6$ . Options: A) -15 B) -2 C) -18 D) 0

See Solution

Problem 26638

Untersuchen Sie die Funktion $f(x)=x^{3}-1$ : a) Zeichnen Sie $f(x)$ für $-2 \leq x \leq 2$ . b) Bestimmen Sie die Steigung bei $x=2$ . c) Geben Sie die Ableitung an. d) Berechnen Sie die Tangentengleichung bei $x=2$ und diskutieren Sie den Anstieg.

See Solution

Problem 26639

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{f(2+\Delta x)-f(2)}{\Delta x}$ for $f(x)=\frac{-4}{5-x}$ .

See Solution

Problem 26640

Determine the truth about $f(x)=\frac{|x+2|}{x+2}$ at $x=-2$ : A) vertical asymptote B) differentiable C) continuous D) discontinuous.

See Solution

Problem 26641

Berechne das Integral $\int_{3}^{6} 2 x \, dx$ und finde den Wert.

See Solution

Problem 26642

Find the limit: $\lim _{x \rightarrow \infty}\left(\frac{2 x^{2}+1}{3 x^{2}+1}\right)^{x-x^{2}}$ .

See Solution

Problem 26643

Finde die Stammfunktion von $f(x)=-6x+7$ .

See Solution

Problem 26644

Finde eine Stammfunktion von $f(x)=\frac{2}{3} x^{2}$ .

See Solution

Problem 26645

Calculate the integral $\int_{-1}^{5} 2x^{2} \, dx$ using the evaluation $\left[\frac{2}{3}x^{3}\right]_{-1}^{5}$ .

See Solution

Problem 26646

Bestimmen Sie die Tangente an den Graphen von $f$ in $P_{0}(x_{0} \mid f(x_{0}))$ für die folgenden Funktionen und Punkte: a) $f(x)=x^{4}, x_{0}=0,5$ b) $f(x)=2 x^{-2}, x_{0}=3$ c) $f(x)=2 x^{3}-3 x^{-2}, x_{0}=2$ d) $f(x)=\frac{1}{x}-x^{\frac{3}{2}}, x_{0}=5$

See Solution

Problem 26647

A roller coaster is launched to a height of $23 \mathrm{~m}$ using a spring.
a. Find the spring constant if compressed to $5.0 \mathrm{~m}$ .
b. Calculate speed at $5.0 \mathrm{~m}$ above the launcher.
c. Determine energy lost going down and its destination.
d. What is the period of oscillation with max mass?

See Solution

Problem 26648

Evaluate the integral from 1 to 2 of the function $4 \sin 3 x - \frac{3}{2 x}$ .

See Solution

Problem 26649

A particle moves along the $x$ axis. When is it moving towards the origin based on the given position and velocity data?

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Problem 26650

Calculate the area under the curve $y=\cos(t)$ from $t=0$ to $t=2\pi$ .

See Solution

Problem 26651

How much will you have after investing \ $4500 at$ 3.8\%$ interest compounded continuously for 20 years?

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Problem 26652

A pizza pan cools from $425^{\circ} \mathrm{F}$ to $130^{\circ} \mathrm{F}$ after $5$ minutes at room temperature $72^{\circ} \mathrm{F}$ . When does this happen? Also, find when it reaches $230^{\circ} \mathrm{F}$ and describe the cooling trend.

See Solution

Problem 26653

Find the elasticity function $E(p)$ for the demand $D(p)=200 e^{-0.8 p}$ and calculate it at price \$3.

See Solution

Problem 26654

Evaluate the integral from 1 to 2 of the function $2x + 4$ .

See Solution

Problem 26655

Find the speed of a $1.5 \mathrm{~kg}$ book dropped from 5.0m just before it hits the ground. Use $mgh$ for energy.

See Solution

Problem 26656

What is the acceleration if velocity is 2 m/s at 0s and 22 m/s at 100s? Choose from: -0.4, -0.2, 0.2, 0.4 m/s².

See Solution

Problem 26657

Find the derivative of $f(x)=-3 x^{8} \ln x$ and evaluate $f^{\prime}(e^{3})$ .

See Solution

Problem 26658

Find the derivative $f'(x)$ of the function $f(x)=4 \log_{9}(x)$ and evaluate $f'(5)$ .

