Calculus

Problem 7201

Find the slope of the tangent line to $y=\frac{1}{x^{2}}-2 x^{1 / 2}$ at $x=1$ .

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Problem 7202

Bestimme alle Werte von $a$ , für die der Graph $G$ der Funktion $f_{a}(x)=\sqrt{a^{2} x}$ (für $x \geq 0$ ) rechtsgekrümmt ist.

See Solution

Problem 7203

Find the slope of the tangent line to $y=\frac{1}{x^{2}}-2 x^{1 / 2}$ at $x=1$ .

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Problem 7204

Find the derivative $f^{\prime}(x)$ for $f(x)=200 x^{4}-5 / x^{1 / 2}$ . Which option is correct?

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Problem 7205

Find the tangent line equation for the curve $y=2x^{2}-5x+1$ at the point $(3,4)$ .

See Solution

Problem 7206

Find the rock's instantaneous velocity after $t=2$ seconds using the function $s(t)=-4.9 t^{2}+400$ .

See Solution

Problem 7207

Find the derivative of $f(x)=4 \cos 2 x+\log x+x$ . What is $f'(x)$ ?

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Problem 7208

Find the limit as $x$ approaches 7 for the expression $\frac{x^{5}-1}{x-1}$ .

See Solution

Problem 7209

Find the limit as $x$ approaches 7 for the expression $\frac{x^{51}-1}{x-1}$ .

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Problem 7210

Bestimmen Sie die Ableitung von $f(x)=x^{2} \cdot \cos (1-x)$ und erläutern Sie den Prozess.

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Problem 7211

Determine the vertical and horizontal asymptotes of $f(x)=\frac{\cos x+7 \sqrt{x}}{\sqrt{x}}$ .

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Problem 7212

Bestimme die Ableitung von $f(x)=0.5x^{2} \cdot \sqrt{4-x}$ und erkläre den Prozess.

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Problem 7213

Find the end behavior of the function $y = \cosh x = \frac{e^{x} + e^{-x}}{2}$ .

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Problem 7214

Bestimmen Sie die Ableitung von $f(x)=(5 x+2)^{7} \cdot \cos (x)$ und erläutern Sie das Vorgehen.

See Solution

Problem 7215

Bestimmen Sie die Funktion $f(x)$ aus $f^{\prime}(x)=\sin (x)+(x+2) \cos (x)$ und erläutern Sie die Lösung.

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Problem 7216

Evaluate the limits $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x)$ , and find horizontal and vertical asymptotes for $f(x)=\frac{4 x^{3}+24 x^{2}+20 x}{x^{3}+5 x^{2}}$ .

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Problem 7217

Find the differential $d y$ for the function $y=\sqrt{2 x-9}$ .

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Problem 7218

Find the differential $d y$ for the function $y=5 \sin (4 x)$ .

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Problem 7219

Find the derivative of the function $f(x)=-5$ .

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Problem 7220

Find the critical numbers of $f(x)=2 x^{3}-24 x^{2}+72 x-9$ where the derivative is 0 or undefined. Identify the smaller and larger critical number.

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Problem 7221

Find $dy$ for $y=-3 \cos(7x)$ when $x=0$ and $dx=-0.3$ .

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Problem 7222

Find the derivative of the function $g(t) = 14 \sqrt{t}$ .

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Problem 7223

Find the derivative of $2 f(x) + 4 x$ at $x = 2$ .

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Problem 7224

Find $d y$ for $y=\sqrt{x+4}$ and evaluate at $x=3$ , $d x=0.3$ .

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Problem 7225

Find the critical numbers of the function $f(x)=2 x^{3}-30 x^{2}+144 x-2$ . What are the smaller and larger critical numbers?

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Problem 7226

Find the absolute max and min of $f(x)=2-3 x^{2}$ for $-3 \leq x \leq 2$ . Where do they occur?

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Problem 7227

Find the critical point $A$ for the function $f(x)=-2 x^{2}+4 x-5$ and determine if it’s LMIN, LMAX, or NEITHER.

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Problem 7228

Find the absolute maximum and minimum of $f(x)=4 x^{2}-6 x+6$ on the interval $0 \leq x \leq 9$ .

