Calculus

Problem 21901

Find the derivative $f'(0)$ for the function $f(x)=2 e^{x+1}+e^{8}$ .

See Solution

Problem 21902

Berechne den Grenzwert: $\lim _{x \rightarrow \infty} \frac{3.8 x^{4}-7.6}{x^{4}}$ und gib das Ergebnis auf zwei Dezimalstellen an!

See Solution

Problem 21903

Bestimme die erste Ableitung von $f(x)=(5 x^{3}-6)^{3}$ bei $x^{*}=1.7$ und gib das Ergebnis auf zwei Dezimalstellen an.

See Solution

Problem 21904

Evaluate the integral $\int_{2}^{5} 3 y(2-5 y^{2}) d y$ .

See Solution

Problem 21905

Evaluate the integral: $\int_{0}^{1} \frac{5 t^{4}}{\left(1+t^{5}\right)^{3}} d t$

See Solution

Problem 21906

Find the derivative of the integral from 2 to $x^2$ of $\cos t$ with respect to $t$ .

See Solution

Problem 21907

Evaluate the integral using the 2nd Fundamental Theorem of Calculus: $\int_{-2}^{3} 3 x^{2} d x$ .

See Solution

Problem 21908

Berechne den Grenzwert der Folge $a_{n} = -1 - \frac{3}{n} - \frac{5n}{n^{2}}$ mit Grenzwertsätzen.

See Solution

Problem 21909

Ibuprofen levels halve every 2.1 hours. Find the decay rate $k$ (to 3 decimal places) for the exponential model.

See Solution

Problem 21910

In an Exponential Decay Model with initial quantity 500 decreasing to 300 in 10 days, how long to reach 200?

See Solution

Problem 21911

A circular oil spill's area increases at $4 \mathrm{~km}^{2}/\mathrm{hr}$ . Find the radius growth rate when $r=16 \mathrm{~km}$ . Use $A=\pi r^{2}$ .

See Solution

Problem 21912

Find $\frac{d y}{d t}$ when $y=6$ , given $y^{2}=7 x+1$ and $\frac{d x}{d t}=-3$ .

See Solution

Problem 21913

Find the derivative of $f(x)=3 x^{4}-5 x^{3}+3$ . Choose the correct option: A, B, C, or D.

See Solution

Problem 21914

See Solution

Problem 21915

Find $\frac{d x}{d t}$ when $x=3$ given that $\frac{x}{y}=9$ and $\frac{d y}{d t}=-\frac{2}{3}$ .

See Solution

Problem 21916

Find the derivative of $f(x)=2(\sqrt[4]{x})$ for $x>0$ . Options: A. $\frac{1}{2} x^{5 / 4}$ B. $-\frac{5}{2} x^{-5 / 4}$ C. $\frac{1}{2} x^{-3 / 4}$ D. $6(\sqrt[3]{x})$

See Solution

Problem 21917

Find the derivative of $f(x)=\sqrt{x}(5x-3)+10x-6$ . What is $f^{\prime}(x)$ ? A. B. C. D.

See Solution

Problem 21918

Find the tangent line equation for $f(x)=\frac{\sqrt{x}}{5}$ at $x=4$ . Options: A. $y=\frac{x}{20}+\frac{1}{5}$ B. $y=13 x-16$ C. $y=-\frac{4 x}{25}+\frac{8}{5}$ D. $y=-39 x-80$

See Solution

Problem 21919

Find the marginal profit from the function $P(x)=1,080+30 x-3 x^{2}$ at $x=16$ . Options: A. -66 B. 384 C. 480 D. 1,080.

See Solution

Problem 21920

A ball is thrown to 50 feet, then bounces to $50/6$ , $50/36$ , etc. What is the total distance traveled when it stops?

See Solution

Problem 21921

Solve the differential equation: $\frac{dy}{dx} = \frac{x^2 + xy}{y^2}$ .

See Solution

Problem 21922

Find the derivative using the quotient rule for $y=\frac{x^{3}}{x-1}$ . Which is $y^{\prime}$ ? A, B, C, or D?

See Solution

Problem 21923

Find the difference quotient for $f(x)=x^{2}-7$ : calculate $\frac{f(x+h)-f(x)}{h}$ and simplify, where $h \neq 0$ .

See Solution

Problem 21924

Find the derivative of $y=(4 x+3)^{5}$ . Choose the correct option: A, B, C, or D.

See Solution

Problem 21925

Find the derivative of $y=4 e^{x^{2}}$ . A. $8 x e$ B. $8 x e^{4 x^{2}}$ C. $8 x e^{2 x}$ D. $8 x e^{x^{2}}$

See Solution

Problem 21926

Find the arc length of $y=\frac{1}{3}(x^{2}+2)^{3/2}$ on $[3,6]$ . The length is $\square$ .

See Solution

Problem 21927

Calculate the half-life of a radioactive substance that decays from $6.10 \mathrm{mg}$ to $1.40 \mathrm{mg}$ in 1.40 minutes. Round to 2 significant digits.