See Solution

Problem 26659

Find the derivative of the function $f(t)=3 t^{4}-7 t+2 e^{t}$ , denoted as $f^{\prime}(t)$ .

See Solution

Problem 26660

Find the derivative of the function $f(x)=\ln(x^{2}-18x+86)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 26661

Differentiate the function $f(x)=e^{2 x}$ . Provide $f^{\prime}(x)=$ .

See Solution

Problem 26662

Differentiate the function $G(x)=e^{-8 x}$ without simplifying the derivative. Find $G^{\prime}(x)=$ .

See Solution

Problem 26663

Find the derivative of $f(x)=3 e^{-5 x-5}$ . What is $f^{\prime}(x)=?$

See Solution

Problem 26664

Differentiate the function $h(x)=\frac{e^{-5 x}}{x^{6}}$ without simplifying the derivative. Find $h^{\prime}(x)=$ .

See Solution

Problem 26665

Find the second derivative of the functions $x(t) = a t^{3}$ and $y(t) = b t$ with respect to $t$ .

See Solution

Problem 26666

Differentiate the function $G(x)=e^{x^{7}+11 x}$ without simplifying the derivative. Find $G^{\prime}(x)=$

See Solution

Problem 26667

Differentiate the function $F(x)=-4 e^{x^{2}}$ without simplifying the derivative. Find $F^{\prime}(x)=$ .

See Solution

Problem 26668

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

See Solution

Problem 26669

Find the derivative of $f(x)=\ln \left(\frac{5 x-1}{3 x-7}\right)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 26670

Determine the intervals where the curve defined by $x(t)=2-t^{2}$ and $y(t)=t^{2}+t^{3}$ is concave up or down.

See Solution

Problem 26671

Find the horizontal asymptote, global max/min, domain, intercepts, and critical points of $\frac{x+2}{e^{x}}$ .

See Solution

Problem 26672

Find the derivative of the function $f(x)=\ln(x^{2}+10x+31)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 26673

How long will it take for a pie cooling from $140^{\circ} \mathrm{F}$ to reach $80^{\circ} \mathrm{F}$ using $T(t)=68 e^{-0.0174 t}+72$ ?

See Solution

Problem 26674

Integrate: $\int\left(2 x^{3}+3-\frac{1}{2 x^{3}}\right) dx$

See Solution

Problem 26675

Find the derivative of the function $f(x)=\ln \left(\sqrt{\frac{4 x-2}{7 x+6}}\right)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 26676

Differentiate: $f(x)=\ln \left(\frac{x^{5}+2}{x^{4}}\right)$ to find $f^{\prime}(x)$ .

See Solution

Problem 26677

Find the derivative $f'(x)$ for the function $f(x)=(\ln x)^{4}$ and evaluate $f'(e^{2})$ .

See Solution

Problem 26678

Find the growth rate of infected people after 12 days using $I(t)=3700 e^{0.086 t}$ . What are the units?

See Solution

Problem 26679

Find the growth rate of infected people after 10 days using $I(t)=5200 e^{0.087 t}$ . What are the units?

See Solution

Problem 26680

A marketing model is given by $f(t)=550+255 \log_{7}(t)$ . Find $f^{\prime}(9)$ and interpret $f(9)=838$ and $f^{\prime}(16)=8$ .

See Solution

Problem 26681

Find the elasticity function $E(p)$ for the demand $D(p)=200 e^{-0.8 p}$ and calculate it at price $\$ 3$ .

See Solution

Problem 26682

Find critical points for $f(x)=x^{3}-x^{2}-9x+9$ :
1. Identify local minima/maxima.
2. Find zeros of $f(x)$ .
Explain differences between zeros of $f(x)$ and its derivative.

See Solution

Problem 26683

Find local extrema and zeros of the function $f(x)=x^{3}-x^{2}-9x+9$ . Explain critical points and the difference between zeros of $f$ and its derivative.

See Solution

Problem 26684

Bestimme die Ableitungen für: a) $f(x)=12 \cdot \sin (x)$ , b) $f(x)=-2 \cdot \cos (x)$ , c) $f(x)=\sqrt{5} \cdot \cos (x)$ , d) $f(x)=\frac{1}{\pi} \cdot \sin (x)$ , e) $f(x)=5 x^{3}-\sin (x)$ , f) $f(x)=2 \cos (x)-\sin (x)$ .

