Calculus

Problem 14301

Find the limit: $\lim _{h \rightarrow 0} \frac{\ln (2+h)-\ln (2)}{h}$ . Choose from: a. $\ln (2)$ , b. $\frac{1}{2}$ , c. $\frac{1}{\ln (2)}$ , d. does not exist.

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Problem 14302

Find the limits: $\lim _{x \rightarrow 2^{-}} h(x)=3 \log (-2 x+4)$ and $\lim _{x \rightarrow-\infty} K(x)=-\ln (-x+6)-1$ .

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Problem 14303

Calculate the half-life of I-10 with a decay rate of $9.82\%$ per day. Round to 1 decimal place.

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Problem 14304

Given $f(x)=-\ln (x+2)-6$ , find its asymptote, domain, range, $y$ -intercept, and limits as $x \to -2^+$ and $x \to \infty$ .

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Problem 14305

Find the limit: $\lim _{h \rightarrow 0} \frac{\ln (2+h)-\ln (2)}{h}$ . Choices: a. $\ln (2)$ b. $\frac{1}{2}$ c. $\frac{1}{\ln (2)}$ d. does not exist.

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Problem 14306

Determine the horizontal asymptote of $g(x)=\left(\frac{1}{3}\right)^{x+3}$ .

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Problem 14307

Find the initial population size and the size after 8 years for $P(t)=\frac{600}{1+3 e^{-0.31 t}}$ . Round to whole numbers.

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Problem 14308

Find the linearization $L(x)$ at $x=-1$ for the function $f(x)=x+\frac{1}{x}$ .

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Problem 14309

Find the derivative of the quadratic function $f(x)=a(x-r)(x-s)$ using first principles.

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Problem 14310

Find the time (in s) to charge a $200 \mathrm{pF}$ capacitor through a $69.0 \mathrm{M} \Omega$ resistor to $92.0 \%$ voltage.

See Solution

Problem 14311

Find the limit as $x$ approaches -1: $\lim _{x \rightarrow-1} \frac{x+1}{\sqrt{5 x+9}-2}$

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Problem 14312

Check if $f(x)=x^{2/11}$ meets the Mean Value Theorem conditions on $[-9,1]$ . Justify your answer.

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Problem 14313

Gegeben sind $f(x)=x^{3}-3 x^{2}-x+4$ und $g(x)=-4 x+5$ . Untersuchen Sie $f$ und $g$ in Bezug auf Graphen, Steigungen und Tangenten.

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Problem 14314

Find the derivative of $5 x^{3}+x y^{2}=5 x^{3} y^{3}$ with respect to $x$ , i.e., compute $\frac{dy}{dx}$ .

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Problem 14315

Differentiate $2 x^{3}=(3 x y+1)^{2}$ with respect to $x$ to find $\frac{dy}{dx}$ .

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Problem 14316

Find the derivative $\frac{dy}{dx}$ for the equation $5 x^{3}=-3 x y+2$ .

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Problem 14317

Express the limit as a definite integral: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} x_{i} \ln(1+x_{i}^{2}) \Delta x$ over $[0,3]$ .

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Problem 14318

Bakterien-Population:
a) Bestimme das Maximum von $f(t)=\frac{1}{100} t^{3}-\frac{3}{5} t^{2}+9 t+10$ für $t \in [0, 21]$ .
b) Finde Zeitpunkte für stärksten Anstieg und Abfall der Bakterienzahl.
c) Bestimme, wann die Bakterienzahl auf 0 sinkt, wenn $f(t)$ tangential fortgesetzt wird.

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Problem 14319

Approximate the integral $\int_{0}^{\pi / 2} 2 \cos ^{5}(x) d x$ using the Midpoint Rule with $n=4$ . Round to four decimal places.

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Problem 14320

Evaluate $\int_{1}^{3} 7 e^{x+4} d x$ using $\int_{1}^{3} e^{x} d x=e^{3}-e$ .

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Problem 14321

Find $\int_{0}^{6}[3 f(x)+5 g(x)] d x$ given $\int_{0}^{6} f(x) d x=37$ and $\int_{0}^{6} g(x) d x=18$ .

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Problem 14322

Untersuchen Sie das Verhalten von $f$ für die Grenzprozesse: a) $f(x)=\frac{2x+1}{x}, x<0$ , $x \rightarrow -\infty$ ; b) $f(x)=\frac{x+1}{x^{2}}, x>0$ , $x \rightarrow \infty$ . Skizzieren Sie die Graphen.

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Problem 14323

Evaluate $\int_{1}^{3}\left(2 e^{x}-3\right) d x$ using $\int_{1}^{3} e^{x} d x=e^{3}-e$ .

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Problem 14324

Find the tangent line equation at the point for $t=-1$ on the curve defined by $x=t^{5}+1$ , $y=t^{6}+t$ .

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Problem 14325

Express the integral $\int_{2}^{4} \sqrt{7+x^{2}} dx$ as a limit of Riemann sums using right endpoints without evaluating it.

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Problem 14326

Express the limit as a definite integral: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{x_{i}^{}}{\left(x_{i}^{}\right)^{2}+4} \Delta x$ on $[1,5]$ .

