Calculus

Problem 4101

Find $A^{\prime}(0)$ for $A=f \cdot g$ given $f(0)=6$ , $g(0)=-8$ , $f^{\prime}(0)=1$ , $g^{\prime}(0)=\frac{1}{3}$ . A. 0 B. -6 C. $\frac{10}{3}$ D. $\frac{1}{3}$

See Solution

Problem 4102

Find the derivative of the function defined by $r^{4}+s^{4}=Q$ as $\frac{d}{d s} r=A\left(\frac{s}{r}\right)^{a}+B(s \cdot r)^{b}+C$ . Determine coefficients $A$ , $a$ , $B$ , $b$ , and $C$ .

See Solution

Problem 4103

Find $A^{\prime}(2)$ for $A=\left(\frac{g}{f}\right)$ using values from the table provided. Choices: A. 23 B. $-\frac{23}{64}$ C. $\frac{23}{64}$ D. -23

See Solution

Problem 4104

Find the derivative of $y$ from the equation $4 x^{3}+8 x y-4 y^{2}=128$ and identify coefficients $a$ to $m$ .

See Solution

Problem 4105

Find the derivative of the function defined by $2 x^{4}+5 x y-4 y^{3}=169$ . Identify coefficients in the expression for $\frac{d}{d x} y$ .

See Solution

Problem 4106

Find $A^{\prime}(-2)$ if $A=\sqrt{g(x)}$ and use the values from the table for $g$ and $g^{\prime}$ .

See Solution

Problem 4107

Find $A^{\prime}(-2)$ if $A=\sqrt{g(x)}$ and use values from the table for $g$ and its derivative.

See Solution

Problem 4108

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(3)$ for the equation $x^{-3} z^4 = 81$ . Round to two digits.

See Solution

Problem 4109

Bestimmen Sie die Grenzwerte für die folgenden Ausdrücke: a) $\lim _{n \rightarrow \infty} \frac{2^{n}-1}{2^{n}}$ b) $\lim _{n \rightarrow \infty} \frac{2^{n}-1}{2^{n-1}}$ c) $\lim _{n \rightarrow \infty} \frac{2^{n}}{1+4^{n}}$ d) $\lim _{n \rightarrow \infty} \frac{2^{n}-3^{n}}{2^{n}+3^{n}}$ e) $\lim _{n \rightarrow \infty} \frac{2^{n}+3^{n+1}}{2 \cdot 3^{n}}$

See Solution

Problem 4110

Find the average rate of change $A V=\frac{s(1)-s(0.5)}{1-0.5}$ for $s(t)=64-16(t-1)^{2}$ . What are its units and geometric meaning?

See Solution

Problem 4111

Find three positive numbers that sum to 200 and maximize their product using Lagrange multipliers.

See Solution

Problem 4112

Maximize the product of three positive numbers that sum to 200 using Lagrange multipliers.

See Solution

Problem 4113

Find the derivative of $y=-3 x^{3}+5 x^{2}-1$ using first principles.

See Solution

Problem 4114

Bestimme die möglichen Extremstellen der Funktion $f(x)=-\frac{1}{4} x^{4}-\frac{1}{6} x^{3}+7 x^{2}$ .

See Solution

Problem 4115

Evaluate the double integral $\int_{0}^{2} \int_{3 y^{2}-6 y}^{2 y-y^{2}} 3 y \, dx \, dy$ .

See Solution

Problem 4116

Find the integral $\int \frac{x^{5}+x-1}{x^{4}-x^{3}} \partial x$ using partial fractions.

See Solution

Problem 4117

Bestimmen Sie die Tangenten- und Normalengleichung von $f(x)=\frac{1}{2} x^{2}$ am Punkt $(1 \mid f(1))$ und den Steigungswinkel.

See Solution

Problem 4118

Bestimmen Sie die ersten beiden Ableitungen von $f(x) = \sqrt{x} + 1$ .

See Solution

Problem 4119

Berechne die ersten zwei Ableitungen von $f(x)=x^{2} \cdot e^{x}$ .

See Solution

Problem 4120

Bestimmen Sie die Ableitung von $f(x) = e^{2x + 3}$ mit der Kettenregel.

See Solution

Problem 4121

Untersuche die Funktion $f(x)=\frac{1}{4} x^{4}+4$ auf Extrem- und Sattelpunkte mittels erster und zweiter Ableitung.

See Solution

Problem 4122

Berechnen Sie die Integrale: a) $\int_{1}^{3}(x+1) d x$ , b) $\int_{-2}^{2}(4-x^{2}) d x$ , e) $\int_{0}^{4}(4-x) d x$ , f) $\int_{1}^{3}(0,5 x-1)^{2} d x$ .

See Solution

Problem 4123

Find where the gradient is zero on the curve $y=(n-2)^{2}(2n+3)$ .

See Solution

Problem 4124

Berechnen Sie das bestimmte Integral $\int^{3}(x+1) d x$ .