See Solution

Problem 26685

An airplane flies at $600 \mathrm{mph}$ at 7 miles altitude. Find the distance change rate from observer when $x=5$ miles.

See Solution

Problem 26686

Find the critical number of the function $f(x)=(3 x+9) e^{2 x}$ .
$x=$

See Solution

Problem 26687

A cylinder's radius decreases at 7 ft/s with a volume of 72 ft³. When height is 2 ft, find the height's rate of change. Use $V=\pi r^{2} h$ . Round to three decimal places.

See Solution

Problem 26688

Find the critical numbers $A$ and $B$ for the function $f(x)=x^{2} e^{14 x}$ over the intervals $(-\infty, A]$ , $[A, B]$ , and $[B, \infty)$ .

See Solution

Problem 26689

Invest \$16,000 at 7.2\% interest. Find: A) Future value after 29 years. B) Rate of change at 6 years. C) Time for \$4,690.31 change. D) Average change from year 29 to 32.

See Solution

Problem 26690

Find the revenue function $R(x)=30 e^{-0.03 x} \cdot x$ . Determine the units sold and price for maximum revenue.

See Solution

Problem 26691

Find $\frac{d y}{d t}$ for $y=x^{3}-2x$ at $x=2$ and $\frac{d x}{d t}=4$ .

See Solution

Problem 26692

Evaluate the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-9}{x+3}$

See Solution

Problem 26693

Find the derivative of the function $f(t)=7 t^{4}-7 t+7 e^{t}$ . What is $f^{\prime}(t)$ ?

See Solution

Problem 26694

Find the tangent line $y=m x+b$ to $f(x)=4 x+10-17 e^{x}$ at $(0,-7)$ ; determine $m$ and $b$ .

See Solution

Problem 26695

Find the radius and interval of convergence for the series $\sum_{n=2}^{\infty} \frac{x^{2 n}}{n(\ln n)^{2}}$ .

See Solution

Problem 26696

Find the derivative $f'(x)$ of $f(x)=4 \ln(3+x)$ and calculate $f'(1)$ .

See Solution

Problem 26697

Find the derivative of the function $f(x)=7 e^{x-6}$ . What is $\frac{d f}{d x}$ ?

See Solution

Problem 26698

Find the absolute maximum and minimum of $f(x)=3 x^{2}-6 x+3$ on the interval $0 \leq x \leq 9$ .

See Solution

Problem 26699

Find the derivative using the product rule of $(-8 x^{6}-2 x^{4})(4 e^{\wedge} x-9)$ .

See Solution

Problem 26700

Find the limit: $\lim _{h \rightarrow 0} \frac{2 \csc \left(\frac{7 \pi}{4}+h\right)-2 \csc \left(\frac{7 \pi}{4}\right)}{h}$

See Solution

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Calculus

Problem 26601

Find the derivative of f(x)=ln⁡(1x2) f(x) = \ln \left( \frac{1}{x^{2}} \right) f(x)=ln(x21​) and explain each step.

Problem 26602

Find the limit: lim⁡x→±∞3−11x310x3\lim _{x \rightarrow \pm \infty} \frac{3-11 x^{3}}{10 x^{3}}limx→±∞​10x33−11x3​

Problem 26603

Logistic growth function f(t)=103,0001+5700e−tf(t)=\frac{103,000}{1+5700 e^{-t}}f(t)=1+5700e−t103,000​ describes flu cases over time. a. Initial cases? b. Cases after 4 weeks? c. Maximum possible cases?

Problem 26604

Find the steepest tangent slope of f(x)=x+2f(x)=\sqrt{x+2}f(x)=x+2​ on [0,1][0,1][0,1], the secant line equation, and compare slopes.

Problem 26605

Find the derivative of f(x)=ln⁡2+cos⁡2x f(x) = \ln \sqrt{2 + \cos^2 x} f(x)=ln2+cos2x​ and explain each step.

Problem 26606

Find the derivatives of these functions: y=6x4y=\frac{6}{\sqrt[4]{x}}y=4x​6​, y=−2x3y=\frac{-2}{\sqrt[3]{x}}y=3x​−2​, f(x)=x3+5xf(x)=\frac{x^{3}+5}{x}f(x)=xx3+5​, g(x)=x3−4xxg(x)=\frac{x^{3}-4 x}{\sqrt{x}}g(x)=x​x3−4x​, g(x)=(8x2−4x)2g(x)=(8 x^{2}-4 x)^{2}g(x)=(8x2−4x)2.

