Calculus

Problem 16601

Find the $x$ -values where the function $f(x)=\frac{3}{20} x^{5}-13 x^{3}$ has an inflection point.

See Solution

Problem 16602

Find the intervals where the function $f(x)=\frac{1}{6} x^{4}-19 x^{2}$ is concave down.

See Solution

Problem 16603

Find all $x$ -values where the function $f(x)=\frac{1}{15} x^{6}+x^{5}$ has an inflection point.

See Solution

Problem 16604

Find the left Riemann sum for $\int_{2}^{3} x^{3} dx$ using 5 equal subintervals. Round to the nearest thousandth.

See Solution

Problem 16605

A cylinder's radius decreases at 2 cm/min, volume at 467 cm³/min. When radius is 4 cm and volume is 347 cm³, find height's rate of change using $V=\pi r^{2} h$ . Round to three decimal places.

See Solution

Problem 16606

A sphere's radius grows at 8 m/s. When the radius is 2 m, find the volume's rate of change using $V=\frac{4}{3} \pi r^{3}$ . Round to three decimals.

See Solution

Problem 16607

Bestimmen Sie die Stammfunktion von $f$ : a) $f(x)=4 x^{3}+2 x^{2}-1$ , b) $f(x)=x^{5}-4 x^{3}$ , e) $f(x)=0,24 x^{7}-5,4 x$ , f) $f(x)=-25 x^{4}+33 x^{2}$ .

See Solution

Problem 16608

Bestimmen Sie eine Stammfunktion für die folgenden Funktionen: a) $f(x)=\frac{1}{4} x^{-2}$ , b) $f(x)=\frac{3}{x^{4}}$ , e) $f(x)=\frac{-5}{x^{6}}$ , f) $f(x)=\frac{x^{2}+2 x}{x^{5}}$ .

See Solution

Problem 16609

Calculate the integral $\int_{\pi / 4}^{\pi / 2} \sin^{3} \alpha \cos \alpha \, d\alpha$ .

See Solution

Problem 16610

Find the gradient of the curve $y=\frac{8}{4x-5}$ at $x=2$ .

See Solution

Problem 16611

Find the initial amount of Carbon14, the amount after 12208 years, and its half-life from $A(t)=236 e^{-0.00012 t}$ .

See Solution

Problem 16612

Die Höhe einer Eisenkugel wird durch $h(t)=3,5 \cdot e^{-0,2 t} \cdot \sin (5 t)$ beschrieben.
a) Finde $h(4)$ und $v(4)$ . b) Bestimme mit dem GTR, wann $v(t)=3 \frac{\mathrm{cm}}{\mathrm{s}}$ .

See Solution

Problem 16613

Berechnen Sie das Integral von $f(x)=(x^{2}+3 x)^{2}$ .

See Solution

Problem 16614

Hassan wants to know how much to invest at $2.3\%$ interest to have $\$1,130$ in 8 years. Calculate the amount needed.

See Solution

Problem 16615

Gegeben ist die Funktion $f(x)=-\frac{1}{4} x^{2}+x$ .
a) Finde den Punkt $P$ mit Steigung -2.
b) Finde alle Stellen mit waagerechter Tangente.
c) Bestimme die Tangentengleichung im Punkt $Q(4 \mid f(4))$ .
d) Finde Stellen, wo die Tangente parallel zu $y=3x+5$ ist.

See Solution

Problem 16616

Bestimmen Sie die Intervalle, in denen $f(x)=\frac{2}{3} x^{3}+x^{2}-12 x+1$ streng monoton wächst.

See Solution

Problem 16617

Ximena wants to invest to reach \$5,100 in 6 years at a 2.2% continuous interest rate. How much does she need to invest?

See Solution

Problem 16618

Hakeem wants to invest for an account to reach \ $3,700 in 15 years at a continuous interest rate of$ 3.5\%$. How much?

See Solution

Problem 16619

How much should London invest at a $5.7\%$ continuous interest to have $\$281,000$ in 19 years? Round to nearest \$10.

See Solution

Problem 16620

Verify that the tangent lines of the bicorn curve at $(0,4)$ and $(0,4/3)$ are horizontal using implicit differentiation.

See Solution

Problem 16621

Levi wants to know how much to invest at $3.4\%$ interest compounded continuously to reach \$620 in 16 years.

See Solution

Problem 16622

Berechnen Sie das unbestimmte Integral der Funktion $0,3 x^{-10}$ .

See Solution

Problem 16623

Berechnen Sie das unbestimmte Integral von $f(x)=\frac{3}{x^{4}}$ .

See Solution

Problem 16624

A biologist has an 8824-gram sample of a radioactive substance. Find its mass after 3 hours with a decay rate of $9\%$ per hour. Round to the nearest tenth.

See Solution

Problem 16625

Find the half-life of a radioactive substance with a decay rate of $1.9\%$ per day. Round to the nearest hundredth.

See Solution

Problem 16626

Find the half-life of a substance with a decay rate of $1.9\%$ per day using the continuous exponential decay model.

See Solution

Problem 16627

Calculate the integral from 0 to 10 of $\sqrt{100 - x^{2}}$ dx.