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Problem 7229

Find the derivative of the function $g(t) = 90 t^{2}$ .

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Problem 7230

Find the min and max of $f(x)=2x^3 + 12x^2 - 126x + 4$ on $[-7,4]$ using the Closed Interval Method.

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Problem 7231

Evaluate the integral $\int_{-1}^{1} \int_{-1}^{1} \int_{-1}^{1} x^{2} y^{2} z^{2} \, d x \, d y \, d z$ .

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Problem 7232

Find the critical numbers of $f(x)=-4 x^{5}+15 x^{4}+20 x^{3}+7$ and classify them with a graph.

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Problem 7233

Find the derivative of the function $f(x) = 13 x^{3} - 30 x + e^{5}$ .

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Problem 7234

Find the volume change $d V$ when a sphere's radius goes from 16 cm to 16.1 cm. Round to the nearest tenth.

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Problem 7235

A shell launched at $2,452 \mathrm{~m/s}$ at 82 degrees. How long was it in the air? Answer in minutes.

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Problem 7236

A projectile is shot upward at $88.1 \mathrm{~m/s}$ . What is its speed at the peak of its path? Answer in m/s.

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Problem 7237

Set up a definite integral using the shell method for the volume of the solid formed by rotating the region bounded by $y=\sqrt{x}$ and $x=2y$ around the line $x=7$ .

See Solution

Problem 7238

Find the derivative of the function $y=-\frac{1}{2} x^{4}+3 x^{\frac{5}{3}}+2 x$ .

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Problem 7239

A 6 ft man walks away from an 18 ft lamppost at 6 ft/sec. Find the shadow's length change rate when he's 30 ft away.

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Problem 7240

Calculate the directional derivative of $f(x, y)=\sin(x^{2} y)$ at $P=(\sqrt{\pi}, 1)$ in the direction $\mathbf{u}=\mathbf{i}$ .

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Problem 7241

Find the rate of water drainage from an inverted cone (radius 3 in, height 6 in) when water is 4 in deep and falling at -1.8 in/s. Options: -33.93 in³/s, -45.24 in³/s, -21.6 in³/s, -22.62 in³/s.

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Problem 7242

Find the derivative of the function $y=17 x^{-2}+11 x^{3}+5 x$ . What is $f^{\prime}(x)$ ?

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Problem 7243

A cone with radius 3 in and height 6 in has water 4 in deep. If the surface falls at $-1.8 \mathrm{in/s}$ , find the drainage rate.

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Problem 7244

Find the sales function $S(t)=0.04 t^{3}+0.2 t^{2}+4 t+9$ .
(A) Calculate $S^{\prime}(t)$ . (B) Determine $S(7)$ and $S^{\prime}(7)$ (two decimal places). (C) Explain $S(12)=154.92$ and $S^{\prime}(12)=26.08$ .

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Problem 7245

Estimate $f(1.02, 1.98)$ using linear approximation given $f(1,2)=3$ , $f_{x}(1,2)=4$ , and $f_{y}(1,2)=5$ .

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Problem 7246

A 6 ft man walks away from an 18 ft lamppost at 6 ft/sec. Find the shadow's length change rate when he's 30 ft away.

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Problem 7247

A marine manufacturer sells $N(x)$ boats after spending \ $x thousand on ads, where$ N(x)=990-\frac{3,840}{x}, 5 \leq x \leq 30$.
(A) Find $N^{\prime}(x)$ . (B) Calculate $N^{\prime}(10)$ and $N^{\prime}(20)$ , then interpret these results.

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Problem 7248

Find the speed of two people walking from a right angle of a triangle if the area changes at $2 \frac{\mathrm{m}^{2}}{\mathrm{~s}}$ when they are $4 \mathrm{~m}$ away.

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Problem 7249

Find the derivative $f^{\prime}(x)$ using logarithmic differentiation for $f(x)=\frac{(4 x^{2}-4) \sqrt{5 x^{3}-6}}{5 x-5}$ .

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Problem 7250

Find the derivative of the function $f(x)=\sin^{-1}\left(e^{-3x}\right)$ .