Problem 26607

Find the tangent line equation to y=(3x−2−2x3)5y=(3 x^{-2}-2 x^{3})^{5}y=(3x−2−2x3)5 at the point (1,1)(1,1)(1,1).

Problem 26608

Differentiate h(x)=(x2+3)4(4x−5)3h(x)=(x^{2}+3)^{4}(4x-5)^{3}h(x)=(x2+3)4(4x−5)3.

Problem 26609

Find the arc length of y=x36+12xy=\frac{x^{3}}{6}+\frac{1}{2 x}y=6x3​+2x1​ from x=12x=\frac{1}{2}x=21​ to x=2x=2x=2.

Problem 26610

Differentiate the function 8−1x3\sqrt{8-\frac{1}{x^{3}}}8−x31​​ with respect to xxx.

Problem 26611

Find the derivative of f(x)=ex(sin⁡x+cos⁡x)f(x) = e^x(\sin x + \cos x)f(x)=ex(sinx+cosx) and explain each step.

Problem 26612

Bestimme die Punkte, an denen die Tangente von f\mathrm{f}f parallel zur Linie y=3x+4y=3x+4y=3x+4 ist für: a) f(x)=x3f(x)=x^{3}f(x)=x3, b) f(x)=x2f(x)=x^{2}f(x)=x2, c) f(x)=x32f(x)=x^{\frac{3}{2}}f(x)=x23​, d) f(x)=x6f(x)=x^{6}f(x)=x6.

Problem 26613

Find ccc where fave=f(c)f_{a v e}=f(c)fave​=f(c) and calculate the arc length of y=x36+12xy=\frac{x^{3}}{6}+\frac{1}{2 x}y=6x3​+2x1​ from x=12x=\frac{1}{2}x=21​ to x=2x=2x=2.

Problem 26614

Find kkk so that the line y=−34x+3y = -\frac{3}{4}x + 3y=−43​x+3 is tangent to f(x)=kxf(x) = \frac{k}{x}f(x)=xk​.

Problem 26615

Problem 26616

Problem 26617

Problem 26618

Problem 26619

Problem 26620

Problem 26621

Für welche Funktionen fff gilt f=f′f = f'f=f′? a) f(x)=−3exf(x)=-3 e^{x}f(x)=−3ex b) f(x)=2ex+3f(x)=2 e^{x+3}f(x)=2ex+3 c) f(x)=1,5ex+3−2ex−2f(x)=1,5 e^{x+3}-2 e^{x-2}f(x)=1,5ex+3−2ex−2 d) f(x)=2e−xf(x)=2 e^{-x}f(x)=2e−x

Problem 26622

Berechnen Sie f′(x)f^{\prime}(x)f′(x) für f(x)=6x+3x−ax2f(x)=\frac{6}{x}+3 \sqrt{x}-a x^{2}f(x)=x6​+3x​−ax2 oder f(x)=−38x−4+ab2x−π2f(x)=-\frac{3}{8} x^{-4}+a b^{2} x-\pi^{2}f(x)=−83​x−4+ab2x−π2.

Problem 26623

Berechne die Steigung der Funktion f(x)=2x2+1f(x)=2 x^{2}+1f(x)=2x2+1 bei x0=3x_{0}=3x0​=3 mit der h-Methode oder der Ableitungsmethode.

Problem 26624

Bestimme die Ableitungsfunktion von f(x)=x15f(x)=x^{15}f(x)=x15. Fasse zuerst mit Potenzgesetzen zusammen.

Problem 26625

Berechnen Sie den Flächeninhalt AAA zwischen den Funktionen f(x)=x2+2f(x)=x^{2}+2f(x)=x2+2 und g(x)=x+1g(x)=x+1g(x)=x+1 im Intervall [−1;2][-1 ; 2][−1;2].

Problem 26626

Find critical points of the function f(x)=x5−4x4+10xxf(x)=\frac{x^{5}-4 x^{4}+10 x}{x}f(x)=xx5−4x4+10x​.

