Calculus

Problem 9801

Find the derivative of the function $f(x)=\sqrt[3]{5 x^{2}+3}$ , i.e., calculate $f^{\prime}(x)$ .

See Solution

Problem 9802

Find the derivative $\frac{d y}{d x}$ of the function $y=\frac{1}{(4 x^{2}-8)^{2}}$ .

See Solution

Problem 9803

Find the derivative of the function: $y=4 x e^{-2 x}-2 e^{x^{2}}$ .

See Solution

Problem 9804

Find the derivative of the function $f(x)=2 \cos (4-4 x)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 9805

Find the derivative of the function $y=3 \sin (2-3 x)$ , i.e., compute $\frac{d y}{d x}$ .

See Solution

Problem 9806

Find the derivative of the function $y=\cos \left(1-6 x^{3}\right)$ , i.e., calculate $\frac{d y}{d x}$ .

See Solution

Problem 9807

Find the derivative of the function $f(x)=3 \sin ^{3}\left(2 x^{2}\right)$ , i.e., calculate $f^{\prime}(x)$ .

See Solution

Problem 9808

Find the derivative $g^{\prime}(3)$ for the function $g(x)=4^{\sqrt{2 x+3}}$ .

See Solution

Problem 9809

Find the derivative $g^{\prime}(3)$ for the function $g(x)=4^{\sqrt{2 x+3}}$ . Options: A. $64 \ln 4$ , B. $\frac{64 \ln 4}{3}$ , C. $\frac{32 \ln 4}{3}$ , D. $\frac{32 \ln 2}{3}$ .

See Solution

Problem 9810

Find the tangent line for $f(x)=(1+\tan x)^{3/2}$ at $x=0$ , approximate $f(0.02)$ , and calculate percent error.

See Solution

Problem 9811

Consider the curve $-8 x^{2}+5 x y+y^{3}=-149$ .
(a) Find $\frac{d y}{d x}$ for $16 x + (5 y + 5 x \frac{d y}{d x}) + 7 y^{2} = 0$ .
(b) Write the tangent line equation at $(4,-1)$ .
(c) Approximate $k$ for the point $(4.2, k)$ using the tangent line from (b).
(d) Write and solve an equation to find the actual value of $k$ for $(4.2, k)$ on the curve.
10. Approximate the fourth root of 20 using local linear approximation.

See Solution

Problem 9812

Differentiate: $y=\sin^{-1}(\sqrt{1-9x^{2}})$

See Solution

Problem 9813

Find the area of the surface formed by rotating $y=\sqrt{9-x^{2}}$ from $x=-2$ to $x=2$ around the x-axis.

See Solution

Problem 9814

Find the volume change rate when the sphere's radius $r$ is $15 \mathrm{~cm}$ and expanding at $60 \mathrm{~cm/min}$ . $V=\frac{4}{3} \pi r^{3}$

See Solution

Problem 9815

Find the derivative of $f(x)=\left(x^{3}+7 x^{2}-8\right)\left(2 x^{-3}+x^{-4}\right)$ using the product rule.

See Solution

Problem 9816

Invest \$150000 at 8% continuous compounding. (a) Value after 9 years? (b) Years to reach \$480000? Round to 3 decimals.

See Solution

Problem 9817

Evaluate the integral $\int_{0}^{7}\left(64-\left(8-\frac{7}{x+1}\right)^{2}\right) d x$ .

See Solution

Problem 9818

Solve the linear differential equation: $y' - 3x^2 y = x^2$ .

See Solution

Problem 9819

A balloon inflates at 40 in³/s. Find how fast the radius changes when $d=10$ and $d=16$ . Which is faster?

See Solution

Problem 9820

Find the derivative $f^{\prime}(4)$ for the function $f(x)=-\frac{6}{x}-\frac{3 \sqrt{x^{3}}}{4}$ .

See Solution

Problem 9821

Calculate the integral $\pi \int_{0}^{7}\left(64-\left(8-\frac{7}{x+1}\right)^{2}\right) d x$ .

See Solution

Problem 9822

Water fills a cylindrical pool at 9 ft³/min. Find the height change rate when water is 3 feet deep. Answer: $\mathrm{ft} / \mathrm{min}$

See Solution

Problem 9823

A hot air balloon is tracked 2 miles away. At angle $\frac{\pi}{3}$ , changing at $0.1 \mathrm{rad/min}$ , find its rise speed. Answer: miles/min

See Solution

Problem 9824

Find the water level rise rate in an inverted cone tank (top radius $3 m$ , height $5 m$ ) when water is $2 m$ deep, pumped at $2 m^{3}/\min$ . What is the rate in $m/\min$ ?