See Solution

Problem 21928

Find the derivative of $y=\ln(7+x^{2})$ . Choose from: A. $\frac{2 x}{x^{2}+7}$ B. $\frac{2}{x}$ C. $\frac{1}{2 x+7}$ D. $\frac{14}{x}$ .

See Solution

Problem 21929

Find the marginal average profit from selling $x$ cookbooks given $P(x)=(9x-4)(5x-2)$ . Choices: A. $45x-38$ , B. $45x-8$ , C. $45-\frac{8}{x^{2}}$ , D. $45-\frac{38}{x^{2}}$ .

See Solution

Problem 21930

Find the rate of change of $A$ w.r.t. $r$ in $A=3,000\left(1+\frac{r}{1200}\right)^{96}$ when $r=4$ .

See Solution

Problem 21931

Find the rate of change of sales given by $S(t)=230-60 e^{-0.8 t}$ at $t=3$ . Round to the nearest tenth.

See Solution

Problem 21932

Find local extrema using the first derivative test for $f(x)=\frac{x^{2}}{x^{2}+3}$ . Identify points and values.

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Problem 21933

Determine the largest interval where the function $f(x)=x^{3}-4x$ is decreasing. Options: A. $\left(-\infty,-\frac{2 \sqrt{3}}{3}\right)$ B. $\left(-\frac{2 \sqrt{3}}{3}, \frac{2 \sqrt{3}}{3}\right)$ C. $(-\infty, \infty)$ D. $\left(\frac{2 \sqrt{3}}{3}, \infty\right)$

See Solution

Problem 21934

Find all values of $c$ in $[1,4]$ for the function $f(x) = x + \frac{4}{x}$ using the Mean Value Theorem (MVT).

See Solution

Problem 21935

Find $f^{\prime \prime}(1)$ for $f(x)=\frac{3x-4}{4x-3}$ . Options: A. 7 B. 44 C. 32 D. -56

See Solution

Problem 21936

Find the function $f(x)$ if $f^{\prime \prime}(x)=7 x+5 \sin (x)$ , $f(0)=4$ , and $f^{\prime}(0)=2$ .

See Solution

Problem 21937

Calculate the tangential velocity of a satellite orbiting at a distance of $4.23 \times 10^{7} \mathrm{~m}$ from Earth's center.

See Solution

Problem 21938

Find the acceleration $a(t)$ from the position function $s(t)=2 t^{3}-4 t^{2}+4 t+6$ . Options: A, B, C, D.

See Solution

Problem 21939

Given the function $f(x)=5 x^{6}-4 x^{5}$ , find critical numbers, intervals of increase/decrease, local maxima/minima, concavity, inflection points, and asymptotes.

See Solution

Problem 21940

Find critical numbers for $f(x)=x^{3}-9 x^{2}+27 x+4$ and determine if they are max, min, or neither.

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Problem 21941

Find the year of maximum profit and its value for the function $f(x)=-28 x^{2}+840 x-4592$ .
(a) Derivative: $f^{\prime}(x)=\square$ ; Year: $\square$ ; (b) Maximum profit: \$\square million.

See Solution

Problem 21942

Find the tangent line equation for $f(x)=2 x^{2}+2 x$ at $x=4$ .

See Solution

Problem 21943

Find the slope of the tangent line for $f(x)=-4 x^{5}-4 x-2$ at $x=-1$ .

See Solution

Problem 21944

Find the production level for the lowest average cost per wheel using $C(x)=0.01 x^{3}-0.5 x^{2}+174 x ;(0,100]$ . What is the minimum average cost?

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Problem 21945

Find the tangent line equation of $f(x)=-3 x^{2}+4 x$ at $x=-3$ .

See Solution

Problem 21946

Given the function $f(x)$ with $f' > 0$ on $(-1,2)$ and $(4,\infty)$ , and $f' < 0$ on $(-\infty,-1)$ and $(2,4)$ , sketch its graph.

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Problem 21947

Find the slope of the tangent line for $f(x)=-2 x^{2}+3$ at $x=-4$ .

See Solution

Problem 21948

Find the derivative $f^{\prime}(x)$ , critical numbers using the quadratic formula, second derivative $f^{\prime \prime}(x)$ , and local extrema years.

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Problem 21949

A marshy area is contaminated. The element's percentage after $x$ months is $f(x)=\frac{x^{2}+25}{4 x}, 1 \leq x \leq 12$ . Find $f'(x)$ , when is the minimum, and what is the minimum percentage?

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Problem 21950

For the function $M(x)=-.015 x^{2}+1.34 x-7.7$ for $30 \leq x \leq 60$ , find max and min MPG and their speeds.

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Problem 21951

Berechne den Anstieg der Funktion $f(x) = \frac{1}{2} x^{2} - 2$ bei $x_0 = 2$ .

See Solution

Problem 21952

How much heat (in kJ) is needed to turn 1802 g of ice at $-15^{\circ} \mathrm{C}$ to steam at $146^{\circ} \mathrm{C}$ ?