See Solution

Problem 4125

Bestimme die Steigung von $\mathrm{f}$ an $\mathrm{x}_{0}$ : a) $f(x)=\frac{1}{2} x^{2}-2$ , $x_{0}=2$ ; b) $f(x)=4-2 x$ , $x_{0}=3$ .

See Solution

Problem 4126

Find the derivative of $f(x)=\frac{1}{8} \cdot\left(\frac{1}{2}-x^{2}\right)^{7}$ .

See Solution

Problem 4127

Find the derivative of $y=\sqrt[3]{7 x^{3}-3}$ using only positive, negative, and fractional exponents.

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Problem 4128

Find the derivative of $g(t)=\frac{2}{(t^{4}-2)^{1/2}}$ using positive, negative, and fractional exponents.

See Solution

Problem 4129

Find the derivative of $g(t)=\frac{7}{(t^{3}-5)^{1/2}}$ using positive, negative, and fractional exponents.

See Solution

Problem 4130

Find $h^{\prime}(7)$ if $f(7)=9$ and $f^{\prime}(7)=5$ for $h(x)=(f(x))^{3}$ .

See Solution

Problem 4131

Find the population change rate in 8 years for $P(t)=20(65+5t)^{2}-1400t$ .

See Solution

Problem 4132

Find $\frac{d y}{d x}$ at $x=2$ for $y=u^{3}-2 u+8$ with $u=5 x^{2}+6 x+8$ .

See Solution

Problem 4133

Find the derivative of $g(x)=\frac{-8 x}{3 x-2}$ using the Product or Quotient Rule.

See Solution

Problem 4134

Find the derivative of $f(x)=x^{5}(5x^{2}+5)$ using the Product or Quotient Rule.

See Solution

Problem 4135

Find the derivative of $f(x)=(5x+5)\left(-4x^{-2}-10\right)$ using the Product or Quotient Rule.

See Solution

Problem 4136

Find the marginal revenue for the demand function $D(x)=\frac{171}{3 x+10}$ at $x=5$ . Round to the nearest cent.

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Problem 4137

Find the derivative of $h(x) = x^{12} f(x)$ , using $F$ for $f'(x)$ . What is $h'(x)$ ?

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Problem 4138

Calculate the derivative $h^{\prime}(2)$ for $h(x)=x f(x)+4 g(x)$ using values from the given table.

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Problem 4139

Find the derivative $h'(x)$ of the function $h(x) = \frac{f(x) g(x)}{3}$ , using $F$ for $f'$ and $G$ for $g'$ .

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Problem 4140

Find the rate of change of force $F=\frac{G m M}{d^{2}}$ with respect to distance $d$ and evaluate for $G=6.67 \times 10^{-11}$ , $m=M=100$ kg, $d=25$ m. Answer in scientific notation.

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Problem 4141

Find the rate of change of force $F$ with $G=6.67 \times 10^{-11} \mathrm{Nm}^{2} / \mathrm{kg}^{2}$ for 100 kg bodies 25 m apart. Answer in scientific notation, 2 decimal places.

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Problem 4142

Find the line equation through point $P(6,-3)$ that is tangent to $f(x)=\frac{12}{x-1}$ for $x>1$ .

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Problem 4143

Find the tangent line equation for $f(x)=3 x^{3}+3 x^{2}+2 x+5$ at $x=1$ . $y=$

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Problem 4144

Find the line's equation that passes through $P(6,-3)$ and is tangent to $f(x)=\frac{12}{x-1}$ for $x>1$ .

See Solution

Problem 4145

Calculate the integral $\int_{-2}^{2} 0.25 x^{3} \, dx$ .

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Problem 4146

Find the line equation through point $P(6,-3)$ that is tangent to $f(x)=\frac{12}{x-1}$ for $x>1$ .

See Solution

Problem 4147

Find $f^{\prime}(0)$ for $f(x)=x^{2}-2x$ , then graph $f(x)$ and its tangent line at $x=0$ .

See Solution

Problem 4148

Calculate the integral: $\int_{-2}^{2}\left(4-x^{2}\right) d x$

See Solution

Problem 4149

Berechnen Sie das Integral $\int_{-1}^{4}\left(3 x^{2}-4 x+1\right) d x$ .

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Problem 4150

Find the derivative of $f(x)=3x(2-x)$ .

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Problem 4151

Find $\frac{d y}{d x}$ for $x^{3}-x y+y^{4}=37$ and the tangent line at $(3,2)$ .

See Solution

Problem 4152

Berechnen Sie die Integrale: a) $\int_{1}^{2} 4 x^{3} dx$ b) $\int_{-1}^{1}(9 x^{2}-1) dx$

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Problem 4153

A ball dropped from a building takes 4 s to hit the ground. What is the height of the building? Use $h = \frac{1}{2} g t^2$ .

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Problem 4154

Trova il valore di a tale che $\lim _{x \rightarrow 0} \frac{\ln (3-x)-\ln 3}{a x}=-\frac{1}{6}$ .

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Problem 4155

Find the derivative of $f(x)=\frac{4}{(2-x)^{2}}$ .