See Solution

Problem 16628

Find the interest rate needed for Jason's \$16,000 to grow to \$26,000 in 7 years with continuous compounding.

See Solution

Problem 16629

Which expression equals $\lim _{n \rightarrow \infty} \sum_{i=1}^{\infty} \sqrt{\frac{3 i}{n}} \frac{3}{n}$ ? Options include various integrals.

See Solution

Problem 16630

Calculate the area under the curve $x^{2}$ from $x=0$ to $x=3$ using the definition of the definite integral.

See Solution

Problem 16631

Given $f(5)=-7$ , $f^{\prime}(5)=0$ , and $f^{\prime \prime}(5)=1$ , what can we conclude about $f$ at 5? A, B, C, or D?

See Solution

Problem 16632

Find critical points of $f(x)=x e^{-x^{2}}$ and use the Second Derivative Test to classify them.

See Solution

Problem 16633

Find intervals where the function is concave up or down, and identify points of inflection for $f(x)=4 x^{3}+4 x^{2}-5 x+9$ .

See Solution

Problem 16634

Determine where the function $f(x)=\frac{3}{x-3}$ is concave up, concave down, and find points of inflection.

See Solution

Problem 16635

Find critical points of $f(x) = x e^{-x^{2}}$ and use the Second Derivative Test to classify them.

See Solution

Problem 16636

Determine the intervals where $f(x)=\frac{3}{x-3}$ is concave up or down, and locate any points of inflection.

See Solution

Problem 16637

Find $R^{\prime \prime}(x)$ and the point of diminishing returns for $R(x)=-x^{3}+57x^{2}+1000$ where $0 \leq x \leq 24$ .

See Solution

Problem 16638

Find the absolute max and min of $f(x)=x^{3}-6x^{2}-63x+11$ on intervals: (A) $[-4,0]$ , (B) $[-1,8]$ , (C) $[-4,8]$ . Use -1000 for non-existent extrema.

See Solution

Problem 16639

Given $R(x)=-x^{3}+57x^{2}+1000$ for $0 \leq x \leq 24$ , find $R^{\prime \prime}(x)$ and the point of diminishing returns.

See Solution

Problem 16640

Find critical numbers of $f(x)=\frac{4-x}{4+x}$ and use the second derivative test for local max/min.

See Solution

Problem 16641

Find critical numbers of $f(x)=-2 x^{3}-3 x^{2}+12 x+3$ and use the second derivative test to classify them. What is $f''(x)$ ?

See Solution

Problem 16642

Find the second derivative of $f(x)=-2x^3-3x^2+12x+3$ and determine the critical numbers and local extrema.

See Solution

Problem 16643

Find the absolute max and min of $f(x)=-x^{4}+32 x^{2}-256$ on $[-5,5]$ . Max at $x=\square$ .

See Solution

Problem 16644

Find the production level $x$ that minimizes the average cost per wheel given $C(x)=0.03 x^{3}-4.5 x^{2}+174 x$ for $x \in (0,100]$ . $x=\square$

See Solution

Problem 16645

Find the absolute extrema of $f(x)=\frac{6+x}{6-x}$ on $[3,5]$ . Max at $x=\square$ .

See Solution

Problem 16646

Find the production level $x$ that minimizes the average cost function $C(x)=0.03 x^{3}-4.5 x^{2}+174 x$ for $0 \leq x \leq 100$ .

See Solution

Problem 16647

Welche Stammfunktion von $f$ hat bei $x=1$ den Wert 2? a) $f(x)=3 x^{2}$ b) $f(x)=x^{3}-2 x^{2}+1$

See Solution

Problem 16648

Find the point of diminishing returns for the function $R(x)=10,000-x^{3}+36 x^{2}+700 x$ in the interval $0 \leq x \leq 20$ .

See Solution

Problem 16649

Find the critical numbers of $g(x)=\frac{x-1}{x^{x-3x+3}}$ .

See Solution

Problem 16650

A graphing calculator is recommended.
The bird population follows the model $n(t)=\frac{5000}{0.5+24.5 e^{-0.044 t}}$ .
(a) Find the initial population. (b) Graph $n(t)$ . (c) What is the population limit as $t \to \infty$ ?

See Solution

Problem 16651

Bestimmen Sie den maximalen Gewinn und die Produktionsmenge für die Gewinnfunktion $G(x)=-2 x^{3}+12 x^{2}-6 x-20$ im Intervall $[0; 8]$ .

See Solution

Problem 16652

Given a fish population in a pond with the logistic growth model $P(t)=\frac{1200}{1+11 e^{-0.2 t}}$ , find:
(a) Initial fish count. (b) Population at 10, 20, and 30 years (round to nearest whole number). (c) Limit of $P(t)$ as $t \to \infty$ .
Does the graph confirm your results? Yes/No

See Solution

Problem 16653

Find the limit expression for the area under $f(x)=3+\sin ^{2}(x)$ from $0$ to $\pi$ using $A=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} f(x) \Delta x$ .