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Problem 21953

Find the max and min of $f(x) = x^2 - 3x + 1$ on the interval $[-3, 2]$ .

See Solution

Problem 21954

Cells in a culture follow $P(t)=3000 e^{0.1 t}$ . Find initial cells, a differential equation, doubling time, and when there are 9,000 cells.

See Solution

Problem 21955

After $t$ hours, the cell count is given by $P(t) = 3000 e^{0.1 t}$ . Find: (a) initial cell count, (b) the differential equation for $P(t)$ .

See Solution

Problem 21956

Find the initial cell count, the differential equation for $P(t)=3000 e^{0.1 t}$ , and when the population doubles.

See Solution

Problem 21957

Find the consumer surplus at equilibrium for demand $D(x) = \frac{4}{\sqrt{x}}$ and supply $S(x) = \sqrt{x}$ .

See Solution

Problem 21958

Find the integral of $\frac{4}{\sqrt{x}}$ with respect to $x$ .

See Solution

Problem 21959

Cells in a culture grow as $P(t)=3000 e^{0.1 t}$ . Find: (a) initial cells, (b) differential equation, (c) doubling time, (d) when 9000 cells.

See Solution

Problem 21960

Find the local extrema of the function. Options: A. (-2,5),(0,0),(2,5) B. (-2,5),(2,5) C. (0,0) D. None.

See Solution

Problem 21961

Calculate the integral: $\int\left(-6+x^{2}+5 x^{-3}\right) dx$ and simplify the answer.

See Solution

Problem 21962

Find the function $f(x)$ if its derivative is $f^{\prime}(x)=-\frac{1}{2} \cdot (x-1)^{-\frac{3}{2}}$ .

See Solution

Problem 21963

Calculate the integral: $\int(4-6 x^{-3}) dx$ and provide the answer in simplest form.

See Solution

Problem 21964

Find the integral and express the result in simplest form: $\int(4 x-3) d x$

See Solution

Problem 21965

Calculate the integral of $5 x^{4}+2 x^{-3}$ and simplify the result.

See Solution

Problem 21966

Calculate the integral: $\int(4-3 x^{3}-5 x^{-2}) \, dx$ and simplify your answer.

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Problem 21967

Given the function $f(x)=\frac{3 x}{x^{2}-16}$ , find critical numbers, intervals of decrease, local maxima/minima, inflection points, concavity, and asymptotes.

See Solution

Problem 21968

Find the indefinite integral: $\int e^{x^{9}} x^{8} d x$ . Use $u=x^{9}$ and $\int e^{u} d u=e^{u}+C$ .

See Solution

Problem 21969

Find the local linear approximation $k$ for $g(1.2)$ given $\frac{d y}{d x} = y^{2} - x$ and $g(1) = 2$ .

See Solution

Problem 21970

Berechne die durchschnittliche Steigung $m$ der Funktion $f(x) = 2 \cdot x^{3} - 4$ im Intervall $[1, 5]$ .

See Solution

Problem 21971

Insect population is $P(t)=400 e^{0.03 t}$ . Find initial count, the differential equation, doubling time, and when $P(t)=2400$ .

See Solution

Problem 21972

Insect population is given by $P(t)=400 e^{0.01 t}$ . Find: (a) initial insects, (b) differential equation, (c) doubling time, (d) time for $P(t)=1200$ .

See Solution

Problem 21973

Bestimme die durchschnittliche Steigung $m$ von der Funktion $f(x)=0,5 x^{3}+x^{2}-1,5 x-2$ im Intervall $[-3; 2]$ .

See Solution

Problem 21974

Insect population is given by $P(t)=400 e^{0.03t}$ . Find initial count, differential equation, doubling time, and when $P(t)=2400$ .

See Solution

Problem 21975

Find where the function $f(x)=4(x+3)^{2}-1$ is increasing and decreasing. List the intervals.

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Problem 21976

Find the local linear approximation for the function $g$ at $x=1$ and determine if $k=5.6$ or $k=2.6$ is an over/underestimate for $g(1.2)$ .

See Solution

Problem 21977

Deposit \ $4000 at 7.5% interest compounded continuously. Find: (a) formula for$ A(t)$, (b) its differential equation, (c) balance after 6 years, (d) when balance hits \$6000, (e) growth rate at \$6000.

See Solution

Problem 21978

Insect population is $P(t)=400 e^{0.03 t}$ . Find: (a) initial count, (b) its differential equation, (c) doubling time, (d) time for 2400 insects.

See Solution

Problem 21979

Berechne die durchschnittliche Steigung der Funktion $f(x)=3x^{2}-2x$ im Intervall $[1; 3]$ .

See Solution

Problem 21980

Deposit \$4000 at 7.5% interest compounded continuously. Find:
(a) Formula for $A(t)$ after $t$ years. (b) Differential equation for $A(t)$ . (c) Balance after 6 years.

See Solution

Problem 21981

Deposit \$2000 at 5.5% interest compounded continuously. Find:
(a) Formula for $A(t)$ after $t$ years. (b) Differential equation for $A(t)$ . (c) Balance after 7 years. (d) Time to reach \$4000. (e) Growth rate when balance is \$4000.

