Calculus

Problem 12901

Find the derivative of $f(x) = \frac{\ln x}{x^{2}}$ .

See Solution

Problem 12902

Logistic growth function $f(t)=\frac{116,000}{1+5900 e^{-t}}$ describes flu cases over time.
a. Initial cases when $t=0$ ? b. Cases at $t=4$ weeks? c. Maximum cases possible?

See Solution

Problem 12903

Invest \ $14,000 at$ 7\% $compounded quarterly or$ 6.93\%$ compounded continuously. Which yields more in 3 years?

See Solution

Problem 12904

How long for $43\%$ of a material to remain if it decays at $0.084\%$ per year?

See Solution

Problem 12905

Find the average rate of change of $f(x)=x^{2}$ from $x=1$ to $x=3$ .

See Solution

Problem 12906

Calculate the average rate of change of $g(x)=-8x^{2}$ from $x=-2$ to $x=-1$ .

See Solution

Problem 12907

Find the critical numbers of $f(x)=-4 x^{5}+15 x^{4}+20 x^{3}-3$ and classify them with a graph.

See Solution

Problem 12908

Find the local max and min of the function $f(x)=4+9x+225x^{-1}$ , given max at $x=-5$ and min at $x=5$ .

See Solution

Problem 12909

Find the surface area of $z=e^{-(x^{2}+y^{2})}$ over the unit disc using cylindrical coordinates.

See Solution

Problem 12910

Berechnen Sie die Ableitungen von: $f(x)=(x^{2}-5)^{2}$ , $g(x)=(x^{2}+5x+1)^{2}$ , $h(x)=\sqrt{x}(x^{2}-3x)$ , $k(x)=(x^{2}-5)(x^{2}+5)$ .

See Solution

Problem 12911

Find the correct integral to compute the surface area of $z=e^{-(x^{2}+y^{2})}$ over the unit disc $x^{2}+y^{2} \leq 1$ .

See Solution

Problem 12912

Does $f(x)=3x^2-4x+2$ satisfy the Mean Value Theorem on $[0,2]$ ? Choose: Yes, No, or Not enough info.

See Solution

Problem 12913

Find the local maximum of the function $f(x)=2 x+4 x^{-1}$ and its value; local minimum occurs at $x=\sqrt{2}$ .

See Solution

Problem 12914

Find the local minimum and maximum of $f(x)=2 x^{3}-45 x^{2}+300 x+7$ and their values.

See Solution

Problem 12915

Find the local min and max of $f(x)=2x^3-45x^2+300x+5$ . Determine the $x$ values and function values using derivatives.

See Solution

Problem 12916

Find the integral for the surface area of $z=e^{-(x^{2}+y^{2})}$ over the unit disc $x^{2}+y^{2} \leq 1$ .

See Solution

Problem 12917

If $3 \leq f^{\prime}(x) \leq 4$ , find the min and max of $f(4) - f(1)$ .

See Solution

Problem 12918

Find the surface area for the parametrization $G(s,t) = (st, s-t, s+t)$ where $s^2 + t^2 \leq 4$ .

See Solution

Problem 12919

Find $\frac{d y}{d x}$ for the equation $-x y=3 y^{3}+2 x^{2}$ in terms of $x$ and $y$ .

See Solution

Problem 12920

Differentiate the function $3 x \ln (x)$ with respect to $x$ .

See Solution

Problem 12921

Given the function $f(x)=-3 x^{2}+4 x-2$ , find the critical number, intervals of increase/decrease, and local extrema type.

See Solution

Problem 12922

Find where the function $f(x)=6x + 2x^{-1}$ is decreasing, given it increases on $\left(-\infty,-\sqrt{\frac{1}{3}}\right) \cup\left(\sqrt{\frac{1}{3}}, \infty\right)$ .

See Solution

Problem 12923

Evaluate the Riemann sum $R_{6}$ for $f(x)=5x^{2}-4x$ using right endpoints with $n=6$ .

See Solution

Problem 12924

Berechnen Sie die 1. und 2. Ableitung von $f_{t}(x)=\frac{5}{2 t} x^{2}-t x+3 t$ ohne Taschenrechner.

See Solution

Problem 12925

Invest \ $15,000 at$ 6\% $compounded monthly or$ 5.93\%$ continuously. Which yields more in 5 years?

See Solution

Problem 12926

Invest \ $10,000 at$ 6\% $daily or$ 5.90\%$ continuously. Which gives more in 1 year?

See Solution

Problem 12927

Find critical points of $f(x)=2x^{3}-21x^{2}+36x+11$ and classify them using the second derivative test.

See Solution

Problem 12928

Find the first and second derivatives of $f_{t}(x)=\frac{5}{2 t} x^{2}-t x+3 t$ .

