Calculus

Problem 6501

Gegeben ist $f(x)=x$ . Finde die Stammfunktion $F$ durch den Punkt $P(1 \mid 1,5)$ .

See Solution

Problem 6502

Gegeben ist $f(x)=x$ . Finde die Stammfunktion $F$ , die durch den Punkt $P(1 \mid 1,5)$ geht.

See Solution

Problem 6503

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=2x+8$ , where $h \neq 0$ .

See Solution

Problem 6504

Find the first and second derivatives $f'(x)$ and $f''(x)$ for these functions:
(a) $f(x)=\left(x^{2}-3 x+5\right) x^{2 x}$ , (b) $f(x)=x^{i x+3}$ , (c) $f(x)=\sin 3 x$ , (d) $f(x)=\tan 4 x$ , (e) $f(x)=\cos \left(5 x^{2}\right)$ , (f) $f(x)=2 e^{3 x} \cos 5 x$ .

See Solution

Problem 6505

Find the end behavior of $f(x)=-6 x^{4}-6 x^{2}+1$ using $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ .

See Solution

Problem 6506

Analyze the end behavior of $f(x)=7 x^{2}-x^{3}+4 x-2$ using $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ .

See Solution

Problem 6507

Find the sum of the series: $36 + 12 + 4 + \cdots$ . Is it a finite sum or does it not exist? Choose A or B.

See Solution

Problem 6508

Find the limit and check continuity at $y=1$ : $\lim _{y \rightarrow 1} \sec \left(y \sec ^{2} y-\tan ^{2} y-1\right)$ .

See Solution

Problem 6509

Find the limit of $f(x)=\frac{2}{x}-4$ as $x \to \infty$ and $x \to -\infty$ . What is $\lim_{x \to \infty} f(x)$ ?

See Solution

Problem 6510

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sin 20 x}{16 x}=\square$ .

See Solution

Problem 6511

Why does $f(x)=0$ have a solution between $x=5$ and $x=7$ if $f(5)<0$ and $f(7)>0$ ? Explain and sketch.

See Solution

Problem 6512

If $h(x)=f(x) \cdot g(x)$ is continuous at $x=0$ , must $f(x)$ and $g(x)$ be continuous at $x=0$ ? Explain.

See Solution

Problem 6513

Calculate the average rate of change of $f(x)=\sqrt{x}$ from $x_{1}=9$ to $x_{2}=64$ . Simplify your answer.

See Solution

Problem 6514

Find the critical number(s) of the function $f(x)=x^{23/11}-x^{12/11}$ .

See Solution

Problem 6515

Find the critical numbers of the function $f(x)=x^{15/7}+x^{8/7}$ .

See Solution

Problem 6516

Calculate the average rate of change for $f(x)=x^{2}+8x$ from $x=4$ to $x=6$ . Simplify your answer.

See Solution

Problem 6517

Find the critical numbers of the function $f(x)=\frac{2}{3}x^3 - 3x^2 - 80x - 10$ .

See Solution

Problem 6518

Find where the cost function $C(x)=x^{3}-2 x^{2}+4 x+40$ is decreasing and increasing for $x>0$ .

See Solution

Problem 6519

Which functions are continuous for all real numbers: $h(x)=\sin(x)$ or $f(x)=\cos(x)$ ? Choose: A) $h$ only, B) $f$ only, C) both, D) neither.

See Solution

Problem 6520

Find the intervals where the profit function $P(x) = R(x) - C(x)$ is increasing, given:
$C(x) = 0.13x^{2} - 0.00006x^{3}$ and $R(x) = 0.886x^{2} - 0.0002x^{3}$ .

See Solution

Problem 6521

Show that for small $x$ , $\sin 3x \cos 4x \approx 3x - 24x^3$ . Approximate $\sin 0.3 \cos 0.4$ and find the percentage error.

See Solution

Problem 6522

Find local and absolute extreme values of a function $f$ with given points $(a,b)$ , $(u,v)$ , $(c,d)$ , $(x,y)$ .

See Solution

Problem 6523

Find intervals where the drug concentration $K(t)=\frac{12 t}{t^{2}+4}$ is increasing and decreasing.

See Solution

Problem 6524

Estimate the increase in metabolic rate $P$ when body mass $m$ changes from 102 kg to 103 kg using $P=73.3 m^{3/4}$ .

See Solution

Problem 6525

Lös differentialekvationen $y^{\prime}(t)=-\frac{1}{10}(y(t)-10)$ med $y(0)=2022$ .

See Solution

Problem 6526

Find the growth rate when $M=125 \mathrm{~kg}$ , given $\frac{d M}{d t}=14$ kg/year at $M=100 \mathrm{~kg}$ .
Also, how much mass must $M=0.7 \mathrm{~kg}$ gain to double its growth rate?

See Solution

Problem 6527

Find $\frac{d B}{d I}$ for $B=k I^{2/3}$ and $\frac{d H}{d W}$ for $H=k W^{3/2}$ . Identify the correct statements.