See Solution

Problem 16654

Given the logistic growth model:
$P(t)=\frac{1200}{1+11 e^{-0.2 t}}$
(a) What is the initial fish population?
(b) Find $P(10)$ , $P(20)$ , and $P(30)$ (round to whole numbers).
(c) What does $P(t)$ approach as $t \to \infty$ ?
Does the graph confirm your results? Yes/No

See Solution

Problem 16655

Finde die Stückzahl $x$ für maximalen Gewinn in der Funktion $G(x)=-0,5 x^{3}+60 x^{2}-1225 x-30.000$ im Intervall $[0 ; 90]$ . Was ist der Gewinn?

See Solution

Problem 16656

Bestimme die Ableitungen der Funktionen: a) $f(x)=3 x^{3}+2 x^{2}$ , b) $f(t)=t^{4}-t+1$ , c) $f(x)=(3+x)(3-x)$ , d) $f(x)=3 x^{3} \cdot 2 x^{2}$ , e) $f(x)=3 x^{2}-4 a^{2}$ , f) $f(a)=3 x^{2}-4 a^{2}$ , g) $f(x)=2(x-a)^{2}$ , h) $f(t)=t^{2}-(2-t)^{2}$ .

See Solution

Problem 16657

Skizzieren Sie die Graphen und bestimmen Sie die lokale Änderungsrate von $f$ an $x_0$ : a) $f(x)=0,5 x^2$ , $x_0=2$ b) $f(x)=1-x^2$ , $x_0=2$ c) $f(x)=\frac{1}{x}$ , $x_0=1$ .

See Solution

Problem 16658

Given $f^{\prime \prime}(x)=x^{2}(x-1)^{2}$ , determine the true statements about inflection points of $f(x)$ .

See Solution

Problem 16659

Snowboarder bewegt sich mit $s(t)=1,5 t^{2}$ . Berechne: a) Weg nach 1s und 5s, b) mittlere Geschwindigkeit in 5s, c) Momentangeschwindigkeit nach 5s.

See Solution

Problem 16660

Find the antiderivative of $f(x)=0.75x^{3}-1.25x^{2}-1$ .

See Solution

Problem 16661

Finde die lokalen Extrempunkte der Funktion $a(x)=0,25x^{4}-x^{3}+1$ durch Ableitungen.

See Solution

Problem 16662

Determine the max and min of $f(\theta)=1-\sin^{2}(\theta)$ on $\left[\frac{\pi}{4}, \pi\right]$ .

See Solution

Problem 16663

Find the function $f(x)$ if $f^{\prime}(x)=7x^{6}$ .

See Solution

Problem 16664

Find the peak concentration of the drug given by $C(t)=20 t \cdot e^{-0.3 t}$ . Options: $13$ , $51$ , $15$ , $25$ , $3$ mg/ml.

See Solution

Problem 16665

If $f(x)$ is continuous on $[a, b]$ with a unique critical point $c \in (a, b)$ and $f'(x)$ doesn't change sign, where does the global max occur?

See Solution

Problem 16666

Find the limit: $\lim _{n \rightarrow \infty}\left((\sin (\pi / 2))^{n}+(2 \sin (\pi / 4))^{n}+(1 / 10)^{n}\right)^{1 / n}$ .

See Solution

Problem 16667

Find the light intensity $I$ that maximizes the photosynthesis rate $P=\frac{80 I}{I^{2}+I+9}$ .

See Solution

Problem 16668

Approximate the integral $\int_{0}^{2}(3 x+7) d x$ using the Midpoint Rule with $n=4$ .

See Solution

Problem 16669

Maximize profit with revenue $R(q)=140 q-0.5 q^{2}$ and cost $C(q)=q^{2}+20 q+1050$ . What is $q$ ? Options: 260, 120, 40, 80.
Model virus spread with logistic equation $P(t)$ , starting with 10 infected, growth rate 1.45, max 4600. What is the equation? Options: $P(t)=\frac{4600}{\left(1+1.45 e^{-10 t}\right)}$ , $P(t)=\frac{4600}{1+10 e^{-1.45 t}}$ , $P(t)=\frac{10}{\left(1+459 e^{-1.45 t}\right)}$ , $P(t)=\frac{4600}{\left(1+459 e^{-1.45 t}\right)}$ .

See Solution

Problem 16670

Mosquito population grows uninhibitedly. Find $N(t)$ , size after 3 days, and time to reach 40,000 mosquitoes.
(a) $N(t)=N_0 \cdot e^{kt}$ (b) Start with 1000, after 1 day 1200, find size after 3 days. (c) Time to reach 40,000 mosquitoes?

See Solution

Problem 16671

Find the area between the $x$ -axis and $f(x)=12-3x^{2}$ on the interval $[0,4]$ . Does the graph cross the $x$ -axis? The area is $\square$ .

See Solution

Problem 16672

Find the derivative of $h(x)=\int_{1}^{e^{x}} 3 \ln (t) dt$ using the fundamental theorem of calculus.

See Solution

Problem 16673

Find the area between the $x$ -axis and $f(x) = 12 - 3x^2$ over the interval $[0,8]$ . Area = $\square$ .

