Calculus

Problem 28801

Calculate the limit: $\lim _{t \rightarrow 0} \frac{e^{2 t}-1}{\sin (t)}$ .

See Solution

Problem 28802

1. Use Newton's Law of Cooling to find $k$ for coffee cooling from $200^{\circ} \mathrm{F}$ to $180^{\circ} \mathrm{F}$ in 7 minutes. Then, find time to reach $140^{\circ} \mathrm{F}$ using $k$ .

See Solution

Problem 28803

Show that the following functions are continuous at specified points: a. $g(x)=x(x^{2}-3x+5)$ at $x=0$ b. $f(x)=\frac{x^{2}-4}{(x^{2}+4x+4)(x^{2}+2x+1)}$ at $x=2$ c. $g(x)=\frac{\sqrt{2-x^{2}}}{3x^{2}-1}$ at $x=-1$

See Solution

Problem 28804

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{2} \cot (x)$ .

See Solution

Problem 28805

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{\sec (x)}{x}$ .

See Solution

Problem 28806

Find the derivative $\frac{d y}{d x}$ for the function $(x+\cos (x))(1-\sin (x))$ .

See Solution

Problem 28807

Find the dimensions of a rectangular field (in $\mathrm{ft}$ ) that minimize fencing costs for 6 million sq ft area, divided in half.

See Solution

Problem 28808

Find the tangent line equation for $y=x \tan (x)$ at $x=\frac{\pi}{4}$ . Use the product rule for the derivative.

See Solution

Problem 28809

Find the value of $\left(\frac{\square \cos ^{2}(x)}{\square \square}\right)$ if $\frac{d}{dx}(\cot(x))=-\csc^2(x)$ .

See Solution

Problem 28810

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the function $y=x-\frac{1}{4} \sin (x)$ .

See Solution

Problem 28811

Find the integral of $\tan ^{7}(x) \sec ^{4}(x)$ with respect to $x$ .

See Solution

Problem 28812

Calculate the average rate of change for $f(x)=\sqrt{x}$ from $x_{1}=81$ to $x_{2}=144$ . Answer: $\square$ .

See Solution

Problem 28813

Explain the discontinuity types for these functions: a. $g(x)=\frac{x^{2}+4}{x^{2}-x-2}$ at $x=2$ b. $h(x)=\frac{x^{2}+4x+3}{x^{2}-x-2}$ at $x=-1$ c. $f(x)=\begin{cases}3x-1, & x \leq 2 \\ 2x, & x>2\end{cases}$ at $x=2$

See Solution

Problem 28814

Jack walked for 2 hours at $3 \mathrm{mi}/\mathrm{hr}$ . Graph it, find the area under the curve, and conclude.

See Solution

Problem 28815

Estimate the area under the curve of $f$ on $[0,8]$ using 4 left endpoints: $f(0), f(2), f(4), f(6)$ .

See Solution

Problem 28816

A kid shoots an arrow with mass $122 \mathrm{~g}$ at $13.75 \mathrm{~m/s}$ . How high does it rise?

See Solution

Problem 28817

Approximate the area under $y=\frac{2}{x^{2}}$ from 1 to 2 using 10 rectangles with right endpoints. Round to the nearest hundredth.

See Solution

Problem 28818

Find the slope of a tangent line for a function $f(x)$ with $f(0)=0$ and $f(5)=10$ on $(0,5)$ . Options: $10$ , $5$ , $2$ , $-5$ .

See Solution

Problem 28819

A girl drops a stone (mass $325 \mathrm{~g}$ ) from a bridge $36.1 \mathrm{~m}$ high. Find its speed before hitting the water.

See Solution

Problem 28820

Find the limit as $x$ approaches 3 for $\frac{2 \tan (6-2 x)}{3 e^{x-3}-3}$ and simplify your answer.

See Solution

Problem 28821

Evaluate the limit: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{4}{n}\left(24+\frac{20}{n} i\right)$ .

See Solution

Problem 28822

Why does the function $f(x)=|x|$ not meet the Mean Value Theorem conditions on $[-5,5]$ ?

See Solution

Problem 28823

The area of a circle grows at 187 sq ft/sec. When the area is $16 \pi$ , find the circumference's rate of change.

See Solution

Problem 28824

Find the integral of $2 x^{3}$ with respect to $x$ . What is $\int 2 x^{3} d x$ ?

See Solution

Problem 28825

Find the limit $v = \lim _{x \rightarrow 2} \frac{2 x^{2}-8}{2 f(2 x)+x}$ using the given values of $f$ and $f'$ .

See Solution

Problem 28826

Find the limit as $x$ approaches 2 for $\frac{3 \tan (4-2 x)-2 x^{3}}{2 \ln (x-1)}$ . Simplify your answer.

See Solution

Problem 28827

Find the simplified form of $\Delta x$ for the Riemann sum of $f(x)=x^{3}+5 x^{2}-2 x+3$ on $[-1,6]$ . Options: $\frac{6}{n}$ , $\frac{7}{n}$ , $-\frac{1}{n}$ , $\frac{5}{n}$ .