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Problem 14327

Find the limit as a definite integral: $\lim_{n \to \infty} \sum_{i=1}^{n} [7(x_i^)^3 - 6x_i^] \Delta x$ on $[2,6]$ .

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Problem 14328

Find the half-life of a substance that decays at a rate of $9.75\%$ continuously over 8 hours. Round to 1 decimal place.

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Problem 14329

Gegeben sind die Funktionen $f(x)=x^{3}-3 x^{2}-x+4$ und $g(x)=-4 x+5$ .
a) Zeichnen Sie die Graphen für $-1 \leq x \leq 3$ . b) Bestimmen Sie $f^{\prime}(x)$ und $g^{\prime}(x)$ . c) Zeigen Sie, dass $f$ und $g$ am Schnittpunkt dieselbe Steigung haben. d) Für welche Punkte sind die Tangenten von $f$ senkrecht zu $g$ ? e) Zeigen Sie, dass $f$ die Linie $y=1$ in Winkeln von ca. $75,96^{\circ}$ und $82,88^{\circ}$ schneidet. f) Bestimmen Sie $f^{\prime}(x)$ mit dem Differentialquotienten. g) Für welche $x$ -Werte ist die Steigung von $f$ > 104? h) Untersuchen Sie $\frac{f(x)}{g(x)}$ für $x \rightarrow \infty$ .

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Problem 14330

Express the area under $y=x^{3}$ from 0 to 1 using right endpoints as a limit: $A=\lim_{n \to \infty} \sum_{j=1}^{n} f(x_{j}) \Delta x$ . Use $1^3 + 2^3 + \cdots + n^3 = \left[\frac{n(n+1)}{2}\right]^2$ to evaluate it.

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Problem 14331

Approximate the integral $\int_{0}^{\pi / 2} 2 \cos ^{5}(x) d x$ using the Midpoint Rule with $n=4$ , rounding to four decimal places.

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Problem 14332

Graph $f(x)=3x^{5}-20x^{3}$ . Find critical points and intervals of increase/decrease. Critical points: $x=$ , $x=$ . Increasing for, decreasing for.

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Problem 14333

Find the sensitivity $S$ by calculating the derivative $R'(x)$ of the reaction $R(x)=\frac{40+24 x^{0.4}}{1+4 x^{0.4}}$ .

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Problem 14334

Erklären Sie die Ableitungs- und Aufleitungsregeln und geben Sie je ein Beispiel für jede Regel.

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Problem 14335

Find intervals where $f(x) = x^{3} + x$ is increasing or decreasing, and analyze its curvature.

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Problem 14336

Find the limits for the piecewise function $f(x)$ defined as:
1. $3+\sin(x)$ for $x<0$
2. $3\cos(x)$ for $0 \leq x \leq \pi$
3. $3\sin(x)$ for $x>\pi$
a) $\lim_{x \to 0^-} f(x)$ b) $\lim_{x \to 0^+} f(x)$ c) $\lim_{x \to 0} f(x)$ d) $\lim_{x \to \pi^-} f(x)$

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Problem 14337

Find the $x$ where the function $g(x)=x e^{-4 x}$ has a local maximum. If none, enter NA.

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Problem 14338

Find the limits of the piecewise function $f(x)$ at specific points:
a) $\lim _{x \rightarrow-1^{-}} f(x)$ , b) $\lim _{x \rightarrow-1^{+}} f(x)$ , c) $\lim _{x \rightarrow-1} f(x)$ , d) $\lim _{x \rightarrow 4^{-}} f(x)$ .

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Problem 14339

Insect population $\mathrm{P}(t)=800 e^{0.03 t}$ : (a) Find $\mathrm{P}(0)$ . (b) Determine growth rate. (c) Find $\mathrm{P}(10)$ . (d) When is $\mathrm{P}=1200$ ? (e) When does population double?

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Problem 14340

Find the tangent line equation for $y=\frac{-6}{\sin x+\cos x}$ at point $(0,-6)$ . $y=$

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Problem 14341

Einstein's mass equation is $m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$ . For $m_{0}=3 \mathrm{~kg}$ , find limits as $v \to 0.9999c$ and $v \to c$ .

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Problem 14342

Einstein's mass equation is $m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$ . Given $m_{0}=3 \mathrm{~kg}$ , find limits as $v \to 0.9999c$ and $v \to c$ .

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Problem 14343

Find the slope of the secant line from $P(1,0)$ to $Q(x, \sin(3 \pi x))$ for $x = 0, 0.9, 0.99, 0.999$ .

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Problem 14344

Was passiert mit der Masse $m$ , wenn die Geschwindigkeit $v$ $99,99\%$ der Lichtgeschwindigkeit erreicht? Berechne $\lim _{v \rightarrow(0.9999 c)} \frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$ für $m_{0}=3 \mathrm{~kg}$ . Runde auf 3 Dezimalstellen.

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Problem 14345

Gegeben ist die Funktion $f_{a}(x)=-a x^{3}+4 a x (a \neq 0)$ . Untersuchen Sie Symmetrie, Punkte, Extremstellen und Wendetangente.

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Problem 14346

Find the formula for the annual growth rate of a bear population that increased by $50\%$ in 4 years using continuous compounding. Use ln in your answer.