See Solution

Problem 16674

A rumor spreads in 400 people given by $N(t)=\frac{400}{1+399 e^{-0.05 t}}$ . When does it spread fastest?
Select one: 15 hours 60 hours 400 hours 120 hours 180 hours
Medication concentration is $C(t)=3 t e^{-2 t}$ . When is peak concentration?
Select one: 2 hours 20 minutes 30 minutes 3 hours

See Solution

Problem 16675

Find $\int_{4} f(x) d x$ for the piecewise function $f(x)=\begin{cases} 8x+5 & x \leq 6 \\ -0.1x+1 & x > 6 \end{cases}$ given that $\int_{4}^{8} f(x) d x=0$ .

See Solution

Problem 16676

Mosquito population grows uninhibitedly. Find $\mathrm{N}(t)$ and the population after 3 days if $\mathrm{N}_0 = 1000$ and $\mathrm{N}(1) = 1200$ .

See Solution

Problem 16677

Find when the rumor spreads fastest for 400 people, given $N(t)=\frac{400}{1+399 e^{-0.05 t}}$ . Round to nearest hour.

See Solution

Problem 16678

Find $\int_{4}^{8} f(x) d x$ for the piecewise function: $f(x)=8x+5$ if $x \leq 6$ and $f(x)=-0.1x+1$ if $x>6$ .

See Solution

Problem 16679

Evaluate the integral from 0 to 1 of $(9 x^{e}+2 e^{x})$ .

See Solution

Problem 16680

Differentiate $y=\sqrt{1-x^{2}} \sin^{-1} x$ . Find $y'$ .

See Solution

Problem 16681

A bicyclist moves north at 60 ft/s and a jogger moves west at 15 ft/s. Find the distance change rate when 120 ft and 50 ft away.

See Solution

Problem 16682

Calculate the indefinite integral of the function: $\int(5+8 x) \, dx$

See Solution

Problem 16683

Set up the definite integral for total pollutant concentration in the lake after 4 years using $f(t)$ .

See Solution

Problem 16684

Evaluate the integral $\int_{e}^{e^{36}} \frac{d x}{x \sqrt{\ln (x)}}$ using substitution.

See Solution

Problem 16685

Bestimmen Sie die Punkte von $f$ , wo die Steigung $m$ ist: a) $f(x)=x^{2}+1 ; m=2$ b) $f(x)=x^{3}-3 x ; m=3$ c) $f(x)=\frac{1}{6} x^{4} ; m=\frac{16}{3}$

See Solution

Problem 16686

Bestimme die Punkte, wo $f$ die Steigung hat: a) $f(x)=x^{2}+1 ; m=2$ , b) $f(x)=x^{3}-3 x ; m=3$ , c) $f(x)=\frac{1}{6} x^{4} ; m=\frac{16}{3}$ .

See Solution

Problem 16687

Verify if $f(x)=x+\frac{1}{x}$ meets Rolle's Theorem on $\left[\frac{1}{3}, 3\right]$ and find all $c$ values.

See Solution

Problem 16688

Find critical numbers for the function $g(x)=\frac{x-1}{x^{2}-3x+3}$ .

See Solution

Problem 16689

Find the line where the function $g(x)=\frac{x^{2}+4 x+4}{x^{2}-1}$ has a horizontal asymptote. $y=1$

See Solution

Problem 16690

Find the general antiderivative of $f(x)=\frac{4 x^{2}-7 x+9}{x^{2}}$ , for $x>0$ .

See Solution

Problem 16691

Radium's half-life is 1690 years. If 70 grams are present, how much remains after 540 years? Use $A(t)=A_{0} e^{k t}$ .

See Solution

Problem 16692

Find intervals where $f(x)=x^{6} e^{-x}$ is increasing/decreasing and determine local max/min values.

See Solution

Problem 16693

Strontium-90 decays as $A(t)=A_{0} e^{-0.0244 t}$ . Given 400g, find: (a) decay rate, (b) amount after 30 yrs, (c) time for 100g, (d) half-life.

See Solution

Problem 16694

Die Höhe einer Kletterpflanze wird durch $h(t)=0,02 \cdot e^{k t}$ beschrieben. Bestimme:
a) Höhe zu Beginn ( $t=0$ ). b) $k$ bei $40 \mathrm{~cm}$ nach 6 Wochen. c) Höhe nach 9 Wochen. d) Zeitpunkt für 3 m Höhe. e) Zeitpunkt für 150 cm Wachstum in einer Woche. f) Zeitpunkt für Wachstumsrate von $1 \frac{m}{\text{Woche}}$ . g) Zeitpunkt für 3 m Höhe mit $k(t)=3,5-8,2 e^{-0,175 t}$ .

See Solution

Problem 16695

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

See Solution

Problem 16696

Calculate the integral $\int_{0}^{3}|x-1| dx$ . Choose the correct answer: A. 3, B. $\frac{9}{2}$ , C. $\frac{5}{2}$ , D. 5, E. $\frac{7}{2}$ .

See Solution

Problem 16697

Evaluate $\frac{g(a+h)-g(a)}{h}$ for the function $g(x)=5-x^{2}$ .