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Problem 7251

Find the $x$ -values where the function $f(x)=(x-8)^{2/5}$ is differentiable.

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Problem 7252

Find the derivative of the function: $f(x)=\frac{1}{3x-1}$ . What is $f'(x)$ ?

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Problem 7253

Find the limit: $\lim _{x \rightarrow-\infty} \frac{3 x^{5}-2 x^{2}+x}{2 x^{3}+x+1}$ .

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Problem 7254

Find the derivative of the function $f(x)=(3x-1)^{-2}$ . What is $f^{\prime}(x)$ ?

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Problem 7255

Find the derivative of the function: $f(x)=(x^{6}-x)^{6}$ . What is $f^{\prime}(x)$ ?

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Problem 7256

Determine the intervals where the function $f(x)=2(x+6)(x-2)$ is increasing.

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Problem 7257

Determine the intervals where the function $f(x)=2(x-2)^{2}-50$ is increasing.

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Problem 7258

Find the derivative of the function $s(x)=\left(\frac{2 x+8}{3 x-1}\right)^{2}$ . What is $s^{\prime}(x)=?$

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Problem 7259

Find the derivative of the function $f(t) = \left(\cos^{-1} t\right)^{17}$ .

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Problem 7260

Find the derivative of the function $f(u)=\sec ^{-1}(3 u+1)$ .

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Problem 7261

Calculate the integral of $k(x)=\frac{14}{x^{2}+49}$ .

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Problem 7262

Find the derivative of the function $f(x)=(7x-1)^{2}$ . What is $f^{\prime}(x)$ ?

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Problem 7263

Find the derivative of $f(x)=\ln(x+7)$ . What is $f'(x)$ ?

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Problem 7264

Find the derivative of $f(x) = \log_{4}(x)$ , denoted as $f'(x) = \square$ .

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Problem 7265

Find the derivative of the function $h(x)=e^{x^{6}-x+3}$ .

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Problem 7266

Find the derivative of $h(x)=e^{x^{6}}-x+3$ .

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Problem 7267

Find the derivative of the function $g(x) = 8^x$ . What is $g'(x)$ ?

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Problem 7268

Find the price $p$ that maximizes revenue given the demand $q=120-3 \cdot p^{2}$ .

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Problem 7269

Find the derivative of the function $f(x)=(x^{2}+9) \ln (x)$ . What is $f'(x)$ ?

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Problem 7270

Given demand $q=2700-p^{2}$ , find elasticity $E$ in terms of $p$ , at $p=30$ , and interpret its meaning.

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Problem 7271

Find the elasticity $E$ in terms of $p$ for the demand function $q=2700-p^{2}$ .

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Problem 7272

Find the derivative of the function $h(x)=\ln [(-4 x+5)(x+5)]$ . What is $h^{\prime}(x)$ ?

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Problem 7273

Find the derivative of $f(x)=(7x^2-x) \ln(x)$ . What is $f'(x)=\square$ ?

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Problem 7274

Find the derivative of the function $r(x)=\ln (x+1)+2 x^{7} e^{x}$ . What is $r^{\prime}(x)=?$

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Problem 7275

Find the derivative of $h(x)=\ln \left(\frac{7 x+1}{5 x-2}\right)$ . What is $h^{\prime}(x)$ ?

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Problem 7276

Find the function $f$ and compute the limit: $\lim _{x \rightarrow 1} \frac{8 x^{2}-8}{x-1}$ .

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Problem 7277

Find the derivative $v^{\prime}(4)$ for the speed function $v(t)=(t-5)^{2}$ , $0 \leq t \leq 5$ . Explain its meaning.

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Problem 7278

Find a function $f$ such that the limit $\lim _{h \rightarrow 0} \frac{\sqrt{7+h}-\sqrt{7}}{h}$ represents its slope. Choose $f$ from A-D.

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Problem 7279

Find the decay equation for Tc-99 to Ru-99. Use $N(t) = N_0 e^{-\lambda t}$ with $\lambda = \frac{\ln(2)}{211000}$ .

