Calculus

Problem 4201

Untersuchen Sie die Funktion $f(x)=2 x \cdot e^{-x}$ auf Nullstellen, Extrema, Wendepunkte und ihr Verhalten für $x \rightarrow \infty$ und $x \rightarrow -\infty$ . Zeichnen Sie den Graphen für $-0,5 \leq x \leq 3$ .

See Solution

Problem 4202

Hilf bei diesen Aufgaben:
1. Ableitungen: a) $a(x)=e^{5x}$ , b) $b(x)=2^{x}$ , c) $c(x)=4^{x}+7x$ , d) $d(x)=2x \cdot 3^{x}$ .
2. Lösungen: a) $e^{x}=5$ , b) $e^{x} \cdot \frac{1}{4}=12$ , c) $30=(x+3)(e^{3x}-1)$ .
3. Skizziere $f(x)=e^{x}$ und erkläre, warum $e^{x}=-2$ und $x=\ln(-2)$ keine Lösungen haben.
4. Ordne die Graphen: $f$ und $f'$ , $g$ und $g'$ .

See Solution

Problem 4203

Calculate the integral of the function: $\int 6 x^{2} dx$ .

See Solution

Problem 4204

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{2+10 x^{2}}}{8+7 x}$ .

See Solution

Problem 4205

Bestimme die Flächeninhalte unter den Graphen der Funktionen: a) $f(x)=x^{2}-2$ für $I=[-2;-1]$ c) $f(x)=2x^{2}-2x$ für $I=[-1;3]$ ohne GTR.

See Solution

Problem 4206

Schüttkegel: Gegeben ist $f(x)=10,6 x^{2}$ . a) Zeichne $f$ für $[0;4]$ . b) Was bedeutet der Flächeninhalt? c) Berechne die Masse für $x=4$ .

See Solution

Problem 4207

Find the derivatives of these functions: a. $f(x)=5 x^{10}+4 x^{3}-2 x^{2}+4 x+5$ b. $f(x)=5 x^{2}+4 \sqrt{x}-\frac{2}{x}-10$ c. $f(x)=\sqrt{x}+\sqrt[3]{x}+1$ d. $f(x)=\sqrt{4 x^{3}+x+1}$ e. $f(x)=\left(x^{4}+4 x+2\right)^{3}$ f. $f(x)=\left(x^{5}+2 x\right)^{1 / 3}$

See Solution

Problem 4208

A diver on a $10 \mathrm{~m}$ platform has height $h(t)=10+2t-4.9t^{2}$ .
a) Find when the diver hits the water. b) Estimate the height change rate as the diver enters the water.

See Solution

Problem 4209

Find the derivatives of these functions: a. $f(x)=5 x^{10}+4 x^{3}-2 x^{2}+4 x+5$ b. $f(x)=5 x^{2}+4 \sqrt{x}-\frac{2}{x}-10$ c. $f(x)=\sqrt{x}+\sqrt[3]{x}+1$ d. $f(x)=\sqrt{4 x^{3}+x+1}$ e. $f(x)=\left(x^{4}+4 x+2\right)^{3}$ f. $f(x)=\left(x^{5}+2 x\right)^{1 / 3}$ g. $f(x)=\sin ^{5}(4 x)$

See Solution

Problem 4210

Find the derivatives of these functions: a. $f(x)=5 x^{6}+4 x^{3}-2 x^{2}+4 x+5$ b. $f(x)=5 x^{2}+4 \sqrt{x}-\frac{2}{x}-10$ c. $f(x)=\sqrt{x}+\sqrt{x}+1$ d. $f(x)=\sqrt{4 x^{3}+x+1}$ e. $f(x)=\left(x^{4}+4 x+2\right)^{3}$ f. $f(x)=\left(x^{3}+2 x\right)^{17}$ g. $f(x)=\sin ^{3}(4 x)$

See Solution

Problem 4211

Untersuchen Sie die Funktion $f(x)=2x \cdot e^{-x}$ auf Nullstellen, Extrema, Wendepunkte und Verhalten für $x \to \pm\infty$ . Graph für $-0,5 \leq x \leq 3$ zeichnen.

See Solution

Problem 4212

Find the derivatives: $f'(x)$ for $f(x)=x^{8}$ , $g'(x)$ for $g(x)=-6 x^{2}$ , and $h'(x)$ for $h(x)=\frac{1}{x^{3}}$ .

See Solution

Problem 4213

Find the derivative of the function $f(x) = -5 x^{8} + 7 x^{4}$ .

See Solution

Problem 4214

Find the linearization of $f(x)=-e^{x}$ at $x=0$ . Use it to approximate $f(0.9)$ and state if it's an overestimate or underestimate.

See Solution

Problem 4215

Given the equation $x^{-5} \cdot z^{3}=125$ , find:
(a) $z(1)=$ (b) $z'(1)=$ (c) $z''(1)=$ (d) $x'(5)=$

See Solution

Problem 4216

Find the derivative of $y$ from $4 x^{5}+4 x y-8 y^{2}=179$ and identify coefficients $a$ to $m$ . Round answers to two digits.