See Solution

Problem 16698

Ordnen Sie die Grenzwertterme den entsprechenden Grenzwerten zu:
$\lim _{x \rightarrow \infty} \frac{2 x}{x-2}, \quad \lim _{x \rightarrow 0} \frac{x^{2}-x}{x}, \quad \lim _{x \rightarrow 2} \frac{2 x^{2}-8}{x-2}, \quad \lim _{x \rightarrow 2}\left(x^{2}+\frac{1}{x}\right), \quad \lim _{x \rightarrow \infty} \frac{x^{2}+1}{x}$
Wählen Sie aus: $-1$ , 0, 2, 8, 4.5, $\infty$ .

See Solution

Problem 16699

Find the average rate of change of $k(x)=7 x^{3}+\frac{3}{x^{2}}$ from $x=-2$ to $x=1$ .

See Solution

Problem 16700

Find relative extrema or saddle points for the function $f(x, y)=3 x^{2}+2 y^{2}-6 x-4 y+16$ .

See Solution

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Calculus

Problem 16601

Find the xxx-values where the function f(x)=320x5−13x3f(x)=\frac{3}{20} x^{5}-13 x^{3}f(x)=203​x5−13x3 has an inflection point.

Problem 16602

Find the intervals where the function f(x)=16x4−19x2f(x)=\frac{1}{6} x^{4}-19 x^{2}f(x)=61​x4−19x2 is concave down.

Problem 16603

Find all xxx-values where the function f(x)=115x6+x5f(x)=\frac{1}{15} x^{6}+x^{5}f(x)=151​x6+x5 has an inflection point.

Problem 16604

Find the left Riemann sum for ∫23x3dx\int_{2}^{3} x^{3} dx∫23​x3dx using 5 equal subintervals. Round to the nearest thousandth.

Problem 16605

A cylinder's radius decreases at 2 cm/min, volume at 467 cm³/min. When radius is 4 cm and volume is 347 cm³, find height's rate of change using V=πr2hV=\pi r^{2} hV=πr2h. Round to three decimal places.

Problem 16606

A sphere's radius grows at 8 m/s. When the radius is 2 m, find the volume's rate of change using V=43πr3V=\frac{4}{3} \pi r^{3}V=34​πr3. Round to three decimals.

Problem 16607

Bestimmen Sie die Stammfunktion von fff: a) f(x)=4x3+2x2−1f(x)=4 x^{3}+2 x^{2}-1f(x)=4x3+2x2−1, b) f(x)=x5−4x3f(x)=x^{5}-4 x^{3}f(x)=x5−4x3, e) f(x)=0,24x7−5,4xf(x)=0,24 x^{7}-5,4 xf(x)=0,24x7−5,4x, f) f(x)=−25x4+33x2f(x)=-25 x^{4}+33 x^{2}f(x)=−25x4+33x2.

Problem 16608

Bestimmen Sie eine Stammfunktion für die folgenden Funktionen: a) f(x)=14x−2f(x)=\frac{1}{4} x^{-2}f(x)=41​x−2, b) f(x)=3x4f(x)=\frac{3}{x^{4}}f(x)=x43​, e) f(x)=−5x6f(x)=\frac{-5}{x^{6}}f(x)=x6−5​, f) f(x)=x2+2xx5f(x)=\frac{x^{2}+2 x}{x^{5}}f(x)=x5x2+2x​.

Problem 16609

Calculate the integral ∫π/4π/2sin⁡3αcos⁡α dα\int_{\pi / 4}^{\pi / 2} \sin^{3} \alpha \cos \alpha \, d\alpha∫π/4π/2​sin3αcosαdα.

Problem 16610

Find the gradient of the curve y=84x−5y=\frac{8}{4x-5}y=4x−58​ at x=2x=2x=2.

Problem 16611

Find the initial amount of Carbon14, the amount after 12208 years, and its half-life from A(t)=236e−0.00012tA(t)=236 e^{-0.00012 t}A(t)=236e−0.00012t.

Problem 16612

Die Höhe einer Eisenkugel wird durch h(t)=3,5⋅e−0,2t⋅sin⁡(5t)h(t)=3,5 \cdot e^{-0,2 t} \cdot \sin (5 t)h(t)=3,5⋅e−0,2t⋅sin(5t) beschrieben. a) Finde h(4)h(4)h(4) und v(4)v(4)v(4). b) Bestimme mit dem GTR, wann v(t)=3cmsv(t)=3 \frac{\mathrm{cm}}{\mathrm{s}}v(t)=3scm​.

Problem 16613

Berechnen Sie das Integral von f(x)=(x2+3x)2f(x)=(x^{2}+3 x)^{2}f(x)=(x2+3x)2.

Problem 16614

Hassan wants to know how much to invest at 2.3%2.3\%2.3% interest to have $1,130\$1,130$1,130 in 8 years. Calculate the amount needed.

Problem 16615

Problem 16616

Bestimmen Sie die Intervalle, in denen f(x)=23x3+x2−12x+1f(x)=\frac{2}{3} x^{3}+x^{2}-12 x+1f(x)=32​x3+x2−12x+1 streng monoton wächst.