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Problem 4156

Wachstumsgeschwindigkeit eines Baumes: a) Was bedeutet $\int_{10}^{50} v(t) d t$ ? Berechne das Integral. b) Höhe eines 10 Jahre alten Baumes nach 10 Jahren?

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Problem 4157

Find $\frac{d y}{d x}$ for $x^{3}-x y+y^{4}=37$ and the tangent line at $(3,2)$ . Show your work.

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Problem 4158

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos ^{2} 2 x}{x \sin x}$

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Problem 4159

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(7)$ for the function defined by $x^{-4} \cdot z^{2}=49$ . Round to two digits.

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Problem 4160

Find the derivative of $f(x)=\sin (2 x)+4 \cos (3 x)+2 \tan (5 x)$ .

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Problem 4161

Find the derivative of $y$ from the equation $4 x^{3}+6 x y-7 y^{3}=235$ and identify coefficients $a$ to $m$ .

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Problem 4162

Find the derivative $\frac{dk}{dl}$ from $k^9 + l^9 = W$ and express it as $\frac{d}{dl} k = A\left(\frac{k}{l}\right)^{a}+B(k \cdot l)^{b}+C$ . Determine coefficients $A$ , $a$ , $B$ , $b$ , and $C$ .

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Problem 4163

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(3)$ for the function defined by $x^{-2} \cdot z^{3}=27$ . Round to two digits.

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Problem 4164

Berechnen Sie die Flächeninhalte zwischen den Funktionen in den angegebenen Intervallen: a) $f(x)=x^{3}, g(x)=x, I=[-2, 5]$ ; b) $f(x)=x^{3}+x, g(x)=x^{2}+1, I=[0, 2]$ ; c) $f(x)=e^{x}, g(x)=x, I=[0, \ln 2]$ ; d) $f(x)=\frac{1}{2} \sin (x), g(x)=-\cos (x), I=[0, \pi]$ ; e) $f(x)=\frac{1}{x^{2}}, g(x)=\frac{17}{4}-x^{2}, I=[1, 3]$ ; f) $f(x)=x+1, g(x)=e^{-x}, I=[-1, 1]$ .

See Solution

Problem 4165

Evaluate the integral: $\int \frac{2}{x \sqrt{25 x^{2}-64}} d x$

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Problem 4166

Find the limit as $x$ approaches $-\infty$ : $\lim _{x \rightarrow-\infty} \frac{9 x-15}{x^{3}+15 x-15}$ .

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Problem 4167

Find first and second order approximations of $f(x)=9 x^{5}$ around $x_{0}=-1$ . Determine coefficients $a$ , $b$ , and $c$ .

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Problem 4168

Bestimme die Stellen, an denen die Funktion $f(x)=x^{3}+\frac{5}{2} x^{2}+1$ eine horizontale Tangente ( $m_{t}=0$ ) hat.

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Problem 4169

Find the marginal cost function for $C(x)=170+5.3 x-0.01 x^{2}$ . What is $C^{\prime}(x)=?$

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Problem 4170

Find the marginal revenue function for $R(x)=x(26-0.04 x)$ . What is $R^{\prime}(x)$ ?

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Problem 4171

Set up an integral for the arc length of $y=x^{5}$ from $x=2$ to $x=3$ . Do not evaluate it.

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Problem 4172

Find $\hat{x}_{0}$ and $\tilde{x}_{0}$ where $L(x)=8x+15$ approximates $f(x)=9x^2+26x+24$ and $g(x)=5\ln(5x)-17x+20$ .

See Solution

Problem 4173

Berechnen Sie die ersten 2 Ableitungen der Funktion $f(t)=5+20 t \cdot e^{-0,2 t}$ zur Medikamenteneinstellung.

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Problem 4174

Set up an integral for the surface area of $y=x^{5}$ from $x=2$ to $x=3$ when revolved around the x-axis.

See Solution

Problem 4175

Find the limit: $\lim _{x \rightarrow 2^{+}} \frac{x^{2}+5 x+6}{x^{2}-4}$

See Solution

Problem 4176

Find the linear approximation for $f(x)=x^{m}$ near $x_{0}=1$ . Use it to estimate (a) $0.98^{26}$ and (b) $\sqrt[6]{1.08}$ . Round to two digits.

See Solution

Problem 4177

Welche Stammfunktion von $f(x)=x^{2}$ verläuft durch den Punkt $P(1, 1)$ ?

See Solution

Problem 4178

Sketch the area between $y=e^{2 x}$ , $y=1-x$ , $x=-1$ , and $x=1$ , then find the exact area without rounding.

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Problem 4179

Gegeben ist die Funktion $f(t)=5+20 t \cdot e^{-0,2 t}$ .
1. Berechne die ersten 2 Ableitungen $f'(t)$ und $f''(t)$ .
2. Bestimme die Anfangskonzentration.
3. Finde den Zeitraum, in dem die Konzentration $10 \mathrm{mg}$ /L erreicht.
4. Zeige, dass die Konzentration langfristig bei $5 \mathrm{mg}$ /L stabilisiert.
5. Berechne Zeitpunkt und Höhe der maximalen Konzentration.
6. Zeige, dass die größte Abnahme der Konzentration nach 10 Stunden erfolgt.
7. Bestimme die Stunde, in der die Konzentration um $1 \mathrm{mg}$ abnimmt.
8. Finde die Konzentrationsveränderungsrate nach 2 Stunden.
9. Zeige, dass die Zunahme der Konzentration bis zum Maximum abnimmt.