See Solution

Problem 9825

At noon, ship A is 180 km west of ship B. A sails south at 20 km/h, B sails north at 40 km/h. Find the speed of distance change at 4 PM in km/h.

See Solution

Problem 9826

Zeigen Sie, dass $x_{0}=0$ ein Minimum von $f(x)=x^{4}$ ist.

See Solution

Problem 9827

Finde die Extrem- und Wendepunkte der Funktion $h(x)=(3-4 x)^{2}+32 x$ .

See Solution

Problem 9828

Finde die Extremstellen der Funktion $f(x)=x^{3}-6 x^{2}+12 x-9$ und bestimme, ob sie Minima oder Maxima sind.

See Solution

Problem 9829

Finde die Extrem- und Wendepunkte der Funktion $g(t) = t^{2} \cdot (2t + 5)$ .

See Solution

Problem 9830

Bestimme $f^{\prime}(1)$ und $f^{\prime}(2)$ für $f(x)=3 x^{2}$ ; sowie $f^{\prime}(3)$ und $f^{\prime}(-2)$ für $f(x)=4 x^{2}$ .

See Solution

Problem 9831

Find values of $c$ that satisfy the Mean Value Theorem for: 5) $y=x^{2}+8 x+15$ on $[-5,-3]$ ; 6) $y=x^{3}-4 x^{2}+7$ on $[0,3]$ .

See Solution

Problem 9832

Find the limit: $\lim _{x \rightarrow 0} \frac{2 x^{2}}{1-\cos (5 x)}$ .

See Solution

Problem 9833

Evaluate the integral: $\int \sqrt{e^{x}-9} \, dx$

See Solution

Problem 9834

Find the volume change rate of a sphere with radius $r$ expanding at $50 \mathrm{~cm/min}$ at $t=2 \mathrm{~min}$ . $\frac{d V}{d t}=$

See Solution

Problem 9835

Find the rate of change of surface area, $A=4 \pi r^{2}$ , for a sphere with $r=10$ cm at $t=2$ min, given $dr/dt=40$ cm/min.

See Solution

Problem 9836

Find the rate of change of volume $\frac{d V}{d t}$ of a sphere when $r=15 \mathrm{~cm}$ , given $r$ expands at $40 \mathrm{~cm/min}$ .

See Solution

Problem 9837

Evaluate the integral: $\int y \sec^{2} y \, dy$

See Solution

Problem 9838

Find $\frac{d h}{d t}$ for a conical tank (height $8 \mathrm{~m}$ , radius $4 \mathrm{~m}$ ) with water inflow $2.4 \mathrm{~m}^{3}/\mathrm{min}$ .

See Solution

Problem 9839

Find the derivative of $f(x) = 5x^7 - 7x^3 + 2$ .

See Solution

Problem 9840

Find the points of inflection for $f(x)=\sin ^{2} x$ in $[0, \pi]$ . Which $x$ -coordinates are correct?

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Problem 9841

Find the value of $k$ for which $f(x)=\frac{x}{x^{2}+k}$ has a relative maximum at $x=0.5$ . Options: A) 1 B) 0.25 C) -0.25 D) 0.5

See Solution

Problem 9842

Evaluate the integral: $\int \frac{\sin ^{3}(y)}{1+\cos (y)} \mathrm{d} y$

See Solution

Problem 9843

Calculate the integral: $\int \frac{d u}{u^{2} \sqrt{4 u^{2}-1}}$ .

See Solution

Problem 9844

Find the derivative of the function $f(x)=x^{2}+6x+9$ .

See Solution

Problem 9845

Find the derivative of $f(x)=\frac{5 x^{2}}{x+47}$ .

See Solution

Problem 9846

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

See Solution

Problem 9847

Find the derivative of the function $f(x)=5 x^{2}(x+47)$ .

See Solution

Problem 9848

Find the unit consumption vector for sector 2. For fixed costs 15 and variable costs $2Q$ , express TC, AC, and MC, then find $Q$ minimizing AC. Verify that AC = MC at this point.

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Problem 9849

Berechnen Sie Hoch- und Tiefpunkte sowie Wendepunkte der Funktionen: a) $f(x)=x^{4}-4 x^{2}+3$ , b) $f(x)=-x^{3}+3 x^{2}-4$ , c) $f(x)=0,5 x^{3}+x^{2}-3,5 x$ , d) $f(x)=\frac{1}{9} x^{3}-3 x+1$ , e) $f(x)=x^{4}-4 x^{2}$ , f) $f(x)=\frac{1}{3} x^{3}-\frac{1}{2} x^{2}+\frac{1}{4}$ .

See Solution

Problem 9850

Evaluate the integral $\int_{-2}^{2}\left(x^{3} \cos \frac{x}{2}+\frac{1}{2}\right) \sqrt{4-x^{2}} d x$ .