Problem 26627

Find the limit: lim⁡Δx→0∣1+Δx−1∣−∣1−1∣Δx\lim _{\Delta x \rightarrow 0} \frac{|1+\Delta x-1|-|1-1|}{\Delta x}limΔx→0​Δx∣1+Δx−1∣−∣1−1∣​.

Problem 26628

Find the limit: lim⁡Δx→0f(2+Δx)−f(2)Δx\lim _{\Delta x \rightarrow 0} \frac{f(2+\Delta x)-f(2)}{\Delta x}limΔx→0​Δxf(2+Δx)−f(2)​ for f(x)=−45−xf(x)=\frac{-4}{5-x}f(x)=5−x−4​.

Problem 26629

Gegeben ist die Funktion f(x)=−12x3+92xf(x) = -\frac{1}{2} x^{3} + \frac{9}{2} xf(x)=−21​x3+29​x. Finde Nullstellen, Symmetrie, Grenzverhalten und skizziere den Graphen.

Problem 26630

Find lim⁡x→c(f(x)g(x))\lim _{x \rightarrow c}(f(x) g(x))limx→c​(f(x)g(x)) given lim⁡x→c(f(x)+g(x))=3\lim _{x \rightarrow c}(f(x)+g(x))=3limx→c​(f(x)+g(x))=3 and lim⁡x→c(f(x)−g(x))=−1\lim _{x \rightarrow c}(f(x)-g(x))=-1limx→c​(f(x)−g(x))=−1.

Problem 26631

Bestimme die Ableitung von f(x)=(x−3)−5f(x)=(x^{-3})^{-5}f(x)=(x−3)−5 und fasse sie zuerst zu einer Potenz zusammen.

Problem 26632

If fff is odd and f′(c)=3f^{\prime}(c)=3f′(c)=3, what is f′(−c)f^{\prime}(-c)f′(−c)? A) -3 B) 13\frac{1}{3}31​ C) 3 D) −13\frac{-1}{3}3−1​

Problem 26633

Given the function f(x)=−45−xf(x) = \frac{-4}{5-x}f(x)=5−x−4​, find its derivative and the limit as xxx approaches 2.

Problem 26634

Find the tangent gradient of y=2x3+15x2−84xy=2 x^{3}+15 x^{2}-84 xy=2x3+15x2−84x at x=−1x=-1x=−1 and identify stationary points.

Problem 26635

Bestimme die Tangentengleichung von f(x)=x6f(x)=x^{6}f(x)=x6 bei x0=2x_{0}=2x0​=2.

Problem 26636

Problem 26637

Problem 26638

Problem 26639

Find the limit: lim⁡Δx→0f(2+Δx)−f(2)Δx\lim _{\Delta x \rightarrow 0} \frac{f(2+\Delta x)-f(2)}{\Delta x}limΔx→0​Δxf(2+Δx)−f(2)​ for f(x)=−45−xf(x)=\frac{-4}{5-x}f(x)=5−x−4​.

Problem 26640

Determine the truth about f(x)=∣x+2∣x+2f(x)=\frac{|x+2|}{x+2}f(x)=x+2∣x+2∣​ at x=−2x=-2x=−2: A) vertical asymptote B) differentiable C) continuous D) discontinuous.

Problem 26641

Berechne das Integral ∫362x dx\int_{3}^{6} 2 x \, dx∫36​2xdx und finde den Wert.

Problem 26642

Find the limit: lim⁡x→∞(2x2+13x2+1)x−x2\lim _{x \rightarrow \infty}\left(\frac{2 x^{2}+1}{3 x^{2}+1}\right)^{x-x^{2}}limx→∞​(3x2+12x2+1​)x−x2.

Problem 26643

Finde die Stammfunktion von f(x)=−6x+7f(x)=-6x+7f(x)=−6x+7.

Problem 26644

Finde eine Stammfunktion von f(x)=23x2f(x)=\frac{2}{3} x^{2}f(x)=32​x2.

Find the derivative of $f(x) = \ln \left( \frac{1}{x^{2}} \right)$ and explain each step.

Find the limit: $\lim _{x \rightarrow \pm \infty} \frac{3-11 x^{3}}{10 x^{3}}$

Logistic growth function $f(t)=\frac{103,000}{1+5700 e^{-t}}$ describes flu cases over time.
a. Initial cases? b. Cases after 4 weeks? c. Maximum possible cases?