See Solution

Problem 4217

Find the linear approximation of $f(3.5)$ using $f(4)=8$ and $f^{\prime}(4)=0.7$ . Is it an overestimate or underestimate?

See Solution

Problem 4218

Find the linear approximation of $f(x)=\sqrt{x}$ at $x=9$ and use it to estimate $\sqrt{8}$ and $\sqrt{11}$ .

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Problem 4219

Ein Glas Tee kühlt ab. Gegeben sind die Funktionen $f(t)=20+75 \cdot e^{-0.11t}$ und $g(t)=20+65 \cdot e^{-0.08t}$ .
a) Skizzieren Sie $f(t)$ . b) Bestimmen Sie $f(0)$ und $f(5)$ . c) Finde $t$ für $f(t)=50$ und die minimale Temperatur. d) Wann sind $f(t)$ und $g(t)$ gleich?

See Solution

Problem 4220

Find the tangent line $y=m x+b$ to $f(x)=10 x+1-13 e^{x}$ at $(0,-12)$ . Calculate $m$ and $b$ .

See Solution

Problem 4221

Find the derivative of $y$ from $3 x^{5}+6 x y-6 y^{2}=185$ and identify coefficients in the expression for $\frac{dy}{dx}$ .

See Solution

Problem 4222

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(8)$ for the equation $x^{-8} \cdot z^{3}=512$ . Round answers to two digits.

See Solution

Problem 4223

Find the tangent line equation for the function $f(x)=-7 x-2+8 e^{x}$ at the point $(0,6)$ .

See Solution

Problem 4224

Given $x^{-8} \cdot z^{2}=81$ , find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(9)$ . Round answers to two digits.

See Solution

Problem 4225

Find the derivative of the function $f(x)=8 e^{x}+x^{e}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 4226

Find the derivative of $f(x)=7^{x}$ . What is $f^{\prime}(x)=?$

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Problem 4227

Find the derivative of $y$ from $2 x^{4}+6 x y-4 y^{4}=260$ and identify coefficients $a$ to $m$ . Round answers to two digits.

See Solution

Problem 4228

Find the derivative of the function $f(x)=7 e^{x}-5 x^{5}+33$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 4229

Find $z$ for $x=1$ from $x^{-5} \cdot z^{3}=216$ , then find $z'$ , $z''$ , and $x'$ at $z=6$ . Round to two digits.

See Solution

Problem 4230

Find the derivative of $y$ from $3x^4 + 8xy - 8y^4 = 271$ and identify coefficients $a, b, c, d, e, f, g, h, j, k, l, m$ .

See Solution

Problem 4231

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

See Solution

Problem 4232

Find the derivative of $y$ from $4x^2 + 7xy - 3y^2 = 163$ and identify coefficients $a$ to $m$ . Round answers to two digits.

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Problem 4233

Given the equation $x^{-4} \cdot z^{3}=27$ , find:
(a) $z(1)=$ (b) $z'(1)=$ (c) $z''(1)=$ (d) $x'(3)=$

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Problem 4234

Given $f(x)=7 x^{3}-3 e^{x}$ , find $f^{\prime}(x)$ , $f^{\prime}(2)$ , $f^{\prime \prime}(x)$ , and $f^{\prime \prime}(2)$ .

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Problem 4235

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(4)$ for the equation $x^{-8} \cdot z^{3}=64$ . Round answers to two digits.

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Problem 4236

Find the derivative of $f(x)=4 \sqrt{x}(x^{3}-8 \sqrt{x}+7)$ and calculate $f^{\prime}(3)$ (rounded to the nearest tenth).

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Problem 4237

Find the slope of the tangent line to the parabola $y=3 x^{2}+2 x+3$ at $(-4,43)$ and write the line as $y=m x+b$ .

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Problem 4238

Find the derivative of $y$ from the equation $3 x^{4}+3 x y-7 y^{3}=279$ and identify coefficients $a$ to $m$ .

See Solution

Problem 4239

Find the derivative of the function $7 x^{1/3} + 5 x^{-10}$ .

See Solution

Problem 4240

Find the derivative of $f(x)=\frac{-40 x^{6}+40 x^{4}+10}{5 x^{2}}$ without negative exponents. Simplify. $f^{\prime}(x)=$

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Problem 4241

Find the derivative of $f(x)=\frac{-90 x^{6}-63 x^{9}-45}{9 x^{2}}$ without negative exponents. Simplify. $f^{\prime}(x)=$

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Problem 4242

Find the derivative of $f(x)=3+\frac{8}{x}+\frac{4}{x^{2}}$ and calculate $f^{\prime}(4)$ rounded to the nearest hundredth.

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Problem 4243

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(4)$ for the equation $x^{-9} \cdot z^{3}=64$ . Round answers to two digits.

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Problem 4244

Find the derivative of $f(x)=\frac{1}{x^{2}-49}$ using the definition. Also, state the domains of $f$ and $f'$ .