Problem 16617

Ximena wants to invest to reach \$5,100 in 6 years at a 2.2% continuous interest rate. How much does she need to invest?

Problem 16618

Hakeem wants to invest for an account to reach \3,700in15yearsatacontinuousinterestrateof3,700 in 15 years at a continuous interest rate of 3,700in15yearsatacontinuousinterestrateof3.5\%$. How much?

Problem 16619

How much should London invest at a 5.7%5.7\%5.7% continuous interest to have $281,000\$281,000$281,000 in 19 years? Round to nearest \$10.

Problem 16620

Verify that the tangent lines of the bicorn curve at (0,4)(0,4)(0,4) and (0,4/3)(0,4/3)(0,4/3) are horizontal using implicit differentiation.

Problem 16621

Levi wants to know how much to invest at 3.4%3.4\%3.4% interest compounded continuously to reach \$620 in 16 years.

Problem 16622

Berechnen Sie das unbestimmte Integral der Funktion 0,3x−100,3 x^{-10}0,3x−10.

Problem 16623

Berechnen Sie das unbestimmte Integral von f(x)=3x4f(x)=\frac{3}{x^{4}}f(x)=x43​.

Problem 16624

A biologist has an 8824-gram sample of a radioactive substance. Find its mass after 3 hours with a decay rate of 9%9\%9% per hour. Round to the nearest tenth.

Problem 16625

Find the half-life of a radioactive substance with a decay rate of 1.9%1.9\%1.9% per day. Round to the nearest hundredth.

Problem 16626

Find the half-life of a substance with a decay rate of 1.9%1.9\%1.9% per day using the continuous exponential decay model.

Problem 16627

Calculate the integral from 0 to 10 of 100−x2\sqrt{100 - x^{2}}100−x2​ dx.

Problem 16628

Find the interest rate needed for Jason's \$16,000 to grow to \$26,000 in 7 years with continuous compounding.

Problem 16629

Which expression equals lim⁡n→∞∑i=1∞3in3n\lim _{n \rightarrow \infty} \sum_{i=1}^{\infty} \sqrt{\frac{3 i}{n}} \frac{3}{n}limn→∞​∑i=1∞​n3i​​n3​? Options include various integrals.

Problem 16630

Calculate the area under the curve x2x^{2}x2 from x=0x=0x=0 to x=3x=3x=3 using the definition of the definite integral.

Problem 16631

Given f(5)=−7f(5)=-7f(5)=−7, f′(5)=0f^{\prime}(5)=0f′(5)=0, and f′′(5)=1f^{\prime \prime}(5)=1f′′(5)=1, what can we conclude about fff at 5? A, B, C, or D?

Problem 16632

Find critical points of f(x)=xe−x2f(x)=x e^{-x^{2}}f(x)=xe−x2 and use the Second Derivative Test to classify them.

Problem 16633

Find intervals where the function is concave up or down, and identify points of inflection for f(x)=4x3+4x2−5x+9f(x)=4 x^{3}+4 x^{2}-5 x+9f(x)=4x3+4x2−5x+9.

Problem 16634

Determine where the function f(x)=3x−3f(x)=\frac{3}{x-3}f(x)=x−33​ is concave up, concave down, and find points of inflection.

Problem 16635

Find critical points of f(x)=xe−x2f(x) = x e^{-x^{2}}f(x)=xe−x2 and use the Second Derivative Test to classify them.

Problem 16636

Determine the intervals where f(x)=3x−3f(x)=\frac{3}{x-3}f(x)=x−33​ is concave up or down, and locate any points of inflection.

Problem 16637

Find R′′(x)R^{\prime \prime}(x)R′′(x) and the point of diminishing returns for R(x)=−x3+57x2+1000R(x)=-x^{3}+57x^{2}+1000R(x)=−x3+57x2+1000 where 0≤x≤240 \leq x \leq 240≤x≤24.

Problem 16638

Find the absolute max and min of f(x)=x3−6x2−63x+11f(x)=x^{3}-6x^{2}-63x+11f(x)=x3−6x2−63x+11 on intervals: (A) [−4,0][-4,0][−4,0], (B) [−1,8][-1,8][−1,8], (C) [−4,8][-4,8][−4,8]. Use -1000 for non-existent extrema.

Problem 16639

Given R(x)=−x3+57x2+1000R(x)=-x^{3}+57x^{2}+1000R(x)=−x3+57x2+1000 for 0≤x≤240 \leq x \leq 240≤x≤24, find R′′(x)R^{\prime \prime}(x)R′′(x) and the point of diminishing returns.

Problem 16640

Find critical numbers of f(x)=4−x4+xf(x)=\frac{4-x}{4+x}f(x)=4+x4−x​ and use the second derivative test for local max/min.

Find the $x$ -values where the function $f(x)=\frac{3}{20} x^{5}-13 x^{3}$ has an inflection point.

Find the intervals where the function $f(x)=\frac{1}{6} x^{4}-19 x^{2}$ is concave down.

Find all $x$ -values where the function $f(x)=\frac{1}{15} x^{6}+x^{5}$ has an inflection point.