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Problem 7280

Find a function $f$ for the limit $\lim _{h \rightarrow 0} \frac{\sqrt{7+h}-\sqrt{7}}{h}$ and determine $a$ . Choose $f$ : A. $f(x)=\sqrt{x}$ , B. $f(x)=\sqrt{h+x}-\sqrt{7}$ , C. $f(x)=\sqrt{7}$ , D. $f(x)=\sqrt{h+x}$ . What is $a$ ? $a=$

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Problem 7281

Find the derivative of the function: $f(t)=8 \sqrt{t}-4 t^{4}+5$ . Use the hint: $\frac{d}{d x}(\sqrt{x})=\frac{1}{2 \sqrt{x}}$ .

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Problem 7282

Find the slope $m$ of the tangent line to the curve $y=2 x^{2}-5 x+1$ at the point $(4,13)$ and its equation.

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Problem 7283

Find a function $f$ and calculate the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{7+h}-\sqrt{7}}{h}$ . Options for $f$ : A. $f(x)=\sqrt{x}$ , B. $f(x)=\sqrt{h+x}-\sqrt{7}$ , C. $f(x)=\sqrt{7}$ , D. $f(x)=\sqrt{h+x}$ . Let $a=7$ . What is the limit?

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Problem 7284

Is $f(x)=\frac{x^{2}-9 x+20}{x-4}$ differentiable at $x=4$ ? Choose the correct justification from options A-D.

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Problem 7285

Find the derivative of the function $y=(7+6 \tan u)^{5.3}$ . What is $\frac{d y}{d u}=$ ?

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Problem 7286

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

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Problem 7287

Differentiate $\sqrt[3]{x}+\sqrt[3]{y^{7}}=2$ to find $\frac{d y}{d x}$ , then find the slope at (1,1).

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Problem 7288

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

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Problem 7289

Complete the derivative of $f(x)=x^{3}$ using the limit definition by expanding $(x+h)(x^{2}+2xh+h^{2})$ .

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Problem 7290

Calculate the work (in Joules) to pump water from a conical tank (14 m height, 6 m radius) out 5 m from the top.

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Problem 7291

Find $\frac{d r}{d h}$ for a spherical cap with volume $\frac{4 \pi}{3}$ . Choose the best method: A, B, C, or D.

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Problem 7292

Find the limit as $h \to 0$ of $x^{3} + 3x^{2} + 3xh + h^{2} - x^{2}$ using the derivative limit definition.

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Problem 7293

Find the limit: $\lim _{x \rightarrow 9^{+}} \frac{x^{2}-81}{\sqrt{x}-3}$ .

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Problem 7294

Find $\frac{dr}{dh}$ using implicit differentiation for $V=\frac{\pi h^{2}(3r-h)}{3}$ . Also, solve for $h$ and find $\frac{dr}{dh}$ .

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Problem 7295

Find $\frac{dr}{dh}$ using implicit differentiation on $V=\frac{\pi h^{2}(3 r-h)}{3}$ . Then find $\frac{dr}{dh}$ for $V=\frac{4\pi}{3}$ .

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Problem 7296

Find $\frac{dr}{dh}$ for a spherical cap with volume $\frac{4\pi}{3}$ given $V=\frac{\pi h^2(3r-h)}{3}$ . Evaluate at $r=8$ , $h=4$ .

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Problem 7297

Find the limit: $\lim _{x \rightarrow 9} \frac{8 x^{2}-648}{x-9}$ . Simplify your answer.

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Problem 7298

Find the function $f$ and the value $a$ , then compute the limit: $\lim _{x \rightarrow 9} \frac{8 x^{2}-648}{x-9}$ .

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Problem 7299

Find $\frac{d r}{d h}$ for a spherical cap with volume $\frac{5 \pi}{3}$ . Use $V=\frac{\pi h^{2}(3 r-h)}{3}$ .

See Solution

Problem 7300

Find the local minimum and maximum of the function $f(x)=2 x^{3}-36 x^{2}+210 x+4$ .

See Solution

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Calculus

Problem 7201

Find the slope of the tangent line to y=1x2−2x1/2y=\frac{1}{x^{2}}-2 x^{1 / 2}y=x21​−2x1/2 at x=1x=1x=1.