Find the steepest tangent slope of $f(x)=\sqrt{x+2}$ on $[0,1]$ , the secant line equation, and compare slopes.

Find the derivative of $f(x) = \ln \sqrt{2 + \cos^2 x}$ and explain each step.

Find the derivatives of these functions: $y=\frac{6}{\sqrt[4]{x}}$ , $y=\frac{-2}{\sqrt[3]{x}}$ , $f(x)=\frac{x^{3}+5}{x}$ , $g(x)=\frac{x^{3}-4 x}{\sqrt{x}}$ , $g(x)=(8 x^{2}-4 x)^{2}$ .

Find the tangent line equation to $y=(3 x^{-2}-2 x^{3})^{5}$ at the point $(1,1)$ .

Differentiate $h(x)=(x^{2}+3)^{4}(4x-5)^{3}$ .

Find the arc length of $y=\frac{x^{3}}{6}+\frac{1}{2 x}$ from $x=\frac{1}{2}$ to $x=2$ .

Differentiate the function $\sqrt{8-\frac{1}{x^{3}}}$ with respect to $x$ .

Find the derivative of $f(x) = e^x(\sin x + \cos x)$ and explain each step.

Bestimme die Punkte, an denen die Tangente von $\mathrm{f}$ parallel zur Linie $y=3x+4$ ist für: a) $f(x)=x^{3}$ , b) $f(x)=x^{2}$ , c) $f(x)=x^{\frac{3}{2}}$ , d) $f(x)=x^{6}$ .

Find $c$ where $f_{a v e}=f(c)$ and calculate the arc length of $y=\frac{x^{3}}{6}+\frac{1}{2 x}$ from $x=\frac{1}{2}$ to $x=2$ .

Find $k$ so that the line $y = -\frac{3}{4}x + 3$ is tangent to $f(x) = \frac{k}{x}$ .

Für welche Funktionen $f$ gilt $f = f'$ ? a) $f(x)=-3 e^{x}$ b) $f(x)=2 e^{x+3}$ c) $f(x)=1,5 e^{x+3}-2 e^{x-2}$ d) $f(x)=2 e^{-x}$

Berechnen Sie $f^{\prime}(x)$ für $f(x)=\frac{6}{x}+3 \sqrt{x}-a x^{2}$ oder $f(x)=-\frac{3}{8} x^{-4}+a b^{2} x-\pi^{2}$ .

Berechne die Steigung der Funktion $f(x)=2 x^{2}+1$ bei $x_{0}=3$ mit der h-Methode oder der Ableitungsmethode.

Bestimme die Ableitungsfunktion von $f(x)=x^{15}$ . Fasse zuerst mit Potenzgesetzen zusammen.

Berechnen Sie den Flächeninhalt $A$ zwischen den Funktionen $f(x)=x^{2}+2$ und $g(x)=x+1$ im Intervall $[-1 ; 2]$ .

Find critical points of the function $f(x)=\frac{x^{5}-4 x^{4}+10 x}{x}$ .

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{|1+\Delta x-1|-|1-1|}{\Delta x}$ .

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{f(2+\Delta x)-f(2)}{\Delta x}$ for $f(x)=\frac{-4}{5-x}$ .

Gegeben ist die Funktion $f(x) = -\frac{1}{2} x^{3} + \frac{9}{2} x$ . Finde Nullstellen, Symmetrie, Grenzverhalten und skizziere den Graphen.

Find $\lim _{x \rightarrow c}(f(x) g(x))$ given $\lim _{x \rightarrow c}(f(x)+g(x))=3$ and $\lim _{x \rightarrow c}(f(x)-g(x))=-1$ .

Bestimme die Ableitung von $f(x)=(x^{-3})^{-5}$ und fasse sie zuerst zu einer Potenz zusammen.

If $f$ is odd and $f^{\prime}(c)=3$ , what is $f^{\prime}(-c)$ ? A) -3 B) $\frac{1}{3}$ C) 3 D) $\frac{-1}{3}$

Given the function $f(x) = \frac{-4}{5-x}$ , find its derivative and the limit as $x$ approaches 2.

Find the tangent gradient of $y=2 x^{3}+15 x^{2}-84 x$ at $x=-1$ and identify stationary points.

Bestimme die Tangentengleichung von $f(x)=x^{6}$ bei $x_{0}=2$ .