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Problem 14347

Gegeben sind die Funktionen $f(x)=x^{3}-3 x^{2}-x+4$ und $g(x)=-4 x+5$ . Untersuchen Sie $f$ und $g$ in Bezug auf Graphen, Ableitungen und Steigungen.

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Problem 14348

Determine the long run behavior of $\frac{x^{3}+1}{x^{2}+2}$ . Options: No asymptote, $y=0$ , or $y=1$ .

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Problem 14349

A jello salad at $49^{\circ} \mathrm{F}$ warms in a $68^{\circ} \mathrm{F}$ room. After 5 min it's $54^{\circ} \mathrm{F}$ . Find its temp after 20 min.

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Problem 14350

Differentiate the function: $y=x^{5} \ln x-\frac{1}{2} x^{2}$ . Find $\frac{d y}{d x}=\square$ .

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Problem 14351

Find the third derivative of $f(x)=(2x+3)^6$ .

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Problem 14352

Strontium-90 decays as $A(t)=A_{0} e^{-0.0244 t}$ . Given 500g, find: (a) decay rate, (b) amount after 10 years, (c) time for 300g, (d) half-life.

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Problem 14353

Gegeben sind die Funktionen $f(x)=x^{3}-3 x^{2}-x+4$ und $g(x)=-4 x+5$ .
a) Graphen von $f$ und $g$ für $-1 \leq x \leq 3$ zeichnen. b) Bestimmen Sie $f^{\prime}(x)$ und $g^{\prime}(x)$ . c) Zeigen Sie, dass $f$ und $g$ im Schnittpunkt dieselbe Steigung haben. d) Für welche Punkte sind Tangenten von $f$ senkrecht zu $g$ ? e) Zeigen Sie, dass $f$ die Linie $y=1$ in ca. $75,96^{\circ}$ und $82,88^{\circ}$ schneidet. f) Bestimmen Sie $f^{\prime}(x)$ mit dem Differentialquotienten. g) Für welche $x$ -Werte ist die Steigung von $f$ > 104? h) Untersuchen Sie $\frac{f(x)}{g(x)}$ für $x \rightarrow \infty$ .

See Solution

Problem 14354

A 600 g sample of lead-210 decays as $A(t)=600 e^{-0.032 t}$ . Find amounts after (a) 4 yr, (b) 6 yr, (c) 20 yr, and (d) half-life.

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Problem 14355

Evaluate the integral $\int x^{2} \ln x \, dx$ using integration by parts with $u=\ln x$ and $dv=x^{2} dx$ .

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Problem 14356

Evaluate the integral: $\int x \cos 5 x \, dx$

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Problem 14357

Find the half-life of a radioactive element with decay function $A(t)=A_{0} e^{-0.0188 t}$ . Round to the nearest tenth.

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Problem 14358

Oil spills spread in a circle with area increasing at $6.5 \mathrm{mi}^{2} / \mathrm{hr}$ . Find the radius increase rate when area is $9 \mathrm{mi}^{2}$ .

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Problem 14359

Find $\frac{d y}{d x}$ using implicit differentiation for $\frac{x y^{3}}{\sec (y)-1}=1+y^{4}$ .

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Problem 14360

A triangle's altitude increases at $2 \mathrm{~cm/min}$ and area at $3 \mathrm{~cm}^2/\mathrm{min}$ . Find the base's rate when altitude = 8 cm and area = 91 cm².

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Problem 14361

Find critical points of $f(x)=x(8-x)^{3}$ . Answer: $x=2,8$ .

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Problem 14362

A 200 g wood block on a table with friction 0.40 is compressed 18 cm by a 10 N force. How far will it stretch after release?

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Problem 14363

A pizza pan cools from $400^{\circ} \mathrm{F}$ to room temp $75^{\circ} \mathrm{F}$ . After 5 min it's $300^{\circ} \mathrm{F}$ .
(a) When is it $125^{\circ} \mathrm{F}$ ?
(b) How long until it reaches $240^{\circ} \mathrm{F}$ ?
(c) What trend do you observe in the temperature over time?
Use: $T(t) = T_s + (T_0 - T_s)e^{-kt}$ , where $T_s = 75$ , $T_0 = 400$ .

See Solution

Problem 14364

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ given $\sqrt{x+3 y^{3}}=2 y$ .

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Problem 14365

A pizza pan cools from $450^{\circ} \mathrm{F}$ to $72^{\circ} \mathrm{F}$ . After 5 min, it's $300^{\circ} \mathrm{F}$ .
(a) When is it $135^{\circ} \mathrm{F}$ ? (b) When is it $190^{\circ} \mathrm{F}$ ? (c) Describe the temperature trend over time.

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Problem 14366

Set up and evaluate the integral for the volume of the solid formed by revolving the region bounded by $y=x^{6}$ , $x=0$ , $y=64$ about the $x$ -axis.

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Problem 14367

A pizza pan cools from $450^{\circ}F$ to $72^{\circ}F$ . After 5 mins it's $300^{\circ}F$ . Find when it's $135^{\circ}F$ and $190^{\circ}F$ . What trend do you observe?

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Problem 14368

Find the volume of the solid from revolving the area between $y=8-3x-x^2$ and $y=x+8$ around (i) $x$ -axis, (ii) $y=4$ .