See Solution

Problem 6528

Given the piecewise function $f(x)$ , determine which statement about its continuity and differentiability at $x=-1$ and $x=2$ is TRUE.

See Solution

Problem 6529

Find the slope of the tangent line to $2y^2=x^4y+c g(x)$ at (1,3) given $g'(1)=-1$ . Choices: $2+c$ , $\frac{c-24}{2}$ , $24-c$ , $-c$ , $\frac{12-c}{11}$ .

See Solution

Problem 6530

Given the function $g(v)=v^{3}-75v+6$ , find $g^{\prime}(v)$ , where $g^{\prime}(v)=0$ , and identify critical numbers.

See Solution

Problem 6531

Find the tangent line equation for $f(x)=2x^{2}+g(x)$ at $x=3$ , given $g(3)=1$ and $g'(3)=2$ .

See Solution

Problem 6532

Consider the piecewise function $f(x)$ defined as follows:
1. $2x^2 + 5$ for $x < -1$
2. $x^3 + 8$ for $-1 \leq x \leq 2$
3. $3x + 10$ for $x > 2$
Which statement about the continuity and differentiability of $f$ at $x = -1$ and $x = 2$ is TRUE?

See Solution

Problem 6533

Find the derivative of the function $f(x)=3 \cos (6 \ln (x))$ , denoted as $f^{\prime}(x)$ .

See Solution

Problem 6534

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{1+9 x^{6}}}{4-x^{3}}$ .

See Solution

Problem 6535

Find the function $f(x)$ and the number $a$ such that $\lim _{h \rightarrow 0} \frac{h^{3}}{h} = f^{\prime}(a)$ . $f(x)=$ $a=$

See Solution

Problem 6536

Find the derivative of the function $f(x)=8 x^{3} \arctan \left(4 x^{4}\right)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 6537

Find the limit: $\lim _{x \rightarrow \infty}\left[\ln \left(4+x^{2}\right)-\ln (7+x)\right]$ .

See Solution

Problem 6538

Find the function $f(x)$ and the number $a$ such that $\operatorname{limit}_{h \rightarrow 0} \frac{h^{3}}{h} = f^{\prime}(a)$ . $f(x)=\square \\ a=\square$

See Solution

Problem 6539

Find the limit: $\lim _{x \rightarrow-\infty}\left(x^{2}+2 x^{5}\right)$ . Enter $\infty$ , $-\infty$ , or DNE if it doesn't exist.

See Solution

Problem 6540

Calculate the limit: $\lim _{x \rightarrow \infty} \frac{5 x-6}{2 x+1}$ . Enter ' $\infty$ ', '- $\infty$ ', or DNE if it doesn't exist.

See Solution

Problem 6541

Find the derivative of $f(x) = \int_{3}^{x} \frac{1}{1+t^{5}} dt$ and evaluate $f'(12)$ .

See Solution

Problem 6542

Find the tangent line equation for $f(x)=(5 x-1)^{\frac{1}{2}}$ at point $P=(1, f(1))$ .

See Solution

Problem 6543

Find the limit: $\lim _{x \rightarrow \infty}\left(\sqrt{25 x^{2}+x}-5 x\right)$ . Enter $\omega$ , - $\infty$ , or DNE if it doesn't exist.

See Solution

Problem 6544

Find when the function $f(x)=\cos (6 x)+3 x$ is decreasing on $[0, \frac{\pi}{3}]$ .

See Solution

Problem 6545

Find the limit: $\lim _{x \rightarrow \infty}\left(e^{-5 x} \cos (x)\right)$ . Enter $\infty$ , $-\infty$ , or DNE if it doesn't exist.

See Solution

Problem 6546

Find the derivative $\frac{d y}{d x}$ of the function $y=x^{10 \cos (x)}$ .

See Solution

Problem 6547

A stone is thrown from a 650 ft roof at 18 ft/s. Find its height after 5 seconds, when it hits the ground, and its impact velocity.

See Solution

Problem 6548

Find the limits as $x \to \infty$ and $x \to -\infty$ for $f(x)=(2-x)(1+x)^{2}(1-x)^{4}$ .

See Solution

Problem 6549

Find the intervals where the polynomial function $f(x)$ , with derivative $f^{\prime}(x)=-2(x+5)^{3}(x-3)^{2}(x-4)$ , is increasing.

See Solution

Problem 6550

Find the function $f(x)$ and the number $a$ such that $\operatorname{limit}_{h \rightarrow 0} \frac{h^{3}}{h} = f^{\prime}(a)$ . $f(x)= \\ a=$

See Solution

Problem 6551

Calculate the limit: $\lim _{t \rightarrow \infty} \frac{\sqrt{t}+t^{2}}{6 t-t^{2}}$ . Enter $\infty$ , $-\infty$ , or DNE if it doesn't exist.

See Solution

Problem 6552

Find the limit: $\lim _{x \rightarrow \infty} \frac{x^{4}-9 x^{2}+x}{x^{3}-x+3}$ . Enter $\infty$ , $-\infty$ , or DNE if it doesn't exist.