Problem 7202

Bestimme alle Werte von aaa, für die der Graph GGG der Funktion fa(x)=a2xf_{a}(x)=\sqrt{a^{2} x}fa​(x)=a2x​ (für x≥0x \geq 0x≥0) rechtsgekrümmt ist.

Problem 7203

Find the slope of the tangent line to y=1x2−2x1/2y=\frac{1}{x^{2}}-2 x^{1 / 2}y=x21​−2x1/2 at x=1x=1x=1.

Problem 7204

Find the derivative f′(x)f^{\prime}(x)f′(x) for f(x)=200x4−5/x1/2f(x)=200 x^{4}-5 / x^{1 / 2}f(x)=200x4−5/x1/2. Which option is correct?

Problem 7205

Find the tangent line equation for the curve y=2x2−5x+1y=2x^{2}-5x+1y=2x2−5x+1 at the point (3,4)(3,4)(3,4).

Problem 7206

Find the rock's instantaneous velocity after t=2t=2t=2 seconds using the function s(t)=−4.9t2+400s(t)=-4.9 t^{2}+400s(t)=−4.9t2+400.

Problem 7207

Find the derivative of f(x)=4cos⁡2x+log⁡x+xf(x)=4 \cos 2 x+\log x+xf(x)=4cos2x+logx+x. What is f′(x)f'(x)f′(x)?

Problem 7208

Find the limit as xxx approaches 7 for the expression x5−1x−1\frac{x^{5}-1}{x-1}x−1x5−1​.

Problem 7209

Find the limit as xxx approaches 7 for the expression x51−1x−1\frac{x^{51}-1}{x-1}x−1x51−1​.

Problem 7210

Bestimmen Sie die Ableitung von f(x)=x2⋅cos⁡(1−x)f(x)=x^{2} \cdot \cos (1-x)f(x)=x2⋅cos(1−x) und erläutern Sie den Prozess.

Problem 7211

Determine the vertical and horizontal asymptotes of f(x)=cos⁡x+7xxf(x)=\frac{\cos x+7 \sqrt{x}}{\sqrt{x}}f(x)=x​cosx+7x​​.

Problem 7212

Bestimme die Ableitung von f(x)=0.5x2⋅4−xf(x)=0.5x^{2} \cdot \sqrt{4-x}f(x)=0.5x2⋅4−x​ und erkläre den Prozess.

Problem 7213

Find the end behavior of the function y=cosh⁡x=ex+e−x2y = \cosh x = \frac{e^{x} + e^{-x}}{2}y=coshx=2ex+e−x​.

Problem 7214

Bestimmen Sie die Ableitung von f(x)=(5x+2)7⋅cos⁡(x)f(x)=(5 x+2)^{7} \cdot \cos (x)f(x)=(5x+2)7⋅cos(x) und erläutern Sie das Vorgehen.

Problem 7215

Bestimmen Sie die Funktion f(x)f(x)f(x) aus f′(x)=sin⁡(x)+(x+2)cos⁡(x)f^{\prime}(x)=\sin (x)+(x+2) \cos (x)f′(x)=sin(x)+(x+2)cos(x) und erläutern Sie die Lösung.

Problem 7216

Problem 7217

Find the differential dyd ydy for the function y=2x−9y=\sqrt{2 x-9}y=2x−9​.

Problem 7218

Find the differential dyd ydy for the function y=5sin⁡(4x)y=5 \sin (4 x)y=5sin(4x).

Problem 7219

Find the derivative of the function f(x)=−5f(x)=-5f(x)=−5.

Problem 7220

Find the critical numbers of f(x)=2x3−24x2+72x−9f(x)=2 x^{3}-24 x^{2}+72 x-9f(x)=2x3−24x2+72x−9 where the derivative is 0 or undefined. Identify the smaller and larger critical number.

Problem 7221

Find dydydy for y=−3cos⁡(7x)y=-3 \cos(7x)y=−3cos(7x) when x=0x=0x=0 and dx=−0.3dx=-0.3dx=−0.3.