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Problem 14369

Find the volume of the solid formed by revolving the area between $y=2x$ , $y=0$ , and $x=4$ around $x=5$ .

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Problem 14370

Find the carrying capacity and growth rate from the logistic model $P(t)=\frac{3000}{1+31.73 e^{-0.327 t}}$ .

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Problem 14371

Find the limit: $\lim _{x \rightarrow 6}(\ln ((7-x)^{\frac{1}{x-6}}))=-1$ and exponentiate it.

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Problem 14372

Find the volume of the solid formed by revolving the area between $y=\cos x$ , $y=0$ , $x=0$ , and $x=\frac{\pi}{3}$ around the $x$ -axis.

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Problem 14373

Find the volume of the solid formed by revolving the region bounded by $y=2-x$ , $y=0$ , $x=0$ around the $x$ -axis.

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Problem 14374

Find the volume of the solid formed by revolving the region between $y=\frac{15}{x^{2}}$ , $y=0$ , $x=1$ , and $x=8$ around the $x$ -axis. Round to two decimal places.

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Problem 14375

Find the volume using the shell method for $y=14x-x^2$ , $x=0$ , and $y=49$ revolved around the $y$ -axis.

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Problem 14376

Find the volume of the solid formed by revolving the area between $y=\frac{1}{x}$ , $y=0$ , $x=8$ , and $x=15$ around the $x$ -axis.

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Problem 14377

A pizza pan cools from $450^{\circ} \mathrm{F}$ to room temp $72^{\circ} \mathrm{F}$ . After 5 min it's $300^{\circ} \mathrm{F}$ .
(a) When is it $135^{\circ} \mathrm{F}$ ? (b) When is it $190^{\circ} \mathrm{F}$ ? (c) Describe the temperature trend over time.

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Problem 14378

The logistic model $P(t)=\frac{3000}{1+31.73 e^{-0.327 t}}$ shows bacteria growth. Find carrying capacity, growth rate, and initial population.

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Problem 14379

Evaluate the line integral using Green's Theorem: $\oint_{C}\left(\frac{1}{2} x y^{2}, 4 y^{2} \cos y\right) \cdot d \vec{r}$ where $C$ is bounded by $y=x^{2}$ and $y=x^{3}$ in the first quadrant.

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Problem 14380

Find the area between $y=e^{x}$ and $y=\frac{1}{2} x$ for $x$ in $[0,8]$ . Choose from the options given.

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Problem 14381

Calculate the area between the curves $y=x^{2}$ and $y=12-x$ by integrating with respect to $x$ and $y$ .

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Problem 14382

Find critical points of $f(x)=(x^{2}-9)^{7}$ , use the first-derivative test for local maxima/minima, graph to verify.
Enter critical points in order. If none, enter NA.
Local maximum at $x=$ , local minimum at $x=$ .

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Problem 14383

The logistic model $\mathrm{P}(\mathrm{t})=\frac{1000}{1+30.28 e^{-0.433 \mathrm{t}}}$ shows bacterial growth. Find (a) carrying capacity, (b) growth rate, (c) initial population.

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Problem 14384

Calculate the orbital period of a satellite at a height of $488000 \mathrm{~m}$ above Mars, with radius $3397 \mathrm{~km}$ .

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Problem 14385

The logistic model $P(t)=\frac{1000}{1+30.28 e^{-0.433 t}}$ shows bacterium growth. Find (b) growth rate, (c) initial size, (d) population after 7 hours.

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Problem 14386

Find the area between $f(x)=\sin 11 x$ and $g(x)=\cos 22 x$ for $-\frac{\pi}{22} \leq x \leq \frac{\pi}{22}$ . Round to three decimal places.

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Problem 14387

Find the limit using L'Hôpital's Rule: $\lim _{x \rightarrow \infty} \frac{x}{e^{x}}=$

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Problem 14388

Calculate the area between $y=3 \ln x$ , $y=5-3 \ln x$ , and $x=5$ . Round to three decimal places.

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Problem 14389

Find the derivative of the function $f(x)=e^{5 x^{2}}$ .

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Problem 14390

Find the derivative of $\sin^{-1}(8x)$ with respect to $x$ .

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Problem 14391

Calculate the average rate of change for $f(x)=mx+c$ from $a=-1$ to $b=7$ .

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Problem 14392

Find $f^{\prime}(0)$ if $f(x)=\ln(x+4+e^{-3x})$ .

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Problem 14393

Find the limit as $x$ approaches 1 for the expression $\frac{5 \ln x^{2}}{x^{2}-1}$ .

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Problem 14394

Evaluate the limit using L'Hôpital's Rule: $\lim _{x \rightarrow 324}\left(\frac{1}{\sqrt{x}-18}-\frac{36}{x-324}\right)=$

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Problem 14395

Find the area of the surface formed by revolving $y=\frac{x}{3}$ from $x=3$ to $x=9$ about the $x$ -axis.

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Problem 14396

Find the derivative of $\log_{2}(x^{3})$ .

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Problem 14397

Logistic growth model $P(t)=\frac{1000}{1+30.28 e^{-0.433 t}}$ . Find initial size, population after 7 hours, and time to reach $700 \mathrm{~g}$ .

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Problem 14398

Find the surface area of the curve $y=\sqrt[3]{x}+16$ from $x=1$ to $x=4,096$ revolved around the $y$ -axis.