See Solution

Problem 28828

Find the Riemann sum for $f(x)=x^{3}+5 x^{2}-2 x+3$ on $[-1,6]$ . Which expression simplifies to $f(x_{i})$ ? Options:
1. $9-\frac{63}{n} i+\frac{98}{n^{2}} i^{2}+\frac{343}{n^{3}} i^{3}$
2. $1+\frac{343}{n^{3}} i^{3}+\frac{35}{n^{2}} i^{2}+3$
3. None of these
4. $i^{3}+5 i^{2}-2 i+3$

See Solution

Problem 28829

Find the average rate of change of $f(x)=(x-2)^{2}+3$ over the interval $1 \leq x \leq 2$ . Options: A. 1 B. 2 C. $\frac{3}{2}$ D. -1

See Solution

Problem 28830

Evaluate the limit: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{6}{n}\left(\left(5+\frac{6}{n} i\right)^{2}+3\left(5+\frac{6}{n} i\right)+9\right)$ and find the answer.

See Solution

Problem 28831

A sphere's volume increases at 1276 cm³/s. Find the surface area change rate when the radius is 4 cm. Use $V=\frac{4}{3} \pi r^{3}$ and $S=4 \pi r^{2}$ . Round to three decimal places.

See Solution

Problem 28832

Given $f$ is differentiable with $f^{\prime}(x)$ increasing, $f(4)=-10$ , $f(6)=6$ . Which statement about $c \in [4,6]$ is true? I. $f^{\prime}(c)=8$ II. $f(c)=0$ III. $f^{\prime}(c)=0$

See Solution

Problem 28833

Find the slope of the secant line for $f(x)=x^{3}-2x^{2}+3$ if $c=2$ satisfies the Mean Value Theorem.

See Solution

Problem 28834

Find $x$ in $(0, \pi)$ where the average rate of change of $f(x)=\sin x$ on $[0, \pi]$ equals the instantaneous rate.

See Solution

Problem 28835

Find the value of $x=c$ guaranteed by the Mean Value Theorem for $y=x^{3}-x^{2}-2x+1$ on $[-1,4]$ .

See Solution

Problem 28836

Find the value of $x=c$ guaranteed by the Mean Value Theorem for $y=x^{3}-x^{2}-2x+1$ on $[-1, 4]$ .

See Solution

Problem 28837

Find the value of $c$ that satisfies the Mean Value Theorem for $f(x)=3 x^{2}-2 x+4$ on $[0,2]$ .

See Solution

Problem 28838

Find the average rate of change of $g(x)=4 x^{2}+x-1$ on the interval $0.5 \leq x \leq 1$ .

See Solution

Problem 28839

For the function $y=x^{3}-x^{2}-2x$ , which statement about the Mean Value Theorem is correct on $[-1,2]$ ?

See Solution

Problem 28840

Estimate the instantaneous rate of change of $f(x)=\frac{1}{2}(x+1)^{2}-3$ at $x=1$ using the interval method.

See Solution

Problem 28841

Show that the function $f(x)=x^{4}+x-4$ has an $x$ -intercept in the interval $(1,2)$ using the Intermediate Value theorem.

See Solution

Problem 28842

Calculate the integral $\int_{1}^{4}(-2 x+4) d x$ .

See Solution

Problem 28843

1. Given $f(0)=7, f'(0)=2, g(0)=6, g'(0)=-10$ , find $(fg)'(0)$ and $\left(\frac{f}{g}\right)'(0)$ .
2. For $f(x)=\left\{\begin{array}{cc}x^{2}+x & x \leq 1 \\ 2x^{2} & x>1\end{array}\right.$ , does $f'(1)$ exist?
3. Find the derivatives: (a) $y=\sin(3+x)^{2}$ (b) $y=\frac{2 \sec x}{1+\sec x}$ (c) $y=\sqrt{\frac{x-2}{x+2}}$ (d) $y=(x^{6}+3)^{2}(2-x)^{4}$
4. Does $y=\sqrt{2x-x^{2}}$ satisfy $y^{3}y''+1=0$ ?
5. Express $f'$ in terms of $g'$ : (a) $f(x)=x^{3}g(x)$ (b) $f(x)=g(x^{2})$ (c) $f(x)=\cos g(x)$
6. Find $\frac{dy}{dx}$ using implicit differentiation: (a) $y^{3}-xy+2\cos xy=2$ ; (b) $\tan\frac{x}{y}=2y-x$ .
7. Find tangent and normal line equations for: (a) $y=x\sqrt{5-x}$ at $(1,2)$ (b) $y=\sqrt{1+4\sin x}$ at $(0,1)$

See Solution

Problem 28844

Find the limit as $x$ approaches infinity for the expression $\frac{2+\frac{1}{x+4}}{3-\frac{1}{x^{2}}$.

See Solution

Problem 28845

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

See Solution

Problem 28846

Differentiate the function $f(x) = x^{6.5} - \frac{1}{x^{7.5}}$ .

See Solution

Problem 28847

Marcus takes $350 \mathrm{mg}$ of medication every $6 \mathrm{h}$ , with $32 \%$ remaining before the next dose. Find:
a) A recursive formula. b) The steady amount of medication. c) The time to reach this steady level.