See Solution

Problem 9851

Find the derivative y' of $y=(3 x-4)(4 x^{3}-x^{2}+1)$ . Show your work.

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Problem 9852

Find the slope of the tangent line to the curve $y=x^{2}+6x$ at the point where $x=-3$ using the limit definition.

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Problem 9853

Find the average rate of change of the function $h(x)=\frac{1}{8} x^{3}-x^{2}$ from $x=-2$ to $x=2$ .

See Solution

Problem 9854

Untersuchen Sie das Verhalten der Funktionen für $x \rightarrow+\infty$ und $x \rightarrow-\infty$ und erklären Sie Konvergenz/Divergenz.

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Problem 9855

Find the interval where the function $h(x)=\frac{1}{8} x^{3}-x^{2}$ has a positive average rate of change. Choose one: (A) $6 \leq x \leq 8$ , (B) $0 \leq x \leq 8$ , (C) $0 \leq x \leq 2$ , (D) $0 \leq x \leq 6$ .

See Solution

Problem 9856

Untersuchen Sie das Verhalten der Funktionen $f$ für $x \rightarrow+\infty$ und $x \rightarrow-\infty$ bezüglich Konvergenz und Divergenz.

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Problem 9857

Show that the polynomial $f(x)=2 x^{4}-9 x^{2}+3$ has a zero between -2 and 0 using the Intermediate Value Theorem.

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Problem 9858

Untersuchen Sie die Konvergenz und Divergenz der Funktionen für $x \rightarrow+\infty$ und $x \rightarrow-\infty$ :
e) $f: x \mapsto \frac{-\gamma}{2 x+3 x^{2}}$
f) $f: x \mapsto \frac{-3 x^{2}-3 x+12}{-0,25 x^{2}+6 x}$
g) $f: x \mapsto \frac{x^{2}+2 x}{-9 x+3}$
h) $f: x \mapsto \frac{x(3-x)}{-8 x-7}$

See Solution

Problem 9859

Un plongeur de 70 kg saute d'une hauteur de 12 m. Quelle est sa vitesse à l'impact? (20 m/s)

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Problem 9860

Un plongeur de 70 kg saute d'une hauteur de 12 m. Quelle est sa vitesse d'impact avec l'eau? (20 m/s)

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Problem 9861

Bestimme die Ableitung der Funktion $f_a(x)=\frac{1}{3a}x^3 - x^2$ für $a \neq 0$ .

See Solution

Problem 9862

Find $f^{\prime}(1)$ if $f(x)=\operatorname{arcsec}\left(e^{2 x}\right)$ .

See Solution

Problem 9863

Bestimmen Sie die Steigung der Funktionen $f_{t}(x)$ für $x_{0}$ : a) $f_{t}(x)=t-\frac{t}{3} x_{i}$ , $x_{0}=-3$ ; b) $f_{t}(x)=\frac{t}{4} x^{3}-t x^{2}+t x+1$ , $x_{0}=1$ ; c) $f_{t}(x)=t \cdot e^{t x}$ , $x_{0}=0$ .

See Solution

Problem 9864

Find the extreme points of the function $f(x)=\frac{1}{3 a} x^{3}-x^{2}$ , where $a \neq 0$ .

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Problem 9865

Find the speed and acceleration of the body at $t=2$ for the function $s=f(t)=7 t^{2}+2 t+8$ .

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Problem 9866

Un plongeur de 70 kg saute d'une tour de 12 m. a) Vitesse à l'impact? b) Vitesse avec 5 m/s vers le haut?

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Problem 9867

Use the Intermediate Value Theorem to show $f(x)=x^{3}+x^{2}-2x+4$ has a zero between -3 and -1.

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Problem 9868

Use the intermediate value theorem to show $f(x)=3x^3-10x+9$ has a zero between -3 and -2.

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Problem 9869

A balloon rises 30 ft away from you. If the angle of elevation changes at $\frac{1}{5} \mathrm{rad/min}$ , find its rise rate at 40 ft.

See Solution

Problem 9870

Find the inflection points of $f_a(x)=\frac{1}{3 a} x^{3}-x^{2}$ where $a \neq 0$ .

See Solution

Problem 9871

Find the velocity of a particle at $t=3$ seconds, given $s(t)=4 t^{2}+26 t$ .

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Problem 9872

A farmer builds a pen of area 66 m² with 2 interior fences. Exterior costs \$12/m, interior \$10/m. Minimize total cost.

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Problem 9873

Untersuchen Sie, wie die Anzahl der Wendepunkte der Funktion $f(x)=t \cdot x^{4}+4 x^{3}+2 x^{2}$ vom Parameter $t$ abhängt.