See Solution

Problem 21982

Berechnen Sie die durchschnittliche Steigung von $f(x) = x^{4} + 3x^{2}$ auf dem Intervall $[-2; 4]$ .

See Solution

Problem 21983

A dog received 15 units of medicine. After 35 min, 9 units remained. Find the formula for $f(t)$ , where $f(t)$ is the amount left.

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Problem 21984

Cigarette demand: $p=100-0.5 q_{d}^{3}-q_{d}^{2}$ ; supply: $p=20+0.5 q_{s}^{3}$ . Find equilibrium, elasticity, revenue, and tax effects.

See Solution

Problem 21985

Find $\lim _{x \rightarrow \frac{\pi}{2}} \frac{3 \cos x}{2 x-\pi}$ . Choose from (A) $-\frac{3}{2}$ , (B) 0, (C) $+\frac{3}{2}$ , (D) nonexistent.

See Solution

Problem 21986

Berechnen Sie die durchschnittliche Steigung von $f(x)=-5x^{3}-2x^{2}+x$ im Intervall $[-7; 4]$ .

See Solution

Problem 21987

Find the area enclosed by the ellipse defined by $x=a \cos \theta$ , $y=b \sin \theta$ and the curve $x=t^{2}-2t$ , $y=\sqrt{t}$ .

See Solution

Problem 21988

Berechne die durchschnittliche Steigung der Funktion $f(x) = -4x^2 + 1$ im Intervall $[-1; 1]$ mit $m_{\text{durchschnitt}} = \frac{f(1) - f(-1)}{2}$ .

See Solution

Problem 21989

Evaluate the limits using L'Hôpital's Rule. Enter INF for $\infty$ , -INF for $-\infty$ , or DNE if the limit does not exist. a) $\lim _{x \rightarrow \pi / 2} 5 \tan x \cos x=5$ b) $\lim _{x \rightarrow \pi / 2} 4 \tan x \sin (2 x)=\square$

See Solution

Problem 21990

Calculate the distance driven between 1:00pm and 2:15pm given $v(t)=-5 t^{4}+44 t^{3}-144 t^{2}+170 t$ and start at 165 miles from home.

See Solution

Problem 21991

Evaluate the integral $A = \int_{0}^{(\sin x)}(0-F(y)) d y$ and simplify to find the result.

See Solution

Problem 21992

Gegeben ist die Funktion $f(x)=x^{3}+x$ mit $f^{\prime}(x)=3 x^{2}+1$ . Berechne $f^{\prime}(-1)$ , $f^{\prime}(2)$ , finde $x$ für $f^{\prime}(x)=4$ und wo die Steigung 13 ist.

See Solution

Problem 21993

Find the semicircle radius for a church window with perimeter $p$ to maximize area. Use $\pi$ for $\pi$ . Radius:

See Solution

Problem 21994

Berechnen Sie die durchschnittliche Steigung von $f(x) = -2x^{5} - 3x^{2}$ im Intervall $[3; 4]$ .

See Solution

Problem 21995

Dr. Acula wants to minimize metal use for cylindrical cans. Find height $h(r)$ for min surface area and $A_{\min}(V)$ for volume $V$ .

See Solution

Problem 21996

Berechne die durchschnittliche Steigung von $f(x) = 0,5 x^{4} - \frac{1}{3} x^{2} + 2$ im Intervall $[0; 5]$ .

See Solution

Problem 21997

Find the critical points, intervals of increase, local max/min, concavity, and inflection points for $f(x)=x^{3}-9x^{2}+15x+1$ .

See Solution

Problem 21998

Find the arc length of the curve $y=2 \ln x-\frac{x^{2}}{16}$ from $x=4$ to $x=24$ . Answer: $\square$

See Solution

Problem 21999

A radioactive substance starts at 33.8 mg and decays every 6 days.
(a) Write the formula $y=\square_{e} \mathbb{D}_{t}$ .
(b) Find the amount after 16 days, rounding to the nearest tenth.

See Solution

Problem 22000

Gegeben ist die Geschwindigkeits-Zeit-Funktion $v(t)=-\frac{1}{1000}(t^{4}-54 t^{3}+956 t^{2}-5754 t-3229)$ .
a) Finde die Anfangsgeschwindigkeit $v(0)$ . b) Bestimme die Zeitpunkte, an denen die Geschwindigkeit extremal ist. c) Finde die Wendepunkte und analysiere die Beschleunigung. d) Berechne $v(5)$ , $v(10)$ , $v(15)$ , $v(20)$ und skizziere den Graphen.

See Solution

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Calculus

Problem 21901

Find the derivative f′(0)f'(0)f′(0) for the function f(x)=2ex+1+e8f(x)=2 e^{x+1}+e^{8}f(x)=2ex+1+e8.