See Solution

Problem 12929

Calculate the derivative of $f(x) = 3x^2 + 4x + 2$ when $x = -1$ .

See Solution

Problem 12930

Një gur hidhet lart me $20 \mathrm{~m/s}$ nga $24 \mathrm{~m}$ . Gjeni shpejtësinë kur godet tokën dhe kohën e udhëtimit.

See Solution

Problem 12931

Find a point $c$ that meets the MVT for $y(x)=8\sqrt{x}$ on $[0,23]$ .

See Solution

Problem 12932

Find the limit of $\frac{f(1+h)-f(1)}{h}$ for $f(x)=x^{3}+2x$ as $h$ approaches 0.

See Solution

Problem 12933

Find the antiderivative $F$ of $f(v)=\frac{2}{5} \sec v \tan v$ with $F(0)=3$ , where $-\frac{\pi}{2}<v<\frac{\pi}{2}$ .

See Solution

Problem 12934

Find the difference quotient for $f(x)=x^{3}+2x$ at $a=1$ : $\frac{f(1+h)-f(1)}{h}$ .

See Solution

Problem 12935

Analyze the vertical motion of a softball launched at $31 \mathrm{~m/s}$ under gravity $g=-9.8 \mathrm{~m/s}^2$ . Find its velocity, position, time to highest point, height, and time to hit the ground.

See Solution

Problem 12936

Find the derivative of $f(x)=x^{3}+2x$ and calculate it at $x=1$ .

See Solution

Problem 12937

Find the tangent line equation for $g(x)$ at $x=3$ , given $g(3)=g'(3)=4$ . Format: $y=mx+b$ .

See Solution

Problem 12938

Find $\lim_{x \to -3^-} f(x)$ , $\lim_{x \to 3^+} f(x)$ , and $\lim_{x \to +\infty} f(x)$ for $f(x)=\frac{3+x}{9-x^2}$ .

See Solution

Problem 12939

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{5}$ .

See Solution

Problem 12940

Find the tangent line equation for $f(x)=x^{2}$ at $x=-1$ . The equation is $\square$ .

See Solution

Problem 12941

Find the derivative $f^{\prime}(x)$ for the constant function $f(x)=-4$ .

See Solution

Problem 12942

Find the derivative $g^{\prime}(x)$ of the function $g(x)=x^{4/9}$ .

See Solution

Problem 12943

Find the derivative $\frac{dy}{dx}$ for the function $y = x^{-7}$ .

See Solution

Problem 12944

Find the derivative of $y=\frac{1}{x^{10}}$ .

See Solution

Problem 12945

Find the derivative $f^{\prime}(x)$ for the function $f(x)=6 x^{4}$ .

See Solution

Problem 12946

Find a point $c$ that meets the MVT for $y(x)=5 x^{2/3}$ on the interval $[0, 64]$ .

See Solution

Problem 12947

Find $f^{\prime}(5)$ for $f(x)=1 / \sqrt{x}$ using the expression $\frac{f(5+h)-f(5)}{h}$ .

See Solution

Problem 12948

Find the derivative $f^{\prime}(x)$ of the function $f(x)=3 x^{5}$ .

See Solution

Problem 12949

Find the absolute minimum and maximum of $f(x)=x^{4}-18 x^{2}+7$ on the interval $-2 \leq x \leq 7$ .

See Solution

Problem 12950

Find the derivative of $\frac{x^{8}}{32}$ with respect to $x$ . What is $\frac{d}{d x}\left(\frac{x^{8}}{32}\right)$ ?

See Solution

Problem 12951

Find the derivative $y^{\prime}$ for $y=\frac{-5}{\sqrt[3]{x}}$ . What is $y^{\prime}$ ?

See Solution

Problem 12952

Calculate the derivative of the function: $\frac{d}{d x}(-3 x-7) = \square$

See Solution

Problem 12953

Find the derivative $f^{\prime}(t)$ for the function $f(t)=3 t^{2}+6 t+9$ .

See Solution

Problem 12954

Find the derivative $y^{\prime}$ of the function $y=8 x^{-6}-4 x^{-1}$ .

See Solution

Problem 12955

Find the derivative $f^{\prime}(x)$ of the function $f(x)=e^{x}+5x-2\ln x$ .

See Solution

Problem 12956

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-11 \ln x + 9 x^{2} - 9$ .

See Solution

Problem 12957

Find the derivative $h^{\prime}(t)$ for the function $h(t)=5.1-6.9 t+0.7 t^{3}$ . What is $h^{\prime}(t)$ ?

See Solution

Problem 12958

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\ln x+2 e^{x}-3 x^{2}$ .