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Problem 9874

Untersuchen Sie, ob $F(x)$ und $F^{}(x)$ zu $f(x)$ gehören, und geben Sie $g g f . f(x)$ an. a) $F(x)=2 e^{x}(e^{x}+1)$ ; $F^{}(x)=e^{2 x}+2 e^{x}+4$ b) $F(x)=(x+1)^{3}$ ; $F^{}(x)=x^{3}+3 x(x+1)$ c) $F(x)=\sin x$ ; $F^{}(x)=\cos (x-\pi)$ d) $F(x)=x^{2}+\frac{1}{x^{2}}$ ; $F^{*}(x)=\frac{(x^{2}-1)^{2}}{x^{2}}$

See Solution

Problem 9875

Untersuchen Sie die Wendestellen von $f(x)=t \cdot x^{4}+4 x^{3}+2 x^{2}$ in Abhängigkeit von $t$ .

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Problem 9876

Estimate the population of country $X$ in 24 years given $N(t)=500 e^{0.02 t}$ with current population 500 million.

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Problem 9877

A baseball machine throws a ball up at $96 \mathrm{ft/sec}$ from $1.5 \mathrm{ft}$ . Find the height equation and max height/time.

See Solution

Problem 9878

Find the derivative of the function $\frac{3 \cos (x-\pi)-3 \cos (\pi)}{x-2 \pi}$ .

See Solution

Problem 9879

Find the first derivative of the function $f(x)=\frac{3 x}{2 x+1}$ .

See Solution

Problem 9880

Find the limit: $=\lim _{x \rightarrow \frac{\pi}{2}} \frac{3 \cos (x-\pi)}{x-\frac{\pi}{2}}.$

See Solution

Problem 9881

A particle's velocity is $v(t)=-2+(t^{2}+3t)^{\frac{6}{5}}-t^{3}$ with $s(0)=10$ .
a. Find $t$ where speed is 2 in $2 \leq t \leq 4$ . b. Determine when it changes direction in $0 \leq t \leq 5$ . c. Is speed increasing or decreasing at $t=4$ ? Explain.

See Solution

Problem 9882

Find the derivative of the function $y=-\frac{\sqrt[3]{x^{2}}+4x}{\sqrt{x}}$ .

See Solution

Problem 9883

Find the protozoa population after 8 days if it starts with 7 members and grows at a rate of 0.469 per day. Use $P(0) = 7$ .

See Solution

Problem 9884

Approximate $\sqrt[3]{64.3}$ using the tangent line to $f(x)=\sqrt[3]{x}$ at $x=64$ . Find $m$ and $b$ for $y=mx+b$ .

See Solution

Problem 9885

Find the $x$ -values where $f(x)$ has horizontal tangents given $f^{\prime}(x)=\frac{5}{3} x^{\frac{2}{3}} \ln x+x^{\frac{2}{3}}$ .

See Solution

Problem 9886

A ball bearing has a diameter of $7.4 \mathrm{~mm}$ with a possible error of $0.05 \mathrm{~mm}$ . Find: a. Max error in volume using differentials. b. Relative error in volume using differentials.

See Solution

Problem 9887

Approximate the change in surface area $d A$ for a sphere when the radius changes from $r$ by $dr$ . $d A =$

See Solution

Problem 9888

Approximate $y$ for $x=1.24$ using the linearization of $x^{3}+y^{5}+2y=4$ at $(1,1)$ . Round to three decimals. $y \approx$

See Solution

Problem 9889

Find the derivative of $y=\frac{1}{16} x^{4} \ln x$ .

See Solution

Problem 9890

Find the production level for max profit with $C(q)=0.04 q^{2}+4 q+50$ and selling price of \$8. What is max profit?

See Solution

Problem 9891

Find the derivative of $y$ where $y=4^{x}$ .

See Solution

Problem 9892

Find the second derivative of $y=\frac{7 x^{3}}{6}-7$ .

See Solution

Problem 9893

Find the differential $d y$ for $y=2x^{2}+2x+3$ and estimate $\Delta y$ using $\Delta x=0.2$ at $x=5$ .

See Solution

Problem 9894

Approximate $3.7^{4}$ using the tangent line of $f(x)=x^{4}$ at $x=4$ . Find $m$ and $b$ for $y=m x+b$ .

See Solution

Problem 9895

Bestimme die Bedeutung von $\int_{10}^{50} v(t) dt$ und berechne $\int_{10}^{50} v(t) dt$ mit $V(t)=\frac{2}{30}(t+4)^{\frac{3}{2}}$ .

See Solution

Problem 9896

Find horizontal or vertical asymptotes for $f(x)=\frac{x^{4}+1}{x^{2}+5 x-14}$ . A. $y=$ , B. No horizontal asymptotes.