Problem 21902

Berechne den Grenzwert: lim⁡x→∞3.8x4−7.6x4\lim _{x \rightarrow \infty} \frac{3.8 x^{4}-7.6}{x^{4}}limx→∞​x43.8x4−7.6​ und gib das Ergebnis auf zwei Dezimalstellen an!

Problem 21903

Bestimme die erste Ableitung von f(x)=(5x3−6)3f(x)=(5 x^{3}-6)^{3}f(x)=(5x3−6)3 bei x∗=1.7x^{*}=1.7x∗=1.7 und gib das Ergebnis auf zwei Dezimalstellen an.

Problem 21904

Evaluate the integral ∫253y(2−5y2)dy\int_{2}^{5} 3 y(2-5 y^{2}) d y∫25​3y(2−5y2)dy.

Problem 21905

Evaluate the integral: ∫015t4(1+t5)3dt\int_{0}^{1} \frac{5 t^{4}}{\left(1+t^{5}\right)^{3}} d t∫01​(1+t5)35t4​dt

Problem 21906

Find the derivative of the integral from 2 to x2x^2x2 of cos⁡t\cos tcost with respect to ttt.

Problem 21907

Evaluate the integral using the 2nd Fundamental Theorem of Calculus: ∫−233x2dx\int_{-2}^{3} 3 x^{2} d x∫−23​3x2dx.

Problem 21908

Berechne den Grenzwert der Folge an=−1−3n−5nn2a_{n} = -1 - \frac{3}{n} - \frac{5n}{n^{2}}an​=−1−n3​−n25n​ mit Grenzwertsätzen.

Problem 21909

Ibuprofen levels halve every 2.1 hours. Find the decay rate kkk (to 3 decimal places) for the exponential model.

Problem 21910

In an Exponential Decay Model with initial quantity 500 decreasing to 300 in 10 days, how long to reach 200?

Problem 21911

A circular oil spill's area increases at 4 km2/hr4 \mathrm{~km}^{2}/\mathrm{hr}4 km2/hr. Find the radius growth rate when r=16 kmr=16 \mathrm{~km}r=16 km. Use A=πr2A=\pi r^{2}A=πr2.

Problem 21912

Find dydt\frac{d y}{d t}dtdy​ when y=6y=6y=6, given y2=7x+1y^{2}=7 x+1y2=7x+1 and dxdt=−3\frac{d x}{d t}=-3dtdx​=−3.

Problem 21913

Find the derivative of f(x)=3x4−5x3+3f(x)=3 x^{4}-5 x^{3}+3f(x)=3x4−5x3+3. Choose the correct option: A, B, C, or D.

Problem 21914

Problem 21915

Find dxdt\frac{d x}{d t}dtdx​ when x=3x=3x=3 given that xy=9\frac{x}{y}=9yx​=9 and dydt=−23\frac{d y}{d t}=-\frac{2}{3}dtdy​=−32​.

Problem 21916

Find the derivative of f(x)=2(x4)f(x)=2(\sqrt[4]{x})f(x)=2(4x​) for x>0x>0x>0. Options: A. 12x5/4\frac{1}{2} x^{5 / 4}21​x5/4 B. −52x−5/4-\frac{5}{2} x^{-5 / 4}−25​x−5/4 C. 12x−3/4\frac{1}{2} x^{-3 / 4}21​x−3/4 D. 6(x3)6(\sqrt[3]{x})6(3x​)

Problem 21917

Find the derivative of f(x)=x(5x−3)+10x−6f(x)=\sqrt{x}(5x-3)+10x-6f(x)=x​(5x−3)+10x−6. What is f′(x)f^{\prime}(x)f′(x)? A. B. C. D.

Problem 21918

Find the tangent line equation for f(x)=x5f(x)=\frac{\sqrt{x}}{5}f(x)=5x​​ at x=4x=4x=4. Options: A. y=x20+15y=\frac{x}{20}+\frac{1}{5}y=20x​+51​ B. y=13x−16y=13 x-16y=13x−16 C. y=−4x25+85y=-\frac{4 x}{25}+\frac{8}{5}y=−254x​+58​ D. y=−39x−80y=-39 x-80y=−39x−80

Problem 21919

Find the marginal profit from the function P(x)=1,080+30x−3x2P(x)=1,080+30 x-3 x^{2}P(x)=1,080+30x−3x2 at x=16x=16x=16. Options: A. -66 B. 384 C. 480 D. 1,080.

Problem 21920

A ball is thrown to 50 feet, then bounces to 50/650/650/6, 50/3650/3650/36, etc. What is the total distance traveled when it stops?

Problem 21921

Solve the differential equation: dydx=x2+xyy2\frac{dy}{dx} = \frac{x^2 + xy}{y^2}dxdy​=y2x2+xy​.

Problem 21922

Find the derivative using the quotient rule for y=x3x−1y=\frac{x^{3}}{x-1}y=x−1x3​. Which is y′y^{\prime}y′? A, B, C, or D?

Problem 21923

Find the difference quotient for f(x)=x2−7f(x)=x^{2}-7f(x)=x2−7: calculate f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ and simplify, where h≠0h \neq 0h=0.