See Solution

Problem 6553

Find the limit: $\lim _{x \rightarrow-\infty} \frac{6 x^{5}-x}{x^{4}+5}$ . Enter $\infty$ , $-\infty$ , or DNE if it doesn't exist.

See Solution

Problem 6554

Let $C(q)$ be cost and $R(q)$ be revenue of producing $q$ items.
(a) Estimate $C(54)$ given $C(50)=4500$ and $C'(50)=23$ .
(b) Find profit from the $51^{\text{st}}$ item using $C'(50)=23$ and $R'(50)=38$ .
(c) Should the company produce the $101^{\text{st}}$ item if $C'(100)=35$ and $R'(100)=38$ ?

See Solution

Problem 6555

Find the derivative of $h(x)=\frac{1}{x^{2}+3 x-1}$ using the chain rule.

See Solution

Problem 6556

Find $g^{\prime}\left(\frac{\pi}{2}\right)$ for $g(x)=\ln((f(x))^{2}+A \sin x)$ given $f\left(\frac{\pi}{2}\right)=4$ and $f^{\prime}\left(\frac{\pi}{2}\right)=3$ .

See Solution

Problem 6557

Determine the roots, asymptotes, or critical points of $h(x)=\frac{1}{x^{2}+3 x-1}$ using the chain rule.

See Solution

Problem 6558

Find $\left.\frac{\partial z}{\partial y}\right|_{\substack{x=1 \\ y=e}}$ for $z=x y^{x}$ : A. 6 B. $\frac{1}{e}$ C. 1 D. $1+\frac{1}{e}$

See Solution

Problem 6559

Find the value of $\lim _{x \rightarrow 0} f(x) \sin \left(\frac{1}{x^{3}}\right)$ if $\lim _{x \rightarrow 0} f(x)=0$ .

See Solution

Problem 6560

Find the derivative of $f(x) = 3(4-9x)^4$ using the chain rule.

See Solution

Problem 6561

Given a continuous function $f$ with $f(-2)=-4$ , $f(2)=1$ , and $\lim_{x \to \infty} f(x)=0$ , which statement is TRUE?

See Solution

Problem 6562

Find the derivative of $y=\csc \sqrt{x}+\sqrt{\tan x}$ using the chain rule.

See Solution

Problem 6563

Find $f\left(\frac{\pi}{4}\right)$ given $f''(x)=-9 \sin(3x)$ , $f'(0)=-1$ , and $f(0)=-1$ .

See Solution

Problem 6564

A stone is thrown from a 650 ft roof at 18 ft/s. Find its height after 5s, when it hits the ground, and its impact velocity.

See Solution

Problem 6565

Determine which limit cannot be found using the Squeeze Theorem for the function $f(x)$ defined by the inequalities.

See Solution

Problem 6566

If $y=f(x)$ is not differentiable at $x=0$ , which statement is always TRUE?

See Solution

Problem 6567

Given a continuous function $f$ on $[0,3]$ with values $f(0)=-2$ , $f(1)=2$ , and $f(2)=k$ , which statement is always TRUE?

See Solution

Problem 6568

Find $g'(3)$ for $g(x)=\frac{A f(x)}{e^{4 x}}$ given $f(3)=1$ , $f'(3)=4$ .

See Solution

Problem 6569

Approximate a root of $3 x^{3}+6 x+2=0$ using Newton's method with initial guess $x_{1}=-1$ . Find $x_{2}$ and $x_{3}$ .

See Solution

Problem 6570

Given a continuous function $f$ on $[0,3]$ with values $f(0)=-2$ , $f(1)=2$ , and $f(2)=k$ , which statement is always TRUE?

See Solution

Problem 6571

An inverted pyramid with a square base (sides $6 \mathrm{~cm}$ ) and height $10 \mathrm{~cm}$ is filled at $75 \mathrm{~cm}^3/\mathrm{s}$ . Find the water level rise rate when it's $6 \mathrm{~cm}$ .

See Solution

Problem 6572

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{1}{x+8}$ , where $h \neq 0$ .

See Solution

Problem 6573

Given the piecewise function $f(x)$ , which statements about its continuity and differentiability at $x=-1$ and $x=2$ are true?

See Solution

Problem 6574

Find the $x$ -intercept of the tangent line to $f(x)=(3x-1)^{4}(2x+2)^{2}$ at the $y$ -axis crossing point.

See Solution

Problem 6575

Find the volume change rate of a cube with side $s$ at $s=6$ . (Answer as a whole number: cubic units/unit increase.)

See Solution

Problem 6576

Find the rate of change of power $P(R)=\frac{2.5 R}{(R+0.5)^{2}}$ W at $R=10 \Omega$ . Round to four decimal places.

See Solution

Problem 6577

Find $dy/dx$ for the equation $4y^{2} - xy - 15 = 0$ at the point $(17,5)$ .

See Solution

Problem 6578

Find the farthest distance to the left of the origin for the particle with position $s(t)=t^{4}-200 t^{2}$ .