See Solution

Problem 28848

Evaluate the integral: $\int \frac{d x}{\sqrt{x}+\sqrt[3]{x^{2}}}$

See Solution

Problem 28849

Use the decay equation $A(t)=A_{0}\left(\frac{1}{2}\right)^{\frac{t}{3.1}}$ to solve these: a) Amount left from 50 mg after 90 s? b) Time to decay to 10% of 50 mg? c) Does initial size affect part b? Explain.

See Solution

Problem 28850

Find the derivative of the loss function
$E(w)=-\sum_{n=1}^{N}\left\{t_{n} \ln y_{n}+\left(1-t_{n}\right) \ln \left(1-y_{n}\right)\right\}$
with respect to $a_{k}$ , showing
$\frac{\partial E}{\partial a_{k}}=\left(1-y_{k}^{2}\right)\left(y_{k}-t_{k}\right)$ .

See Solution

Problem 28851

Find the limit: $\lim _{x \rightarrow \infty} \frac{2 \sqrt{11}(8)^{x}+15,000}{7(8)^{x}-9}$ .

See Solution

Problem 28852

Find the limit: $\lim _{x \rightarrow-\infty} \frac{9}{5^{x}-7}$ .

See Solution

Problem 28853

Find the derivative of the function $f(x)=\ln \left[x^{6}(x+5)^{9}(x^{3}+1)^{10}\right]$ .

See Solution

Problem 28854

Estimate the limit as $x$ approaches -2 for $\frac{8x + 16}{x^2 - 6x - 16}$ using values $x=-1.9,-1.99,-1.999,-1.9999,-2.1,-2.01,-2.001,-2.0001$ . Enter DNE if it doesn't exist.

See Solution

Problem 28855

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\arcsin^{6}(2x+4)$ .

See Solution

Problem 28856

Evaluate the integral: $\int_{1}^{e^{5}} \frac{2}{x} \, dx$ exactly.

See Solution

Problem 28857

Sketch the graph of $f(x)=\left\{\begin{array}{ll}6-x, & \text { if } x<-3 \\ x, & \text { if }-3 \leq x<3 \\ (x-3)^{2} & \text { if } x \geq 3\end{array}\right.$ and find these limits: $\lim _{x \rightarrow-3^{-}} f(x)$ , $\lim _{x \rightarrow-3^{+}} f(x)$ , $\lim _{x \rightarrow 3^{-}} f(x)$ , $\lim _{x \rightarrow 3^{+}} f(x)$ , $\lim _{x \rightarrow 3} f(x)$ .

See Solution

Problem 28858

Find the derivative of the loss function $E(\mathbf{w})=-\sum_{n=1}^{N}\left\{t_{n} \ln y_{n}+\left(1-t_{n}\right) \ln \left(1-y_{n}\right)\right\}$ with respect to $a_{k}$ , showing $\frac{\partial E}{\partial a_{k}}=\left(1-y_{k}^{2}\right)\left(y_{k}-t_{k}\right)$ .

See Solution

Problem 28859

Find $\delta$ for $\mathrm{f}(x)=x^{\frac{2}{3}}$ as $x \rightarrow 8$ with $\varepsilon = 0.01$ such that $|f(x)-L|<\varepsilon$ .

See Solution

Problem 28860

Find $\delta$ for $\varepsilon=0.1$ in the limit: $\lim _{x \rightarrow 6} 4 x+6=30$ .

See Solution

Problem 28861

Estimate $f(2.03)$ using linearization with $f^{\prime}(x)=\frac{4 x^{3}}{x^{2}-2}$ and $f(2)=5$ .

See Solution

Problem 28862

Find the integral of $x^{2} \cdot 5^{2x^{3}}$ with respect to $x$ .

See Solution

Problem 28863

Lavaughn invested \ $310 at$ 3 \frac{7}{8} \% $continuous interest. Jeriel invested \$310 at$ 4 \frac{1}{4} \%$ quarterly. How much longer for Lavaughn's money to triple than Jeriel's?

See Solution

Problem 28864

Check if the function $y=\sqrt{2x-x^{2}}$ satisfies the equation $y^{3} y^{\prime \prime}+1=0$ .

See Solution

Problem 28865

Find the limit: $\lim _{x \rightarrow \infty}\left(\frac{2 x+5}{2 x}\right)^{3 x-7}$ .

See Solution

Problem 28866

Berechne die mittlere Änderungsrate von $f(x)=x^{2}$ im Intervall $[a-1 ; a+1]$ und vergleiche sie mit der Änderungsrate bei $a$ .

See Solution

Problem 28867

Find the limit as $x$ approaches 0 for the expression $\frac{1-\cos x}{x^{2}}$ .

See Solution

Problem 28868

Berechnen Sie die mittlere Änderungsrate von $f(x)=x^{2}$ auf $[a-1; a+1]$ und vergleichen Sie sie mit der Änderungsrate bei $a$ .

See Solution

Problem 28869

Evaluate $6^{\prime} \times 4=24^{\prime}$ and find the derivative of $y=x \sqrt{5-x}$ , i.e., $y^{\prime}$ .

See Solution

Problem 28870

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(\frac{(x+3)-4}{x+3}\right)^{5-4 x}$ .