See Solution

Problem 4180

Calculate the integral: $\int 3 \cdot \sin (3 x) \, dx$

See Solution

Problem 4181

Bestimme die Stammfunktion von $f(x)=x^{2}$ , die durch $P(1 \mid 1)$ geht. Wo schneidet $f(x)=1-x^{2}$ die $y$ -Achse?

See Solution

Problem 4182

Finde die Stammfunktion von $f(x)=1-x^{2}$ , die die $y$ -Achse bei $y=4$ schneidet.

See Solution

Problem 4183

Find the volume when the area between $y=2x$ , $x=0$ , and $y=4$ is rotated around the $x$ -axis.

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Problem 4184

Gegeben ist die Funktion $f(t) = 5+20t \cdot e^{-0,2t}$ .
1. Berechne die ersten 2 Ableitungen von $f$ .
2. Bestimme die Anfangskonzentration.
3. Bestimme den Zeitraum, in dem die Konzentration $10 \mathrm{mg}$ /L erreicht.
4. Zeige, dass die Konzentration langfristig bei 5 mg/L bleibt.
5. Berechne Zeitpunkt und Menge der höchsten Konzentration.
6. Nachweis: stärkste Abnahme 10 Stunden nach Einnahme.
7. Bestimme die Stunde, in der die Konzentration um $1 \mathrm{mg}$ abnimmt.
8. Berechne die Konzentrationsrate 2 Stunden nach Einnahme.
9. Zeige, dass die Zunahme bis zum Maximum abnimmt.

See Solution

Problem 4185

Find the limit: $\lim _{x \rightarrow-2} \frac{x^{3}+3 x^{2}-4}{x^{2}-4}$ using long division.

See Solution

Problem 4186

Finde die Stammfunktion von $f(x)=2+x$ , die bei $x=1$ eine Nullstelle hat.

See Solution

Problem 4187

Total cost for $x$ frames is $C(x)=70,000+800 x$ . Find average cost for 500 frames, marginal cost at 500, and estimate for 501 frames.

See Solution

Problem 4188

Find $\frac{d y}{d x}$ for $x^{3}-x y+y^{4}=37$ and the tangent line at $(3,2)$ . Show work using implicit differentiation.

See Solution

Problem 4189

Find the $x$ -coordinates where the tangent line to $f(x)=\sqrt{x^{2}+10 x+26}$ is horizontal.

See Solution

Problem 4190

Find the $\mathrm{x}$ -coordinates where the tangent line to $f(x)=\sqrt{x^{2}+10 x+26}$ is horizontal.

See Solution

Problem 4191

Find $\lim_{x \to 4} f(x)$ for the piecewise function $f(x)=\{2x, x>4; 5, x=4; x^2-8, x<4\}$ .

See Solution

Problem 4192

Find the integral for the volume of the solid formed by revolving the region bounded by $y=\sqrt{x}, y=x+4, x=0$ , and $x=1$ around $x=2$ . Specify disk or shell method. No evaluation needed.

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Problem 4193

Find the integral of the function $x + \frac{3}{x^{2}}$ .

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Problem 4194

Determine if these functions are continuous and identify any discontinuities:
a. $f(x)=\left\{\begin{array}{cc}5-x & x<-2 \\ 7 & x=-2 \\ x^{2}-5 & x>-2\end{array}\right.$ b. $f(x)=\left\{\begin{array}{cc}x^{2}, & \text { if } x<-1 \\ x+2, & \text { if } x>-1\end{array}\right.$ c. $f(x)=\left\{\begin{array}{lr}5+x, & x<-1 \\ 4 x, & -1<x<0 \\ 2 x, & 0 \leq x \leq 2 \\ \frac{4 x}{x-3}, & x>2\end{array}\right.$

See Solution

Problem 4195

A jolly jumper with a $10 \mathrm{~kg}$ baby stretches a spring by $0.30 \mathrm{~m}$ . Find the speed at equilibrium after an extra $0.45 \mathrm{~m}$ drop.

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Problem 4196

Calculate the integral of the function $e^{x} e^{x+2}$ .

See Solution

Problem 4197

Find the volume of the solid formed by rotating the area between the $x$ -axis and $y=\frac{1}{1+x^{2}}$ , $x=0$ , $x=8$ around the $y$ -axis.

See Solution

Problem 4198

Find the volume when the area between $y=5 x^{3}$ , $y=0$ , and $x=3$ is rotated around the $y$ -axis.

See Solution

Problem 4199

Calculate the integral: $\int \frac{4}{e^{x}} d x$

See Solution

Problem 4200

Estimate $y(1)$ using Euler's method with step size 0.2 for $y^{\prime}=x^{2} y-\frac{1}{2} y^{2}$ , $y(0)=1$ .