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{f(2+\Delta x)-f(2)}{\Delta x}$ for $f(x)=\frac{-4}{5-x}$ .

Determine the truth about $f(x)=\frac{|x+2|}{x+2}$ at $x=-2$ : A) vertical asymptote B) differentiable C) continuous D) discontinuous.

Berechne das Integral $\int_{3}^{6} 2 x \, dx$ und finde den Wert.

Find the limit: $\lim _{x \rightarrow \infty}\left(\frac{2 x^{2}+1}{3 x^{2}+1}\right)^{x-x^{2}}$ .

Finde die Stammfunktion von $f(x)=-6x+7$ .

Finde eine Stammfunktion von $f(x)=\frac{2}{3} x^{2}$ .

Calculate the integral $\int_{-1}^{5} 2x^{2} \, dx$ using the evaluation $\left[\frac{2}{3}x^{3}\right]_{-1}^{5}$ .

Evaluate the integral from 1 to 2 of the function $4 \sin 3 x - \frac{3}{2 x}$ .

A particle moves along the $x$ axis. When is it moving towards the origin based on the given position and velocity data?

Calculate the area under the curve $y=\cos(t)$ from $t=0$ to $t=2\pi$ .

How much will you have after investing \ $4500 at$ 3.8\%$ interest compounded continuously for 20 years?

A pizza pan cools from $425^{\circ} \mathrm{F}$ to $130^{\circ} \mathrm{F}$ after $5$ minutes at room temperature $72^{\circ} \mathrm{F}$ . When does this happen? Also, find when it reaches $230^{\circ} \mathrm{F}$ and describe the cooling trend.

Find the elasticity function $E(p)$ for the demand $D(p)=200 e^{-0.8 p}$ and calculate it at price \$3.

Evaluate the integral from 1 to 2 of the function $2x + 4$ .

Find the speed of a $1.5 \mathrm{~kg}$ book dropped from 5.0m just before it hits the ground. Use $mgh$ for energy.

Find the derivative of $f(x)=-3 x^{8} \ln x$ and evaluate $f^{\prime}(e^{3})$ .

Find the derivative $f'(x)$ of the function $f(x)=4 \log_{9}(x)$ and evaluate $f'(5)$ .

Find the derivative of the function $f(t)=3 t^{4}-7 t+2 e^{t}$ , denoted as $f^{\prime}(t)$ .

Find the derivative of the function $f(x)=\ln(x^{2}-18x+86)$ . What is $f^{\prime}(x)$ ?

Differentiate the function $f(x)=e^{2 x}$ . Provide $f^{\prime}(x)=$ .

Differentiate the function $G(x)=e^{-8 x}$ without simplifying the derivative. Find $G^{\prime}(x)=$ .

Find the derivative of $f(x)=3 e^{-5 x-5}$ . What is $f^{\prime}(x)=?$

Differentiate the function $h(x)=\frac{e^{-5 x}}{x^{6}}$ without simplifying the derivative. Find $h^{\prime}(x)=$ .

Find the second derivative of the functions $x(t) = a t^{3}$ and $y(t) = b t$ with respect to $t$ .

Differentiate the function $G(x)=e^{x^{7}+11 x}$ without simplifying the derivative. Find $G^{\prime}(x)=$

Differentiate the function $F(x)=-4 e^{x^{2}}$ without simplifying the derivative. Find $F^{\prime}(x)=$ .

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

Find the derivative of $f(x)=\ln \left(\frac{5 x-1}{3 x-7}\right)$ . What is $f^{\prime}(x)$ ?

Determine the intervals where the curve defined by $x(t)=2-t^{2}$ and $y(t)=t^{2}+t^{3}$ is concave up or down.

Find the horizontal asymptote, global max/min, domain, intercepts, and critical points of $\frac{x+2}{e^{x}}$ .

Find the derivative of the function $f(x)=\ln(x^{2}+10x+31)$ . What is $f^{\prime}(x)$ ?

How long will it take for a pie cooling from $140^{\circ} \mathrm{F}$ to reach $80^{\circ} \mathrm{F}$ using $T(t)=68 e^{-0.0174 t}+72$ ?

Integrate: $\int\left(2 x^{3}+3-\frac{1}{2 x^{3}}\right) dx$