See Solution

Problem 28871

Find the derivative of the function $y=x \sqrt{5-x}$ , denoted as $y^{\prime}$ .

See Solution

Problem 28872

Find points where the gradient is zero for these curves and determine if they are maxima or minima: $y=4 x^{2}+6 x$ , $y=9+x-x^{2}$ , $y=x^{3}-x^{2}-x+1$ .

See Solution

Problem 28873

Analyze if the graphs of the functions are increasing/decreasing and concave up/down. Write limit statements for end behavior. a) $f(x)=2(4)^{x}$ b) $g(x)=2(0.4)^{x}$ c) $f(x)=-2(4)^{x}$ d) $g(x)=-2(0.4)^{x}$

See Solution

Problem 28874

1. Find the polynomial $f(x)$ that fits the points (-2,1), (-1,4), (2,6).
2. Estimate $f(1)$ .
3. Calculate $\int_{-1}^{2} f(x) \, dx$ .
4. Estimate $\left.\frac{d}{dx} f(x)\right|_{x=1}$ .

See Solution

Problem 28875

Untersuche das Wachstum einer Ameisenkolonie mit $N(t)=\frac{1}{64} t^{3}-\frac{9}{32} t^{2}+\frac{3}{2} t+1$ für $0 \leq t \leq 9$ . a) Graph zeichnen. b) Zeitpunkt max. Koloniegröße? c) Zeitraum des Schrumpfens? d) Mittlere Wachstumsrate? e) Momentane Wachstumsrate bei $t=5$ ? Maximal und minimal?

See Solution

Problem 28876

A curve has the equation $y=\frac{12+z^{2} \sqrt{2}}{2}, x>0$ .
a. Express $\frac{12+x^{2} \sqrt{2}}{x}$ as $12 x^{3}+x^{4}$ .
b. Find $\frac{d y}{d x}$ .
i. Find the normal equation at $x=4$ .
ii. Show the $x$ -coordinate of stationary point $P$ can be written as $k$ , where $k$ is rational.

See Solution

Problem 28877

A curve is given by $y=\frac{12+z^{2} \sqrt{2}}{2}$ for $x>0$ .
a) Express $\frac{12+z^{2} \sqrt{x}}{z}$ as $[2 z^{3}+x^{4}$ . b) I) Find $\frac{d y}{d x}$ .
1. Determine the normal at $z=4$ . 百) Show the $z$ -coordinate of stationary point $P$ can be written as $rt$ , where $k$ is rational.

See Solution

Problem 28878

Find points where the gradient is zero for these curves and determine if they are max or min using the second derivative:
1. $y=4x^2+6x$
2. $y=x(x^2-4x-3)$
3. $y=x-3\sqrt{x}$
4. $y=9+x-x^2$
5. $y=x^3-x^2-x+1$
6. $y=x+\frac{1}{x}$
7. $y=x^2+\frac{54}{x}$
8. $y=x^{\frac{1}{2}}(x-6)$
9. $y=x^4-12x^2$

See Solution

Problem 28879

Evaluate the integral: $\int \frac{\sin x \cos x \, dx}{\cos^{2} x - \sin^{2} x}$ .

See Solution

Problem 28880

Calculate the integral: $\int \frac{3 x+2}{(3 x^{2}+4 x+2)^{4}} d x$

See Solution

Problem 28881

Bestimme die Tangentengleichung der Funktion $k(x)=x^{3}+2 \ln x$ im Punkt $x=1$ .

See Solution

Problem 28882

Sophie hat Fieber, beschrieben durch $f(t)=1,5 \cdot t \cdot e^{-0,5 t+1}+37$ . Beantworte a)-g) zur Temperaturentwicklung.

See Solution

Problem 28883

Evaluate the integral: $\int(4+\sin x)^{5} \cos x \, dx$

See Solution

Problem 28884

Find the derivative $\frac{d y}{d x}$ for these functions: a) $y=\frac{4}{x}+\frac{x^{2}}{(x+2)}$ , b) $y=\sqrt{x-2}(4 x^{2}+1)+2 x^{3}$ .

See Solution

Problem 28885

Find the integral of $x e^{\ln x}$ with respect to $x$ : $\int x e^{\ln x} dx$ .

See Solution

Problem 28886

Evaluate the integral: $\int \frac{x \, dx}{(4x^{2}+1)^{1/5}}$

See Solution

Problem 28887

Ergänzen Sie die Ableitungen: a) $f'(x)=\square \cdot \sin (x)+x^{2}$ , b) $f'(x)=\square \cdot \cos (4 x)-(2 x-3)$ .

See Solution

Problem 28888

Bestimmen Sie die Ableitung: a) $f(x)=x^{2} \cdot \sin (x)$ ; b) $f(x)=0 x-3$ .

See Solution

Problem 28889

Gegeben ist $f(x)=(x-1)(x-3)^{2}$ . Finde Nullstellen, Tangente bei $P(1 \mid f(1))$ , Punkte mit waagerechter Tangente und skizziere den Graphen.

See Solution

Problem 28890

Find the absolute extrema of $f(x)=2 x^{3}-6 x$ on the interval $[-2,0]$ .