See Solution

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Calculus

Problem 4101

Find A′(0)A^{\prime}(0)A′(0) for A=f⋅gA=f \cdot gA=f⋅g given f(0)=6f(0)=6f(0)=6, g(0)=−8g(0)=-8g(0)=−8, f′(0)=1f^{\prime}(0)=1f′(0)=1, g′(0)=13g^{\prime}(0)=\frac{1}{3}g′(0)=31​. A. 0 B. -6 C. 103\frac{10}{3}310​ D. 13\frac{1}{3}31​

Problem 4102

Find the derivative of the function defined by r4+s4=Qr^{4}+s^{4}=Qr4+s4=Q as ddsr=A(sr)a+B(s⋅r)b+C\frac{d}{d s} r=A\left(\frac{s}{r}\right)^{a}+B(s \cdot r)^{b}+Cdsd​r=A(rs​)a+B(s⋅r)b+C. Determine coefficients AAA, aaa, BBB, bbb, and CCC.

Problem 4103

Find A′(2)A^{\prime}(2)A′(2) for A=(gf)A=\left(\frac{g}{f}\right)A=(fg​) using values from the table provided. Choices: A. 23 B. −2364-\frac{23}{64}−6423​ C. 2364\frac{23}{64}6423​ D. -23

Problem 4104

Find the derivative of yyy from the equation 4x3+8xy−4y2=1284 x^{3}+8 x y-4 y^{2}=1284x3+8xy−4y2=128 and identify coefficients aaa to mmm.

Problem 4105

Find the derivative of the function defined by 2x4+5xy−4y3=1692 x^{4}+5 x y-4 y^{3}=1692x4+5xy−4y3=169. Identify coefficients in the expression for ddxy\frac{d}{d x} ydxd​y.

Problem 4106

Find A′(−2)A^{\prime}(-2)A′(−2) if A=g(x)A=\sqrt{g(x)}A=g(x)​ and use the values from the table for ggg and g′g^{\prime}g′.

Problem 4107

Find A′(−2)A^{\prime}(-2)A′(−2) if A=g(x)A=\sqrt{g(x)}A=g(x)​ and use values from the table for ggg and its derivative.

Problem 4108

Find z(1)z(1)z(1), z′(1)z'(1)z′(1), z′′(1)z''(1)z′′(1), and x′(3)x'(3)x′(3) for the equation x−3z4=81x^{-3} z^4 = 81x−3z4=81. Round to two digits.

Problem 4109

Problem 4110

Find the average rate of change AV=s(1)−s(0.5)1−0.5A V=\frac{s(1)-s(0.5)}{1-0.5}AV=1−0.5s(1)−s(0.5)​ for s(t)=64−16(t−1)2s(t)=64-16(t-1)^{2}s(t)=64−16(t−1)2. What are its units and geometric meaning?

Problem 4111

Find three positive numbers that sum to 200 and maximize their product using Lagrange multipliers.

Problem 4112

Maximize the product of three positive numbers that sum to 200 using Lagrange multipliers.

Problem 4113

Find the derivative of y=−3x3+5x2−1y=-3 x^{3}+5 x^{2}-1y=−3x3+5x2−1 using first principles.

Problem 4114

Bestimme die möglichen Extremstellen der Funktion f(x)=−14x4−16x3+7x2f(x)=-\frac{1}{4} x^{4}-\frac{1}{6} x^{3}+7 x^{2}f(x)=−41​x4−61​x3+7x2.

Problem 4115

Evaluate the double integral ∫02∫3y2−6y2y−y23y dx dy\int_{0}^{2} \int_{3 y^{2}-6 y}^{2 y-y^{2}} 3 y \, dx \, dy∫02​∫3y2−6y2y−y2​3ydxdy.

Problem 4116

Find the integral ∫x5+x−1x4−x3∂x\int \frac{x^{5}+x-1}{x^{4}-x^{3}} \partial x∫x4−x3x5+x−1​∂x using partial fractions.

Problem 4117

Bestimmen Sie die Tangenten- und Normalengleichung von f(x)=12x2f(x)=\frac{1}{2} x^{2}f(x)=21​x2 am Punkt (1∣f(1))(1 \mid f(1))(1∣f(1)) und den Steigungswinkel.

Problem 4118

Bestimmen Sie die ersten beiden Ableitungen von f(x)=x+1f(x) = \sqrt{x} + 1f(x)=x​+1.

Problem 4119

Berechne die ersten zwei Ableitungen von f(x)=x2⋅exf(x)=x^{2} \cdot e^{x}f(x)=x2⋅ex.

Problem 4120

Bestimmen Sie die Ableitung von f(x)=e2x+3f(x) = e^{2x + 3}f(x)=e2x+3 mit der Kettenregel.

Problem 4121

Untersuche die Funktion f(x)=14x4+4f(x)=\frac{1}{4} x^{4}+4f(x)=41​x4+4 auf Extrem- und Sattelpunkte mittels erster und zweiter Ableitung.