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Problem 4245

Find the velocity and acceleration of the particle given by $s(t)=2 t^{3}+6 t+9$ at $t=3$ seconds.

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Problem 4246

Find the derivative of the function defined by $5 x^{4}+6 x y-7 y^{4}=139$ and identify coefficients in the expression for $\frac{d}{d x} y$ .

See Solution

Problem 4247

Find the first and second derivatives of $f(x)=x^{7}-5 x^{5}+3 x^{3}-4 x-2$ and evaluate them at $x=4$ .

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Problem 4248

Die Funktion $f(t)=-0,32 t^{3}+4,8 t^{2}+18,4$ beschreibt die Temperatur eines Backofens.
a) Finde den Zeitpunkt der stärksten Temperaturänderung. b) Bestimme die Temperatur und Änderungsrate zu diesem Zeitpunkt.

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Problem 4249

What is the derivative of $\cos x \sin x$ with respect to $x$ ?

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Problem 4250

Find $z$ at $x=1$ , $z'$ at $x=1$ , $z''$ at $x=1$ , and $x'$ at $z=8$ from $x^{-4} \cdot z^{2}=64$ .

See Solution

Problem 4251

Find the derivative of $y$ from $2 x^{5}+5 x y-8 y^{2}=122$ and determine coefficients $a$ to $m$ . Round answers to two digits.

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Problem 4252

Find the derivative of $F(v)=\frac{v}{v+6}$ and state the domains of $F$ and $F'(v)$ in interval notation.

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Problem 4253

Find first and second-order approximations of $f(x)=5x^{4}$ around $x_{0}=-2$ . Determine coefficients $a$ , $b$ , and $c$ .

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Problem 4254

Calculate the limit: $\lim _{h \rightarrow 0^{+}} \frac{\sqrt{h^{2}+4 h+13}-\sqrt{13}}{h}$ .

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Problem 4255

Find the limit of the function $f(x)=\left(\frac{7 x}{x+1}\right)\left(\frac{3 x+8}{x^{2}+x}\right)$ as $x$ approaches 8.

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Problem 4256

Find the limit: $\lim _{t \rightarrow 0} \frac{\sin k t}{t}$ using $\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1$ .

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Problem 4257

Find linear and quadratic approximations of $f(x)=5x^4$ at $x_{0}=-2$ , then estimate $f(-1.9)$ .

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Problem 4258

Find the limit: $\lim _{\theta \rightarrow 0} \frac{9 \sin \sqrt{6} \theta}{\sqrt{6} \theta}$ . What is the answer?

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Problem 4259

Find the limit: $\lim _{y \rightarrow 0} \frac{\sin 17 y}{18 y}$ . Is it A. $\square$ or B. Does not exist?

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Problem 4260

Find the derivative $f'(x)$ of the function $f(x)=-6 x^{4} \sqrt{x}-\frac{2}{x^{3} \sqrt{x}}$ .

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Problem 4261

Find the limit: $\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\sin (7 \theta)}$ . What is the answer? A. $\square$ B. Limit does not exist.

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Problem 4262

Find the limit: $\lim _{x \rightarrow 0} \frac{x \csc 7 x}{\cos 9 x}=\square$ or it doesn't exist.

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Problem 4263

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

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Problem 4264

Find the linear approximation of $f(x)=e^{3x}+5x$ at $x_0=2$ and $x_0=2.3$ . Then approximate $f(2.3)$ and $f(2)$ .

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Problem 4265

Find the linear approximation of $f(x)=e^{3x}+5x$ at $x_0=2$ as $L(x)=A+B(x-2)$ . Calculate $A$ , $B$ , and $f(2.3) \approx L(2.3)$ .

See Solution

Problem 4266

Find the limit: $\lim _{\theta \rightarrow 0} \frac{\tan \theta}{\theta^{2} \cot 10 \theta}$ . A. = $\square$ (Type an integer or fraction.) B. Limit does not exist.

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Problem 4267

Given $f(x)=3 \sqrt{x}(x^{3}-7 \sqrt{x}+4)$ , find $f^{\prime}(x)$ and then $f^{\prime}(3)$ (round to the nearest tenth).

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Problem 4268

If $\lim _{x \rightarrow c} f(x)$ exists, can you find it using $\lim _{x \rightarrow c^{+}} f(x)$ ? Explain. Choose A, B, C, or D.

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Problem 4269

Find $\int_{1}^{e} \frac{1}{x}(3+2 \ln x)^{3} d x$ using $u=3+2 \ln x$ . Also, sketch $y^{2}=9(x+1)$ and $y=-x+9$ , then find the area between them.

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Problem 4270

Find the derivative $f'(x)$ of $f(x)=6\sqrt{x}(x^3-4\sqrt{x}+4)$ and calculate $f'(3)$ (round to the nearest tenth).

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Problem 4271

Graph the function $f(x)=\left\{\begin{array}{ll}1-x^{2}, & x \neq 1 \\ 2, & x=1\end{array}\right.$ and find the limits as $x \rightarrow 1^{-}$ and $x \rightarrow 1^{+}$ . Does $\lim _{x \rightarrow 1} f(x)$ exist?