Find the left Riemann sum for $\int_{2}^{3} x^{3} dx$ using 5 equal subintervals. Round to the nearest thousandth.

A cylinder's radius decreases at 2 cm/min, volume at 467 cm³/min. When radius is 4 cm and volume is 347 cm³, find height's rate of change using $V=\pi r^{2} h$ . Round to three decimal places.

A sphere's radius grows at 8 m/s. When the radius is 2 m, find the volume's rate of change using $V=\frac{4}{3} \pi r^{3}$ . Round to three decimals.

Bestimmen Sie die Stammfunktion von $f$ : a) $f(x)=4 x^{3}+2 x^{2}-1$ , b) $f(x)=x^{5}-4 x^{3}$ , e) $f(x)=0,24 x^{7}-5,4 x$ , f) $f(x)=-25 x^{4}+33 x^{2}$ .

Bestimmen Sie eine Stammfunktion für die folgenden Funktionen: a) $f(x)=\frac{1}{4} x^{-2}$ , b) $f(x)=\frac{3}{x^{4}}$ , e) $f(x)=\frac{-5}{x^{6}}$ , f) $f(x)=\frac{x^{2}+2 x}{x^{5}}$ .

Calculate the integral $\int_{\pi / 4}^{\pi / 2} \sin^{3} \alpha \cos \alpha \, d\alpha$ .

Find the gradient of the curve $y=\frac{8}{4x-5}$ at $x=2$ .

Find the initial amount of Carbon14, the amount after 12208 years, and its half-life from $A(t)=236 e^{-0.00012 t}$ .

Die Höhe einer Eisenkugel wird durch $h(t)=3,5 \cdot e^{-0,2 t} \cdot \sin (5 t)$ beschrieben.
a) Finde $h(4)$ und $v(4)$ . b) Bestimme mit dem GTR, wann $v(t)=3 \frac{\mathrm{cm}}{\mathrm{s}}$ .

Berechnen Sie das Integral von $f(x)=(x^{2}+3 x)^{2}$ .

Hassan wants to know how much to invest at $2.3\%$ interest to have $\$1,130$ in 8 years. Calculate the amount needed.

Bestimmen Sie die Intervalle, in denen $f(x)=\frac{2}{3} x^{3}+x^{2}-12 x+1$ streng monoton wächst.

Hakeem wants to invest for an account to reach \ $3,700 in 15 years at a continuous interest rate of$ 3.5\%$. How much?

How much should London invest at a $5.7\%$ continuous interest to have $\$281,000$ in 19 years? Round to nearest \$10.

Verify that the tangent lines of the bicorn curve at $(0,4)$ and $(0,4/3)$ are horizontal using implicit differentiation.

Levi wants to know how much to invest at $3.4\%$ interest compounded continuously to reach \$620 in 16 years.

Berechnen Sie das unbestimmte Integral der Funktion $0,3 x^{-10}$ .

Berechnen Sie das unbestimmte Integral von $f(x)=\frac{3}{x^{4}}$ .

A biologist has an 8824-gram sample of a radioactive substance. Find its mass after 3 hours with a decay rate of $9\%$ per hour. Round to the nearest tenth.

Find the half-life of a radioactive substance with a decay rate of $1.9\%$ per day. Round to the nearest hundredth.

Find the half-life of a substance with a decay rate of $1.9\%$ per day using the continuous exponential decay model.

Calculate the integral from 0 to 10 of $\sqrt{100 - x^{2}}$ dx.

Which expression equals $\lim _{n \rightarrow \infty} \sum_{i=1}^{\infty} \sqrt{\frac{3 i}{n}} \frac{3}{n}$ ? Options include various integrals.

Calculate the area under the curve $x^{2}$ from $x=0$ to $x=3$ using the definition of the definite integral.

Given $f(5)=-7$ , $f^{\prime}(5)=0$ , and $f^{\prime \prime}(5)=1$ , what can we conclude about $f$ at 5? A, B, C, or D?

Find critical points of $f(x)=x e^{-x^{2}}$ and use the Second Derivative Test to classify them.

Find intervals where the function is concave up or down, and identify points of inflection for $f(x)=4 x^{3}+4 x^{2}-5 x+9$ .

Determine where the function $f(x)=\frac{3}{x-3}$ is concave up, concave down, and find points of inflection.

Find critical points of $f(x) = x e^{-x^{2}}$ and use the Second Derivative Test to classify them.

Determine the intervals where $f(x)=\frac{3}{x-3}$ is concave up or down, and locate any points of inflection.

Find $R^{\prime \prime}(x)$ and the point of diminishing returns for $R(x)=-x^{3}+57x^{2}+1000$ where $0 \leq x \leq 24$ .

Find the absolute max and min of $f(x)=x^{3}-6x^{2}-63x+11$ on intervals: (A) $[-4,0]$ , (B) $[-1,8]$ , (C) $[-4,8]$ . Use -1000 for non-existent extrema.

Given $R(x)=-x^{3}+57x^{2}+1000$ for $0 \leq x \leq 24$ , find $R^{\prime \prime}(x)$ and the point of diminishing returns.