Problem 4122

Berechnen Sie die Integrale: a) ∫13(x+1)dx\int_{1}^{3}(x+1) d x∫13​(x+1)dx, b) ∫−22(4−x2)dx\int_{-2}^{2}(4-x^{2}) d x∫−22​(4−x2)dx, e) ∫04(4−x)dx\int_{0}^{4}(4-x) d x∫04​(4−x)dx, f) ∫13(0,5x−1)2dx\int_{1}^{3}(0,5 x-1)^{2} d x∫13​(0,5x−1)2dx.

Problem 4123

Find where the gradient is zero on the curve y=(n−2)2(2n+3)y=(n-2)^{2}(2n+3)y=(n−2)2(2n+3).

Problem 4124

Berechnen Sie das bestimmte Integral ∫3(x+1)dx\int^{3}(x+1) d x∫3(x+1)dx.

Problem 4125

Bestimme die Steigung von f\mathrm{f}f an x0\mathrm{x}_{0}x0​: a) f(x)=12x2−2f(x)=\frac{1}{2} x^{2}-2f(x)=21​x2−2, x0=2x_{0}=2x0​=2; b) f(x)=4−2xf(x)=4-2 xf(x)=4−2x, x0=3x_{0}=3x0​=3.

Problem 4126

Find the derivative of f(x)=18⋅(12−x2)7f(x)=\frac{1}{8} \cdot\left(\frac{1}{2}-x^{2}\right)^{7}f(x)=81​⋅(21​−x2)7.

Problem 4127

Find the derivative of y=7x3−33y=\sqrt[3]{7 x^{3}-3}y=37x3−3​ using only positive, negative, and fractional exponents.

Problem 4128

Find the derivative of g(t)=2(t4−2)1/2g(t)=\frac{2}{(t^{4}-2)^{1/2}}g(t)=(t4−2)1/22​ using positive, negative, and fractional exponents.

Problem 4129

Find the derivative of g(t)=7(t3−5)1/2g(t)=\frac{7}{(t^{3}-5)^{1/2}}g(t)=(t3−5)1/27​ using positive, negative, and fractional exponents.

Problem 4130

Find h′(7)h^{\prime}(7)h′(7) if f(7)=9f(7)=9f(7)=9 and f′(7)=5f^{\prime}(7)=5f′(7)=5 for h(x)=(f(x))3h(x)=(f(x))^{3}h(x)=(f(x))3.

Problem 4131

Find the population change rate in 8 years for P(t)=20(65+5t)2−1400tP(t)=20(65+5t)^{2}-1400tP(t)=20(65+5t)2−1400t.

Problem 4132

Find dydx\frac{d y}{d x}dxdy​ at x=2x=2x=2 for y=u3−2u+8y=u^{3}-2 u+8y=u3−2u+8 with u=5x2+6x+8u=5 x^{2}+6 x+8u=5x2+6x+8.

Problem 4133

Find the derivative of g(x)=−8x3x−2g(x)=\frac{-8 x}{3 x-2}g(x)=3x−2−8x​ using the Product or Quotient Rule.

Problem 4134

Find the derivative of f(x)=x5(5x2+5)f(x)=x^{5}(5x^{2}+5)f(x)=x5(5x2+5) using the Product or Quotient Rule.

Problem 4135

Find the derivative of f(x)=(5x+5)(−4x−2−10)f(x)=(5x+5)\left(-4x^{-2}-10\right)f(x)=(5x+5)(−4x−2−10) using the Product or Quotient Rule.

Problem 4136

Find the marginal revenue for the demand function D(x)=1713x+10D(x)=\frac{171}{3 x+10}D(x)=3x+10171​ at x=5x=5x=5. Round to the nearest cent.

Problem 4137

Find the derivative of h(x)=x12f(x)h(x) = x^{12} f(x)h(x)=x12f(x), using FFF for f′(x)f'(x)f′(x). What is h′(x)h'(x)h′(x)?

Problem 4138

Calculate the derivative h′(2)h^{\prime}(2)h′(2) for h(x)=xf(x)+4g(x)h(x)=x f(x)+4 g(x)h(x)=xf(x)+4g(x) using values from the given table.

Problem 4139

Find the derivative h′(x)h'(x)h′(x) of the function h(x)=f(x)g(x)3h(x) = \frac{f(x) g(x)}{3}h(x)=3f(x)g(x)​, using FFF for f′f'f′ and GGG for g′g'g′.

Problem 4140

Find the rate of change of force F=GmMd2F=\frac{G m M}{d^{2}}F=d2GmM​ with respect to distance ddd and evaluate for G=6.67×10−11G=6.67 \times 10^{-11}G=6.67×10−11, m=M=100m=M=100m=M=100 kg, d=25d=25d=25 m. Answer in scientific notation.

Find $A^{\prime}(0)$ for $A=f \cdot g$ given $f(0)=6$ , $g(0)=-8$ , $f^{\prime}(0)=1$ , $g^{\prime}(0)=\frac{1}{3}$ . A. 0 B. -6 C. $\frac{10}{3}$ D. $\frac{1}{3}$

Find the derivative of the function defined by $r^{4}+s^{4}=Q$ as $\frac{d}{d s} r=A\left(\frac{s}{r}\right)^{a}+B(s \cdot r)^{b}+C$ . Determine coefficients $A$ , $a$ , $B$ , $b$ , and $C$ .