See Solution

Problem 6579

Find the rate of change of $q$ with respect to $p$ given $p=\frac{20}{q^{2}+5}$ .

See Solution

Problem 6580

Find $F^{\prime}(2)$ for $F(x)=f(g(x))$ using $f(3)=7, f^{\prime}(3)=-5, g(2)=3, g^{\prime}(2)=-3$ .

See Solution

Problem 6581

Find the population change rate in 8 years for $P(t)=30(45+4t)^{2}-1900t$ .

See Solution

Problem 6582

Find the derivative of $y=((x+6)(x^{2}+4))^{9}$ .

See Solution

Problem 6583

Find $d y / d x$ using implicit differentiation for the equation $y \sqrt{x+1}=4$ .

See Solution

Problem 6584

Find the derivative of $y=e^{4 x}(2 x-7)^{5}$ using the chain rule.

See Solution

Problem 6585

Find the derivative $\frac{d y}{d x}$ for the equation $(x^{2}+y^{2})(x^{2}+y^{2}+x)=8 x y^{2}$ .

See Solution

Problem 6586

Find the derivative of $y=\tan(4xe^{x})$ using the chain rule.

See Solution

Problem 6587

Find $f^{\prime \prime \prime}(1)$ for $f(x)=2 \ln (x+1)$ .

See Solution

Problem 6588

Find the rate of change of voltage $\frac{d V}{d v}$ for $B=8$ , $l=0.8$ , where $V(v)=-B l v$ . Answer to one decimal place. $\frac{d V}{d v}=$

See Solution

Problem 6589

Find the limit of $f(x)=3-\frac{4}{x-2}$ as $x$ approaches $2$ from the left.

See Solution

Problem 6590

Find the derivative $f'(x)$ of the function $f(x)=\frac{5 \sqrt{x^{3}}}{6}+\frac{3}{\sqrt{x}}$ and calculate $f'(6)$ .

See Solution

Problem 6591

Find the derivative of $f(x)=\ln \sqrt{\frac{1+2 x^{2}}{1-2 x^{2}}}$ .

See Solution

Problem 6592

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

See Solution

Problem 6593

Using Ohm's Law $V=IR$ with $V=15 \mathrm{~V}$ , find the average rate of change of $I$ from $R=8$ to $R=8.1$ . Then, find the rate of change of $I$ at $R=8$ and the rate of change of $R$ at $I=2.3$ . Provide answers to three decimal places.

See Solution

Problem 6594

Find the rate of change of $q$ with respect to $p$ for $p=\frac{20}{q^{2}+5}$ .

See Solution

Problem 6595

Find the average rate of change of $f(x)=\frac{x-8}{x+4}$ on $[3,5]$ and $[x, x+h]$ as reduced fractions.

See Solution

Problem 6596

Is the function $f(x)=\sqrt[8]{x^{8}+0.01}$ differentiable at $x=0$ ? True or False?

See Solution

Problem 6597

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

See Solution

Problem 6598

正の整数 $k$ に対し、次の不等式を示せ： $\frac{1}{\sqrt{(k+1) \pi}} \leq A_{k} \leq \frac{1}{\sqrt{k \pi}}$ で $A_{k}=\int_{\sqrt{k \pi}}^{\sqrt{(k+1) \pi}}\left|\sin \left(x^{2}\right)\right| d x$ 。

See Solution

Problem 6599

Find $g^{\prime \prime}(2)$ given $g(2)=3$ and $g^{\prime}(x)=x^{3} g(x)$ . Choices: a. 228 b. 114 c. 456 d. 57 e. 342

See Solution

Problem 6600

Find the tangent line to the curve $y \sin 6 x = x \cos 6 y$ at the point $\left(\frac{\pi}{6}, \frac{\pi}{12}\right)$ .

See Solution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337

Calculus

Problem 6501

Gegeben ist f(x)=xf(x)=xf(x)=x. Finde die Stammfunktion FFF durch den Punkt P(1∣1,5)P(1 \mid 1,5)P(1∣1,5).

Problem 6502

Gegeben ist f(x)=xf(x)=xf(x)=x. Finde die Stammfunktion FFF, die durch den Punkt P(1∣1,5)P(1 \mid 1,5)P(1∣1,5) geht.

Problem 6503

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for the function f(x)=2x+8f(x)=2x+8f(x)=2x+8, where h≠0h \neq 0h=0.

Problem 6504

Problem 6505

Find the end behavior of f(x)=−6x4−6x2+1f(x)=-6 x^{4}-6 x^{2}+1f(x)=−6x4−6x2+1 using lim⁡x→∞f(x)\lim _{x \rightarrow \infty} f(x)limx→∞​f(x) and lim⁡x→−∞f(x)\lim _{x \rightarrow-\infty} f(x)limx→−∞​f(x).