See Solution

Problem 12959

A penny falls from a $425 \mathrm{~m}$ high skyscraper. How long does it take to reach the ground?

See Solution

Problem 12960

Berechne den Differenzenquotienten der Funktion $f(x)=\frac{1}{2 x}$ im Intervall $[1 ; 2]$ und $[1 ; 1,5]$ .

See Solution

Problem 12961

Given $f(x)=3 x^{4}-8 x^{3}+9$ , find $f'(x)$ , slope at $x=-1$ , tangent line at $x=-1$ , and where tangent is horizontal.

See Solution

Problem 12962

Find the marginal-cost function for $C(x)=0.04 x^{3}+0.6 x^{2}+30 x+140$ at $x=800$ .

See Solution

Problem 12963

Graph the drug concentration $c(t)=\frac{3t}{t^2+1}$ .
(a) What is the maximum concentration? (b) What happens to the concentration over time? (c) When does it drop below $0.3 \mathrm{mg}/L$ ? $\mathrm{t}=\square \mathrm{hr}$

See Solution

Problem 12964

Ein Radfahrer hat die Geschwindigkeit $v(t)=-0,006 t^{3}+0,21 t^{2}-1,92 t+10$ .
a) Erkläre die Fläche unter dem Geschwindigkeitsdiagramm.
b) Finde die maximale Geschwindigkeit zwischen 5 und 20 Sekunden in $\mathrm{km/h}$ .
c) Berechne die durchschnittliche Geschwindigkeit in den ersten 10 Sekunden.

See Solution

Problem 12965

Find (a) the rate of change of $y$ with respect to $x$ , (b) the relative rate of change, and (c) at $x=6$ , find $y=x^{2}+x-2$ .

See Solution

Problem 12966

Construisez la table des valeurs de $f(x)=\frac{x}{\sqrt{1+3x}-1}$ près de 0 et conjecturez la limite.

See Solution

Problem 12967

Find the minimum slope of the tangent line to the graph of $f(x)=6+2x-4x^{2}+\frac{1}{3}x^{3}$ .

See Solution

Problem 12968

The drug concentration $C=15 \cdot\left(\frac{t}{60+t^{2}}\right)$ ; find $C$ after 1 hour, time to reach 0.5\%, and end behavior.

See Solution

Problem 12969

Find the maximum area of a rectangle with corners on the axes and one on the graph of $y=\frac{6}{x^{2}+1}$ .

See Solution

Problem 12970

A circle's radius increases by $5 \mathrm{~mm/s}$ . Find the area change rate when the radius is $29 \mathrm{~mm}$ .

See Solution

Problem 12971

Find the Riemann sum for $f(x)=e^{x}-1$ over $[0, 2]$ with $n=4$ using midpoints, accurate to six decimal places.

See Solution

Problem 12972

Find the critical points of $y(x)=x(x-7)^{5}$ . Provide your answer as a comma-separated list.

See Solution

Problem 12973

Find critical points $c$ and intervals where $g(x)=3 x^{5}+3 x^{3}+3 x$ is increasing or decreasing. Identify local min/max.

See Solution

Problem 12974

Find critical points and intervals of increase/decrease for $f(x)=6 x^{4}+9 x^{3}$ . Provide a list of results.

See Solution

Problem 12975

Find the Riemann sum for $f(x)=6 \sin x$ over $[0, \frac{3\pi}{2}]$ using 6 right endpoints. Round to six decimal places.

See Solution

Problem 12976

Find critical points and intervals of increase/decrease for $f(x)=7 e^{-x} \cos (x)$ on $[-\frac{\pi}{2}, \frac{\pi}{2}]$ . Use First Derivative Test.

See Solution

Problem 12977

Hochbehälter: Gegeben ist $h(t)=\frac{1}{8} t^{2}-2 t+8$ . Bestimme a) Graph, b) Leerzeit, c) Halbleer und $1/4$ voll, d) Sinkgeschwindigkeit.

See Solution

Problem 12978

Find the rates of change of revenue $R(x)=4x$ and cost $C(x)=0.01x^2+0.4x+5$ at $x=25$ when $\frac{dx}{dt}=7$ .

See Solution

Problem 12979

Zeichnen Sie den Graphen der Funktionen $f$ und berechnen Sie die Fläche zwischen dem Graphen und der $x$ -Achse für: a) $f(x)=x^{3}-4 x^{2}+4 x$ , b) $f(x)=0,5 x^{3}-4,5 x^{2}$ , c) $f(x)=6 x^{3}(x^{2}-1)$ , d) $f(x)=\frac{5}{9} x^{4}-5 x^{2}$ , e) $f(x)=6 x(x^{2}-1)^{2}$ , f) $f(x)=-\frac{1}{6} x^{4}+\frac{1}{2} x^{2}+\frac{2}{3}$ .

