Calculus

Problem 12601

Find the tangent line equation for $g(x)=(3x^{2}-4)f(x)$ at $x=-2$ given $f(-2)=-2$ and $f'(-2)=-3$ .

See Solution

Problem 12602

Find the derivative of $y=\frac{x^{2}}{e^{x}}$ .

See Solution

Problem 12603

Find the rate of change of the surface area $A=4 \pi r^{2}$ with respect to the radius $r$ .

See Solution

Problem 12604

Check if the function is continuous, differentiable, both, or neither at the point where it changes:
$f(x)=\left\{\begin{array}{ll} x^{2}+3 x+24, & x \leq-4 \\ -x+20, & x>-4 \end{array}\right.$

See Solution

Problem 12605

Check if the function is continuous or differentiable at the point where its definition changes:
$f(x)=\left\{\begin{array}{ll} x^{2}-13 x-7, & x<1 \\ -17 x-11, & x \geq 1 \end{array}\right.$

See Solution

Problem 12606

Evaluate the limit and simplify your answer: $\lim _{h \rightarrow 0} \frac{\sqrt{-3+h+4}-\sqrt{-3+4}}{h}$

See Solution

Problem 12607

Evaluate the limit: $\lim_{h \rightarrow 0} \frac{3 \csc (x+h) - 3 \csc (x)}{h}$

See Solution

Problem 12608

Find the limit as $h$ approaches 0 for $\frac{\frac{1}{x+h-1}-\frac{1}{x-1}}{h}$ .

See Solution

Problem 12609

Evaluate the double integral: $\int_{0}^{2} \int_{0}^{2}(u+2 v) \cos (2u) 4 \, du \, dv$ .

See Solution

Problem 12610

Calculate the limit: $\lim _{n \rightarrow \infty}\left(\sqrt{n^{2}+5 n}-\sqrt{n^{2}-n}\right)$ .

See Solution

Problem 12611

Find values of $a$ and $b$ for the function $f(x)$ to be differentiable at $x=2$ :
$f(x)=\begin{cases} 2 a x-7 & \text{for } x<2 \\ 2 b x^{2}+4 x-1 & \text{for } x \geq 2 \end{cases}$

See Solution

Problem 12612

Find the derivative $f^{\prime}(6)$ for the function $f(x)=\frac{5 \sqrt{x^{3}}}{6}+\frac{3}{\sqrt{x}}$ .

See Solution

Problem 12613

Find values of $a$ and $b$ so that the function $f(x)$ is differentiable at $x=4$ :
$f(x)=\left\{\begin{array}{lll} 2 a x+5 & \text { for } & x \leq 4 \\ 2 b x^{2}-x-6 & \text { for } & x>4 \end{array}\right.$

See Solution

Problem 12614

Find the derivatives: (a) $\frac{d}{d y}(A y^{b+1})$ (b) $\frac{d f}{d x}$ for $f(x)=x^{2}(x^{4}+1)$ (c) $\frac{d g}{d p}$ for $g(p)=\frac{p+1}{p-1}$ Also, for $f(x)=100+2x$ and $g(x)=x^{3}$ : (a) Find $f(g(x))$ and $g(f(x))$ . (b) Derivative of $f(g(x))$ using the chain rule. (c) Derivative of $g(f(x))$ using the chain rule.

See Solution

Problem 12615

Bestimmen Sie die Ableitung von $f(x) = \frac{2}{\sqrt{x}} + \frac{\sqrt{x}}{2}$ .

See Solution

Problem 12616

Find the derivative of $y = \frac{x^{2}}{e^{x}}$ .

See Solution

Problem 12617

A \$6,000 deposit in an IRA earns 7\% interest compounded continuously. What is its value in 15 years? Round to the nearest cent.

See Solution

Problem 12618

Find the antiderivative of $f^{\prime}(t)=\frac{1}{\sqrt{2 t+1}}$ .

See Solution

Problem 12619

Evaluate the integral $\int_{C} f(x, y, z) d s=\int_{0}^{1}\left(8 e^{2 t}+32 t\right) \sqrt{4 e^{2 t}+16 t^{2}+16} d t$ .

See Solution

Problem 12620

Evaluate the line integral using $u$ -substitution:
$\int_{C} f(x, y, z) d s = \int_{0}^{1}(8 e^{2 t}+32 t) \sqrt{4 e^{2 t}+16 t^{2}+16} d t$

See Solution

Problem 12621

Differentiate the function: $y=x^{7} \ln x-\frac{1}{3} x^{3}$ . Find $\frac{d y}{d x}$ .

See Solution

Problem 12622

Find the derivative of $y=\ln \left(\frac{(x^{2}+7)^{5}}{\sqrt{3-x}}\right)$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 12623

Find the number of cars that pass by from 8 to 9 AM given $q(t)=1400+2200 t-270 t^{2}$ .

See Solution

Problem 12624

Find the displacement of an object with velocity $v(t)=-18t+150$ over the intervals: a) $[1,7]$ , b) $[0,10]$ .