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Problem 14399

Find the expectation $\mathbb{E}$ and variance $\mathbb{V}$ of $\rho(x)=\frac{2}{\pi} \frac{1}{x^{2}+1}$ on $[0,+\infty)$ . Check for convergence.

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Problem 14400

Calculate the average rate of change of $h(x)=2-x^{2}$ from $x=3$ to $x=4$ .

See Solution

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Calculus

Problem 14301

Find the limit: lim⁡h→0ln⁡(2+h)−ln⁡(2)h\lim _{h \rightarrow 0} \frac{\ln (2+h)-\ln (2)}{h}limh→0​hln(2+h)−ln(2)​. Choose from: a. ln⁡(2)\ln (2)ln(2), b. 12\frac{1}{2}21​, c. 1ln⁡(2)\frac{1}{\ln (2)}ln(2)1​, d. does not exist.

Problem 14302

Find the limits: lim⁡x→2−h(x)=3log⁡(−2x+4)\lim _{x \rightarrow 2^{-}} h(x)=3 \log (-2 x+4)limx→2−​h(x)=3log(−2x+4) and lim⁡x→−∞K(x)=−ln⁡(−x+6)−1\lim _{x \rightarrow-\infty} K(x)=-\ln (-x+6)-1limx→−∞​K(x)=−ln(−x+6)−1.

Problem 14303

Calculate the half-life of I-10 with a decay rate of 9.82%9.82\%9.82% per day. Round to 1 decimal place.

Problem 14304

Given f(x)=−ln⁡(x+2)−6f(x)=-\ln (x+2)-6f(x)=−ln(x+2)−6, find its asymptote, domain, range, yyy-intercept, and limits as x→−2+x \to -2^+x→−2+ and x→∞x \to \inftyx→∞.

Problem 14305

Find the limit: lim⁡h→0ln⁡(2+h)−ln⁡(2)h\lim _{h \rightarrow 0} \frac{\ln (2+h)-\ln (2)}{h}limh→0​hln(2+h)−ln(2)​. Choices: a. ln⁡(2)\ln (2)ln(2) b. 12\frac{1}{2}21​ c. 1ln⁡(2)\frac{1}{\ln (2)}ln(2)1​ d. does not exist.

Problem 14306

Determine the horizontal asymptote of g(x)=(13)x+3g(x)=\left(\frac{1}{3}\right)^{x+3}g(x)=(31​)x+3.

Problem 14307

Find the initial population size and the size after 8 years for P(t)=6001+3e−0.31tP(t)=\frac{600}{1+3 e^{-0.31 t}}P(t)=1+3e−0.31t600​. Round to whole numbers.

Problem 14308

Find the linearization L(x)L(x)L(x) at x=−1x=-1x=−1 for the function f(x)=x+1xf(x)=x+\frac{1}{x}f(x)=x+x1​.

Problem 14309

Find the derivative of the quadratic function f(x)=a(x−r)(x−s)f(x)=a(x-r)(x-s)f(x)=a(x−r)(x−s) using first principles.

Problem 14310

Find the time (in s) to charge a 200pF200 \mathrm{pF}200pF capacitor through a 69.0MΩ69.0 \mathrm{M} \Omega69.0MΩ resistor to 92.0%92.0 \%92.0% voltage.

Problem 14311

Find the limit as xxx approaches -1: lim⁡x→−1x+15x+9−2\lim _{x \rightarrow-1} \frac{x+1}{\sqrt{5 x+9}-2}x→−1lim​5x+9​−2x+1​

Problem 14312

Check if f(x)=x2/11f(x)=x^{2/11}f(x)=x2/11 meets the Mean Value Theorem conditions on [−9,1][-9,1][−9,1]. Justify your answer.

Problem 14313

Gegeben sind f(x)=x3−3x2−x+4f(x)=x^{3}-3 x^{2}-x+4f(x)=x3−3x2−x+4 und g(x)=−4x+5g(x)=-4 x+5g(x)=−4x+5. Untersuchen Sie fff und ggg in Bezug auf Graphen, Steigungen und Tangenten.

Problem 14314

Find the derivative of 5x3+xy2=5x3y35 x^{3}+x y^{2}=5 x^{3} y^{3}5x3+xy2=5x3y3 with respect to xxx, i.e., compute dydx\frac{dy}{dx}dxdy​.

Problem 14315

Differentiate 2x3=(3xy+1)22 x^{3}=(3 x y+1)^{2}2x3=(3xy+1)2 with respect to xxx to find dydx\frac{dy}{dx}dxdy​.

Problem 14316

Find the derivative dydx\frac{dy}{dx}dxdy​ for the equation 5x3=−3xy+25 x^{3}=-3 x y+25x3=−3xy+2.

Problem 14317

Express the limit as a definite integral: lim⁡n→∞∑i=1nxiln⁡(1+xi2)Δx\lim _{n \rightarrow \infty} \sum_{i=1}^{n} x_{i} \ln(1+x_{i}^{2}) \Delta xn→∞lim​i=1∑n​xi​ln(1+xi2​)Δx over [0,3][0,3][0,3].