Problem 6506

Analyze the end behavior of f(x)=7x2−x3+4x−2f(x)=7 x^{2}-x^{3}+4 x-2f(x)=7x2−x3+4x−2 using lim⁡x→∞f(x)\lim _{x \rightarrow \infty} f(x)limx→∞​f(x) and lim⁡x→−∞f(x)\lim _{x \rightarrow-\infty} f(x)limx→−∞​f(x).

Problem 6507

Find the sum of the series: 36+12+4+⋯36 + 12 + 4 + \cdots36+12+4+⋯. Is it a finite sum or does it not exist? Choose A or B.

Problem 6508

Find the limit and check continuity at y=1y=1y=1: lim⁡y→1sec⁡(ysec⁡2y−tan⁡2y−1)\lim _{y \rightarrow 1} \sec \left(y \sec ^{2} y-\tan ^{2} y-1\right)y→1lim​sec(ysec2y−tan2y−1).

Problem 6509

Find the limit of f(x)=2x−4f(x)=\frac{2}{x}-4f(x)=x2​−4 as x→∞x \to \inftyx→∞ and x→−∞x \to -\inftyx→−∞. What is lim⁡x→∞f(x)\lim_{x \to \infty} f(x)limx→∞​f(x)?

Problem 6510

Find the limit as xxx approaches infinity: lim⁡x→∞sin⁡20x16x=□\lim _{x \rightarrow \infty} \frac{\sin 20 x}{16 x}=\squarelimx→∞​16xsin20x​=□.

Problem 6511

Why does f(x)=0f(x)=0f(x)=0 have a solution between x=5x=5x=5 and x=7x=7x=7 if f(5)<0f(5)<0f(5)<0 and f(7)>0f(7)>0f(7)>0? Explain and sketch.

Problem 6512

If h(x)=f(x)⋅g(x)h(x)=f(x) \cdot g(x)h(x)=f(x)⋅g(x) is continuous at x=0x=0x=0, must f(x)f(x)f(x) and g(x)g(x)g(x) be continuous at x=0x=0x=0? Explain.

Problem 6513

Calculate the average rate of change of f(x)=xf(x)=\sqrt{x}f(x)=x​ from x1=9x_{1}=9x1​=9 to x2=64x_{2}=64x2​=64. Simplify your answer.

Problem 6514

Find the critical number(s) of the function f(x)=x23/11−x12/11f(x)=x^{23/11}-x^{12/11}f(x)=x23/11−x12/11.

Problem 6515

Find the critical numbers of the function f(x)=x15/7+x8/7f(x)=x^{15/7}+x^{8/7}f(x)=x15/7+x8/7.

Problem 6516

Calculate the average rate of change for f(x)=x2+8xf(x)=x^{2}+8xf(x)=x2+8x from x=4x=4x=4 to x=6x=6x=6. Simplify your answer.

Problem 6517

Find the critical numbers of the function f(x)=23x3−3x2−80x−10f(x)=\frac{2}{3}x^3 - 3x^2 - 80x - 10f(x)=32​x3−3x2−80x−10.

Problem 6518

Find where the cost function C(x)=x3−2x2+4x+40C(x)=x^{3}-2 x^{2}+4 x+40C(x)=x3−2x2+4x+40 is decreasing and increasing for x>0x>0x>0.

Problem 6519

Which functions are continuous for all real numbers: h(x)=sin⁡(x)h(x)=\sin(x)h(x)=sin(x) or f(x)=cos⁡(x)f(x)=\cos(x)f(x)=cos(x)? Choose: A) hhh only, B) fff only, C) both, D) neither.

Problem 6520

Find the intervals where the profit function P(x)=R(x)−C(x)P(x) = R(x) - C(x)P(x)=R(x)−C(x) is increasing, given: C(x)=0.13x2−0.00006x3C(x) = 0.13x^{2} - 0.00006x^{3}C(x)=0.13x2−0.00006x3 and R(x)=0.886x2−0.0002x3R(x) = 0.886x^{2} - 0.0002x^{3}R(x)=0.886x2−0.0002x3.

Problem 6521

Show that for small xxx, sin⁡3xcos⁡4x≈3x−24x3\sin 3x \cos 4x \approx 3x - 24x^3sin3xcos4x≈3x−24x3. Approximate sin⁡0.3cos⁡0.4\sin 0.3 \cos 0.4sin0.3cos0.4 and find the percentage error.

Problem 6522

Find local and absolute extreme values of a function fff with given points (a,b)(a,b)(a,b), (u,v)(u,v)(u,v), (c,d)(c,d)(c,d), (x,y)(x,y)(x,y).

Problem 6523

Find intervals where the drug concentration K(t)=12tt2+4K(t)=\frac{12 t}{t^{2}+4}K(t)=t2+412t​ is increasing and decreasing.

Problem 6524

Estimate the increase in metabolic rate PPP when body mass mmm changes from 102 kg to 103 kg using P=73.3m3/4P=73.3 m^{3/4}P=73.3m3/4.

Problem 6525

Lös differentialekvationen y′(t)=−110(y(t)−10)y^{\prime}(t)=-\frac{1}{10}(y(t)-10)y′(t)=−101​(y(t)−10) med y(0)=2022y(0)=2022y(0)=2022.