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Problem 4272

Find the derivative of the function $f(x)=\frac{2}{x^{2}}$ at the point $x=5$ . What is $f^{\prime}(5)$ ?

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Problem 4273

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

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Problem 4274

Find the derivative $f^{\prime}(7)$ given that $f(x)=15$ .

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Problem 4275

Differentiate the equation $8 x^{6}+y^{7}=3 x$ implicitly to find $y^{\prime}$ , then solve for $y$ and differentiate again. Check consistency.

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Problem 4276

Differentiate the equation $8 x^{6}+y^{7}=3 x$ implicitly to find $y^{\prime}$ .

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Problem 4277

Find the derivative $f'(x)$ of $f(x)=5 \sqrt{x}(x^{3}-4 \sqrt{x}+5)$ and calculate $f'(2)$ (round to the nearest tenth).

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Problem 4278

Find the linear approximation of $f(x)=x^{m}$ near $x_{0}=1$ . Use it to estimate (a) $0.97^{23}$ and (b) $\sqrt[8]{1.08}$ .

See Solution

Problem 4279

Find first and second-order approximations of $f(x) = 6x^6$ at $x_0 = 2$ . Round answers to two digits.

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Problem 4280

If a person spends 1.00 hours on a report and gets a 59, with a derivative of 36, what grade for 1.20 hours?

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Problem 4281

Find the first-order approximation of $f(x)=6 x^{6}$ around $x_{0}=2$ and express it as $f(x) \approx a+b \cdot(x-2)$ . What are $a$ and $b$ ? Then rewrite it as $f(x) \approx a+b x$ . What are the new $a$ and $b$ ?

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Problem 4282

Find the second-order approximation of $f(x)=6x^6$ near $x_0=2$ . Determine $a, b, c$ for both forms:
1. $f(x) \approx a + b \cdot (x-2) + \frac{1}{2} c \cdot (x-2)^2$
2. $f(x) \approx a + b x + c x^2$

See Solution

Problem 4283

Find $f(a)$ , $f(a+h)$ , and the difference quotient $\frac{f(a+h)-f(a)}{h}$ for $f(x)=2x^2+7$ .

See Solution

Problem 4284

Find first and second-order approximations of $f(x)=6x^6$ around $x_0=2$ . Use $x=2.1$ for evaluations.

See Solution

Problem 4285

Differentiate $8 x^{6}+y^{7}=3 x$ to find $y^{\prime}$ ; solve for $y$ and check consistency of derivatives.

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Problem 4286

Find the linear approximation of $f(x)=x^{4}+3 \cdot x$ at $x_{0}=3$ and $x_{0}=3.3$ , and approximate $f(3.3)$ and $f(3)$ .

See Solution

Problem 4287

Given $\sqrt{x}+\sqrt{y}=8$ , find $y'$ using implicit differentiation and then solve for $y$ to find $y'$ in terms of $x$ .

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Problem 4288

Find the points $\hat{x}_{0}$ and $\tilde{x}_{0}$ for the linear functions $L(x)=9x+14$ approximating $f(x)$ and $g(x)$ . Round to two digits.

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Problem 4289

Find points $\hat{x}_{0}$ and $\tilde{x}_{0}$ where $L(x)=9x+14$ approximates $f(x)=9x^{4}-279x+446$ and $g(x)=8\ln(2x)-7x+22$ . Round to two digits.

See Solution

Problem 4290

Find the first and second-order approximations of $f(x)=6 x^{6}$ around $x_{0}=2$ and evaluate at $x=2.1$ .

See Solution

Problem 4291

Find the linear approximation of $f(x)=x^{m}$ near $x_{0}=1$ . Use it to estimate (a) $0.97^{25}$ and (b) $\sqrt[6]{1.15}$ .

See Solution

Problem 4292

Find $y^{\prime}$ using implicit differentiation for $\frac{5}{x}-\frac{1}{y}=8$ . Then solve for $y$ and differentiate.

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Problem 4293

Find the number of units $x$ (max 110) that minimizes the average cost per unit $\bar{C}(x) = \frac{C(x)}{x}$ for $C(x)=0.2 x^{3}-26 x^{2}+1516 x$ .

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Problem 4294

Differentiate implicitly the equation $e^{x / y} = 8x - y$ to find $\frac{dy}{dx}$ .

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Problem 4295

Find $f^{\prime}(1)$ given $f(x)+x^{2}[f(x)]^{3}=10$ and $f(1)=2$ . $f^{\prime}(1)=$

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Problem 4296

Differentiate $x^{2}+4xy+8y^{2}=20$ to find $y'$ and the tangent line at $(2,1)$ .

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Problem 4297

Find the slope $m$ of the tangent line to the curve $y=3 x^{2}-11 x+1$ at the point $(4,5)$ and its equation.

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Problem 4298

Bestimmen Sie die Extrempunkte von $f$ mit dem zweiten Extremstellenkriterium für die Funktionen a) bis d).