See Solution

Problem 12980

Estimate the coffee's temperature after 10 minutes if it cools at a rate of $f^{\prime}(t)=-6(0.6)^{t}$ .

See Solution

Problem 12981

Show that the polynomial $P(x)=3x^2-2x-6$ has a real zero between 1 and 2 using the Intermediate Value Theorem.

See Solution

Problem 12982

Find the total cost for producing 25 units if $C^{\prime}(q)=2 q^{2}-13 q+68$ and $C(0)=440$ .

See Solution

Problem 12983

Analyze the limits as $x \rightarrow 0$ for these functions: (a) $U(x)=0$ if $x<0$ , $1$ if $x \geq 0$ ; (b) $g(x)=1/x$ if $x \neq 0$ , $0$ if $x=0$ ; (c) $f(x)=0$ if $x \leq 0$ , $\sin(1/x)$ if $x>0$ .

See Solution

Problem 12984

Leite die Funktionen mit der Produktregel ab:
1) $f(x)=-x^{4} \cdot e^{x}$
2) $f(x)=\sqrt{x} \cdot e^{x}$

See Solution

Problem 12985

A bike manufacturer has a marginal cost function $C^{\prime}(q)=\frac{700}{0.7 q+4}$ .
a) If fixed costs are \$2200, find total cost for 35 bikes.
b) If each bike sells for \$150, what is the profit/loss on the first 35 bikes?
c) What is the marginal profit for bike 36?

See Solution

Problem 12986

Find the derivative of $f(x)=\frac{1}{\sqrt{x}}$ at $x=4$ .

See Solution

Problem 12987

Kostenfunktion $K(x)$ : $K(x)=0,0003 x^{3}-0,017 x^{2}+0,4 x+40$ , Nachfrage $p_{N}(x)=1,5$ . Bestimme maximalen Erlös, Gewinnfunktion $G$ und Cournot-Punkt.

See Solution

Problem 12988

Find the derivative of $f(x)=(x^{2}+x+1)(x^{3}+1)$ using the product rule.

See Solution

Problem 12989

Bestimmen Sie die Ableitung von $f(x)=e^{x} \cdot\left(\frac{1}{x^{3}}+x\right)$ mit der Produktregel.

See Solution

Problem 12990

Differentiate $f(x)=(3 x+2)^{2} \cdot e^{x}$ using product and chain rules.

See Solution

Problem 12991

Bestimmen Sie die Ableitung von $f(x) = \sqrt{x} \cdot e^{-x^{2}}$ mit Produkt- und Kettenregel.

See Solution

Problem 12992

A particle moves with $s=f(t)=t^{3}-9t^{2}+24t$ . Find velocity $v(t)$ , when it's at rest, moving positively, and acceleration.

See Solution

Problem 12993

Find the derivative of $f(x)=(x^{2}+2)^{2} \cdot e^{-4 x}$ using product and chain rule.

See Solution

Problem 12994

Find the derivative of $f(x)=\left(x^{2}+2\right)^{2} \cdot e^{-4 x}$ using product and chain rules.

See Solution

Problem 12995

Berechne den Differenzenquotienten für $f(x)=\frac{1}{2 x}$ im Intervall $[1 ; 2]$ und $[1 ; 1,5]$ . Finde geometrisch den Differenzenquotienten im Intervall $[2 ; 4]$ und $[0 ; 2]$ . Bestimme ein Intervall, wo der Differenzenquotient 0 ist.

See Solution

Problem 12996

Find the marginal-revenue function for the demand equation $p=\frac{5 q+6}{9 q+8}$ , where revenue $=p q$ .

See Solution

Problem 12997

Find $\lambda$ so that $\rho(x)=\lambda x^{2} e^{-x}$ is a probability density function on $[0,+\infty)$ . Round to three decimal places.

See Solution

Problem 12998

Berechnen Sie den Differenzenquotienten der Funktion $f(x)=\frac{1}{2 x}$ für die Intervalle $[1; 2]$ und $[1; 1,5]$ .

See Solution

Problem 12999

If a hammer falls from a roof twice as tall, how does its kinetic energy change? Explain using $KE = \frac{1}{2}mv^2$ .

See Solution

Problem 13000

Find the marginal-revenue function from the demand equation $p=\frac{6 q+7}{7 q+4}$ , where revenue $=p q$ .

See Solution

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Calculus

Problem 12901

Find the derivative of f(x)=ln⁡xx2f(x) = \frac{\ln x}{x^{2}}f(x)=x2lnx​.