Problem 7222

Find the derivative of the function g(t)=14tg(t) = 14 \sqrt{t}g(t)=14t​.

Problem 7223

Find the derivative of 2f(x)+4x2 f(x) + 4 x2f(x)+4x at x=2x = 2x=2.

Problem 7224

Find dyd ydy for y=x+4y=\sqrt{x+4}y=x+4​ and evaluate at x=3x=3x=3, dx=0.3d x=0.3dx=0.3.

Problem 7225

Find the critical numbers of the function f(x)=2x3−30x2+144x−2f(x)=2 x^{3}-30 x^{2}+144 x-2f(x)=2x3−30x2+144x−2. What are the smaller and larger critical numbers?

Problem 7226

Find the absolute max and min of f(x)=2−3x2f(x)=2-3 x^{2}f(x)=2−3x2 for −3≤x≤2-3 \leq x \leq 2−3≤x≤2. Where do they occur?

Problem 7227

Find the critical point AAA for the function f(x)=−2x2+4x−5f(x)=-2 x^{2}+4 x-5f(x)=−2x2+4x−5 and determine if it’s LMIN, LMAX, or NEITHER.

Problem 7228

Find the absolute maximum and minimum of f(x)=4x2−6x+6f(x)=4 x^{2}-6 x+6f(x)=4x2−6x+6 on the interval 0≤x≤90 \leq x \leq 90≤x≤9.

Problem 7229

Find the derivative of the function g(t)=90t2g(t) = 90 t^{2}g(t)=90t2.

Problem 7230

Find the min and max of f(x)=2x3+12x2−126x+4f(x)=2x^3 + 12x^2 - 126x + 4f(x)=2x3+12x2−126x+4 on [−7,4][-7,4][−7,4] using the Closed Interval Method.

Problem 7231

Evaluate the integral ∫−11∫−11∫−11x2y2z2 dx dy dz\int_{-1}^{1} \int_{-1}^{1} \int_{-1}^{1} x^{2} y^{2} z^{2} \, d x \, d y \, d z∫−11​∫−11​∫−11​x2y2z2dxdydz.

Problem 7232

Find the critical numbers of f(x)=−4x5+15x4+20x3+7f(x)=-4 x^{5}+15 x^{4}+20 x^{3}+7f(x)=−4x5+15x4+20x3+7 and classify them with a graph.

Problem 7233

Find the derivative of the function f(x)=13x3−30x+e5f(x) = 13 x^{3} - 30 x + e^{5}f(x)=13x3−30x+e5.

Problem 7234

Find the volume change dVd VdV when a sphere's radius goes from 16 cm to 16.1 cm. Round to the nearest tenth.

Problem 7235

A shell launched at 2,452 m/s2,452 \mathrm{~m/s}2,452 m/s at 82 degrees. How long was it in the air? Answer in minutes.

Problem 7236

A projectile is shot upward at 88.1 m/s88.1 \mathrm{~m/s}88.1 m/s. What is its speed at the peak of its path? Answer in m/s.

Problem 7237

Set up a definite integral using the shell method for the volume of the solid formed by rotating the region bounded by y=xy=\sqrt{x}y=x​ and x=2yx=2yx=2y around the line x=7x=7x=7.

Problem 7238

Find the derivative of the function y=−12x4+3x53+2xy=-\frac{1}{2} x^{4}+3 x^{\frac{5}{3}}+2 xy=−21​x4+3x35​+2x.

Problem 7239

A 6 ft man walks away from an 18 ft lamppost at 6 ft/sec. Find the shadow's length change rate when he's 30 ft away.

Problem 7240

Calculate the directional derivative of f(x,y)=sin⁡(x2y)f(x, y)=\sin(x^{2} y)f(x,y)=sin(x2y) at P=(π,1)P=(\sqrt{\pi}, 1)P=(π​,1) in the direction u=i\mathbf{u}=\mathbf{i}u=i.

Find the slope of the tangent line to $y=\frac{1}{x^{2}}-2 x^{1 / 2}$ at $x=1$ .