See Solution

Problem 12625

Find the displacement of an object with velocity $v(t)=18 \sqrt[5]{t}$ over the interval $[0,243]$ .

See Solution

Problem 12626

Find the total number of antibodies produced in 20 minutes given $f(t)=\frac{11 t}{3 t^{2}+10}$ . Round to two decimal places.

See Solution

Problem 12627

At $t=0$ , species $A$ and $B$ have equal populations. Find which has a larger population after 5, 10, and 25 years using:
$R_{1}=19+110 \ln(1+2 t^{2})$
$R_{2}=19 e^{0.2 t+1}$

See Solution

Problem 12628

Differentiate $y=\ln \left[(x+8)^{5}(x+4)^{3}(x+6)^{6}\right]$ and find $\frac{d}{d x}[\ln \left[(x+8)^{5}(x+4)^{3}(x+6)^{6}\right]]$ .

See Solution

Problem 12629

Differentiate the function $y=19 \ln \left(x^{23} \sqrt{5 x+6}\right)$ . Find $y'=\square$ .

See Solution

Problem 12630

Find the total oil leaked in the first hour using $f(t)=A e^{-k t}$ and the integral $\int_{0}^{60} f(t) \, dt$ . What are the units?

See Solution

Problem 12631

Find the marginal-revenue function for the demand $p=\frac{30}{\ln (q+2)}$ . What is $\frac{d r}{d q}$ ?

See Solution

Problem 12632

Find the limit of the sequence $a_{n}=n\left(\sqrt[3]{n^{3}+n+2}-n\right)$ as $n$ approaches infinity.

See Solution

Problem 12633

Find the marginal cost of the total-cost function $c=24 \ln (q+1)+13$ when $q=4$ . The answer is $\square$ .

See Solution

Problem 12634

Approximate the area under $f(x)=-0.2 x^{2}-x+12$ on $[-1,3]$ using trapezoids with $n=4$ subintervals.

See Solution

Problem 12635

Oil leaks at $r=17 e^{4 t}$ gallons/hour. Find the integral for 0 to 3 hours. Estimate using a left sum with 3 parts.

See Solution

Problem 12636

Find the limit: $\lim _{m \rightarrow-\infty} m\left(\sqrt[3]{m^{3}+m+2}\right)$ .

See Solution

Problem 12637

Differentiate the function $y=17 \ln \left(x^{3} \sqrt[3]{4 x+7}\right)$ . Find $y'=\square$ .

See Solution

Problem 12638

Find the rate of change of supply with respect to price given $q(p)=22+8 \ln (3 p+1)$ . Calculate $\frac{d q}{d p}$ .

See Solution

Problem 12639

Find the limit: $\lim _{n \rightarrow \infty} n \sqrt[3]{n^{3}+n+2}-n$ .

See Solution

Problem 12640

Differentiate $y=17 \ln \left(x^{3} \sqrt[3]{7 x+5}\right)$ . Find $\mathbf{y}^{\prime}=\square$ .

See Solution

Problem 12641

Find the tangent line equation to $y=\ln(x^{2}-9x-9)$ at $x=10$ . $\mathbf{y}=\square$ (Simplify your answer.)

See Solution

Problem 12642

Differentiate the function $y=e^{x^{6}+6 x-1}$ . Find $y'=\frac{d}{dx}\left(e^{x^{6}+6 x-1}\right)=\square$ .

See Solution

Problem 12643

Differentiate the function $y=\frac{e^{x}-4 e^{-x}}{e^{x}+4 e^{-x}}$ . Find $y'$ .

See Solution

Problem 12644

Find the derivative of $y$ where $y=\frac{\ln x}{4+\ln x}$ .

See Solution

Problem 12645

Differentiate the function $y=e^{x^{6}+3 x+10}$ . Find $y' = \frac{d}{dx}(e^{x^{6}+3 x+10}) = \square$ .

See Solution

Problem 12646

A particle moves along a line with $s(t)=2t^2-20t$ . Find velocity, acceleration, when stationary, and motion intervals.

See Solution

Problem 12647

Find the derivative of $y$ with respect to $x$ for $y=\frac{\ln x}{2+3 \ln x}$ .

See Solution

Problem 12648

Bestimmen Sie den Grenzwert $L = \lim _{x \rightarrow \infty} \frac{1-2 x}{x+1}$ mithilfe einer Wertetabelle.

See Solution

Problem 12649

Find the derivative of $y=\frac{-8 e^{9 x}}{3 x-2}$ . What is $\frac{d}{d x}\left(\frac{-8 e^{9 x}}{3 x-2}\right)$ ?

See Solution

Problem 12650

Differentiate the function $f(x)=(x+6) e^{-5 x+3}$ and express $f^{\prime}(x)$ in factored form.

See Solution

Problem 12651

Find the derivative of $y$ with respect to $x$ for $y=\frac{\ln x}{1+4 \ln x}$ .