Problem 12902

Logistic growth function f(t)=116,0001+5900e−tf(t)=\frac{116,000}{1+5900 e^{-t}}f(t)=1+5900e−t116,000​ describes flu cases over time. a. Initial cases when t=0t=0t=0? b. Cases at t=4t=4t=4 weeks? c. Maximum cases possible?

Problem 12903

Invest \14,000at14,000 at 14,000at7\%compoundedquarterlyor compounded quarterly or compoundedquarterlyor6.93\%$ compounded continuously. Which yields more in 3 years?

Problem 12904

How long for 43%43\%43% of a material to remain if it decays at 0.084%0.084\%0.084% per year?

Problem 12905

Find the average rate of change of f(x)=x2f(x)=x^{2}f(x)=x2 from x=1x=1x=1 to x=3x=3x=3.

Problem 12906

Calculate the average rate of change of g(x)=−8x2g(x)=-8x^{2}g(x)=−8x2 from x=−2x=-2x=−2 to x=−1x=-1x=−1.

Problem 12907

Find the critical numbers of f(x)=−4x5+15x4+20x3−3f(x)=-4 x^{5}+15 x^{4}+20 x^{3}-3f(x)=−4x5+15x4+20x3−3 and classify them with a graph.

Problem 12908

Find the local max and min of the function f(x)=4+9x+225x−1f(x)=4+9x+225x^{-1}f(x)=4+9x+225x−1, given max at x=−5x=-5x=−5 and min at x=5x=5x=5.

Problem 12909

Find the surface area of z=e−(x2+y2)z=e^{-(x^{2}+y^{2})}z=e−(x2+y2) over the unit disc using cylindrical coordinates.

Problem 12910

Berechnen Sie die Ableitungen von: f(x)=(x2−5)2f(x)=(x^{2}-5)^{2}f(x)=(x2−5)2, g(x)=(x2+5x+1)2g(x)=(x^{2}+5x+1)^{2}g(x)=(x2+5x+1)2, h(x)=x(x2−3x)h(x)=\sqrt{x}(x^{2}-3x)h(x)=x​(x2−3x), k(x)=(x2−5)(x2+5)k(x)=(x^{2}-5)(x^{2}+5)k(x)=(x2−5)(x2+5).

Problem 12911

Find the correct integral to compute the surface area of z=e−(x2+y2)z=e^{-(x^{2}+y^{2})}z=e−(x2+y2) over the unit disc x2+y2≤1x^{2}+y^{2} \leq 1x2+y2≤1.

Problem 12912

Does f(x)=3x2−4x+2f(x)=3x^2-4x+2f(x)=3x2−4x+2 satisfy the Mean Value Theorem on [0,2][0,2][0,2]? Choose: Yes, No, or Not enough info.

Problem 12913

Find the local maximum of the function f(x)=2x+4x−1f(x)=2 x+4 x^{-1}f(x)=2x+4x−1 and its value; local minimum occurs at x=2x=\sqrt{2}x=2​.

Problem 12914

Find the local minimum and maximum of f(x)=2x3−45x2+300x+7f(x)=2 x^{3}-45 x^{2}+300 x+7f(x)=2x3−45x2+300x+7 and their values.

Problem 12915

Find the local min and max of f(x)=2x3−45x2+300x+5f(x)=2x^3-45x^2+300x+5f(x)=2x3−45x2+300x+5. Determine the xxx values and function values using derivatives.

Problem 12916

Find the integral for the surface area of z=e−(x2+y2)z=e^{-(x^{2}+y^{2})}z=e−(x2+y2) over the unit disc x2+y2≤1x^{2}+y^{2} \leq 1x2+y2≤1.

Problem 12917

If 3≤f′(x)≤43 \leq f^{\prime}(x) \leq 43≤f′(x)≤4, find the min and max of f(4)−f(1)f(4) - f(1)f(4)−f(1).

Problem 12918

Find the surface area for the parametrization G(s,t)=(st,s−t,s+t)G(s,t) = (st, s-t, s+t)G(s,t)=(st,s−t,s+t) where s2+t2≤4s^2 + t^2 \leq 4s2+t2≤4.

Problem 12919

Find dydx\frac{d y}{d x}dxdy​ for the equation −xy=3y3+2x2-x y=3 y^{3}+2 x^{2}−xy=3y3+2x2 in terms of xxx and yyy.

Problem 12920

Differentiate the function 3xln⁡(x)3 x \ln (x)3xln(x) with respect to xxx.

Problem 12921

Given the function f(x)=−3x2+4x−2f(x)=-3 x^{2}+4 x-2f(x)=−3x2+4x−2, find the critical number, intervals of increase/decrease, and local extrema type.