See Solution

Problem 4299

Find the difference quotient, $\frac{f(x+h)-f(x)}{h}$ , for the function $f(x)=-7 x^{2}-6 x-7$ .

See Solution

Problem 4300

Berechne die Steigung von $g(x)=x^{3}+x$ an Punkt $P(\boldsymbol{\Lambda} / 2)$ mit der $h$ -Methode.

See Solution

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Calculus

Problem 4201

Problem 4202

Problem 4203

Calculate the integral of the function: ∫6x2dx\int 6 x^{2} dx∫6x2dx.

Problem 4204

Find the limit: lim⁡x→∞2+10x28+7x\lim _{x \rightarrow \infty} \frac{\sqrt{2+10 x^{2}}}{8+7 x}limx→∞​8+7x2+10x2​​.

Problem 4205

Bestimme die Flächeninhalte unter den Graphen der Funktionen: a) f(x)=x2−2f(x)=x^{2}-2f(x)=x2−2 für I=[−2;−1]I=[-2;-1]I=[−2;−1] c) f(x)=2x2−2xf(x)=2x^{2}-2xf(x)=2x2−2x für I=[−1;3]I=[-1;3]I=[−1;3] ohne GTR.

Problem 4206

Schüttkegel: Gegeben ist f(x)=10,6x2f(x)=10,6 x^{2}f(x)=10,6x2. a) Zeichne fff für [0;4][0;4][0;4]. b) Was bedeutet der Flächeninhalt? c) Berechne die Masse für x=4x=4x=4.

Problem 4207

Problem 4208

A diver on a 10 m10 \mathrm{~m}10 m platform has height h(t)=10+2t−4.9t2h(t)=10+2t-4.9t^{2}h(t)=10+2t−4.9t2. a) Find when the diver hits the water. b) Estimate the height change rate as the diver enters the water.

Problem 4209

Problem 4210

Problem 4211

Untersuchen Sie die Funktion f(x)=2x⋅e−xf(x)=2x \cdot e^{-x}f(x)=2x⋅e−x auf Nullstellen, Extrema, Wendepunkte und Verhalten für x→±∞x \to \pm\inftyx→±∞. Graph für −0,5≤x≤3-0,5 \leq x \leq 3−0,5≤x≤3 zeichnen.

Problem 4212

Find the derivatives: f′(x)f'(x)f′(x) for f(x)=x8f(x)=x^{8}f(x)=x8, g′(x)g'(x)g′(x) for g(x)=−6x2g(x)=-6 x^{2}g(x)=−6x2, and h′(x)h'(x)h′(x) for h(x)=1x3h(x)=\frac{1}{x^{3}}h(x)=x31​.

Problem 4213

Find the derivative of the function f(x)=−5x8+7x4f(x) = -5 x^{8} + 7 x^{4}f(x)=−5x8+7x4.

Problem 4214

Find the linearization of f(x)=−exf(x)=-e^{x}f(x)=−ex at x=0x=0x=0. Use it to approximate f(0.9)f(0.9)f(0.9) and state if it's an overestimate or underestimate.

Problem 4215

Given the equation x−5⋅z3=125x^{-5} \cdot z^{3}=125x−5⋅z3=125, find:(a) z(1)=z(1)=z(1)= (b) z′(1)=z'(1)=z′(1)= (c) z′′(1)=z''(1)=z′′(1)= (d) x′(5)=x'(5)=x′(5)=

Problem 4216

Find the derivative of yyy from 4x5+4xy−8y2=1794 x^{5}+4 x y-8 y^{2}=1794x5+4xy−8y2=179 and identify coefficients aaa to mmm. Round answers to two digits.

Problem 4217

Find the linear approximation of f(3.5)f(3.5)f(3.5) using f(4)=8f(4)=8f(4)=8 and f′(4)=0.7f^{\prime}(4)=0.7f′(4)=0.7. Is it an overestimate or underestimate?

Problem 4218

Find the linear approximation of f(x)=xf(x)=\sqrt{x}f(x)=x​ at x=9x=9x=9 and use it to estimate 8\sqrt{8}8​ and 11\sqrt{11}11​.

Problem 4219

Problem 4220

Find the tangent line y=mx+by=m x+by=mx+b to f(x)=10x+1−13exf(x)=10 x+1-13 e^{x}f(x)=10x+1−13ex at (0,−12)(0,-12)(0,−12). Calculate mmm and bbb.

Problem 4221

Find the derivative of yyy from 3x5+6xy−6y2=1853 x^{5}+6 x y-6 y^{2}=1853x5+6xy−6y2=185 and identify coefficients in the expression for dydx\frac{dy}{dx}dxdy​.

Problem 4222

Find z(1)z(1)z(1), z′(1)z'(1)z′(1), z′′(1)z''(1)z′′(1), and x′(8)x'(8)x′(8) for the equation x−8⋅z3=512x^{-8} \cdot z^{3}=512x−8⋅z3=512. Round answers to two digits.

Problem 4223

Find the tangent line equation for the function f(x)=−7x−2+8exf(x)=-7 x-2+8 e^{x}f(x)=−7x−2+8ex at the point (0,6)(0,6)(0,6).