Bestimme alle Werte von $a$ , für die der Graph $G$ der Funktion $f_{a}(x)=\sqrt{a^{2} x}$ (für $x \geq 0$ ) rechtsgekrümmt ist.

Find the slope of the tangent line to $y=\frac{1}{x^{2}}-2 x^{1 / 2}$ at $x=1$ .

Find the derivative $f^{\prime}(x)$ for $f(x)=200 x^{4}-5 / x^{1 / 2}$ . Which option is correct?

Find the tangent line equation for the curve $y=2x^{2}-5x+1$ at the point $(3,4)$ .

Find the rock's instantaneous velocity after $t=2$ seconds using the function $s(t)=-4.9 t^{2}+400$ .

Find the derivative of $f(x)=4 \cos 2 x+\log x+x$ . What is $f'(x)$ ?

Find the limit as $x$ approaches 7 for the expression $\frac{x^{5}-1}{x-1}$ .

Find the limit as $x$ approaches 7 for the expression $\frac{x^{51}-1}{x-1}$ .

Bestimmen Sie die Ableitung von $f(x)=x^{2} \cdot \cos (1-x)$ und erläutern Sie den Prozess.

Determine the vertical and horizontal asymptotes of $f(x)=\frac{\cos x+7 \sqrt{x}}{\sqrt{x}}$ .

Bestimme die Ableitung von $f(x)=0.5x^{2} \cdot \sqrt{4-x}$ und erkläre den Prozess.

Find the end behavior of the function $y = \cosh x = \frac{e^{x} + e^{-x}}{2}$ .

Bestimmen Sie die Ableitung von $f(x)=(5 x+2)^{7} \cdot \cos (x)$ und erläutern Sie das Vorgehen.

Bestimmen Sie die Funktion $f(x)$ aus $f^{\prime}(x)=\sin (x)+(x+2) \cos (x)$ und erläutern Sie die Lösung.

Find the differential $d y$ for the function $y=\sqrt{2 x-9}$ .

Find the differential $d y$ for the function $y=5 \sin (4 x)$ .

Find the derivative of the function $f(x)=-5$ .

Find the critical numbers of $f(x)=2 x^{3}-24 x^{2}+72 x-9$ where the derivative is 0 or undefined. Identify the smaller and larger critical number.

Find $dy$ for $y=-3 \cos(7x)$ when $x=0$ and $dx=-0.3$ .

Find the derivative of the function $g(t) = 14 \sqrt{t}$ .

Find the derivative of $2 f(x) + 4 x$ at $x = 2$ .

Find $d y$ for $y=\sqrt{x+4}$ and evaluate at $x=3$ , $d x=0.3$ .

Find the critical numbers of the function $f(x)=2 x^{3}-30 x^{2}+144 x-2$ . What are the smaller and larger critical numbers?

Find the absolute max and min of $f(x)=2-3 x^{2}$ for $-3 \leq x \leq 2$ . Where do they occur?

Find the critical point $A$ for the function $f(x)=-2 x^{2}+4 x-5$ and determine if it’s LMIN, LMAX, or NEITHER.

Find the absolute maximum and minimum of $f(x)=4 x^{2}-6 x+6$ on the interval $0 \leq x \leq 9$ .

Find the derivative of the function $g(t) = 90 t^{2}$ .

Find the min and max of $f(x)=2x^3 + 12x^2 - 126x + 4$ on $[-7,4]$ using the Closed Interval Method.

Evaluate the integral $\int_{-1}^{1} \int_{-1}^{1} \int_{-1}^{1} x^{2} y^{2} z^{2} \, d x \, d y \, d z$ .

Find the critical numbers of $f(x)=-4 x^{5}+15 x^{4}+20 x^{3}+7$ and classify them with a graph.

Find the derivative of the function $f(x) = 13 x^{3} - 30 x + e^{5}$ .

Find the volume change $d V$ when a sphere's radius goes from 16 cm to 16.1 cm. Round to the nearest tenth.

A shell launched at $2,452 \mathrm{~m/s}$ at 82 degrees. How long was it in the air? Answer in minutes.

A projectile is shot upward at $88.1 \mathrm{~m/s}$ . What is its speed at the peak of its path? Answer in m/s.