Problem 6526

Find the growth rate when M=125 kgM=125 \mathrm{~kg}M=125 kg, given dMdt=14\frac{d M}{d t}=14dtdM​=14 kg/year at M=100 kgM=100 \mathrm{~kg}M=100 kg. Also, how much mass must M=0.7 kgM=0.7 \mathrm{~kg}M=0.7 kg gain to double its growth rate?

Problem 6527

Find dBdI\frac{d B}{d I}dIdB​ for B=kI2/3B=k I^{2/3}B=kI2/3 and dHdW\frac{d H}{d W}dWdH​ for H=kW3/2H=k W^{3/2}H=kW3/2. Identify the correct statements.

Problem 6528

Given the piecewise function f(x)f(x)f(x), determine which statement about its continuity and differentiability at x=−1x=-1x=−1 and x=2x=2x=2 is TRUE.

Problem 6529

Find the slope of the tangent line to 2y2=x4y+cg(x)2y^2=x^4y+c g(x)2y2=x4y+cg(x) at (1,3) given g′(1)=−1g'(1)=-1g′(1)=−1. Choices: 2+c2+c2+c, c−242\frac{c-24}{2}2c−24​, 24−c24-c24−c, −c-c−c, 12−c11\frac{12-c}{11}1112−c​.

Problem 6530

Given the function g(v)=v3−75v+6g(v)=v^{3}-75v+6g(v)=v3−75v+6, find g′(v)g^{\prime}(v)g′(v), where g′(v)=0g^{\prime}(v)=0g′(v)=0, and identify critical numbers.

Problem 6531

Find the tangent line equation for f(x)=2x2+g(x)f(x)=2x^{2}+g(x)f(x)=2x2+g(x) at x=3x=3x=3, given g(3)=1g(3)=1g(3)=1 and g′(3)=2g'(3)=2g′(3)=2.

Problem 6532

Problem 6533

Find the derivative of the function f(x)=3cos⁡(6ln⁡(x))f(x)=3 \cos (6 \ln (x))f(x)=3cos(6ln(x)), denoted as f′(x)f^{\prime}(x)f′(x).

Problem 6534

Find the limit: lim⁡x→∞1+9x64−x3\lim _{x \rightarrow \infty} \frac{\sqrt{1+9 x^{6}}}{4-x^{3}}limx→∞​4−x31+9x6​​.

Problem 6535

Find the function f(x)f(x)f(x) and the number aaa such that lim⁡h→0h3h=f′(a)\lim _{h \rightarrow 0} \frac{h^{3}}{h} = f^{\prime}(a)limh→0​hh3​=f′(a). f(x)= f(x)= f(x)= a= a= a=

Problem 6536

Find the derivative of the function f(x)=8x3arctan⁡(4x4)f(x)=8 x^{3} \arctan \left(4 x^{4}\right)f(x)=8x3arctan(4x4). What is f′(x)f^{\prime}(x)f′(x)?

Problem 6537

Find the limit: lim⁡x→∞[ln⁡(4+x2)−ln⁡(7+x)]\lim _{x \rightarrow \infty}\left[\ln \left(4+x^{2}\right)-\ln (7+x)\right]x→∞lim​[ln(4+x2)−ln(7+x)].

Problem 6538

Find the function f(x)f(x)f(x) and the number aaa such that limit⁡h→0h3h=f′(a)\operatorname{limit}_{h \rightarrow 0} \frac{h^{3}}{h} = f^{\prime}(a)limith→0​hh3​=f′(a). f(x)=□a=□ f(x)=\square \\ a=\square f(x)=□a=□

Problem 6539

Find the limit: lim⁡x→−∞(x2+2x5)\lim _{x \rightarrow-\infty}\left(x^{2}+2 x^{5}\right)x→−∞lim​(x2+2x5). Enter ∞\infty∞, −∞-\infty−∞, or DNE if it doesn't exist.

Problem 6540

Calculate the limit: lim⁡x→∞5x−62x+1\lim _{x \rightarrow \infty} \frac{5 x-6}{2 x+1}x→∞lim​2x+15x−6​. Enter ' ∞\infty∞ ', '- ∞\infty∞ ', or DNE if it doesn't exist.

Problem 6541

Gegeben ist $f(x)=x$ . Finde die Stammfunktion $F$ durch den Punkt $P(1 \mid 1,5)$ .

Gegeben ist $f(x)=x$ . Finde die Stammfunktion $F$ , die durch den Punkt $P(1 \mid 1,5)$ geht.

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=2x+8$ , where $h \neq 0$ .

Find the end behavior of $f(x)=-6 x^{4}-6 x^{2}+1$ using $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ .

Analyze the end behavior of $f(x)=7 x^{2}-x^{3}+4 x-2$ using $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ .

Find the sum of the series: $36 + 12 + 4 + \cdots$ . Is it a finite sum or does it not exist? Choose A or B.