Problem 4224

Given x−8⋅z2=81x^{-8} \cdot z^{2}=81x−8⋅z2=81, find z(1)z(1)z(1), z′(1)z'(1)z′(1), z′′(1)z''(1)z′′(1), and x′(9)x'(9)x′(9). Round answers to two digits.

Problem 4225

Find the derivative of the function f(x)=8ex+xef(x)=8 e^{x}+x^{e}f(x)=8ex+xe. What is f′(x)f^{\prime}(x)f′(x)?

Problem 4226

Find the derivative of f(x)=7xf(x)=7^{x}f(x)=7x. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 4227

Find the derivative of yyy from 2x4+6xy−4y4=2602 x^{4}+6 x y-4 y^{4}=2602x4+6xy−4y4=260 and identify coefficients aaa to mmm. Round answers to two digits.

Problem 4228

Find the derivative of the function f(x)=7ex−5x5+33f(x)=7 e^{x}-5 x^{5}+33f(x)=7ex−5x5+33. What is f′(x)f^{\prime}(x)f′(x)?

Problem 4229

Find zzz for x=1x=1x=1 from x−5⋅z3=216x^{-5} \cdot z^{3}=216x−5⋅z3=216, then find z′z'z′, z′′z''z′′, and x′x'x′ at z=6z=6z=6. Round to two digits.

Problem 4230

Find the derivative of yyy from 3x4+8xy−8y4=2713x^4 + 8xy - 8y^4 = 2713x4+8xy−8y4=271 and identify coefficients a,b,c,d,e,f,g,h,j,k,l,ma, b, c, d, e, f, g, h, j, k, l, ma,b,c,d,e,f,g,h,j,k,l,m.

Problem 4231

Find the derivative of the function f(x)=5xf(x)=5^{x}f(x)=5x. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 4232

Find the derivative of yyy from 4x2+7xy−3y2=1634x^2 + 7xy - 3y^2 = 1634x2+7xy−3y2=163 and identify coefficients aaa to mmm. Round answers to two digits.

Problem 4233

Given the equation x−4⋅z3=27x^{-4} \cdot z^{3}=27x−4⋅z3=27, find:(a) z(1)=z(1)=z(1)= (b) z′(1)=z'(1)=z′(1)= (c) z′′(1)=z''(1)=z′′(1)= (d) x′(3)=x'(3)=x′(3)=

Problem 4234

Given f(x)=7x3−3exf(x)=7 x^{3}-3 e^{x}f(x)=7x3−3ex, find f′(x)f^{\prime}(x)f′(x), f′(2)f^{\prime}(2)f′(2), f′′(x)f^{\prime \prime}(x)f′′(x), and f′′(2)f^{\prime \prime}(2)f′′(2).

Problem 4235

Find z(1)z(1)z(1), z′(1)z'(1)z′(1), z′′(1)z''(1)z′′(1), and x′(4)x'(4)x′(4) for the equation x−8⋅z3=64x^{-8} \cdot z^{3}=64x−8⋅z3=64. Round answers to two digits.

Problem 4236

Find the derivative of f(x)=4x(x3−8x+7)f(x)=4 \sqrt{x}(x^{3}-8 \sqrt{x}+7)f(x)=4x​(x3−8x​+7) and calculate f′(3)f^{\prime}(3)f′(3) (rounded to the nearest tenth).

Problem 4237

Find the slope of the tangent line to the parabola y=3x2+2x+3y=3 x^{2}+2 x+3y=3x2+2x+3 at (−4,43)(-4,43)(−4,43) and write the line as y=mx+by=m x+by=mx+b.

Problem 4238

Find the derivative of yyy from the equation 3x4+3xy−7y3=2793 x^{4}+3 x y-7 y^{3}=2793x4+3xy−7y3=279 and identify coefficients aaa to mmm.

Problem 4239

Find the derivative of the function 7x1/3+5x−107 x^{1/3} + 5 x^{-10}7x1/3+5x−10.

Problem 4240

Find the derivative of f(x)=−40x6+40x4+105x2f(x)=\frac{-40 x^{6}+40 x^{4}+10}{5 x^{2}}f(x)=5x2−40x6+40x4+10​ without negative exponents. Simplify. f′(x)=f^{\prime}(x)=f′(x)=

Problem 4241

Find the derivative of f(x)=−90x6−63x9−459x2f(x)=\frac{-90 x^{6}-63 x^{9}-45}{9 x^{2}}f(x)=9x2−90x6−63x9−45​ without negative exponents. Simplify. f′(x)= f^{\prime}(x)= f′(x)=

Problem 4242

Find the derivative of f(x)=3+8x+4x2f(x)=3+\frac{8}{x}+\frac{4}{x^{2}}f(x)=3+x8​+x24​ and calculate f′(4)f^{\prime}(4)f′(4) rounded to the nearest hundredth.

Problem 4243

Calculate the integral of the function: $\int 6 x^{2} dx$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{2+10 x^{2}}}{8+7 x}$ .