Problem 21924

Find the derivative of y=(4x+3)5y=(4 x+3)^{5}y=(4x+3)5. Choose the correct option: A, B, C, or D.

Problem 21925

Find the derivative of y=4ex2y=4 e^{x^{2}}y=4ex2. A. 8xe8 x e8xe B. 8xe4x28 x e^{4 x^{2}}8xe4x2 C. 8xe2x8 x e^{2 x}8xe2x D. 8xex28 x e^{x^{2}}8xex2

Problem 21926

Find the arc length of y=13(x2+2)3/2y=\frac{1}{3}(x^{2}+2)^{3/2}y=31​(x2+2)3/2 on [3,6][3,6][3,6]. The length is □\square□.

Problem 21927

Calculate the half-life of a radioactive substance that decays from 6.10mg6.10 \mathrm{mg}6.10mg to 1.40mg1.40 \mathrm{mg}1.40mg in 1.40 minutes. Round to 2 significant digits.

Problem 21928

Find the derivative of y=ln⁡(7+x2)y=\ln(7+x^{2})y=ln(7+x2). Choose from: A. 2xx2+7\frac{2 x}{x^{2}+7}x2+72x​ B. 2x\frac{2}{x}x2​ C. 12x+7\frac{1}{2 x+7}2x+71​ D. 14x\frac{14}{x}x14​.

Problem 21929

Find the marginal average profit from selling xxx cookbooks given P(x)=(9x−4)(5x−2)P(x)=(9x-4)(5x-2)P(x)=(9x−4)(5x−2). Choices: A. 45x−3845x-3845x−38, B. 45x−845x-845x−8, C. 45−8x245-\frac{8}{x^{2}}45−x28​, D. 45−38x245-\frac{38}{x^{2}}45−x238​.

Problem 21930

Find the rate of change of AAA w.r.t. rrr in A=3,000(1+r1200)96A=3,000\left(1+\frac{r}{1200}\right)^{96}A=3,000(1+1200r​)96 when r=4r=4r=4.

Problem 21931

Find the rate of change of sales given by S(t)=230−60e−0.8tS(t)=230-60 e^{-0.8 t}S(t)=230−60e−0.8t at t=3t=3t=3. Round to the nearest tenth.

Problem 21932

Find local extrema using the first derivative test for f(x)=x2x2+3f(x)=\frac{x^{2}}{x^{2}+3}f(x)=x2+3x2​. Identify points and values.

Problem 21933

Problem 21934

Find all values of ccc in [1,4][1,4][1,4] for the function f(x)=x+4xf(x) = x + \frac{4}{x}f(x)=x+x4​ using the Mean Value Theorem (MVT).

Problem 21935

Find f′′(1)f^{\prime \prime}(1)f′′(1) for f(x)=3x−44x−3f(x)=\frac{3x-4}{4x-3}f(x)=4x−33x−4​. Options: A. 7 B. 44 C. 32 D. -56

Problem 21936

Find the function f(x)f(x)f(x) if f′′(x)=7x+5sin⁡(x)f^{\prime \prime}(x)=7 x+5 \sin (x)f′′(x)=7x+5sin(x), f(0)=4f(0)=4f(0)=4, and f′(0)=2f^{\prime}(0)=2f′(0)=2.

Problem 21937

Calculate the tangential velocity of a satellite orbiting at a distance of 4.23×107 m4.23 \times 10^{7} \mathrm{~m}4.23×107 m from Earth's center.

Problem 21938

Find the acceleration a(t)a(t)a(t) from the position function s(t)=2t3−4t2+4t+6s(t)=2 t^{3}-4 t^{2}+4 t+6s(t)=2t3−4t2+4t+6. Options: A, B, C, D.

Problem 21939

Given the function f(x)=5x6−4x5f(x)=5 x^{6}-4 x^{5}f(x)=5x6−4x5, find critical numbers, intervals of increase/decrease, local maxima/minima, concavity, inflection points, and asymptotes.

Problem 21940

Find critical numbers for f(x)=x3−9x2+27x+4f(x)=x^{3}-9 x^{2}+27 x+4f(x)=x3−9x2+27x+4 and determine if they are max, min, or neither.

Problem 21941

Find the derivative $f'(0)$ for the function $f(x)=2 e^{x+1}+e^{8}$ .

Berechne den Grenzwert: $\lim _{x \rightarrow \infty} \frac{3.8 x^{4}-7.6}{x^{4}}$ und gib das Ergebnis auf zwei Dezimalstellen an!

Bestimme die erste Ableitung von $f(x)=(5 x^{3}-6)^{3}$ bei $x^{*}=1.7$ und gib das Ergebnis auf zwei Dezimalstellen an.

Evaluate the integral $\int_{2}^{5} 3 y(2-5 y^{2}) d y$ .

Evaluate the integral: $\int_{0}^{1} \frac{5 t^{4}}{\left(1+t^{5}\right)^{3}} d t$

Find the derivative of the integral from 2 to $x^2$ of $\cos t$ with respect to $t$ .