Find the limit and check continuity at $y=1$ : $\lim _{y \rightarrow 1} \sec \left(y \sec ^{2} y-\tan ^{2} y-1\right)$ .

Find the limit of $f(x)=\frac{2}{x}-4$ as $x \to \infty$ and $x \to -\infty$ . What is $\lim_{x \to \infty} f(x)$ ?

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sin 20 x}{16 x}=\square$ .

Why does $f(x)=0$ have a solution between $x=5$ and $x=7$ if $f(5)<0$ and $f(7)>0$ ? Explain and sketch.

If $h(x)=f(x) \cdot g(x)$ is continuous at $x=0$ , must $f(x)$ and $g(x)$ be continuous at $x=0$ ? Explain.

Calculate the average rate of change of $f(x)=\sqrt{x}$ from $x_{1}=9$ to $x_{2}=64$ . Simplify your answer.

Find the critical number(s) of the function $f(x)=x^{23/11}-x^{12/11}$ .

Find the critical numbers of the function $f(x)=x^{15/7}+x^{8/7}$ .

Calculate the average rate of change for $f(x)=x^{2}+8x$ from $x=4$ to $x=6$ . Simplify your answer.

Find the critical numbers of the function $f(x)=\frac{2}{3}x^3 - 3x^2 - 80x - 10$ .

Find where the cost function $C(x)=x^{3}-2 x^{2}+4 x+40$ is decreasing and increasing for $x>0$ .

Which functions are continuous for all real numbers: $h(x)=\sin(x)$ or $f(x)=\cos(x)$ ? Choose: A) $h$ only, B) $f$ only, C) both, D) neither.

Find the intervals where the profit function $P(x) = R(x) - C(x)$ is increasing, given:
$C(x) = 0.13x^{2} - 0.00006x^{3}$ and $R(x) = 0.886x^{2} - 0.0002x^{3}$ .

Show that for small $x$ , $\sin 3x \cos 4x \approx 3x - 24x^3$ . Approximate $\sin 0.3 \cos 0.4$ and find the percentage error.

Find local and absolute extreme values of a function $f$ with given points $(a,b)$ , $(u,v)$ , $(c,d)$ , $(x,y)$ .

Find intervals where the drug concentration $K(t)=\frac{12 t}{t^{2}+4}$ is increasing and decreasing.

Estimate the increase in metabolic rate $P$ when body mass $m$ changes from 102 kg to 103 kg using $P=73.3 m^{3/4}$ .

Lös differentialekvationen $y^{\prime}(t)=-\frac{1}{10}(y(t)-10)$ med $y(0)=2022$ .

Find the growth rate when $M=125 \mathrm{~kg}$ , given $\frac{d M}{d t}=14$ kg/year at $M=100 \mathrm{~kg}$ .
Also, how much mass must $M=0.7 \mathrm{~kg}$ gain to double its growth rate?

Find $\frac{d B}{d I}$ for $B=k I^{2/3}$ and $\frac{d H}{d W}$ for $H=k W^{3/2}$ . Identify the correct statements.

Given the piecewise function $f(x)$ , determine which statement about its continuity and differentiability at $x=-1$ and $x=2$ is TRUE.

Find the slope of the tangent line to $2y^2=x^4y+c g(x)$ at (1,3) given $g'(1)=-1$ . Choices: $2+c$ , $\frac{c-24}{2}$ , $24-c$ , $-c$ , $\frac{12-c}{11}$ .

Given the function $g(v)=v^{3}-75v+6$ , find $g^{\prime}(v)$ , where $g^{\prime}(v)=0$ , and identify critical numbers.

Find the tangent line equation for $f(x)=2x^{2}+g(x)$ at $x=3$ , given $g(3)=1$ and $g'(3)=2$ .

Find the derivative of the function $f(x)=3 \cos (6 \ln (x))$ , denoted as $f^{\prime}(x)$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{1+9 x^{6}}}{4-x^{3}}$ .

Find the function $f(x)$ and the number $a$ such that $\lim _{h \rightarrow 0} \frac{h^{3}}{h} = f^{\prime}(a)$ . $f(x)=$ $a=$

Find the derivative of the function $f(x)=8 x^{3} \arctan \left(4 x^{4}\right)$ . What is $f^{\prime}(x)$ ?

Find the limit: $\lim _{x \rightarrow \infty}\left[\ln \left(4+x^{2}\right)-\ln (7+x)\right]$ .

Find the function $f(x)$ and the number $a$ such that $\operatorname{limit}_{h \rightarrow 0} \frac{h^{3}}{h} = f^{\prime}(a)$ . $f(x)=\square \\ a=\square$

Find the limit: $\lim _{x \rightarrow-\infty}\left(x^{2}+2 x^{5}\right)$ . Enter $\infty$ , $-\infty$ , or DNE if it doesn't exist.

Calculate the limit: $\lim _{x \rightarrow \infty} \frac{5 x-6}{2 x+1}$ . Enter ' $\infty$ ', '- $\infty$ ', or DNE if it doesn't exist.