Bestimme die Flächeninhalte unter den Graphen der Funktionen: a) $f(x)=x^{2}-2$ für $I=[-2;-1]$ c) $f(x)=2x^{2}-2x$ für $I=[-1;3]$ ohne GTR.

Schüttkegel: Gegeben ist $f(x)=10,6 x^{2}$ . a) Zeichne $f$ für $[0;4]$ . b) Was bedeutet der Flächeninhalt? c) Berechne die Masse für $x=4$ .

A diver on a $10 \mathrm{~m}$ platform has height $h(t)=10+2t-4.9t^{2}$ .
a) Find when the diver hits the water. b) Estimate the height change rate as the diver enters the water.

Untersuchen Sie die Funktion $f(x)=2x \cdot e^{-x}$ auf Nullstellen, Extrema, Wendepunkte und Verhalten für $x \to \pm\infty$ . Graph für $-0,5 \leq x \leq 3$ zeichnen.

Find the derivatives: $f'(x)$ for $f(x)=x^{8}$ , $g'(x)$ for $g(x)=-6 x^{2}$ , and $h'(x)$ for $h(x)=\frac{1}{x^{3}}$ .

Find the derivative of the function $f(x) = -5 x^{8} + 7 x^{4}$ .

Find the linearization of $f(x)=-e^{x}$ at $x=0$ . Use it to approximate $f(0.9)$ and state if it's an overestimate or underestimate.

Given the equation $x^{-5} \cdot z^{3}=125$ , find:
(a) $z(1)=$ (b) $z'(1)=$ (c) $z''(1)=$ (d) $x'(5)=$

Find the derivative of $y$ from $4 x^{5}+4 x y-8 y^{2}=179$ and identify coefficients $a$ to $m$ . Round answers to two digits.

Find the linear approximation of $f(3.5)$ using $f(4)=8$ and $f^{\prime}(4)=0.7$ . Is it an overestimate or underestimate?

Find the linear approximation of $f(x)=\sqrt{x}$ at $x=9$ and use it to estimate $\sqrt{8}$ and $\sqrt{11}$ .

Find the tangent line $y=m x+b$ to $f(x)=10 x+1-13 e^{x}$ at $(0,-12)$ . Calculate $m$ and $b$ .

Find the derivative of $y$ from $3 x^{5}+6 x y-6 y^{2}=185$ and identify coefficients in the expression for $\frac{dy}{dx}$ .

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(8)$ for the equation $x^{-8} \cdot z^{3}=512$ . Round answers to two digits.

Find the tangent line equation for the function $f(x)=-7 x-2+8 e^{x}$ at the point $(0,6)$ .

Given $x^{-8} \cdot z^{2}=81$ , find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(9)$ . Round answers to two digits.

Find the derivative of the function $f(x)=8 e^{x}+x^{e}$ . What is $f^{\prime}(x)$ ?

Find the derivative of $f(x)=7^{x}$ . What is $f^{\prime}(x)=?$

Find the derivative of $y$ from $2 x^{4}+6 x y-4 y^{4}=260$ and identify coefficients $a$ to $m$ . Round answers to two digits.

Find the derivative of the function $f(x)=7 e^{x}-5 x^{5}+33$ . What is $f^{\prime}(x)$ ?

Find $z$ for $x=1$ from $x^{-5} \cdot z^{3}=216$ , then find $z'$ , $z''$ , and $x'$ at $z=6$ . Round to two digits.

Find the derivative of $y$ from $3x^4 + 8xy - 8y^4 = 271$ and identify coefficients $a, b, c, d, e, f, g, h, j, k, l, m$ .

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

Find the derivative of $y$ from $4x^2 + 7xy - 3y^2 = 163$ and identify coefficients $a$ to $m$ . Round answers to two digits.

Given the equation $x^{-4} \cdot z^{3}=27$ , find:
(a) $z(1)=$ (b) $z'(1)=$ (c) $z''(1)=$ (d) $x'(3)=$

Given $f(x)=7 x^{3}-3 e^{x}$ , find $f^{\prime}(x)$ , $f^{\prime}(2)$ , $f^{\prime \prime}(x)$ , and $f^{\prime \prime}(2)$ .

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(4)$ for the equation $x^{-8} \cdot z^{3}=64$ . Round answers to two digits.

Find the derivative of $f(x)=4 \sqrt{x}(x^{3}-8 \sqrt{x}+7)$ and calculate $f^{\prime}(3)$ (rounded to the nearest tenth).

Find the slope of the tangent line to the parabola $y=3 x^{2}+2 x+3$ at $(-4,43)$ and write the line as $y=m x+b$ .

Find the derivative of $y$ from the equation $3 x^{4}+3 x y-7 y^{3}=279$ and identify coefficients $a$ to $m$ .

Find the derivative of the function $7 x^{1/3} + 5 x^{-10}$ .