Problem 12922

Find where the function f(x)=6x+2x−1f(x)=6x + 2x^{-1}f(x)=6x+2x−1 is decreasing, given it increases on (−∞,−13)∪(13,∞)\left(-\infty,-\sqrt{\frac{1}{3}}\right) \cup\left(\sqrt{\frac{1}{3}}, \infty\right)(−∞,−31​​)∪(31​​,∞).

Problem 12923

Evaluate the Riemann sum R6R_{6}R6​ for f(x)=5x2−4xf(x)=5x^{2}-4xf(x)=5x2−4x using right endpoints with n=6n=6n=6.

Problem 12924

Berechnen Sie die 1. und 2. Ableitung von ft(x)=52tx2−tx+3tf_{t}(x)=\frac{5}{2 t} x^{2}-t x+3 tft​(x)=2t5​x2−tx+3t ohne Taschenrechner.

Problem 12925

Invest \15,000at15,000 at 15,000at6\%compoundedmonthlyor compounded monthly or compoundedmonthlyor5.93\%$ continuously. Which yields more in 5 years?

Problem 12926

Invest \10,000at10,000 at 10,000at6\%dailyor daily or dailyor5.90\%$ continuously. Which gives more in 1 year?

Problem 12927

Find critical points of f(x)=2x3−21x2+36x+11f(x)=2x^{3}-21x^{2}+36x+11f(x)=2x3−21x2+36x+11 and classify them using the second derivative test.

Problem 12928

Find the first and second derivatives of ft(x)=52tx2−tx+3tf_{t}(x)=\frac{5}{2 t} x^{2}-t x+3 tft​(x)=2t5​x2−tx+3t.

Problem 12929

Calculate the derivative of f(x)=3x2+4x+2f(x) = 3x^2 + 4x + 2f(x)=3x2+4x+2 when x=−1x = -1x=−1.

Problem 12930

Një gur hidhet lart me 20 m/s20 \mathrm{~m/s}20 m/s nga 24 m24 \mathrm{~m}24 m. Gjeni shpejtësinë kur godet tokën dhe kohën e udhëtimit.

Problem 12931

Find a point ccc that meets the MVT for y(x)=8xy(x)=8\sqrt{x}y(x)=8x​ on [0,23][0,23][0,23].

Problem 12932

Find the limit of f(1+h)−f(1)h\frac{f(1+h)-f(1)}{h}hf(1+h)−f(1)​ for f(x)=x3+2xf(x)=x^{3}+2xf(x)=x3+2x as hhh approaches 0.

Problem 12933

Find the antiderivative FFF of f(v)=25sec⁡vtan⁡vf(v)=\frac{2}{5} \sec v \tan vf(v)=52​secvtanv with F(0)=3F(0)=3F(0)=3, where −π2<v<π2-\frac{\pi}{2}<v<\frac{\pi}{2}−2π​<v<2π​.

Problem 12934

Find the difference quotient for f(x)=x3+2xf(x)=x^{3}+2xf(x)=x3+2x at a=1a=1a=1: f(1+h)−f(1)h\frac{f(1+h)-f(1)}{h}hf(1+h)−f(1)​.

Problem 12935

Analyze the vertical motion of a softball launched at 31 m/s31 \mathrm{~m/s}31 m/s under gravity g=−9.8 m/s2g=-9.8 \mathrm{~m/s}^2g=−9.8 m/s2. Find its velocity, position, time to highest point, height, and time to hit the ground.

Problem 12936

Find the derivative of f(x)=x3+2xf(x)=x^{3}+2xf(x)=x3+2x and calculate it at x=1x=1x=1.

Problem 12937

Find the tangent line equation for g(x)g(x)g(x) at x=3x=3x=3, given g(3)=g′(3)=4g(3)=g'(3)=4g(3)=g′(3)=4. Format: y=mx+by=mx+by=mx+b.

Problem 12938

Find lim⁡x→−3−f(x)\lim_{x \to -3^-} f(x)limx→−3−​f(x), lim⁡x→3+f(x)\lim_{x \to 3^+} f(x)limx→3+​f(x), and lim⁡x→+∞f(x)\lim_{x \to +\infty} f(x)limx→+∞​f(x) for f(x)=3+x9−x2f(x)=\frac{3+x}{9-x^2}f(x)=9−x23+x​.

Problem 12939

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=x5y=x^{5}y=x5.

Problem 12940

Find the derivative of $f(x) = \frac{\ln x}{x^{2}}$ .

Logistic growth function $f(t)=\frac{116,000}{1+5900 e^{-t}}$ describes flu cases over time.
a. Initial cases when $t=0$ ? b. Cases at $t=4$ weeks? c. Maximum cases possible?