Find the derivative of $f(x) = \int_{3}^{x} \frac{1}{1+t^{5}} dt$ and evaluate $f'(12)$ .

Find the tangent line equation for $f(x)=(5 x-1)^{\frac{1}{2}}$ at point $P=(1, f(1))$ .

Find the limit: $\lim _{x \rightarrow \infty}\left(\sqrt{25 x^{2}+x}-5 x\right)$ . Enter $\omega$ , - $\infty$ , or DNE if it doesn't exist.

Find when the function $f(x)=\cos (6 x)+3 x$ is decreasing on $[0, \frac{\pi}{3}]$ .

Find the limit: $\lim _{x \rightarrow \infty}\left(e^{-5 x} \cos (x)\right)$ . Enter $\infty$ , $-\infty$ , or DNE if it doesn't exist.

Find the derivative $\frac{d y}{d x}$ of the function $y=x^{10 \cos (x)}$ .

Find the limits as $x \to \infty$ and $x \to -\infty$ for $f(x)=(2-x)(1+x)^{2}(1-x)^{4}$ .

Find the intervals where the polynomial function $f(x)$ , with derivative $f^{\prime}(x)=-2(x+5)^{3}(x-3)^{2}(x-4)$ , is increasing.

Find the function $f(x)$ and the number $a$ such that $\operatorname{limit}_{h \rightarrow 0} \frac{h^{3}}{h} = f^{\prime}(a)$ . $f(x)= \\ a=$

Calculate the limit: $\lim _{t \rightarrow \infty} \frac{\sqrt{t}+t^{2}}{6 t-t^{2}}$ . Enter $\infty$ , $-\infty$ , or DNE if it doesn't exist.

Find the limit: $\lim _{x \rightarrow \infty} \frac{x^{4}-9 x^{2}+x}{x^{3}-x+3}$ . Enter $\infty$ , $-\infty$ , or DNE if it doesn't exist.

Find the limit: $\lim _{x \rightarrow-\infty} \frac{6 x^{5}-x}{x^{4}+5}$ . Enter $\infty$ , $-\infty$ , or DNE if it doesn't exist.

Find the derivative of $h(x)=\frac{1}{x^{2}+3 x-1}$ using the chain rule.

Find $g^{\prime}\left(\frac{\pi}{2}\right)$ for $g(x)=\ln((f(x))^{2}+A \sin x)$ given $f\left(\frac{\pi}{2}\right)=4$ and $f^{\prime}\left(\frac{\pi}{2}\right)=3$ .

Determine the roots, asymptotes, or critical points of $h(x)=\frac{1}{x^{2}+3 x-1}$ using the chain rule.

Find $\left.\frac{\partial z}{\partial y}\right|_{\substack{x=1 \\ y=e}}$ for $z=x y^{x}$ : A. 6 B. $\frac{1}{e}$ C. 1 D. $1+\frac{1}{e}$

Find the value of $\lim _{x \rightarrow 0} f(x) \sin \left(\frac{1}{x^{3}}\right)$ if $\lim _{x \rightarrow 0} f(x)=0$ .

Find the derivative of $f(x) = 3(4-9x)^4$ using the chain rule.

Given a continuous function $f$ with $f(-2)=-4$ , $f(2)=1$ , and $\lim_{x \to \infty} f(x)=0$ , which statement is TRUE?

Find the derivative of $y=\csc \sqrt{x}+\sqrt{\tan x}$ using the chain rule.

Find $f\left(\frac{\pi}{4}\right)$ given $f''(x)=-9 \sin(3x)$ , $f'(0)=-1$ , and $f(0)=-1$ .

Determine which limit cannot be found using the Squeeze Theorem for the function $f(x)$ defined by the inequalities.

If $y=f(x)$ is not differentiable at $x=0$ , which statement is always TRUE?

Given a continuous function $f$ on $[0,3]$ with values $f(0)=-2$ , $f(1)=2$ , and $f(2)=k$ , which statement is always TRUE?

Find $g'(3)$ for $g(x)=\frac{A f(x)}{e^{4 x}}$ given $f(3)=1$ , $f'(3)=4$ .

Approximate a root of $3 x^{3}+6 x+2=0$ using Newton's method with initial guess $x_{1}=-1$ . Find $x_{2}$ and $x_{3}$ .

Given a continuous function $f$ on $[0,3]$ with values $f(0)=-2$ , $f(1)=2$ , and $f(2)=k$ , which statement is always TRUE?

An inverted pyramid with a square base (sides $6 \mathrm{~cm}$ ) and height $10 \mathrm{~cm}$ is filled at $75 \mathrm{~cm}^3/\mathrm{s}$ . Find the water level rise rate when it's $6 \mathrm{~cm}$ .

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{1}{x+8}$ , where $h \neq 0$ .

Given the piecewise function $f(x)$ , which statements about its continuity and differentiability at $x=-1$ and $x=2$ are true?

Find the $x$ -intercept of the tangent line to $f(x)=(3x-1)^{4}(2x+2)^{2}$ at the $y$ -axis crossing point.