Find the derivative of $f(x)=\frac{-40 x^{6}+40 x^{4}+10}{5 x^{2}}$ without negative exponents. Simplify. $f^{\prime}(x)=$

Find the derivative of $f(x)=\frac{-90 x^{6}-63 x^{9}-45}{9 x^{2}}$ without negative exponents. Simplify. $f^{\prime}(x)=$

Find the derivative of $f(x)=3+\frac{8}{x}+\frac{4}{x^{2}}$ and calculate $f^{\prime}(4)$ rounded to the nearest hundredth.

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(4)$ for the equation $x^{-9} \cdot z^{3}=64$ . Round answers to two digits.

Find the derivative of $f(x)=\frac{1}{x^{2}-49}$ using the definition. Also, state the domains of $f$ and $f'$ .

Find the velocity and acceleration of the particle given by $s(t)=2 t^{3}+6 t+9$ at $t=3$ seconds.

Find the derivative of the function defined by $5 x^{4}+6 x y-7 y^{4}=139$ and identify coefficients in the expression for $\frac{d}{d x} y$ .

Find the first and second derivatives of $f(x)=x^{7}-5 x^{5}+3 x^{3}-4 x-2$ and evaluate them at $x=4$ .

Die Funktion $f(t)=-0,32 t^{3}+4,8 t^{2}+18,4$ beschreibt die Temperatur eines Backofens.
a) Finde den Zeitpunkt der stärksten Temperaturänderung. b) Bestimme die Temperatur und Änderungsrate zu diesem Zeitpunkt.

What is the derivative of $\cos x \sin x$ with respect to $x$ ?

Find $z$ at $x=1$ , $z'$ at $x=1$ , $z''$ at $x=1$ , and $x'$ at $z=8$ from $x^{-4} \cdot z^{2}=64$ .

Find the derivative of $y$ from $2 x^{5}+5 x y-8 y^{2}=122$ and determine coefficients $a$ to $m$ . Round answers to two digits.

Find the derivative of $F(v)=\frac{v}{v+6}$ and state the domains of $F$ and $F'(v)$ in interval notation.

Find first and second-order approximations of $f(x)=5x^{4}$ around $x_{0}=-2$ . Determine coefficients $a$ , $b$ , and $c$ .

Calculate the limit: $\lim _{h \rightarrow 0^{+}} \frac{\sqrt{h^{2}+4 h+13}-\sqrt{13}}{h}$ .

Find the limit of the function $f(x)=\left(\frac{7 x}{x+1}\right)\left(\frac{3 x+8}{x^{2}+x}\right)$ as $x$ approaches 8.

Find the limit: $\lim _{t \rightarrow 0} \frac{\sin k t}{t}$ using $\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1$ .

Find linear and quadratic approximations of $f(x)=5x^4$ at $x_{0}=-2$ , then estimate $f(-1.9)$ .

Find the limit: $\lim _{\theta \rightarrow 0} \frac{9 \sin \sqrt{6} \theta}{\sqrt{6} \theta}$ . What is the answer?

Find the limit: $\lim _{y \rightarrow 0} \frac{\sin 17 y}{18 y}$ . Is it A. $\square$ or B. Does not exist?

Find the derivative $f'(x)$ of the function $f(x)=-6 x^{4} \sqrt{x}-\frac{2}{x^{3} \sqrt{x}}$ .

Find the limit: $\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\sin (7 \theta)}$ . What is the answer? A. $\square$ B. Limit does not exist.

Find the limit: $\lim _{x \rightarrow 0} \frac{x \csc 7 x}{\cos 9 x}=\square$ or it doesn't exist.

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

Find the linear approximation of $f(x)=e^{3x}+5x$ at $x_0=2$ and $x_0=2.3$ . Then approximate $f(2.3)$ and $f(2)$ .

Find the linear approximation of $f(x)=e^{3x}+5x$ at $x_0=2$ as $L(x)=A+B(x-2)$ . Calculate $A$ , $B$ , and $f(2.3) \approx L(2.3)$ .

Find the limit: $\lim _{\theta \rightarrow 0} \frac{\tan \theta}{\theta^{2} \cot 10 \theta}$ . A. = $\square$ (Type an integer or fraction.) B. Limit does not exist.

Given $f(x)=3 \sqrt{x}(x^{3}-7 \sqrt{x}+4)$ , find $f^{\prime}(x)$ and then $f^{\prime}(3)$ (round to the nearest tenth).

If $\lim _{x \rightarrow c} f(x)$ exists, can you find it using $\lim _{x \rightarrow c^{+}} f(x)$ ? Explain. Choose A, B, C, or D.

Find $\int_{1}^{e} \frac{1}{x}(3+2 \ln x)^{3} d x$ using $u=3+2 \ln x$ . Also, sketch $y^{2}=9(x+1)$ and $y=-x+9$ , then find the area between them.

Find the derivative $f'(x)$ of $f(x)=6\sqrt{x}(x^3-4\sqrt{x}+4)$ and calculate $f'(3)$ (round to the nearest tenth).

Find the derivative of the function $f(x)=\frac{2}{x^{2}}$ at the point $x=5$ . What is $f^{\prime}(5)$ ?

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

Find the derivative $f^{\prime}(7)$ given that $f(x)=15$ .