See Solution

Problem 12652

Differentiate the function $y=(9 x+8)^{x}$ . Find $\frac{d y}{d x}=\square$ .

See Solution

Problem 12653

Find the limit of the sequence $a_{n}=\frac{n+(-1)^{n}}{n}$ as $n \to \infty$ .

See Solution

Problem 12654

Differentiate the function $y=(2x+7)^{x}$ . Find $\frac{dy}{dx}$ .

See Solution

Problem 12655

Evaluate the double integral: $4 \int_{0}^{2}\left(\int_{0}^{2} u \cos (2 u) d v\right) d u$ .

See Solution

Problem 12656

Find the grams of carbon-14 left after 5403 years using the model $A=16 e^{-0.000121 t}$ . Answer: $\square$ grams.

See Solution

Problem 12657

Differentiate the function $y=(8 x+1)^{x}$ . Find $\frac{d y}{d x}$ .

See Solution

Problem 12658

Find $\frac{dy}{dx}$ if $y=\frac{\ln x}{1+\ln x}$ .

See Solution

Problem 12659

Find the rate of change of price $p$ with respect to quantity $q$ for $p=e^{-0.003 q}$ when $q=300$ . Answer: $\square$ .

See Solution

Problem 12660

Find the marginal-cost function and its values for $\bar{c}=\frac{2250 e^{\frac{q}{450}}}{q}$ at $q=225$ and $q=450$ .

See Solution

Problem 12661

Evaluate the expression $[492 t-576 \cos (t)-(54) \sin (2 t)]_{0}^{2 \pi}$ .

See Solution

Problem 12662

Find the marginal-cost function from $\bar{c}=\frac{2250 e^{\frac{q}{450}}}{q}$ and calculate it for $q=225$ and $q=450$ .

See Solution

Problem 12663

Check if the sequence $a_{n}=(-1)^{n}\left(1-\frac{1}{n}\right)$ converges.

See Solution

Problem 12664

Find $\frac{dy}{dt}$ for $y=e^{(2 \cos t+\ln t)}$ .

See Solution

Problem 12665

Find the derivative of $y$ with respect to $t$ for $y=e^{(11 \sin t+\ln t)}$ .

See Solution

Problem 12666

Analyze the sequence $a_{n}=\frac{n !}{n^{n}}$ : Is it monotonic, bounded, and does it converge?

See Solution

Problem 12667

Determine the truth of these statements about absolute and local maxima of a function $f(x)$ :
I. Absolute maximum is the largest $y$ -value of $f(x)$ . II. A function can have multiple absolute maxima. III. Local maximum $f(c)$ is the largest $y$ -value near $c$ . IV. A function can have multiple local maxima. V. A function can't have both local and absolute maxima at $c$ . VI. A function can have no absolute max or min. VII. Continuous functions on closed intervals have absolute max and min. VIII. Only continuous functions have absolute max and min on closed intervals.

See Solution

Problem 12668

Is each of the following statements true or false based on Fermat's Theorem?
I. If $f(x)$ has a local max at $x=c$ , then $c$ is a critical point. II. If $c$ is not a critical point, then no local max/min exists at $c$ . III. If $c$ is a critical point, then a local max/min exists at $c$ . IV. If no local extremum at $c$ , then $c$ is not a critical point. V. If $f(x)$ has a local max at $c$ , then $f'(c)=0$ .

See Solution

Problem 12669

Find the derivative of $s(t) = \frac{e^{2t}}{1 + e^{2t}}$ .

See Solution

Problem 12670

Calculate the limit: $\lim _{n \rightarrow \infty}\left(\frac{n !}{n^{n}}\right)$ .

See Solution

Problem 12671

Find the curve equation with slope $f^{\prime}(x)=x^{\frac{2}{9}}$ that passes through $(1, \frac{9}{11})$ .

See Solution

Problem 12672

Find the critical numbers of the function $f(x)=x^{3}-48x$ . Enter answers as a comma-separated list.

See Solution

Problem 12673

Find the critical numbers of the function $f(x)=x^{3}+6 x^{2}-15 x-10$ . Provide your answers as a comma-separated list.

See Solution

Problem 12674

Find critical points of $f(x)=\frac{e^{x}+e^{-x}}{2}$ : domain, $f'(x)$ , where $f'(x)=0$ , and where $f'(x)=D N E$ .

See Solution

Problem 12675

Find the derivative of $k(x)=\frac{3}{\cot^{7}(x)}$ and confirm if $f(x)=k(x)$ . Is there a specific point for evaluation?

See Solution

Problem 12676

Is it possible for a function on a finite interval $(a, b)$ to lack an absolute maximum or minimum?

See Solution

Problem 12677

Find $f^{\prime \prime}(2)$ for the Taylor polynomial $T_{4}(x)=1+4(x-2)-5(x-2)^{2}+18(x-2)^{3}-(x-2)^{4}$ . Options: (a) -10 (b) -1 (c) -5 (d) 18 (e) 4.