See Solution

Problem 9897

Find the quantity $q$ that maximizes profit using $R(q)=450q$ and $C(q)=9500+2q^{2}$ . Round answers. a) Max profit quantity: $q=$ b) Total profit: profit $=\$$

See Solution

Problem 9898

Find horizontal or vertical asymptotes of $f(x)=\frac{x^{4}+1}{x^{2}+5 x-14}$ .

See Solution

Problem 9899

Approximate $y$ when $x=1.24$ using the linearization of $x^{3}+y^{5}+2y=4$ at $(1,1)$ , rounding to three decimals.

See Solution

Problem 9900

Find the derivative of the function $f(s)=\frac{\sqrt{s}-5}{\sqrt{s}+1}$ .

See Solution

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Calculus

Problem 9801

Find the derivative of the function f(x)=5x2+33f(x)=\sqrt[3]{5 x^{2}+3}f(x)=35x2+3​, i.e., calculate f′(x)f^{\prime}(x)f′(x).

Problem 9802

Find the derivative dydx\frac{d y}{d x}dxdy​ of the function y=1(4x2−8)2y=\frac{1}{(4 x^{2}-8)^{2}}y=(4x2−8)21​.

Problem 9803

Find the derivative of the function: y=4xe−2x−2ex2y=4 x e^{-2 x}-2 e^{x^{2}}y=4xe−2x−2ex2.

Problem 9804

Find the derivative of the function f(x)=2cos⁡(4−4x)f(x)=2 \cos (4-4 x)f(x)=2cos(4−4x). What is f′(x)f^{\prime}(x)f′(x)?

Problem 9805

Find the derivative of the function y=3sin⁡(2−3x)y=3 \sin (2-3 x)y=3sin(2−3x), i.e., compute dydx\frac{d y}{d x}dxdy​.

Problem 9806

Find the derivative of the function y=cos⁡(1−6x3)y=\cos \left(1-6 x^{3}\right)y=cos(1−6x3), i.e., calculate dydx\frac{d y}{d x}dxdy​.

Problem 9807

Find the derivative of the function f(x)=3sin⁡3(2x2)f(x)=3 \sin ^{3}\left(2 x^{2}\right)f(x)=3sin3(2x2), i.e., calculate f′(x)f^{\prime}(x)f′(x).

Problem 9808

Find the derivative g′(3)g^{\prime}(3)g′(3) for the function g(x)=42x+3g(x)=4^{\sqrt{2 x+3}}g(x)=42x+3​.

Problem 9809

Find the derivative g′(3)g^{\prime}(3)g′(3) for the function g(x)=42x+3g(x)=4^{\sqrt{2 x+3}}g(x)=42x+3​. Options: A. 64ln⁡464 \ln 464ln4, B. 64ln⁡43\frac{64 \ln 4}{3}364ln4​, C. 32ln⁡43\frac{32 \ln 4}{3}332ln4​, D. 32ln⁡23\frac{32 \ln 2}{3}332ln2​.

Problem 9810

Find the tangent line for f(x)=(1+tan⁡x)3/2f(x)=(1+\tan x)^{3/2}f(x)=(1+tanx)3/2 at x=0x=0x=0, approximate f(0.02)f(0.02)f(0.02), and calculate percent error.

Problem 9811

Problem 9812

Differentiate: y=sin⁡−1(1−9x2)y=\sin^{-1}(\sqrt{1-9x^{2}})y=sin−1(1−9x2​)

Problem 9813

Find the area of the surface formed by rotating y=9−x2y=\sqrt{9-x^{2}}y=9−x2​ from x=−2x=-2x=−2 to x=2x=2x=2 around the x-axis.

Problem 9814

Find the volume change rate when the sphere's radius rrr is 15 cm15 \mathrm{~cm}15 cm and expanding at 60 cm/min60 \mathrm{~cm/min}60 cm/min. V=43πr3V=\frac{4}{3} \pi r^{3}V=34​πr3

Problem 9815

Find the derivative of f(x)=(x3+7x2−8)(2x−3+x−4)f(x)=\left(x^{3}+7 x^{2}-8\right)\left(2 x^{-3}+x^{-4}\right)f(x)=(x3+7x2−8)(2x−3+x−4) using the product rule.

Problem 9816

Invest \$150000 at 8% continuous compounding. (a) Value after 9 years? (b) Years to reach \$480000? Round to 3 decimals.

Problem 9817

Evaluate the integral ∫07(64−(8−7x+1)2)dx\int_{0}^{7}\left(64-\left(8-\frac{7}{x+1}\right)^{2}\right) d x∫07​(64−(8−x+17​)2)dx.