Evaluate the integral using the 2nd Fundamental Theorem of Calculus: $\int_{-2}^{3} 3 x^{2} d x$ .

Berechne den Grenzwert der Folge $a_{n} = -1 - \frac{3}{n} - \frac{5n}{n^{2}}$ mit Grenzwertsätzen.

Ibuprofen levels halve every 2.1 hours. Find the decay rate $k$ (to 3 decimal places) for the exponential model.

A circular oil spill's area increases at $4 \mathrm{~km}^{2}/\mathrm{hr}$ . Find the radius growth rate when $r=16 \mathrm{~km}$ . Use $A=\pi r^{2}$ .

Find $\frac{d y}{d t}$ when $y=6$ , given $y^{2}=7 x+1$ and $\frac{d x}{d t}=-3$ .

Find the derivative of $f(x)=3 x^{4}-5 x^{3}+3$ . Choose the correct option: A, B, C, or D.

Find $\frac{d x}{d t}$ when $x=3$ given that $\frac{x}{y}=9$ and $\frac{d y}{d t}=-\frac{2}{3}$ .

Find the derivative of $f(x)=2(\sqrt[4]{x})$ for $x>0$ . Options: A. $\frac{1}{2} x^{5 / 4}$ B. $-\frac{5}{2} x^{-5 / 4}$ C. $\frac{1}{2} x^{-3 / 4}$ D. $6(\sqrt[3]{x})$

Find the derivative of $f(x)=\sqrt{x}(5x-3)+10x-6$ . What is $f^{\prime}(x)$ ? A. B. C. D.

Find the tangent line equation for $f(x)=\frac{\sqrt{x}}{5}$ at $x=4$ . Options: A. $y=\frac{x}{20}+\frac{1}{5}$ B. $y=13 x-16$ C. $y=-\frac{4 x}{25}+\frac{8}{5}$ D. $y=-39 x-80$

Find the marginal profit from the function $P(x)=1,080+30 x-3 x^{2}$ at $x=16$ . Options: A. -66 B. 384 C. 480 D. 1,080.

A ball is thrown to 50 feet, then bounces to $50/6$ , $50/36$ , etc. What is the total distance traveled when it stops?

Solve the differential equation: $\frac{dy}{dx} = \frac{x^2 + xy}{y^2}$ .

Find the derivative using the quotient rule for $y=\frac{x^{3}}{x-1}$ . Which is $y^{\prime}$ ? A, B, C, or D?

Find the difference quotient for $f(x)=x^{2}-7$ : calculate $\frac{f(x+h)-f(x)}{h}$ and simplify, where $h \neq 0$ .

Find the derivative of $y=(4 x+3)^{5}$ . Choose the correct option: A, B, C, or D.

Find the derivative of $y=4 e^{x^{2}}$ . A. $8 x e$ B. $8 x e^{4 x^{2}}$ C. $8 x e^{2 x}$ D. $8 x e^{x^{2}}$

Find the arc length of $y=\frac{1}{3}(x^{2}+2)^{3/2}$ on $[3,6]$ . The length is $\square$ .

Calculate the half-life of a radioactive substance that decays from $6.10 \mathrm{mg}$ to $1.40 \mathrm{mg}$ in 1.40 minutes. Round to 2 significant digits.

Find the derivative of $y=\ln(7+x^{2})$ . Choose from: A. $\frac{2 x}{x^{2}+7}$ B. $\frac{2}{x}$ C. $\frac{1}{2 x+7}$ D. $\frac{14}{x}$ .

Find the marginal average profit from selling $x$ cookbooks given $P(x)=(9x-4)(5x-2)$ . Choices: A. $45x-38$ , B. $45x-8$ , C. $45-\frac{8}{x^{2}}$ , D. $45-\frac{38}{x^{2}}$ .

Find the rate of change of $A$ w.r.t. $r$ in $A=3,000\left(1+\frac{r}{1200}\right)^{96}$ when $r=4$ .

Find the rate of change of sales given by $S(t)=230-60 e^{-0.8 t}$ at $t=3$ . Round to the nearest tenth.

Find local extrema using the first derivative test for $f(x)=\frac{x^{2}}{x^{2}+3}$ . Identify points and values.

Find all values of $c$ in $[1,4]$ for the function $f(x) = x + \frac{4}{x}$ using the Mean Value Theorem (MVT).

Find $f^{\prime \prime}(1)$ for $f(x)=\frac{3x-4}{4x-3}$ . Options: A. 7 B. 44 C. 32 D. -56

Find the function $f(x)$ if $f^{\prime \prime}(x)=7 x+5 \sin (x)$ , $f(0)=4$ , and $f^{\prime}(0)=2$ .

Calculate the tangential velocity of a satellite orbiting at a distance of $4.23 \times 10^{7} \mathrm{~m}$ from Earth's center.

Find the acceleration $a(t)$ from the position function $s(t)=2 t^{3}-4 t^{2}+4 t+6$ . Options: A, B, C, D.