See Solution

Problem 12678

Find the values of the squares and evaluate the integral $\int_{0}^{2 \pi} (8 + (\square) \sin^2(t) + (\square) du \sin^3(t)) \cos(t) dt$ .

See Solution

Problem 12679

Evaluate the integral of $8(1-\sin^2(t))\cos(t) + 16\sin^3(t)\cos(t) + 40\sin^2(t)\cos(t)$ from $0$ to $2\pi$ .

See Solution

Problem 12680

Create a sign diagram for the derivative of $f(x)=x^{3}-6 x^{2}-15 x+5$ . Identify intervals of increase and decrease.

See Solution

Problem 12681

Find the limit as $x$ approaches -∞: $\lim _{x \rightarrow-\infty} \frac{\cos 18 x}{5 x} = \square($ Simplify your answer.)

See Solution

Problem 12682

Calculate the total compression $\Delta L$ (in $m$ ) of an $8 \mathrm{~m}$ sandstone pillar with density $2250 \mathrm{~kg/m}^3$ and Young's Modulus $12.5 \mathrm{GPa}$ .

See Solution

Problem 12683

Evaluate the limit: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(\frac{3 i}{n}\right)^{3} \cdot \frac{3}{n}$ using the formula for the sum of cubes.

See Solution

Problem 12684

Find the limit of $f(x)=\frac{6 x^{5}+9}{x^{5}-x^{2}+x+8}$ as $x \rightarrow \infty$ and $x \rightarrow -\infty$ .

See Solution

Problem 12685

Find the limit: $\lim _{x \rightarrow \infty} \frac{8 \sqrt{x}+x^{-8}}{9 x-3}$ . Simplify your answer.

See Solution

Problem 12686

Find the limit as $x$ approaches $-\infty$ : $\lim _{x \rightarrow-\infty}\left(\frac{1-x^{4}}{x^{2}+9 x}\right)^{5}$ . Simplify your answer.

See Solution

Problem 12687

Find the limits of $h(x)=\frac{6 x^{4}}{13 x^{4}+17 x^{3}+20 x^{2}}$ as $x \rightarrow \infty$ and $x \rightarrow -\infty$ .

See Solution

Problem 12688

Find the limit as $x$ approaches $-\infty$ : $\lim _{x \rightarrow-\infty}\left(\frac{1-x^{4}}{x^{2}+9 x}\right)^{5}$ . Write $\infty$ or $-\infty$ as needed.

See Solution

Problem 12689

Find the limit:
$\lim _{x \rightarrow-\infty} \frac{\sqrt[3]{x}-4 x+6}{2 x+x^{\frac{2}{3}}-1}$
Simplify your answer to $\infty$ or $-\infty$ .

See Solution

Problem 12690

Find the limit of $f(x)=\frac{9 x^{4}+5 x^{3}+8}{4 x^{5}}$ as $x \rightarrow \infty$ and $x \rightarrow -\infty$ .

See Solution

Problem 12691

Find the limit: $\lim _{x \rightarrow 7} \frac{1}{(x-7)^{2}}=\square($ Simplify your answer. $)$

See Solution

Problem 12692

Find the derivative of $f(x)=\ln \left(7 x^{3}-3 x^{2}+5\right)^{4}$ .

See Solution

Problem 12693

Find the limit as $x$ approaches 7 from the right: $\lim _{x \rightarrow 7^{+}} \frac{4}{x-7}=\square$ .

See Solution

Problem 12694

Find the limit as $x$ approaches 0 from the right: $\lim _{x \rightarrow 0^{+}} \frac{1}{8 x}=\square($ Simplify your answer.)

See Solution

Problem 12695

Find the limit as $\theta$ approaches $0$ from the right: $\lim _{\theta \rightarrow 0^{+}}(5+\csc \theta)$ .

See Solution

Problem 12696

Find the limit: $\lim _{x \rightarrow\left(\frac{11 \pi}{2}\right)^{-}}-8 \tan x = \square$ (simplify your answer).

See Solution

Problem 12697

Find the absolute extreme values of the function $f(x)=x^{4}+4x^{3}+4x^{2}$ on the interval $[-2,1]$ .

See Solution

Problem 12698

Approximate $f^{\prime}(-7.2)$ using $f(-7.2)=-6.2$ and $f(-7.1)=-4.6$ . $f^{\prime}(-7.2) \approx$

See Solution

Problem 12699

Find the critical numbers of the function $f(x) = x^{3} + 6x^{2} - 14x - 70$ .

See Solution

Problem 12700

Find the derivative of the inverse function of $f(x)=\frac{e^{x}-e^{-x}}{2}$ using $\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}$ . Then, find the tangent line to $y^{2}(y^{2}-4)=x^{2}(x^{2}-5)$ at $(0,-2)$ .

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 12601

Find the tangent line equation for g(x)=(3x2−4)f(x)g(x)=(3x^{2}-4)f(x)g(x)=(3x2−4)f(x) at x=−2x=-2x=−2 given f(−2)=−2f(-2)=-2f(−2)=−2 and f′(−2)=−3f'(-2)=-3f′(−2)=−3.