Invest \ $14,000 at$ 7\% $compounded quarterly or$ 6.93\%$ compounded continuously. Which yields more in 3 years?

How long for $43\%$ of a material to remain if it decays at $0.084\%$ per year?

Find the average rate of change of $f(x)=x^{2}$ from $x=1$ to $x=3$ .

Calculate the average rate of change of $g(x)=-8x^{2}$ from $x=-2$ to $x=-1$ .

Find the critical numbers of $f(x)=-4 x^{5}+15 x^{4}+20 x^{3}-3$ and classify them with a graph.

Find the local max and min of the function $f(x)=4+9x+225x^{-1}$ , given max at $x=-5$ and min at $x=5$ .

Find the surface area of $z=e^{-(x^{2}+y^{2})}$ over the unit disc using cylindrical coordinates.

Berechnen Sie die Ableitungen von: $f(x)=(x^{2}-5)^{2}$ , $g(x)=(x^{2}+5x+1)^{2}$ , $h(x)=\sqrt{x}(x^{2}-3x)$ , $k(x)=(x^{2}-5)(x^{2}+5)$ .

Find the correct integral to compute the surface area of $z=e^{-(x^{2}+y^{2})}$ over the unit disc $x^{2}+y^{2} \leq 1$ .

Does $f(x)=3x^2-4x+2$ satisfy the Mean Value Theorem on $[0,2]$ ? Choose: Yes, No, or Not enough info.

Find the local maximum of the function $f(x)=2 x+4 x^{-1}$ and its value; local minimum occurs at $x=\sqrt{2}$ .

Find the local minimum and maximum of $f(x)=2 x^{3}-45 x^{2}+300 x+7$ and their values.

Find the local min and max of $f(x)=2x^3-45x^2+300x+5$ . Determine the $x$ values and function values using derivatives.

Find the integral for the surface area of $z=e^{-(x^{2}+y^{2})}$ over the unit disc $x^{2}+y^{2} \leq 1$ .

If $3 \leq f^{\prime}(x) \leq 4$ , find the min and max of $f(4) - f(1)$ .

Find the surface area for the parametrization $G(s,t) = (st, s-t, s+t)$ where $s^2 + t^2 \leq 4$ .

Find $\frac{d y}{d x}$ for the equation $-x y=3 y^{3}+2 x^{2}$ in terms of $x$ and $y$ .

Differentiate the function $3 x \ln (x)$ with respect to $x$ .

Given the function $f(x)=-3 x^{2}+4 x-2$ , find the critical number, intervals of increase/decrease, and local extrema type.

Find where the function $f(x)=6x + 2x^{-1}$ is decreasing, given it increases on $\left(-\infty,-\sqrt{\frac{1}{3}}\right) \cup\left(\sqrt{\frac{1}{3}}, \infty\right)$ .

Evaluate the Riemann sum $R_{6}$ for $f(x)=5x^{2}-4x$ using right endpoints with $n=6$ .

Berechnen Sie die 1. und 2. Ableitung von $f_{t}(x)=\frac{5}{2 t} x^{2}-t x+3 t$ ohne Taschenrechner.

Invest \ $15,000 at$ 6\% $compounded monthly or$ 5.93\%$ continuously. Which yields more in 5 years?

Invest \ $10,000 at$ 6\% $daily or$ 5.90\%$ continuously. Which gives more in 1 year?

Find critical points of $f(x)=2x^{3}-21x^{2}+36x+11$ and classify them using the second derivative test.

Find the first and second derivatives of $f_{t}(x)=\frac{5}{2 t} x^{2}-t x+3 t$ .

Calculate the derivative of $f(x) = 3x^2 + 4x + 2$ when $x = -1$ .

Një gur hidhet lart me $20 \mathrm{~m/s}$ nga $24 \mathrm{~m}$ . Gjeni shpejtësinë kur godet tokën dhe kohën e udhëtimit.

Find a point $c$ that meets the MVT for $y(x)=8\sqrt{x}$ on $[0,23]$ .

Find the limit of $\frac{f(1+h)-f(1)}{h}$ for $f(x)=x^{3}+2x$ as $h$ approaches 0.

Find the antiderivative $F$ of $f(v)=\frac{2}{5} \sec v \tan v$ with $F(0)=3$ , where $-\frac{\pi}{2}<v<\frac{\pi}{2}$ .

Find the difference quotient for $f(x)=x^{3}+2x$ at $a=1$ : $\frac{f(1+h)-f(1)}{h}$ .

Analyze the vertical motion of a softball launched at $31 \mathrm{~m/s}$ under gravity $g=-9.8 \mathrm{~m/s}^2$ . Find its velocity, position, time to highest point, height, and time to hit the ground.