Problem 14318

Problem 14319

Approximate the integral ∫0π/22cos⁡5(x)dx\int_{0}^{\pi / 2} 2 \cos ^{5}(x) d x∫0π/2​2cos5(x)dx using the Midpoint Rule with n=4n=4n=4. Round to four decimal places.

Problem 14320

Evaluate ∫137ex+4dx\int_{1}^{3} 7 e^{x+4} d x∫13​7ex+4dx using ∫13exdx=e3−e\int_{1}^{3} e^{x} d x=e^{3}-e∫13​exdx=e3−e.

Problem 14321

Find ∫06[3f(x)+5g(x)]dx\int_{0}^{6}[3 f(x)+5 g(x)] d x∫06​[3f(x)+5g(x)]dx given ∫06f(x)dx=37\int_{0}^{6} f(x) d x=37∫06​f(x)dx=37 and ∫06g(x)dx=18\int_{0}^{6} g(x) d x=18∫06​g(x)dx=18.

Problem 14322

Untersuchen Sie das Verhalten von fff für die Grenzprozesse: a) f(x)=2x+1x,x<0f(x)=\frac{2x+1}{x}, x<0f(x)=x2x+1​,x<0, x→−∞x \rightarrow -\inftyx→−∞; b) f(x)=x+1x2,x>0f(x)=\frac{x+1}{x^{2}}, x>0f(x)=x2x+1​,x>0, x→∞x \rightarrow \inftyx→∞. Skizzieren Sie die Graphen.

Problem 14323

Evaluate ∫13(2ex−3)dx\int_{1}^{3}\left(2 e^{x}-3\right) d x∫13​(2ex−3)dx using ∫13exdx=e3−e\int_{1}^{3} e^{x} d x=e^{3}-e∫13​exdx=e3−e.

Problem 14324

Find the tangent line equation at the point for t=−1t=-1t=−1 on the curve defined by x=t5+1x=t^{5}+1x=t5+1, y=t6+ty=t^{6}+ty=t6+t.

Problem 14325

Express the integral ∫247+x2dx\int_{2}^{4} \sqrt{7+x^{2}} dx∫24​7+x2​dx as a limit of Riemann sums using right endpoints without evaluating it.

Problem 14326

Express the limit as a definite integral: lim⁡n→∞∑i=1nxi∗(xi∗)2+4Δx\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{x_{i}^{*}}{\left(x_{i}^{*}\right)^{2}+4} \Delta xn→∞lim​i=1∑n​(xi∗​)2+4xi∗​​Δx on [1,5][1,5][1,5].

Problem 14327

Find the limit as a definite integral: lim⁡n→∞∑i=1n[7(xi∗)3−6xi∗]Δx\lim_{n \to \infty} \sum_{i=1}^{n} [7(x_i^*)^3 - 6x_i^*] \Delta xn→∞lim​i=1∑n​[7(xi∗​)3−6xi∗​]Δx on [2,6][2,6][2,6].

Problem 14328

Find the half-life of a substance that decays at a rate of 9.75%9.75\%9.75% continuously over 8 hours. Round to 1 decimal place.

Problem 14329

Problem 14330

Problem 14331

Approximate the integral ∫0π/22cos⁡5(x)dx\int_{0}^{\pi / 2} 2 \cos ^{5}(x) d x∫0π/2​2cos5(x)dx using the Midpoint Rule with n=4n=4n=4, rounding to four decimal places.

Problem 14332

Graph f(x)=3x5−20x3f(x)=3x^{5}-20x^{3}f(x)=3x5−20x3. Find critical points and intervals of increase/decrease. Critical points: x=x=x=, x=x=x=. Increasing for, decreasing for.

Problem 14333

Find the sensitivity SSS by calculating the derivative R′(x)R'(x)R′(x) of the reaction R(x)=40+24x0.41+4x0.4R(x)=\frac{40+24 x^{0.4}}{1+4 x^{0.4}}R(x)=1+4x0.440+24x0.4​.

Problem 14334

Erklären Sie die Ableitungs- und Aufleitungsregeln und geben Sie je ein Beispiel für jede Regel.

Problem 14335

Find intervals where f(x)=x3+xf(x) = x^{3} + xf(x)=x3+x is increasing or decreasing, and analyze its curvature.

Problem 14336

Problem 14337

Find the xxx where the function g(x)=xe−4xg(x)=x e^{-4 x}g(x)=xe−4x has a local maximum. If none, enter NA.

Problem 14338

Problem 14339

Insect population P(t)=800e0.03t\mathrm{P}(t)=800 e^{0.03 t}P(t)=800e0.03t: (a) Find P(0)\mathrm{P}(0)P(0). (b) Determine growth rate. (c) Find P(10)\mathrm{P}(10)P(10). (d) When is P=1200\mathrm{P}=1200P=1200? (e) When does population double?

Problem 14340

Find the tangent line equation for y=−6sin⁡x+cos⁡xy=\frac{-6}{\sin x+\cos x}y=sinx+cosx−6​ at point (0,−6)(0,-6)(0,−6). y=y=y=

Problem 14341

Einstein's mass equation is m=m01−v2c2m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}m=1−c2v2​​m0​​. For m0=3 kgm_{0}=3 \mathrm{~kg}m0​=3 kg, find limits as v→0.9999cv \to 0.9999cv→0.9999c and v→cv \to cv→c.