Problem 12602

Find the derivative of y=x2exy=\frac{x^{2}}{e^{x}}y=exx2​.

Problem 12603

Find the rate of change of the surface area A=4πr2A=4 \pi r^{2}A=4πr2 with respect to the radius rrr.

Problem 12604

Check if the function is continuous, differentiable, both, or neither at the point where it changes: f(x)={x2+3x+24,x≤−4−x+20,x>−4 f(x)=\left\{\begin{array}{ll} x^{2}+3 x+24, & x \leq-4 \\ -x+20, & x>-4 \end{array}\right. f(x)={x2+3x+24,−x+20,​x≤−4x>−4​

Problem 12605

Check if the function is continuous or differentiable at the point where its definition changes: f(x)={x2−13x−7,x<1−17x−11,x≥1 f(x)=\left\{\begin{array}{ll} x^{2}-13 x-7, & x<1 \\ -17 x-11, & x \geq 1 \end{array}\right. f(x)={x2−13x−7,−17x−11,​x<1x≥1​

Problem 12606

Evaluate the limit and simplify your answer: lim⁡h→0−3+h+4−−3+4h\lim _{h \rightarrow 0} \frac{\sqrt{-3+h+4}-\sqrt{-3+4}}{h}h→0lim​h−3+h+4​−−3+4​​

Problem 12607

Evaluate the limit: lim⁡h→03csc⁡(x+h)−3csc⁡(x)h\lim_{h \rightarrow 0} \frac{3 \csc (x+h) - 3 \csc (x)}{h}limh→0​h3csc(x+h)−3csc(x)​

Problem 12608

Find the limit as hhh approaches 0 for 1x+h−1−1x−1h\frac{\frac{1}{x+h-1}-\frac{1}{x-1}}{h}hx+h−11​−x−11​​.

Problem 12609

Evaluate the double integral: ∫02∫02(u+2v)cos⁡(2u)4 du dv\int_{0}^{2} \int_{0}^{2}(u+2 v) \cos (2u) 4 \, du \, dv∫02​∫02​(u+2v)cos(2u)4dudv.

Problem 12610

Calculate the limit: lim⁡n→∞(n2+5n−n2−n)\lim _{n \rightarrow \infty}\left(\sqrt{n^{2}+5 n}-\sqrt{n^{2}-n}\right)limn→∞​(n2+5n​−n2−n​).

Problem 12611

Find values of aaa and bbb for the function f(x)f(x)f(x) to be differentiable at x=2x=2x=2: f(x)={2ax−7for x<22bx2+4x−1for x≥2 f(x)=\begin{cases} 2 a x-7 & \text{for } x<2 \\ 2 b x^{2}+4 x-1 & \text{for } x \geq 2 \end{cases} f(x)={2ax−72bx2+4x−1​for x<2for x≥2​

Problem 12612

Find the derivative f′(6)f^{\prime}(6)f′(6) for the function f(x)=5x36+3xf(x)=\frac{5 \sqrt{x^{3}}}{6}+\frac{3}{\sqrt{x}}f(x)=65x3​​+x​3​.

Problem 12613

Problem 12614

Problem 12615

Bestimmen Sie die Ableitung von f(x)=2x+x2f(x) = \frac{2}{\sqrt{x}} + \frac{\sqrt{x}}{2}f(x)=x​2​+2x​​.

Problem 12616

Find the derivative of y=x2exy = \frac{x^{2}}{e^{x}}y=exx2​.

Problem 12617

A \$6,000 deposit in an IRA earns 7\% interest compounded continuously. What is its value in 15 years? Round to the nearest cent.

Problem 12618

Find the antiderivative of f′(t)=12t+1f^{\prime}(t)=\frac{1}{\sqrt{2 t+1}}f′(t)=2t+1​1​.

Problem 12619

Evaluate the integral ∫Cf(x,y,z)ds=∫01(8e2t+32t)4e2t+16t2+16dt\int_{C} f(x, y, z) d s=\int_{0}^{1}\left(8 e^{2 t}+32 t\right) \sqrt{4 e^{2 t}+16 t^{2}+16} d t∫C​f(x,y,z)ds=∫01​(8e2t+32t)4e2t+16t2+16​dt.

Problem 12620

Evaluate the line integral using uuu-substitution: ∫Cf(x,y,z)ds=∫01(8e2t+32t)4e2t+16t2+16dt\int_{C} f(x, y, z) d s = \int_{0}^{1}(8 e^{2 t}+32 t) \sqrt{4 e^{2 t}+16 t^{2}+16} d t∫C​f(x,y,z)ds=∫01​(8e2t+32t)4e2t+16t2+16​dt

Problem 12621

Differentiate the function: y=x7ln⁡x−13x3y=x^{7} \ln x-\frac{1}{3} x^{3}y=x7lnx−31​x3. Find dydx\frac{d y}{d x}dxdy​.