Find $A^{\prime}(2)$ for $A=\left(\frac{g}{f}\right)$ using values from the table provided. Choices: A. 23 B. $-\frac{23}{64}$ C. $\frac{23}{64}$ D. -23

Find the derivative of $y$ from the equation $4 x^{3}+8 x y-4 y^{2}=128$ and identify coefficients $a$ to $m$ .

Find the derivative of the function defined by $2 x^{4}+5 x y-4 y^{3}=169$ . Identify coefficients in the expression for $\frac{d}{d x} y$ .

Find $A^{\prime}(-2)$ if $A=\sqrt{g(x)}$ and use the values from the table for $g$ and $g^{\prime}$ .

Find $A^{\prime}(-2)$ if $A=\sqrt{g(x)}$ and use values from the table for $g$ and its derivative.

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(3)$ for the equation $x^{-3} z^4 = 81$ . Round to two digits.

Find the average rate of change $A V=\frac{s(1)-s(0.5)}{1-0.5}$ for $s(t)=64-16(t-1)^{2}$ . What are its units and geometric meaning?

Find the derivative of $y=-3 x^{3}+5 x^{2}-1$ using first principles.

Bestimme die möglichen Extremstellen der Funktion $f(x)=-\frac{1}{4} x^{4}-\frac{1}{6} x^{3}+7 x^{2}$ .

Evaluate the double integral $\int_{0}^{2} \int_{3 y^{2}-6 y}^{2 y-y^{2}} 3 y \, dx \, dy$ .

Find the integral $\int \frac{x^{5}+x-1}{x^{4}-x^{3}} \partial x$ using partial fractions.

Bestimmen Sie die Tangenten- und Normalengleichung von $f(x)=\frac{1}{2} x^{2}$ am Punkt $(1 \mid f(1))$ und den Steigungswinkel.

Bestimmen Sie die ersten beiden Ableitungen von $f(x) = \sqrt{x} + 1$ .

Berechne die ersten zwei Ableitungen von $f(x)=x^{2} \cdot e^{x}$ .

Bestimmen Sie die Ableitung von $f(x) = e^{2x + 3}$ mit der Kettenregel.

Untersuche die Funktion $f(x)=\frac{1}{4} x^{4}+4$ auf Extrem- und Sattelpunkte mittels erster und zweiter Ableitung.

Berechnen Sie die Integrale: a) $\int_{1}^{3}(x+1) d x$ , b) $\int_{-2}^{2}(4-x^{2}) d x$ , e) $\int_{0}^{4}(4-x) d x$ , f) $\int_{1}^{3}(0,5 x-1)^{2} d x$ .

Find where the gradient is zero on the curve $y=(n-2)^{2}(2n+3)$ .

Berechnen Sie das bestimmte Integral $\int^{3}(x+1) d x$ .

Bestimme die Steigung von $\mathrm{f}$ an $\mathrm{x}_{0}$ : a) $f(x)=\frac{1}{2} x^{2}-2$ , $x_{0}=2$ ; b) $f(x)=4-2 x$ , $x_{0}=3$ .

Find the derivative of $f(x)=\frac{1}{8} \cdot\left(\frac{1}{2}-x^{2}\right)^{7}$ .

Find the derivative of $y=\sqrt[3]{7 x^{3}-3}$ using only positive, negative, and fractional exponents.

Find the derivative of $g(t)=\frac{2}{(t^{4}-2)^{1/2}}$ using positive, negative, and fractional exponents.

Find the derivative of $g(t)=\frac{7}{(t^{3}-5)^{1/2}}$ using positive, negative, and fractional exponents.

Find $h^{\prime}(7)$ if $f(7)=9$ and $f^{\prime}(7)=5$ for $h(x)=(f(x))^{3}$ .

Find the population change rate in 8 years for $P(t)=20(65+5t)^{2}-1400t$ .

Find $\frac{d y}{d x}$ at $x=2$ for $y=u^{3}-2 u+8$ with $u=5 x^{2}+6 x+8$ .

Find the derivative of $g(x)=\frac{-8 x}{3 x-2}$ using the Product or Quotient Rule.

Find the derivative of $f(x)=x^{5}(5x^{2}+5)$ using the Product or Quotient Rule.

Find the derivative of $f(x)=(5x+5)\left(-4x^{-2}-10\right)$ using the Product or Quotient Rule.

Find the marginal revenue for the demand function $D(x)=\frac{171}{3 x+10}$ at $x=5$ . Round to the nearest cent.

Find the derivative of $h(x) = x^{12} f(x)$ , using $F$ for $f'(x)$ . What is $h'(x)$ ?

Calculate the derivative $h^{\prime}(2)$ for $h(x)=x f(x)+4 g(x)$ using values from the given table.