Find critical numbers of $f(x)=\frac{4-x}{4+x}$ and use the second derivative test for local max/min.

Find critical numbers of $f(x)=-2 x^{3}-3 x^{2}+12 x+3$ and use the second derivative test to classify them. What is $f''(x)$ ?

Find the second derivative of $f(x)=-2x^3-3x^2+12x+3$ and determine the critical numbers and local extrema.

Find the absolute max and min of $f(x)=-x^{4}+32 x^{2}-256$ on $[-5,5]$ . Max at $x=\square$ .

Find the production level $x$ that minimizes the average cost per wheel given $C(x)=0.03 x^{3}-4.5 x^{2}+174 x$ for $x \in (0,100]$ . $x=\square$

Find the absolute extrema of $f(x)=\frac{6+x}{6-x}$ on $[3,5]$ . Max at $x=\square$ .

Find the production level $x$ that minimizes the average cost function $C(x)=0.03 x^{3}-4.5 x^{2}+174 x$ for $0 \leq x \leq 100$ .

Welche Stammfunktion von $f$ hat bei $x=1$ den Wert 2? a) $f(x)=3 x^{2}$ b) $f(x)=x^{3}-2 x^{2}+1$

Find the point of diminishing returns for the function $R(x)=10,000-x^{3}+36 x^{2}+700 x$ in the interval $0 \leq x \leq 20$ .

Find the critical numbers of $g(x)=\frac{x-1}{x^{x-3x+3}}$ .

A graphing calculator is recommended.
The bird population follows the model $n(t)=\frac{5000}{0.5+24.5 e^{-0.044 t}}$ .
(a) Find the initial population. (b) Graph $n(t)$ . (c) What is the population limit as $t \to \infty$ ?

Bestimmen Sie den maximalen Gewinn und die Produktionsmenge für die Gewinnfunktion $G(x)=-2 x^{3}+12 x^{2}-6 x-20$ im Intervall $[0; 8]$ .

Find the limit expression for the area under $f(x)=3+\sin ^{2}(x)$ from $0$ to $\pi$ using $A=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} f(x) \Delta x$ .

Finde die Stückzahl $x$ für maximalen Gewinn in der Funktion $G(x)=-0,5 x^{3}+60 x^{2}-1225 x-30.000$ im Intervall $[0 ; 90]$ . Was ist der Gewinn?

Skizzieren Sie die Graphen und bestimmen Sie die lokale Änderungsrate von $f$ an $x_0$ : a) $f(x)=0,5 x^2$ , $x_0=2$ b) $f(x)=1-x^2$ , $x_0=2$ c) $f(x)=\frac{1}{x}$ , $x_0=1$ .

Given $f^{\prime \prime}(x)=x^{2}(x-1)^{2}$ , determine the true statements about inflection points of $f(x)$ .

Snowboarder bewegt sich mit $s(t)=1,5 t^{2}$ . Berechne: a) Weg nach 1s und 5s, b) mittlere Geschwindigkeit in 5s, c) Momentangeschwindigkeit nach 5s.

Find the antiderivative of $f(x)=0.75x^{3}-1.25x^{2}-1$ .

Finde die lokalen Extrempunkte der Funktion $a(x)=0,25x^{4}-x^{3}+1$ durch Ableitungen.

Determine the max and min of $f(\theta)=1-\sin^{2}(\theta)$ on $\left[\frac{\pi}{4}, \pi\right]$ .

Find the function $f(x)$ if $f^{\prime}(x)=7x^{6}$ .

Find the peak concentration of the drug given by $C(t)=20 t \cdot e^{-0.3 t}$ . Options: $13$ , $51$ , $15$ , $25$ , $3$ mg/ml.

If $f(x)$ is continuous on $[a, b]$ with a unique critical point $c \in (a, b)$ and $f'(x)$ doesn't change sign, where does the global max occur?

Find the limit: $\lim _{n \rightarrow \infty}\left((\sin (\pi / 2))^{n}+(2 \sin (\pi / 4))^{n}+(1 / 10)^{n}\right)^{1 / n}$ .

Find the light intensity $I$ that maximizes the photosynthesis rate $P=\frac{80 I}{I^{2}+I+9}$ .

Approximate the integral $\int_{0}^{2}(3 x+7) d x$ using the Midpoint Rule with $n=4$ .

Mosquito population grows uninhibitedly. Find $N(t)$ , size after 3 days, and time to reach 40,000 mosquitoes.
(a) $N(t)=N_0 \cdot e^{kt}$ (b) Start with 1000, after 1 day 1200, find size after 3 days. (c) Time to reach 40,000 mosquitoes?

Find the area between the $x$ -axis and $f(x)=12-3x^{2}$ on the interval $[0,4]$ . Does the graph cross the $x$ -axis? The area is $\square$ .

Find the derivative of $h(x)=\int_{1}^{e^{x}} 3 \ln (t) dt$ using the fundamental theorem of calculus.

Find the area between the $x$ -axis and $f(x) = 12 - 3x^2$ over the interval $[0,8]$ . Area = $\square$ .