Problem 12622

Find the derivative of y=ln⁡((x2+7)53−x)y=\ln \left(\frac{(x^{2}+7)^{5}}{\sqrt{3-x}}\right)y=ln(3−x​(x2+7)5​). What is dydx\frac{d y}{d x}dxdy​?

Problem 12623

Find the number of cars that pass by from 8 to 9 AM given q(t)=1400+2200t−270t2q(t)=1400+2200 t-270 t^{2}q(t)=1400+2200t−270t2.

Problem 12624

Find the displacement of an object with velocity v(t)=−18t+150v(t)=-18t+150v(t)=−18t+150 over the intervals: a) [1,7][1,7][1,7], b) [0,10][0,10][0,10].

Problem 12625

Find the displacement of an object with velocity v(t)=18t5v(t)=18 \sqrt[5]{t}v(t)=185t​ over the interval [0,243][0,243][0,243].

Problem 12626

Find the total number of antibodies produced in 20 minutes given f(t)=11t3t2+10f(t)=\frac{11 t}{3 t^{2}+10}f(t)=3t2+1011t​. Round to two decimal places.

Problem 12627

At t=0t=0t=0, species AAA and BBB have equal populations. Find which has a larger population after 5, 10, and 25 years using:R1=19+110ln⁡(1+2t2)R_{1}=19+110 \ln(1+2 t^{2})R1​=19+110ln(1+2t2) R2=19e0.2t+1R_{2}=19 e^{0.2 t+1}R2​=19e0.2t+1

Problem 12628

Differentiate y=ln⁡[(x+8)5(x+4)3(x+6)6]y=\ln \left[(x+8)^{5}(x+4)^{3}(x+6)^{6}\right]y=ln[(x+8)5(x+4)3(x+6)6] and find ddx[ln⁡[(x+8)5(x+4)3(x+6)6]]\frac{d}{d x}[\ln \left[(x+8)^{5}(x+4)^{3}(x+6)^{6}\right]]dxd​[ln[(x+8)5(x+4)3(x+6)6]].

Problem 12629

Differentiate the function y=19ln⁡(x235x+6)y=19 \ln \left(x^{23} \sqrt{5 x+6}\right)y=19ln(x235x+6​). Find y′=□y'=\squarey′=□.

Problem 12630

Find the total oil leaked in the first hour using f(t)=Ae−ktf(t)=A e^{-k t}f(t)=Ae−kt and the integral ∫060f(t) dt\int_{0}^{60} f(t) \, dt∫060​f(t)dt. What are the units?

Problem 12631

Find the marginal-revenue function for the demand p=30ln⁡(q+2)p=\frac{30}{\ln (q+2)}p=ln(q+2)30​. What is drdq\frac{d r}{d q}dqdr​?

Problem 12632

Find the limit of the sequence an=n(n3+n+23−n)a_{n}=n\left(\sqrt[3]{n^{3}+n+2}-n\right)an​=n(3n3+n+2​−n) as nnn approaches infinity.

Problem 12633

Find the marginal cost of the total-cost function c=24ln⁡(q+1)+13c=24 \ln (q+1)+13c=24ln(q+1)+13 when q=4q=4q=4. The answer is □\square□.

Problem 12634

Approximate the area under f(x)=−0.2x2−x+12f(x)=-0.2 x^{2}-x+12f(x)=−0.2x2−x+12 on [−1,3][-1,3][−1,3] using trapezoids with n=4n=4n=4 subintervals.

Problem 12635

Oil leaks at r=17e4tr=17 e^{4 t}r=17e4t gallons/hour. Find the integral for 0 to 3 hours. Estimate using a left sum with 3 parts.

Problem 12636

Find the limit: lim⁡m→−∞m(m3+m+23)\lim _{m \rightarrow-\infty} m\left(\sqrt[3]{m^{3}+m+2}\right)limm→−∞​m(3m3+m+2​).

Problem 12637

Differentiate the function y=17ln⁡(x34x+73)y=17 \ln \left(x^{3} \sqrt[3]{4 x+7}\right)y=17ln(x334x+7​). Find y′=□y'=\squarey′=□.

Problem 12638

Find the rate of change of supply with respect to price given q(p)=22+8ln⁡(3p+1)q(p)=22+8 \ln (3 p+1)q(p)=22+8ln(3p+1). Calculate dqdp\frac{d q}{d p}dpdq​.

Problem 12639

Find the limit: lim⁡n→∞nn3+n+23−n\lim _{n \rightarrow \infty} n \sqrt[3]{n^{3}+n+2}-nlimn→∞​n3n3+n+2​−n.

Problem 12640

Differentiate y=17ln⁡(x37x+53)y=17 \ln \left(x^{3} \sqrt[3]{7 x+5}\right)y=17ln(x337x+5​). Find y′=□\mathbf{y}^{\prime}=\squarey′=□.

Problem 12641

Find the tangent line equation for $g(x)=(3x^{2}-4)f(x)$ at $x=-2$ given $f(-2)=-2$ and $f'(-2)=-3$ .