Find the derivative of $f(x)=x^{3}+2x$ and calculate it at $x=1$ .

Find the tangent line equation for $g(x)$ at $x=3$ , given $g(3)=g'(3)=4$ . Format: $y=mx+b$ .

Find $\lim_{x \to -3^-} f(x)$ , $\lim_{x \to 3^+} f(x)$ , and $\lim_{x \to +\infty} f(x)$ for $f(x)=\frac{3+x}{9-x^2}$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{5}$ .

Find the tangent line equation for $f(x)=x^{2}$ at $x=-1$ . The equation is $\square$ .

Find the derivative $f^{\prime}(x)$ for the constant function $f(x)=-4$ .

Find the derivative $g^{\prime}(x)$ of the function $g(x)=x^{4/9}$ .

Find the derivative $\frac{dy}{dx}$ for the function $y = x^{-7}$ .

Find the derivative of $y=\frac{1}{x^{10}}$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=6 x^{4}$ .

Find a point $c$ that meets the MVT for $y(x)=5 x^{2/3}$ on the interval $[0, 64]$ .

Find $f^{\prime}(5)$ for $f(x)=1 / \sqrt{x}$ using the expression $\frac{f(5+h)-f(5)}{h}$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=3 x^{5}$ .

Find the absolute minimum and maximum of $f(x)=x^{4}-18 x^{2}+7$ on the interval $-2 \leq x \leq 7$ .

Find the derivative of $\frac{x^{8}}{32}$ with respect to $x$ . What is $\frac{d}{d x}\left(\frac{x^{8}}{32}\right)$ ?

Find the derivative $y^{\prime}$ for $y=\frac{-5}{\sqrt[3]{x}}$ . What is $y^{\prime}$ ?

Calculate the derivative of the function: $\frac{d}{d x}(-3 x-7) = \square$

Find the derivative $f^{\prime}(t)$ for the function $f(t)=3 t^{2}+6 t+9$ .

Find the derivative $y^{\prime}$ of the function $y=8 x^{-6}-4 x^{-1}$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=e^{x}+5x-2\ln x$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-11 \ln x + 9 x^{2} - 9$ .

Find the derivative $h^{\prime}(t)$ for the function $h(t)=5.1-6.9 t+0.7 t^{3}$ . What is $h^{\prime}(t)$ ?

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\ln x+2 e^{x}-3 x^{2}$ .

A penny falls from a $425 \mathrm{~m}$ high skyscraper. How long does it take to reach the ground?

Berechne den Differenzenquotienten der Funktion $f(x)=\frac{1}{2 x}$ im Intervall $[1 ; 2]$ und $[1 ; 1,5]$ .

Given $f(x)=3 x^{4}-8 x^{3}+9$ , find $f'(x)$ , slope at $x=-1$ , tangent line at $x=-1$ , and where tangent is horizontal.

Find the marginal-cost function for $C(x)=0.04 x^{3}+0.6 x^{2}+30 x+140$ at $x=800$ .

Graph the drug concentration $c(t)=\frac{3t}{t^2+1}$ .
(a) What is the maximum concentration? (b) What happens to the concentration over time? (c) When does it drop below $0.3 \mathrm{mg}/L$ ? $\mathrm{t}=\square \mathrm{hr}$

Find (a) the rate of change of $y$ with respect to $x$ , (b) the relative rate of change, and (c) at $x=6$ , find $y=x^{2}+x-2$ .

Construisez la table des valeurs de $f(x)=\frac{x}{\sqrt{1+3x}-1}$ près de 0 et conjecturez la limite.

Find the minimum slope of the tangent line to the graph of $f(x)=6+2x-4x^{2}+\frac{1}{3}x^{3}$ .

The drug concentration $C=15 \cdot\left(\frac{t}{60+t^{2}}\right)$ ; find $C$ after 1 hour, time to reach 0.5\%, and end behavior.

Find the maximum area of a rectangle with corners on the axes and one on the graph of $y=\frac{6}{x^{2}+1}$ .

A circle's radius increases by $5 \mathrm{~mm/s}$ . Find the area change rate when the radius is $29 \mathrm{~mm}$ .

Find the Riemann sum for $f(x)=e^{x}-1$ over $[0, 2]$ with $n=4$ using midpoints, accurate to six decimal places.

Find the critical points of $y(x)=x(x-7)^{5}$ . Provide your answer as a comma-separated list.

Find critical points $c$ and intervals where $g(x)=3 x^{5}+3 x^{3}+3 x$ is increasing or decreasing. Identify local min/max.

Find critical points and intervals of increase/decrease for $f(x)=6 x^{4}+9 x^{3}$ . Provide a list of results.