Problem 9818

Solve the linear differential equation: y′−3x2y=x2y' - 3x^2 y = x^2y′−3x2y=x2.

Problem 9819

A balloon inflates at 40 in³/s. Find how fast the radius changes when d=10d=10d=10 and d=16d=16d=16. Which is faster?

Problem 9820

Find the derivative f′(4)f^{\prime}(4)f′(4) for the function f(x)=−6x−3x34f(x)=-\frac{6}{x}-\frac{3 \sqrt{x^{3}}}{4}f(x)=−x6​−43x3​​.

Problem 9821

Calculate the integral π∫07(64−(8−7x+1)2)dx\pi \int_{0}^{7}\left(64-\left(8-\frac{7}{x+1}\right)^{2}\right) d xπ∫07​(64−(8−x+17​)2)dx.

Problem 9822

Water fills a cylindrical pool at 9 ft³/min. Find the height change rate when water is 3 feet deep. Answer: ft/min\mathrm{ft} / \mathrm{min}ft/min

Problem 9823

A hot air balloon is tracked 2 miles away. At angle π3\frac{\pi}{3}3π​, changing at 0.1rad/min0.1 \mathrm{rad/min}0.1rad/min, find its rise speed. Answer: miles/min

Problem 9824

Find the water level rise rate in an inverted cone tank (top radius 3m3 m3m, height 5m5 m5m) when water is 2m2 m2m deep, pumped at 2m3/min⁡2 m^{3}/\min2m3/min. What is the rate in m/min⁡m/\minm/min?

Problem 9825

At noon, ship A is 180 km west of ship B. A sails south at 20 km/h, B sails north at 40 km/h. Find the speed of distance change at 4 PM in km/h.

Problem 9826

Zeigen Sie, dass x0=0x_{0}=0x0​=0 ein Minimum von f(x)=x4f(x)=x^{4}f(x)=x4 ist.

Problem 9827

Finde die Extrem- und Wendepunkte der Funktion h(x)=(3−4x)2+32xh(x)=(3-4 x)^{2}+32 xh(x)=(3−4x)2+32x.

Problem 9828

Finde die Extremstellen der Funktion f(x)=x3−6x2+12x−9f(x)=x^{3}-6 x^{2}+12 x-9f(x)=x3−6x2+12x−9 und bestimme, ob sie Minima oder Maxima sind.

Problem 9829

Finde die Extrem- und Wendepunkte der Funktion g(t)=t2⋅(2t+5)g(t) = t^{2} \cdot (2t + 5)g(t)=t2⋅(2t+5).

Problem 9830

Bestimme f′(1)f^{\prime}(1)f′(1) und f′(2)f^{\prime}(2)f′(2) für f(x)=3x2f(x)=3 x^{2}f(x)=3x2; sowie f′(3)f^{\prime}(3)f′(3) und f′(−2)f^{\prime}(-2)f′(−2) für f(x)=4x2f(x)=4 x^{2}f(x)=4x2.

Problem 9831

Find values of ccc that satisfy the Mean Value Theorem for: 5) y=x2+8x+15y=x^{2}+8 x+15y=x2+8x+15 on [−5,−3][-5,-3][−5,−3]; 6) y=x3−4x2+7y=x^{3}-4 x^{2}+7y=x3−4x2+7 on [0,3][0,3][0,3].

Problem 9832

Find the limit: lim⁡x→02x21−cos⁡(5x)\lim _{x \rightarrow 0} \frac{2 x^{2}}{1-\cos (5 x)}limx→0​1−cos(5x)2x2​.

Problem 9833

Evaluate the integral: ∫ex−9 dx\int \sqrt{e^{x}-9} \, dx∫ex−9​dx

Problem 9834

Find the volume change rate of a sphere with radius rrr expanding at 50 cm/min50 \mathrm{~cm/min}50 cm/min at t=2 mint=2 \mathrm{~min}t=2 min. dVdt=\frac{d V}{d t}=dtdV​=

Problem 9835

Find the rate of change of surface area, A=4πr2A=4 \pi r^{2}A=4πr2, for a sphere with r=10r=10r=10 cm at t=2t=2t=2 min, given dr/dt=40dr/dt=40dr/dt=40 cm/min.

Problem 9836

Find the rate of change of volume dVdt\frac{d V}{d t}dtdV​ of a sphere when r=15 cmr=15 \mathrm{~cm}r=15 cm, given rrr expands at 40 cm/min40 \mathrm{~cm/min}40 cm/min.

Problem 9837

Evaluate the integral: ∫ysec⁡2y dy\int y \sec^{2} y \, dy∫ysec2ydy

Problem 9838

Find dhdt\frac{d h}{d t}dtdh​ for a conical tank (height 8 m8 \mathrm{~m}8 m, radius 4 m4 \mathrm{~m}4 m) with water inflow 2.4 m3/min2.4 \mathrm{~m}^{3}/\mathrm{min}2.4 m3/min.