Find the derivative of $y=\frac{x^{2}}{e^{x}}$ .

Find the rate of change of the surface area $A=4 \pi r^{2}$ with respect to the radius $r$ .

Check if the function is continuous, differentiable, both, or neither at the point where it changes:
$f(x)=\left\{\begin{array}{ll} x^{2}+3 x+24, & x \leq-4 \\ -x+20, & x>-4 \end{array}\right.$

Check if the function is continuous or differentiable at the point where its definition changes:
$f(x)=\left\{\begin{array}{ll} x^{2}-13 x-7, & x<1 \\ -17 x-11, & x \geq 1 \end{array}\right.$

Evaluate the limit and simplify your answer: $\lim _{h \rightarrow 0} \frac{\sqrt{-3+h+4}-\sqrt{-3+4}}{h}$

Evaluate the limit: $\lim_{h \rightarrow 0} \frac{3 \csc (x+h) - 3 \csc (x)}{h}$

Find the limit as $h$ approaches 0 for $\frac{\frac{1}{x+h-1}-\frac{1}{x-1}}{h}$ .

Evaluate the double integral: $\int_{0}^{2} \int_{0}^{2}(u+2 v) \cos (2u) 4 \, du \, dv$ .

Calculate the limit: $\lim _{n \rightarrow \infty}\left(\sqrt{n^{2}+5 n}-\sqrt{n^{2}-n}\right)$ .

Find values of $a$ and $b$ for the function $f(x)$ to be differentiable at $x=2$ :
$f(x)=\begin{cases} 2 a x-7 & \text{for } x<2 \\ 2 b x^{2}+4 x-1 & \text{for } x \geq 2 \end{cases}$

Find the derivative $f^{\prime}(6)$ for the function $f(x)=\frac{5 \sqrt{x^{3}}}{6}+\frac{3}{\sqrt{x}}$ .

Bestimmen Sie die Ableitung von $f(x) = \frac{2}{\sqrt{x}} + \frac{\sqrt{x}}{2}$ .

Find the derivative of $y = \frac{x^{2}}{e^{x}}$ .

Find the antiderivative of $f^{\prime}(t)=\frac{1}{\sqrt{2 t+1}}$ .

Evaluate the integral $\int_{C} f(x, y, z) d s=\int_{0}^{1}\left(8 e^{2 t}+32 t\right) \sqrt{4 e^{2 t}+16 t^{2}+16} d t$ .

Evaluate the line integral using $u$ -substitution:
$\int_{C} f(x, y, z) d s = \int_{0}^{1}(8 e^{2 t}+32 t) \sqrt{4 e^{2 t}+16 t^{2}+16} d t$

Differentiate the function: $y=x^{7} \ln x-\frac{1}{3} x^{3}$ . Find $\frac{d y}{d x}$ .

Find the derivative of $y=\ln \left(\frac{(x^{2}+7)^{5}}{\sqrt{3-x}}\right)$ . What is $\frac{d y}{d x}$ ?

Find the number of cars that pass by from 8 to 9 AM given $q(t)=1400+2200 t-270 t^{2}$ .

Find the displacement of an object with velocity $v(t)=-18t+150$ over the intervals: a) $[1,7]$ , b) $[0,10]$ .

Find the displacement of an object with velocity $v(t)=18 \sqrt[5]{t}$ over the interval $[0,243]$ .

Find the total number of antibodies produced in 20 minutes given $f(t)=\frac{11 t}{3 t^{2}+10}$ . Round to two decimal places.

At $t=0$ , species $A$ and $B$ have equal populations. Find which has a larger population after 5, 10, and 25 years using:
$R_{1}=19+110 \ln(1+2 t^{2})$
$R_{2}=19 e^{0.2 t+1}$

Differentiate $y=\ln \left[(x+8)^{5}(x+4)^{3}(x+6)^{6}\right]$ and find $\frac{d}{d x}[\ln \left[(x+8)^{5}(x+4)^{3}(x+6)^{6}\right]]$ .

Differentiate the function $y=19 \ln \left(x^{23} \sqrt{5 x+6}\right)$ . Find $y'=\square$ .

Find the total oil leaked in the first hour using $f(t)=A e^{-k t}$ and the integral $\int_{0}^{60} f(t) \, dt$ . What are the units?

Find the marginal-revenue function for the demand $p=\frac{30}{\ln (q+2)}$ . What is $\frac{d r}{d q}$ ?

Find the limit of the sequence $a_{n}=n\left(\sqrt[3]{n^{3}+n+2}-n\right)$ as $n$ approaches infinity.

Find the marginal cost of the total-cost function $c=24 \ln (q+1)+13$ when $q=4$ . The answer is $\square$ .

Approximate the area under $f(x)=-0.2 x^{2}-x+12$ on $[-1,3]$ using trapezoids with $n=4$ subintervals.

Oil leaks at $r=17 e^{4 t}$ gallons/hour. Find the integral for 0 to 3 hours. Estimate using a left sum with 3 parts.