Problem 14342

Einstein's mass equation is m=m01−v2c2m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}m=1−c2v2​​m0​​. Given m0=3 kgm_{0}=3 \mathrm{~kg}m0​=3 kg, find limits as v→0.9999cv \to 0.9999cv→0.9999c and v→cv \to cv→c.

Find the limit: $\lim _{h \rightarrow 0} \frac{\ln (2+h)-\ln (2)}{h}$ . Choose from: a. $\ln (2)$ , b. $\frac{1}{2}$ , c. $\frac{1}{\ln (2)}$ , d. does not exist.

Find the limits: $\lim _{x \rightarrow 2^{-}} h(x)=3 \log (-2 x+4)$ and $\lim _{x \rightarrow-\infty} K(x)=-\ln (-x+6)-1$ .

Calculate the half-life of I-10 with a decay rate of $9.82\%$ per day. Round to 1 decimal place.

Given $f(x)=-\ln (x+2)-6$ , find its asymptote, domain, range, $y$ -intercept, and limits as $x \to -2^+$ and $x \to \infty$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{\ln (2+h)-\ln (2)}{h}$ . Choices: a. $\ln (2)$ b. $\frac{1}{2}$ c. $\frac{1}{\ln (2)}$ d. does not exist.

Determine the horizontal asymptote of $g(x)=\left(\frac{1}{3}\right)^{x+3}$ .

Find the initial population size and the size after 8 years for $P(t)=\frac{600}{1+3 e^{-0.31 t}}$ . Round to whole numbers.

Find the linearization $L(x)$ at $x=-1$ for the function $f(x)=x+\frac{1}{x}$ .

Find the derivative of the quadratic function $f(x)=a(x-r)(x-s)$ using first principles.

Find the time (in s) to charge a $200 \mathrm{pF}$ capacitor through a $69.0 \mathrm{M} \Omega$ resistor to $92.0 \%$ voltage.

Find the limit as $x$ approaches -1: $\lim _{x \rightarrow-1} \frac{x+1}{\sqrt{5 x+9}-2}$

Check if $f(x)=x^{2/11}$ meets the Mean Value Theorem conditions on $[-9,1]$ . Justify your answer.

Gegeben sind $f(x)=x^{3}-3 x^{2}-x+4$ und $g(x)=-4 x+5$ . Untersuchen Sie $f$ und $g$ in Bezug auf Graphen, Steigungen und Tangenten.

Find the derivative of $5 x^{3}+x y^{2}=5 x^{3} y^{3}$ with respect to $x$ , i.e., compute $\frac{dy}{dx}$ .

Differentiate $2 x^{3}=(3 x y+1)^{2}$ with respect to $x$ to find $\frac{dy}{dx}$ .

Find the derivative $\frac{dy}{dx}$ for the equation $5 x^{3}=-3 x y+2$ .

Express the limit as a definite integral: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} x_{i} \ln(1+x_{i}^{2}) \Delta x$ over $[0,3]$ .

Approximate the integral $\int_{0}^{\pi / 2} 2 \cos ^{5}(x) d x$ using the Midpoint Rule with $n=4$ . Round to four decimal places.

Evaluate $\int_{1}^{3} 7 e^{x+4} d x$ using $\int_{1}^{3} e^{x} d x=e^{3}-e$ .

Find $\int_{0}^{6}[3 f(x)+5 g(x)] d x$ given $\int_{0}^{6} f(x) d x=37$ and $\int_{0}^{6} g(x) d x=18$ .

Untersuchen Sie das Verhalten von $f$ für die Grenzprozesse: a) $f(x)=\frac{2x+1}{x}, x<0$ , $x \rightarrow -\infty$ ; b) $f(x)=\frac{x+1}{x^{2}}, x>0$ , $x \rightarrow \infty$ . Skizzieren Sie die Graphen.

Evaluate $\int_{1}^{3}\left(2 e^{x}-3\right) d x$ using $\int_{1}^{3} e^{x} d x=e^{3}-e$ .

Find the tangent line equation at the point for $t=-1$ on the curve defined by $x=t^{5}+1$ , $y=t^{6}+t$ .

Express the integral $\int_{2}^{4} \sqrt{7+x^{2}} dx$ as a limit of Riemann sums using right endpoints without evaluating it.

Express the limit as a definite integral: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{x_{i}^{}}{\left(x_{i}^{}\right)^{2}+4} \Delta x$ on $[1,5]$ .

Find the limit as a definite integral: $\lim_{n \to \infty} \sum_{i=1}^{n} [7(x_i^)^3 - 6x_i^] \Delta x$ on $[2,6]$ .

Find the half-life of a substance that decays at a rate of $9.75\%$ continuously over 8 hours. Round to 1 decimal place.

Approximate the integral $\int_{0}^{\pi / 2} 2 \cos ^{5}(x) d x$ using the Midpoint Rule with $n=4$ , rounding to four decimal places.

Graph $f(x)=3x^{5}-20x^{3}$ . Find critical points and intervals of increase/decrease. Critical points: $x=$ , $x=$ . Increasing for, decreasing for.

Find the sensitivity $S$ by calculating the derivative $R'(x)$ of the reaction $R(x)=\frac{40+24 x^{0.4}}{1+4 x^{0.4}}$ .