Problem 9839

Find the derivative of f(x)=5x7−7x3+2f(x) = 5x^7 - 7x^3 + 2f(x)=5x7−7x3+2.

Problem 9840

Find the points of inflection for f(x)=sin⁡2xf(x)=\sin ^{2} xf(x)=sin2x in [0,π][0, \pi][0,π]. Which xxx-coordinates are correct?

Find the derivative of the function $f(x)=\sqrt[3]{5 x^{2}+3}$ , i.e., calculate $f^{\prime}(x)$ .

Find the derivative $\frac{d y}{d x}$ of the function $y=\frac{1}{(4 x^{2}-8)^{2}}$ .

Find the derivative of the function: $y=4 x e^{-2 x}-2 e^{x^{2}}$ .

Find the derivative of the function $f(x)=2 \cos (4-4 x)$ . What is $f^{\prime}(x)$ ?

Find the derivative of the function $y=3 \sin (2-3 x)$ , i.e., compute $\frac{d y}{d x}$ .

Find the derivative of the function $y=\cos \left(1-6 x^{3}\right)$ , i.e., calculate $\frac{d y}{d x}$ .

Find the derivative of the function $f(x)=3 \sin ^{3}\left(2 x^{2}\right)$ , i.e., calculate $f^{\prime}(x)$ .

Find the derivative $g^{\prime}(3)$ for the function $g(x)=4^{\sqrt{2 x+3}}$ .

Find the derivative $g^{\prime}(3)$ for the function $g(x)=4^{\sqrt{2 x+3}}$ . Options: A. $64 \ln 4$ , B. $\frac{64 \ln 4}{3}$ , C. $\frac{32 \ln 4}{3}$ , D. $\frac{32 \ln 2}{3}$ .

Find the tangent line for $f(x)=(1+\tan x)^{3/2}$ at $x=0$ , approximate $f(0.02)$ , and calculate percent error.

Differentiate: $y=\sin^{-1}(\sqrt{1-9x^{2}})$

Find the area of the surface formed by rotating $y=\sqrt{9-x^{2}}$ from $x=-2$ to $x=2$ around the x-axis.

Find the volume change rate when the sphere's radius $r$ is $15 \mathrm{~cm}$ and expanding at $60 \mathrm{~cm/min}$ . $V=\frac{4}{3} \pi r^{3}$

Find the derivative of $f(x)=\left(x^{3}+7 x^{2}-8\right)\left(2 x^{-3}+x^{-4}\right)$ using the product rule.

Evaluate the integral $\int_{0}^{7}\left(64-\left(8-\frac{7}{x+1}\right)^{2}\right) d x$ .

Solve the linear differential equation: $y' - 3x^2 y = x^2$ .

A balloon inflates at 40 in³/s. Find how fast the radius changes when $d=10$ and $d=16$ . Which is faster?

Find the derivative $f^{\prime}(4)$ for the function $f(x)=-\frac{6}{x}-\frac{3 \sqrt{x^{3}}}{4}$ .

Calculate the integral $\pi \int_{0}^{7}\left(64-\left(8-\frac{7}{x+1}\right)^{2}\right) d x$ .

Water fills a cylindrical pool at 9 ft³/min. Find the height change rate when water is 3 feet deep. Answer: $\mathrm{ft} / \mathrm{min}$

A hot air balloon is tracked 2 miles away. At angle $\frac{\pi}{3}$ , changing at $0.1 \mathrm{rad/min}$ , find its rise speed. Answer: miles/min

Find the water level rise rate in an inverted cone tank (top radius $3 m$ , height $5 m$ ) when water is $2 m$ deep, pumped at $2 m^{3}/\min$ . What is the rate in $m/\min$ ?

Zeigen Sie, dass $x_{0}=0$ ein Minimum von $f(x)=x^{4}$ ist.

Finde die Extrem- und Wendepunkte der Funktion $h(x)=(3-4 x)^{2}+32 x$ .

Finde die Extremstellen der Funktion $f(x)=x^{3}-6 x^{2}+12 x-9$ und bestimme, ob sie Minima oder Maxima sind.

Finde die Extrem- und Wendepunkte der Funktion $g(t) = t^{2} \cdot (2t + 5)$ .

Bestimme $f^{\prime}(1)$ und $f^{\prime}(2)$ für $f(x)=3 x^{2}$ ; sowie $f^{\prime}(3)$ und $f^{\prime}(-2)$ für $f(x)=4 x^{2}$ .