Set up a definite integral using the shell method for the volume of the solid formed by rotating the region bounded by $y=\sqrt{x}$ and $x=2y$ around the line $x=7$ .

Find the derivative of the function $y=-\frac{1}{2} x^{4}+3 x^{\frac{5}{3}}+2 x$ .

Calculate the directional derivative of $f(x, y)=\sin(x^{2} y)$ at $P=(\sqrt{\pi}, 1)$ in the direction $\mathbf{u}=\mathbf{i}$ .

Find the derivative of the function $y=17 x^{-2}+11 x^{3}+5 x$ . What is $f^{\prime}(x)$ ?

A cone with radius 3 in and height 6 in has water 4 in deep. If the surface falls at $-1.8 \mathrm{in/s}$ , find the drainage rate.

Estimate $f(1.02, 1.98)$ using linear approximation given $f(1,2)=3$ , $f_{x}(1,2)=4$ , and $f_{y}(1,2)=5$ .

Find the speed of two people walking from a right angle of a triangle if the area changes at $2 \frac{\mathrm{m}^{2}}{\mathrm{~s}}$ when they are $4 \mathrm{~m}$ away.

Find the derivative $f^{\prime}(x)$ using logarithmic differentiation for $f(x)=\frac{(4 x^{2}-4) \sqrt{5 x^{3}-6}}{5 x-5}$ .

Find the derivative of the function $f(x)=\sin^{-1}\left(e^{-3x}\right)$ .

Find the $x$ -values where the function $f(x)=(x-8)^{2/5}$ is differentiable.

Find the derivative of the function: $f(x)=\frac{1}{3x-1}$ . What is $f'(x)$ ?

Find the limit: $\lim _{x \rightarrow-\infty} \frac{3 x^{5}-2 x^{2}+x}{2 x^{3}+x+1}$ .

Find the derivative of the function $f(x)=(3x-1)^{-2}$ . What is $f^{\prime}(x)$ ?

Find the derivative of the function: $f(x)=(x^{6}-x)^{6}$ . What is $f^{\prime}(x)$ ?

Determine the intervals where the function $f(x)=2(x+6)(x-2)$ is increasing.

Determine the intervals where the function $f(x)=2(x-2)^{2}-50$ is increasing.

Find the derivative of the function $s(x)=\left(\frac{2 x+8}{3 x-1}\right)^{2}$ . What is $s^{\prime}(x)=?$

Find the derivative of the function $f(t) = \left(\cos^{-1} t\right)^{17}$ .

Find the derivative of the function $f(u)=\sec ^{-1}(3 u+1)$ .

Calculate the integral of $k(x)=\frac{14}{x^{2}+49}$ .

Find the derivative of the function $f(x)=(7x-1)^{2}$ . What is $f^{\prime}(x)$ ?

Find the derivative of $f(x)=\ln(x+7)$ . What is $f'(x)$ ?

Find the derivative of $f(x) = \log_{4}(x)$ , denoted as $f'(x) = \square$ .

Find the derivative of the function $h(x)=e^{x^{6}-x+3}$ .

Find the derivative of $h(x)=e^{x^{6}}-x+3$ .

Find the derivative of the function $g(x) = 8^x$ . What is $g'(x)$ ?

Find the price $p$ that maximizes revenue given the demand $q=120-3 \cdot p^{2}$ .

Find the derivative of the function $f(x)=(x^{2}+9) \ln (x)$ . What is $f'(x)$ ?

Given demand $q=2700-p^{2}$ , find elasticity $E$ in terms of $p$ , at $p=30$ , and interpret its meaning.

Find the elasticity $E$ in terms of $p$ for the demand function $q=2700-p^{2}$ .

Find the derivative of the function $h(x)=\ln [(-4 x+5)(x+5)]$ . What is $h^{\prime}(x)$ ?

Find the derivative of $f(x)=(7x^2-x) \ln(x)$ . What is $f'(x)=\square$ ?

Find the derivative of the function $r(x)=\ln (x+1)+2 x^{7} e^{x}$ . What is $r^{\prime}(x)=?$