See Solution

Problem 28891

Find the derivative of the function: $f(x) = \sqrt[3]{x} - (\sqrt{x} + 5)$ .

See Solution

Problem 28892

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

See Solution

Problem 28893

Find the derivative $\frac{dy}{dx}$ for: $y = x^{3}(x^{2} + 5)^{4}$ and $y = x^{-5} + \frac{1}{x} - \frac{x-1}{x+2}$ .

See Solution

Problem 28894

Find the value of the series: $\sum_{n=1}^{\infty} \frac{\ln n}{n}$ .

See Solution

Problem 28895

Differentiate the function: $f(x) = \frac{7x}{2} \left( \frac{1}{x^2 + 1} \right)$ .

See Solution

Problem 28896

Let $f(x)=|x|$ . Which statements are true on $[-2,1]$ ? I: EVT applies, II: IVT applies with $f(c)=1.5$ , III: MVT applies.

See Solution

Problem 28897

Bestimme die Koordinaten der Punkte mit waagerechter Tangente für die Funktionen $f(x)=(2x-3) \cdot \cos(x)$ und $f(x)=\frac{1}{x} \cdot \sin(x)$ . Skizziere den Graphen und leite $f$ ab.

See Solution

Problem 28898

Gegeben ist die Funktion $f(x) = (x-1)(x-3)^2$ . Finde Punkte mit waagerechter Tangente und skizziere den Graphen.

See Solution

Problem 28899

Resuelve la integral $\int \cos^2(x) \sin(x) \, dx$ .

See Solution

Problem 28900

Find the value(s) of $x$ where $f(x)$ has a point of inflection given $f^{\prime}(x)=(3 x+1)^{\frac{2}{3}}$ .

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 28801

Calculate the limit: lim⁡t→0e2t−1sin⁡(t)\lim _{t \rightarrow 0} \frac{e^{2 t}-1}{\sin (t)}limt→0​sin(t)e2t−1​.

Problem 28802

1. Use Newton's Law of Cooling to find kkk for coffee cooling from 200∘F200^{\circ} \mathrm{F}200∘F to 180∘F180^{\circ} \mathrm{F}180∘F in 7 minutes. Then, find time to reach 140∘F140^{\circ} \mathrm{F}140∘F using kkk.

Problem 28803

Problem 28804

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=x2cot⁡(x)y=x^{2} \cot (x)y=x2cot(x).

Problem 28805

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=sec⁡(x)xy=\frac{\sec (x)}{x}y=xsec(x)​.

Problem 28806

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function (x+cos⁡(x))(1−sin⁡(x))(x+\cos (x))(1-\sin (x))(x+cos(x))(1−sin(x)).

Problem 28807

Find the dimensions of a rectangular field (in ft\mathrm{ft}ft) that minimize fencing costs for 6 million sq ft area, divided in half.

Problem 28808

Find the tangent line equation for y=xtan⁡(x)y=x \tan (x)y=xtan(x) at x=π4x=\frac{\pi}{4}x=4π​. Use the product rule for the derivative.

Problem 28809

Find the value of (□cos⁡2(x)□□)\left(\frac{\square \cos ^{2}(x)}{\square \square}\right)(□□□cos2(x)​) if ddx(cot⁡(x))=−csc⁡2(x)\frac{d}{dx}(\cot(x))=-\csc^2(x)dxd​(cot(x))=−csc2(x).

Problem 28810

Find the second derivative d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ for the function y=x−14sin⁡(x)y=x-\frac{1}{4} \sin (x)y=x−41​sin(x).

Problem 28811

Find the integral of tan⁡7(x)sec⁡4(x)\tan ^{7}(x) \sec ^{4}(x)tan7(x)sec4(x) with respect to xxx.

Problem 28812

Calculate the average rate of change for f(x)=xf(x)=\sqrt{x}f(x)=x​ from x1=81x_{1}=81x1​=81 to x2=144x_{2}=144x2​=144. Answer: □\square□.

Problem 28813

Problem 28814

Jack walked for 2 hours at 3mi/hr3 \mathrm{mi}/\mathrm{hr}3mi/hr. Graph it, find the area under the curve, and conclude.

Problem 28815

Estimate the area under the curve of f f f on [0,8] [0,8] [0,8] using 4 left endpoints: f(0),f(2),f(4),f(6) f(0), f(2), f(4), f(6) f(0),f(2),f(4),f(6).

Problem 28816

A kid shoots an arrow with mass 122 g122 \mathrm{~g}122 g at 13.75 m/s13.75 \mathrm{~m/s}13.75 m/s. How high does it rise?

Problem 28817

Approximate the area under y=2x2y=\frac{2}{x^{2}}y=x22​ from 1 to 2 using 10 rectangles with right endpoints. Round to the nearest hundredth.

Problem 28818

Find the slope of a tangent line for a function f(x)f(x)f(x) with f(0)=0f(0)=0f(0)=0 and f(5)=10f(5)=10f(5)=10 on (0,5)(0,5)(0,5). Options: 101010, 555, 222, −5-5−5.