Find the limit: $\lim _{m \rightarrow-\infty} m\left(\sqrt[3]{m^{3}+m+2}\right)$ .

Differentiate the function $y=17 \ln \left(x^{3} \sqrt[3]{4 x+7}\right)$ . Find $y'=\square$ .

Find the rate of change of supply with respect to price given $q(p)=22+8 \ln (3 p+1)$ . Calculate $\frac{d q}{d p}$ .

Find the limit: $\lim _{n \rightarrow \infty} n \sqrt[3]{n^{3}+n+2}-n$ .

Differentiate $y=17 \ln \left(x^{3} \sqrt[3]{7 x+5}\right)$ . Find $\mathbf{y}^{\prime}=\square$ .

Find the tangent line equation to $y=\ln(x^{2}-9x-9)$ at $x=10$ . $\mathbf{y}=\square$ (Simplify your answer.)

Differentiate the function $y=e^{x^{6}+6 x-1}$ . Find $y'=\frac{d}{dx}\left(e^{x^{6}+6 x-1}\right)=\square$ .

Differentiate the function $y=\frac{e^{x}-4 e^{-x}}{e^{x}+4 e^{-x}}$ . Find $y'$ .

Find the derivative of $y$ where $y=\frac{\ln x}{4+\ln x}$ .

Differentiate the function $y=e^{x^{6}+3 x+10}$ . Find $y' = \frac{d}{dx}(e^{x^{6}+3 x+10}) = \square$ .

A particle moves along a line with $s(t)=2t^2-20t$ . Find velocity, acceleration, when stationary, and motion intervals.

Find the derivative of $y$ with respect to $x$ for $y=\frac{\ln x}{2+3 \ln x}$ .

Bestimmen Sie den Grenzwert $L = \lim _{x \rightarrow \infty} \frac{1-2 x}{x+1}$ mithilfe einer Wertetabelle.

Find the derivative of $y=\frac{-8 e^{9 x}}{3 x-2}$ . What is $\frac{d}{d x}\left(\frac{-8 e^{9 x}}{3 x-2}\right)$ ?

Differentiate the function $f(x)=(x+6) e^{-5 x+3}$ and express $f^{\prime}(x)$ in factored form.

Find the derivative of $y$ with respect to $x$ for $y=\frac{\ln x}{1+4 \ln x}$ .

Differentiate the function $y=(9 x+8)^{x}$ . Find $\frac{d y}{d x}=\square$ .

Find the limit of the sequence $a_{n}=\frac{n+(-1)^{n}}{n}$ as $n \to \infty$ .

Differentiate the function $y=(2x+7)^{x}$ . Find $\frac{dy}{dx}$ .

Evaluate the double integral: $4 \int_{0}^{2}\left(\int_{0}^{2} u \cos (2 u) d v\right) d u$ .

Find the grams of carbon-14 left after 5403 years using the model $A=16 e^{-0.000121 t}$ . Answer: $\square$ grams.

Differentiate the function $y=(8 x+1)^{x}$ . Find $\frac{d y}{d x}$ .

Find $\frac{dy}{dx}$ if $y=\frac{\ln x}{1+\ln x}$ .

Find the rate of change of price $p$ with respect to quantity $q$ for $p=e^{-0.003 q}$ when $q=300$ . Answer: $\square$ .

Find the marginal-cost function and its values for $\bar{c}=\frac{2250 e^{\frac{q}{450}}}{q}$ at $q=225$ and $q=450$ .

Evaluate the expression $[492 t-576 \cos (t)-(54) \sin (2 t)]_{0}^{2 \pi}$ .

Find the marginal-cost function from $\bar{c}=\frac{2250 e^{\frac{q}{450}}}{q}$ and calculate it for $q=225$ and $q=450$ .

Check if the sequence $a_{n}=(-1)^{n}\left(1-\frac{1}{n}\right)$ converges.

Find $\frac{dy}{dt}$ for $y=e^{(2 \cos t+\ln t)}$ .

Find the derivative of $y$ with respect to $t$ for $y=e^{(11 \sin t+\ln t)}$ .

Analyze the sequence $a_{n}=\frac{n !}{n^{n}}$ : Is it monotonic, bounded, and does it converge?

Find the derivative of $s(t) = \frac{e^{2t}}{1 + e^{2t}}$ .

Calculate the limit: $\lim _{n \rightarrow \infty}\left(\frac{n !}{n^{n}}\right)$ .

Find the curve equation with slope $f^{\prime}(x)=x^{\frac{2}{9}}$ that passes through $(1, \frac{9}{11})$ .

Find the critical numbers of the function $f(x)=x^{3}-48x$ . Enter answers as a comma-separated list.

Find the critical numbers of the function $f(x)=x^{3}+6 x^{2}-15 x-10$ . Provide your answers as a comma-separated list.

Find critical points of $f(x)=\frac{e^{x}+e^{-x}}{2}$ : domain, $f'(x)$ , where $f'(x)=0$ , and where $f'(x)=D N E$ .