Find values of $c$ that satisfy the Mean Value Theorem for: 5) $y=x^{2}+8 x+15$ on $[-5,-3]$ ; 6) $y=x^{3}-4 x^{2}+7$ on $[0,3]$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{2 x^{2}}{1-\cos (5 x)}$ .

Evaluate the integral: $\int \sqrt{e^{x}-9} \, dx$

Find the volume change rate of a sphere with radius $r$ expanding at $50 \mathrm{~cm/min}$ at $t=2 \mathrm{~min}$ . $\frac{d V}{d t}=$

Find the rate of change of surface area, $A=4 \pi r^{2}$ , for a sphere with $r=10$ cm at $t=2$ min, given $dr/dt=40$ cm/min.

Find the rate of change of volume $\frac{d V}{d t}$ of a sphere when $r=15 \mathrm{~cm}$ , given $r$ expands at $40 \mathrm{~cm/min}$ .

Evaluate the integral: $\int y \sec^{2} y \, dy$

Find $\frac{d h}{d t}$ for a conical tank (height $8 \mathrm{~m}$ , radius $4 \mathrm{~m}$ ) with water inflow $2.4 \mathrm{~m}^{3}/\mathrm{min}$ .

Find the derivative of $f(x) = 5x^7 - 7x^3 + 2$ .

Find the points of inflection for $f(x)=\sin ^{2} x$ in $[0, \pi]$ . Which $x$ -coordinates are correct?

Find the value of $k$ for which $f(x)=\frac{x}{x^{2}+k}$ has a relative maximum at $x=0.5$ . Options: A) 1 B) 0.25 C) -0.25 D) 0.5

Evaluate the integral: $\int \frac{\sin ^{3}(y)}{1+\cos (y)} \mathrm{d} y$

Calculate the integral: $\int \frac{d u}{u^{2} \sqrt{4 u^{2}-1}}$ .

Find the derivative of the function $f(x)=x^{2}+6x+9$ .

Find the derivative of $f(x)=\frac{5 x^{2}}{x+47}$ .

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

Find the derivative of the function $f(x)=5 x^{2}(x+47)$ .

Find the unit consumption vector for sector 2. For fixed costs 15 and variable costs $2Q$ , express TC, AC, and MC, then find $Q$ minimizing AC. Verify that AC = MC at this point.

Evaluate the integral $\int_{-2}^{2}\left(x^{3} \cos \frac{x}{2}+\frac{1}{2}\right) \sqrt{4-x^{2}} d x$ .

Find the derivative y' of $y=(3 x-4)(4 x^{3}-x^{2}+1)$ . Show your work.

Find the slope of the tangent line to the curve $y=x^{2}+6x$ at the point where $x=-3$ using the limit definition.

Find the average rate of change of the function $h(x)=\frac{1}{8} x^{3}-x^{2}$ from $x=-2$ to $x=2$ .

Untersuchen Sie das Verhalten der Funktionen für $x \rightarrow+\infty$ und $x \rightarrow-\infty$ und erklären Sie Konvergenz/Divergenz.

Untersuchen Sie das Verhalten der Funktionen $f$ für $x \rightarrow+\infty$ und $x \rightarrow-\infty$ bezüglich Konvergenz und Divergenz.

Show that the polynomial $f(x)=2 x^{4}-9 x^{2}+3$ has a zero between -2 and 0 using the Intermediate Value Theorem.

Bestimme die Ableitung der Funktion $f_a(x)=\frac{1}{3a}x^3 - x^2$ für $a \neq 0$ .

Find $f^{\prime}(1)$ if $f(x)=\operatorname{arcsec}\left(e^{2 x}\right)$ .

Find the extreme points of the function $f(x)=\frac{1}{3 a} x^{3}-x^{2}$ , where $a \neq 0$ .

Find the speed and acceleration of the body at $t=2$ for the function $s=f(t)=7 t^{2}+2 t+8$ .

Use the Intermediate Value Theorem to show $f(x)=x^{3}+x^{2}-2x+4$ has a zero between -3 and -1.

Use the intermediate value theorem to show $f(x)=3x^3-10x+9$ has a zero between -3 and -2.

A balloon rises 30 ft away from you. If the angle of elevation changes at $\frac{1}{5} \mathrm{rad/min}$ , find its rise rate at 40 ft.

Find the inflection points of $f_a(x)=\frac{1}{3 a} x^{3}-x^{2}$ where $a \neq 0$ .

Find the velocity of a particle at $t=3$ seconds, given $s(t)=4 t^{2}+26 t$ .

Untersuchen Sie, wie die Anzahl der Wendepunkte der Funktion $f(x)=t \cdot x^{4}+4 x^{3}+2 x^{2}$ vom Parameter $t$ abhängt.