Problem 28819

A girl drops a stone (mass 325 g325 \mathrm{~g}325 g) from a bridge 36.1 m36.1 \mathrm{~m}36.1 m high. Find its speed before hitting the water.

Problem 28820

Find the limit as xxx approaches 3 for 2tan⁡(6−2x)3ex−3−3\frac{2 \tan (6-2 x)}{3 e^{x-3}-3}3ex−3−32tan(6−2x)​ and simplify your answer.

Problem 28821

Evaluate the limit: lim⁡n→∞∑i=1n4n(24+20ni)\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{4}{n}\left(24+\frac{20}{n} i\right)n→∞lim​i=1∑n​n4​(24+n20​i).

Problem 28822

Why does the function f(x)=∣x∣f(x)=|x|f(x)=∣x∣ not meet the Mean Value Theorem conditions on [−5,5][-5,5][−5,5]?

Problem 28823

The area of a circle grows at 187 sq ft/sec. When the area is 16π16 \pi16π, find the circumference's rate of change.

Problem 28824

Find the integral of 2x32 x^{3}2x3 with respect to xxx. What is ∫2x3dx\int 2 x^{3} d x∫2x3dx?

Problem 28825

Find the limit v=lim⁡x→22x2−82f(2x)+xv = \lim _{x \rightarrow 2} \frac{2 x^{2}-8}{2 f(2 x)+x}v=limx→2​2f(2x)+x2x2−8​ using the given values of fff and f′f'f′.

Problem 28826

Find the limit as xxx approaches 2 for 3tan⁡(4−2x)−2x32ln⁡(x−1)\frac{3 \tan (4-2 x)-2 x^{3}}{2 \ln (x-1)}2ln(x−1)3tan(4−2x)−2x3​. Simplify your answer.

Problem 28827

Find the simplified form of Δx\Delta xΔx for the Riemann sum of f(x)=x3+5x2−2x+3f(x)=x^{3}+5 x^{2}-2 x+3f(x)=x3+5x2−2x+3 on [−1,6][-1,6][−1,6]. Options: 6n\frac{6}{n}n6​, 7n\frac{7}{n}n7​, −1n-\frac{1}{n}−n1​, 5n\frac{5}{n}n5​.

Problem 28828

Problem 28829

Find the average rate of change of f(x)=(x−2)2+3f(x)=(x-2)^{2}+3f(x)=(x−2)2+3 over the interval 1≤x≤21 \leq x \leq 21≤x≤2. Options: A. 1 B. 2 C. 32\frac{3}{2}23​ D. -1

Problem 28830

Evaluate the limit: lim⁡n→∞∑i=1n6n((5+6ni)2+3(5+6ni)+9)\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{6}{n}\left(\left(5+\frac{6}{n} i\right)^{2}+3\left(5+\frac{6}{n} i\right)+9\right)n→∞lim​i=1∑n​n6​((5+n6​i)2+3(5+n6​i)+9) and find the answer.

Problem 28831

A sphere's volume increases at 1276 cm³/s. Find the surface area change rate when the radius is 4 cm. Use V=43πr3V=\frac{4}{3} \pi r^{3}V=34​πr3 and S=4πr2S=4 \pi r^{2}S=4πr2. Round to three decimal places.

Problem 28832

Given fff is differentiable with f′(x)f^{\prime}(x)f′(x) increasing, f(4)=−10f(4)=-10f(4)=−10, f(6)=6f(6)=6f(6)=6. Which statement about c∈[4,6]c \in [4,6]c∈[4,6] is true? I. f′(c)=8f^{\prime}(c)=8f′(c)=8 II. f(c)=0f(c)=0f(c)=0 III. f′(c)=0f^{\prime}(c)=0f′(c)=0

Problem 28833

Find the slope of the secant line for f(x)=x3−2x2+3f(x)=x^{3}-2x^{2}+3f(x)=x3−2x2+3 if c=2c=2c=2 satisfies the Mean Value Theorem.

Problem 28834

Find xxx in (0,π)(0, \pi)(0,π) where the average rate of change of f(x)=sin⁡xf(x)=\sin xf(x)=sinx on [0,π][0, \pi][0,π] equals the instantaneous rate.

Problem 28835

Find the value of x=cx=cx=c guaranteed by the Mean Value Theorem for y=x3−x2−2x+1y=x^{3}-x^{2}-2x+1y=x3−x2−2x+1 on [−1,4][-1,4][−1,4].

Problem 28836

Find the value of x=cx=cx=c guaranteed by the Mean Value Theorem for y=x3−x2−2x+1y=x^{3}-x^{2}-2x+1y=x3−x2−2x+1 on [−1,4][-1, 4][−1,4].

Problem 28837

Find the value of ccc that satisfies the Mean Value Theorem for f(x)=3x2−2x+4f(x)=3 x^{2}-2 x+4f(x)=3x2−2x+4 on [0,2][0,2][0,2].

Problem 28838

Find the average rate of change of g(x)=4x2+x−1g(x)=4 x^{2}+x-1g(x)=4x2+x−1 on the interval 0.5≤x≤10.5 \leq x \leq 10.5≤x≤1.