Find the derivative $h'(x)$ of the function $h(x) = \frac{f(x) g(x)}{3}$ , using $F$ for $f'$ and $G$ for $g'$ .

Find the rate of change of force $F=\frac{G m M}{d^{2}}$ with respect to distance $d$ and evaluate for $G=6.67 \times 10^{-11}$ , $m=M=100$ kg, $d=25$ m. Answer in scientific notation.

Find the rate of change of force $F$ with $G=6.67 \times 10^{-11} \mathrm{Nm}^{2} / \mathrm{kg}^{2}$ for 100 kg bodies 25 m apart. Answer in scientific notation, 2 decimal places.

Find the line equation through point $P(6,-3)$ that is tangent to $f(x)=\frac{12}{x-1}$ for $x>1$ .

Find the tangent line equation for $f(x)=3 x^{3}+3 x^{2}+2 x+5$ at $x=1$ . $y=$

Find the line's equation that passes through $P(6,-3)$ and is tangent to $f(x)=\frac{12}{x-1}$ for $x>1$ .

Calculate the integral $\int_{-2}^{2} 0.25 x^{3} \, dx$ .

Find the line equation through point $P(6,-3)$ that is tangent to $f(x)=\frac{12}{x-1}$ for $x>1$ .

Find $f^{\prime}(0)$ for $f(x)=x^{2}-2x$ , then graph $f(x)$ and its tangent line at $x=0$ .

Calculate the integral: $\int_{-2}^{2}\left(4-x^{2}\right) d x$

Berechnen Sie das Integral $\int_{-1}^{4}\left(3 x^{2}-4 x+1\right) d x$ .

Find the derivative of $f(x)=3x(2-x)$ .

Find $\frac{d y}{d x}$ for $x^{3}-x y+y^{4}=37$ and the tangent line at $(3,2)$ .

Berechnen Sie die Integrale: a) $\int_{1}^{2} 4 x^{3} dx$ b) $\int_{-1}^{1}(9 x^{2}-1) dx$

A ball dropped from a building takes 4 s to hit the ground. What is the height of the building? Use $h = \frac{1}{2} g t^2$ .

Trova il valore di a tale che $\lim _{x \rightarrow 0} \frac{\ln (3-x)-\ln 3}{a x}=-\frac{1}{6}$ .

Find the derivative of $f(x)=\frac{4}{(2-x)^{2}}$ .

Wachstumsgeschwindigkeit eines Baumes: a) Was bedeutet $\int_{10}^{50} v(t) d t$ ? Berechne das Integral. b) Höhe eines 10 Jahre alten Baumes nach 10 Jahren?

Find $\frac{d y}{d x}$ for $x^{3}-x y+y^{4}=37$ and the tangent line at $(3,2)$ . Show your work.

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos ^{2} 2 x}{x \sin x}$

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(7)$ for the function defined by $x^{-4} \cdot z^{2}=49$ . Round to two digits.

Find the derivative of $f(x)=\sin (2 x)+4 \cos (3 x)+2 \tan (5 x)$ .

Find the derivative of $y$ from the equation $4 x^{3}+6 x y-7 y^{3}=235$ and identify coefficients $a$ to $m$ .

Find the derivative $\frac{dk}{dl}$ from $k^9 + l^9 = W$ and express it as $\frac{d}{dl} k = A\left(\frac{k}{l}\right)^{a}+B(k \cdot l)^{b}+C$ . Determine coefficients $A$ , $a$ , $B$ , $b$ , and $C$ .

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(3)$ for the function defined by $x^{-2} \cdot z^{3}=27$ . Round to two digits.

Evaluate the integral: $\int \frac{2}{x \sqrt{25 x^{2}-64}} d x$

Find the limit as $x$ approaches $-\infty$ : $\lim _{x \rightarrow-\infty} \frac{9 x-15}{x^{3}+15 x-15}$ .

Find first and second order approximations of $f(x)=9 x^{5}$ around $x_{0}=-1$ . Determine coefficients $a$ , $b$ , and $c$ .

Bestimme die Stellen, an denen die Funktion $f(x)=x^{3}+\frac{5}{2} x^{2}+1$ eine horizontale Tangente ( $m_{t}=0$ ) hat.

Find the marginal cost function for $C(x)=170+5.3 x-0.01 x^{2}$ . What is $C^{\prime}(x)=?$

Find the marginal revenue function for $R(x)=x(26-0.04 x)$ . What is $R^{\prime}(x)$ ?

Set up an integral for the arc length of $y=x^{5}$ from $x=2$ to $x=3$ . Do not evaluate it.

Find $\hat{x}_{0}$ and $\tilde{x}_{0}$ where $L(x)=8x+15$ approximates $f(x)=9x^2+26x+24$ and $g(x)=5\ln(5x)-17x+20$ .

Berechnen Sie die ersten 2 Ableitungen der Funktion $f(t)=5+20 t \cdot e^{-0,2 t}$ zur Medikamenteneinstellung.

Set up an integral for the surface area of $y=x^{5}$ from $x=2$ to $x=3$ when revolved around the x-axis.