Given the function $f(x)=5 x^{6}-4 x^{5}$ , find critical numbers, intervals of increase/decrease, local maxima/minima, concavity, inflection points, and asymptotes.

Find critical numbers for $f(x)=x^{3}-9 x^{2}+27 x+4$ and determine if they are max, min, or neither.

Find the year of maximum profit and its value for the function $f(x)=-28 x^{2}+840 x-4592$ .
(a) Derivative: $f^{\prime}(x)=\square$ ; Year: $\square$ ; (b) Maximum profit: \$\square million.

Find the tangent line equation for $f(x)=2 x^{2}+2 x$ at $x=4$ .

Find the slope of the tangent line for $f(x)=-4 x^{5}-4 x-2$ at $x=-1$ .

Find the production level for the lowest average cost per wheel using $C(x)=0.01 x^{3}-0.5 x^{2}+174 x ;(0,100]$ . What is the minimum average cost?

Find the tangent line equation of $f(x)=-3 x^{2}+4 x$ at $x=-3$ .

Given the function $f(x)$ with $f' > 0$ on $(-1,2)$ and $(4,\infty)$ , and $f' < 0$ on $(-\infty,-1)$ and $(2,4)$ , sketch its graph.

Find the slope of the tangent line for $f(x)=-2 x^{2}+3$ at $x=-4$ .

Find the derivative $f^{\prime}(x)$ , critical numbers using the quadratic formula, second derivative $f^{\prime \prime}(x)$ , and local extrema years.

A marshy area is contaminated. The element's percentage after $x$ months is $f(x)=\frac{x^{2}+25}{4 x}, 1 \leq x \leq 12$ . Find $f'(x)$ , when is the minimum, and what is the minimum percentage?

For the function $M(x)=-.015 x^{2}+1.34 x-7.7$ for $30 \leq x \leq 60$ , find max and min MPG and their speeds.

Berechne den Anstieg der Funktion $f(x) = \frac{1}{2} x^{2} - 2$ bei $x_0 = 2$ .

How much heat (in kJ) is needed to turn 1802 g of ice at $-15^{\circ} \mathrm{C}$ to steam at $146^{\circ} \mathrm{C}$ ?

Find the max and min of $f(x) = x^2 - 3x + 1$ on the interval $[-3, 2]$ .

Cells in a culture follow $P(t)=3000 e^{0.1 t}$ . Find initial cells, a differential equation, doubling time, and when there are 9,000 cells.

After $t$ hours, the cell count is given by $P(t) = 3000 e^{0.1 t}$ . Find: (a) initial cell count, (b) the differential equation for $P(t)$ .

Find the initial cell count, the differential equation for $P(t)=3000 e^{0.1 t}$ , and when the population doubles.

Find the consumer surplus at equilibrium for demand $D(x) = \frac{4}{\sqrt{x}}$ and supply $S(x) = \sqrt{x}$ .

Find the integral of $\frac{4}{\sqrt{x}}$ with respect to $x$ .

Cells in a culture grow as $P(t)=3000 e^{0.1 t}$ . Find: (a) initial cells, (b) differential equation, (c) doubling time, (d) when 9000 cells.

Calculate the integral: $\int\left(-6+x^{2}+5 x^{-3}\right) dx$ and simplify the answer.

Find the function $f(x)$ if its derivative is $f^{\prime}(x)=-\frac{1}{2} \cdot (x-1)^{-\frac{3}{2}}$ .

Calculate the integral: $\int(4-6 x^{-3}) dx$ and provide the answer in simplest form.

Find the integral and express the result in simplest form: $\int(4 x-3) d x$

Calculate the integral of $5 x^{4}+2 x^{-3}$ and simplify the result.

Calculate the integral: $\int(4-3 x^{3}-5 x^{-2}) \, dx$ and simplify your answer.

Given the function $f(x)=\frac{3 x}{x^{2}-16}$ , find critical numbers, intervals of decrease, local maxima/minima, inflection points, concavity, and asymptotes.

Find the indefinite integral: $\int e^{x^{9}} x^{8} d x$ . Use $u=x^{9}$ and $\int e^{u} d u=e^{u}+C$ .

Find the local linear approximation $k$ for $g(1.2)$ given $\frac{d y}{d x} = y^{2} - x$ and $g(1) = 2$ .

Berechne die durchschnittliche Steigung $m$ der Funktion $f(x) = 2 \cdot x^{3} - 4$ im Intervall $[1, 5]$ .

Insect population is $P(t)=400 e^{0.03 t}$ . Find initial count, the differential equation, doubling time, and when $P(t)=2400$ .

Insect population is given by $P(t)=400 e^{0.01 t}$ . Find: (a) initial insects, (b) differential equation, (c) doubling time, (d) time for $P(t)=1200$ .

Bestimme die durchschnittliche Steigung $m$ von der Funktion $f(x)=0,5 x^{3}+x^{2}-1,5 x-2$ im Intervall $[-3; 2]$ .

Insect population is given by $P(t)=400 e^{0.03t}$ . Find initial count, differential equation, doubling time, and when $P(t)=2400$ .