Problem 28839

For the function y=x3−x2−2xy=x^{3}-x^{2}-2xy=x3−x2−2x, which statement about the Mean Value Theorem is correct on [−1,2][-1,2][−1,2]?

Problem 28840

Estimate the instantaneous rate of change of f(x)=12(x+1)2−3f(x)=\frac{1}{2}(x+1)^{2}-3f(x)=21​(x+1)2−3 at x=1x=1x=1 using the interval method.

Problem 28841

Show that the function f(x)=x4+x−4f(x)=x^{4}+x-4f(x)=x4+x−4 has an xxx-intercept in the interval (1,2)(1,2)(1,2) using the Intermediate Value theorem.

Calculate the limit: $\lim _{t \rightarrow 0} \frac{e^{2 t}-1}{\sin (t)}$ .

1. Use Newton's Law of Cooling to find $k$ for coffee cooling from $200^{\circ} \mathrm{F}$ to $180^{\circ} \mathrm{F}$ in 7 minutes. Then, find time to reach $140^{\circ} \mathrm{F}$ using $k$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{2} \cot (x)$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{\sec (x)}{x}$ .

Find the derivative $\frac{d y}{d x}$ for the function $(x+\cos (x))(1-\sin (x))$ .

Find the dimensions of a rectangular field (in $\mathrm{ft}$ ) that minimize fencing costs for 6 million sq ft area, divided in half.

Find the tangent line equation for $y=x \tan (x)$ at $x=\frac{\pi}{4}$ . Use the product rule for the derivative.

Find the value of $\left(\frac{\square \cos ^{2}(x)}{\square \square}\right)$ if $\frac{d}{dx}(\cot(x))=-\csc^2(x)$ .

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the function $y=x-\frac{1}{4} \sin (x)$ .

Find the integral of $\tan ^{7}(x) \sec ^{4}(x)$ with respect to $x$ .

Calculate the average rate of change for $f(x)=\sqrt{x}$ from $x_{1}=81$ to $x_{2}=144$ . Answer: $\square$ .

Jack walked for 2 hours at $3 \mathrm{mi}/\mathrm{hr}$ . Graph it, find the area under the curve, and conclude.

Estimate the area under the curve of $f$ on $[0,8]$ using 4 left endpoints: $f(0), f(2), f(4), f(6)$ .

A kid shoots an arrow with mass $122 \mathrm{~g}$ at $13.75 \mathrm{~m/s}$ . How high does it rise?

Approximate the area under $y=\frac{2}{x^{2}}$ from 1 to 2 using 10 rectangles with right endpoints. Round to the nearest hundredth.

Find the slope of a tangent line for a function $f(x)$ with $f(0)=0$ and $f(5)=10$ on $(0,5)$ . Options: $10$ , $5$ , $2$ , $-5$ .

A girl drops a stone (mass $325 \mathrm{~g}$ ) from a bridge $36.1 \mathrm{~m}$ high. Find its speed before hitting the water.

Find the limit as $x$ approaches 3 for $\frac{2 \tan (6-2 x)}{3 e^{x-3}-3}$ and simplify your answer.

Evaluate the limit: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{4}{n}\left(24+\frac{20}{n} i\right)$ .

Why does the function $f(x)=|x|$ not meet the Mean Value Theorem conditions on $[-5,5]$ ?

The area of a circle grows at 187 sq ft/sec. When the area is $16 \pi$ , find the circumference's rate of change.

Find the integral of $2 x^{3}$ with respect to $x$ . What is $\int 2 x^{3} d x$ ?

Find the limit $v = \lim _{x \rightarrow 2} \frac{2 x^{2}-8}{2 f(2 x)+x}$ using the given values of $f$ and $f'$ .

Find the limit as $x$ approaches 2 for $\frac{3 \tan (4-2 x)-2 x^{3}}{2 \ln (x-1)}$ . Simplify your answer.

Find the simplified form of $\Delta x$ for the Riemann sum of $f(x)=x^{3}+5 x^{2}-2 x+3$ on $[-1,6]$ . Options: $\frac{6}{n}$ , $\frac{7}{n}$ , $-\frac{1}{n}$ , $\frac{5}{n}$ .

Find the average rate of change of $f(x)=(x-2)^{2}+3$ over the interval $1 \leq x \leq 2$ . Options: A. 1 B. 2 C. $\frac{3}{2}$ D. -1

Evaluate the limit: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{6}{n}\left(\left(5+\frac{6}{n} i\right)^{2}+3\left(5+\frac{6}{n} i\right)+9\right)$ and find the answer.

A sphere's volume increases at 1276 cm³/s. Find the surface area change rate when the radius is 4 cm. Use $V=\frac{4}{3} \pi r^{3}$ and $S=4 \pi r^{2}$ . Round to three decimal places.

Given $f$ is differentiable with $f^{\prime}(x)$ increasing, $f(4)=-10$ , $f(6)=6$ . Which statement about $c \in [4,6]$ is true? I. $f^{\prime}(c)=8$ II. $f(c)=0$ III. $f^{\prime}(c)=0$

Find the slope of the secant line for $f(x)=x^{3}-2x^{2}+3$ if $c=2$ satisfies the Mean Value Theorem.