Find intervals where $f(x) = x^{3} + x$ is increasing or decreasing, and analyze its curvature.

Find the $x$ where the function $g(x)=x e^{-4 x}$ has a local maximum. If none, enter NA.

Insect population $\mathrm{P}(t)=800 e^{0.03 t}$ : (a) Find $\mathrm{P}(0)$ . (b) Determine growth rate. (c) Find $\mathrm{P}(10)$ . (d) When is $\mathrm{P}=1200$ ? (e) When does population double?

Find the tangent line equation for $y=\frac{-6}{\sin x+\cos x}$ at point $(0,-6)$ . $y=$

Einstein's mass equation is $m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$ . For $m_{0}=3 \mathrm{~kg}$ , find limits as $v \to 0.9999c$ and $v \to c$ .

Einstein's mass equation is $m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$ . Given $m_{0}=3 \mathrm{~kg}$ , find limits as $v \to 0.9999c$ and $v \to c$ .

Find the slope of the secant line from $P(1,0)$ to $Q(x, \sin(3 \pi x))$ for $x = 0, 0.9, 0.99, 0.999$ .

Gegeben ist die Funktion $f_{a}(x)=-a x^{3}+4 a x (a \neq 0)$ . Untersuchen Sie Symmetrie, Punkte, Extremstellen und Wendetangente.

Find the formula for the annual growth rate of a bear population that increased by $50\%$ in 4 years using continuous compounding. Use ln in your answer.

Gegeben sind die Funktionen $f(x)=x^{3}-3 x^{2}-x+4$ und $g(x)=-4 x+5$ . Untersuchen Sie $f$ und $g$ in Bezug auf Graphen, Ableitungen und Steigungen.

Determine the long run behavior of $\frac{x^{3}+1}{x^{2}+2}$ . Options: No asymptote, $y=0$ , or $y=1$ .

A jello salad at $49^{\circ} \mathrm{F}$ warms in a $68^{\circ} \mathrm{F}$ room. After 5 min it's $54^{\circ} \mathrm{F}$ . Find its temp after 20 min.

Differentiate the function: $y=x^{5} \ln x-\frac{1}{2} x^{2}$ . Find $\frac{d y}{d x}=\square$ .

Find the third derivative of $f(x)=(2x+3)^6$ .

Strontium-90 decays as $A(t)=A_{0} e^{-0.0244 t}$ . Given 500g, find: (a) decay rate, (b) amount after 10 years, (c) time for 300g, (d) half-life.

A 600 g sample of lead-210 decays as $A(t)=600 e^{-0.032 t}$ . Find amounts after (a) 4 yr, (b) 6 yr, (c) 20 yr, and (d) half-life.

Evaluate the integral $\int x^{2} \ln x \, dx$ using integration by parts with $u=\ln x$ and $dv=x^{2} dx$ .

Evaluate the integral: $\int x \cos 5 x \, dx$

Find the half-life of a radioactive element with decay function $A(t)=A_{0} e^{-0.0188 t}$ . Round to the nearest tenth.

Oil spills spread in a circle with area increasing at $6.5 \mathrm{mi}^{2} / \mathrm{hr}$ . Find the radius increase rate when area is $9 \mathrm{mi}^{2}$ .

Find $\frac{d y}{d x}$ using implicit differentiation for $\frac{x y^{3}}{\sec (y)-1}=1+y^{4}$ .

A triangle's altitude increases at $2 \mathrm{~cm/min}$ and area at $3 \mathrm{~cm}^2/\mathrm{min}$ . Find the base's rate when altitude = 8 cm and area = 91 cm².

Find critical points of $f(x)=x(8-x)^{3}$ . Answer: $x=2,8$ .

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ given $\sqrt{x+3 y^{3}}=2 y$ .

Set up and evaluate the integral for the volume of the solid formed by revolving the region bounded by $y=x^{6}$ , $x=0$ , $y=64$ about the $x$ -axis.

A pizza pan cools from $450^{\circ}F$ to $72^{\circ}F$ . After 5 mins it's $300^{\circ}F$ . Find when it's $135^{\circ}F$ and $190^{\circ}F$ . What trend do you observe?

Find the volume of the solid from revolving the area between $y=8-3x-x^2$ and $y=x+8$ around (i) $x$ -axis, (ii) $y=4$ .

Find the volume of the solid formed by revolving the area between $y=2x$ , $y=0$ , and $x=4$ around $x=5$ .

Find the carrying capacity and growth rate from the logistic model $P(t)=\frac{3000}{1+31.73 e^{-0.327 t}}$ .

Find the limit: $\lim _{x \rightarrow 6}(\ln ((7-x)^{\frac{1}{x-6}}))=-1$ and exponentiate it.

Find the volume of the solid formed by revolving the area between $y=\cos x$ , $y=0$ , $x=0$ , and $x=\frac{\pi}{3}$ around the $x$ -axis.

Find the volume of the solid formed by revolving the region bounded by $y=2-x$ , $y=0$ , $x=0$ around the $x$ -axis.

Find the volume of the solid formed by revolving the region between $y=\frac{15}{x^{2}}$ , $y=0$ , $x=1$ , and $x=8$ around the $x$ -axis. Round to two decimal places.