Find $x$ in $(0, \pi)$ where the average rate of change of $f(x)=\sin x$ on $[0, \pi]$ equals the instantaneous rate.

Find the value of $x=c$ guaranteed by the Mean Value Theorem for $y=x^{3}-x^{2}-2x+1$ on $[-1,4]$ .

Find the value of $x=c$ guaranteed by the Mean Value Theorem for $y=x^{3}-x^{2}-2x+1$ on $[-1, 4]$ .

Find the value of $c$ that satisfies the Mean Value Theorem for $f(x)=3 x^{2}-2 x+4$ on $[0,2]$ .

Find the average rate of change of $g(x)=4 x^{2}+x-1$ on the interval $0.5 \leq x \leq 1$ .

For the function $y=x^{3}-x^{2}-2x$ , which statement about the Mean Value Theorem is correct on $[-1,2]$ ?

Estimate the instantaneous rate of change of $f(x)=\frac{1}{2}(x+1)^{2}-3$ at $x=1$ using the interval method.

Show that the function $f(x)=x^{4}+x-4$ has an $x$ -intercept in the interval $(1,2)$ using the Intermediate Value theorem.

Calculate the integral $\int_{1}^{4}(-2 x+4) d x$ .

Find the limit as $x$ approaches infinity for the expression $\frac{2+\frac{1}{x+4}}{3-\frac{1}{x^{2}}$.

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

Differentiate the function $f(x) = x^{6.5} - \frac{1}{x^{7.5}}$ .

Marcus takes $350 \mathrm{mg}$ of medication every $6 \mathrm{h}$ , with $32 \%$ remaining before the next dose. Find:
a) A recursive formula. b) The steady amount of medication. c) The time to reach this steady level.

Evaluate the integral: $\int \frac{d x}{\sqrt{x}+\sqrt[3]{x^{2}}}$

Use the decay equation $A(t)=A_{0}\left(\frac{1}{2}\right)^{\frac{t}{3.1}}$ to solve these: a) Amount left from 50 mg after 90 s? b) Time to decay to 10% of 50 mg? c) Does initial size affect part b? Explain.

Find the limit: $\lim _{x \rightarrow \infty} \frac{2 \sqrt{11}(8)^{x}+15,000}{7(8)^{x}-9}$ .

Find the limit: $\lim _{x \rightarrow-\infty} \frac{9}{5^{x}-7}$ .

Find the derivative of the function $f(x)=\ln \left[x^{6}(x+5)^{9}(x^{3}+1)^{10}\right]$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\arcsin^{6}(2x+4)$ .

Evaluate the integral: $\int_{1}^{e^{5}} \frac{2}{x} \, dx$ exactly.

Find $\delta$ for $\mathrm{f}(x)=x^{\frac{2}{3}}$ as $x \rightarrow 8$ with $\varepsilon = 0.01$ such that $|f(x)-L|<\varepsilon$ .

Find $\delta$ for $\varepsilon=0.1$ in the limit: $\lim _{x \rightarrow 6} 4 x+6=30$ .

Estimate $f(2.03)$ using linearization with $f^{\prime}(x)=\frac{4 x^{3}}{x^{2}-2}$ and $f(2)=5$ .

Find the integral of $x^{2} \cdot 5^{2x^{3}}$ with respect to $x$ .

Lavaughn invested \ $310 at$ 3 \frac{7}{8} \% $continuous interest. Jeriel invested \$310 at$ 4 \frac{1}{4} \%$ quarterly. How much longer for Lavaughn's money to triple than Jeriel's?

Check if the function $y=\sqrt{2x-x^{2}}$ satisfies the equation $y^{3} y^{\prime \prime}+1=0$ .

Find the limit: $\lim _{x \rightarrow \infty}\left(\frac{2 x+5}{2 x}\right)^{3 x-7}$ .

Berechne die mittlere Änderungsrate von $f(x)=x^{2}$ im Intervall $[a-1 ; a+1]$ und vergleiche sie mit der Änderungsrate bei $a$ .

Find the limit as $x$ approaches 0 for the expression $\frac{1-\cos x}{x^{2}}$ .

Berechnen Sie die mittlere Änderungsrate von $f(x)=x^{2}$ auf $[a-1; a+1]$ und vergleichen Sie sie mit der Änderungsrate bei $a$ .

Evaluate $6^{\prime} \times 4=24^{\prime}$ and find the derivative of $y=x \sqrt{5-x}$ , i.e., $y^{\prime}$ .

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(\frac{(x+3)-4}{x+3}\right)^{5-4 x}$ .

Find the derivative of the function $y=x \sqrt{5-x}$ , denoted as $y^{\prime}$ .

Find points where the gradient is zero for these curves and determine if they are maxima or minima: $y=4 x^{2}+6 x$ , $y=9+x-x^{2}$ , $y=x^{3}-x^{2}-x+1$ .

1. Find the polynomial $f(x)$ that fits the points (-2,1), (-1,4), (2,6).
2. Estimate $f(1)$ .
3. Calculate $\int_{-1}^{2} f(x) \, dx$ .
4. Estimate $\left.\frac{d}{dx} f(x)\right|_{x=1}$ .