Calculus

Problem 7601

Find the acceleration of a car at $t=2$ seconds, where its position is given by $s(t)=\frac{1}{3}(t^{2}+5)^{3/2}$ .

See Solution

Problem 7602

Find the rate of water draining from a 2000-gallon tank after 10 minutes, using $V=2000\left(1-\frac{t}{40}\right)^{2}$ .

See Solution

Problem 7603

Analyze the concavity of $f(x)=3(x-2)^{5/3}$ and identify any points of inflection.

See Solution

Problem 7604

Given the polynomial $f(x)=x^{3}+A x^{2}+B x$ , if $x=2$ is a critical point and $x=-3$ is an inflection point, find $f(1)$ . Options: $-38$ , $-36$ , $-57$ , $-35$ , None.

See Solution

Problem 7605

Find the first and second derivatives of $u(x)=e^{x}+5$ .

See Solution

Problem 7606

Analyze the concavity of $f(x)=\frac{1}{2} x^{4}-12 x^{2}$ and identify any points of inflection.

See Solution

Problem 7607

Find x-coordinates of inflection points for $f(x)$ given $f''(x)=\frac{x^{2}-25}{(x-6)^{3}(x-10)^{3}}$ .

See Solution

Problem 7608

Find the $x$ -coordinates of points of inflection for the function with $f^{\prime \prime}(x)=\frac{x^{2}-25}{(x-6)^{3}(x-10)^{3}}$ .

See Solution

Problem 7609

Determine the concavity of $f(x)=\frac{x}{x-3}$ and find any points of inflection.

See Solution

Problem 7610

Find $c$ such that the function $f(x)=c x^{2}+x^{-2}$ has an inflection point at $(2, f(2))$ .

See Solution

Problem 7611

Calculate the integral: $\int 8 x^{2} \cos (x) \, dx =$

See Solution

Problem 7612

Which statements about the graph of $f(x)=6 x^{3}-12 x^{2}+6 x+4$ are true regarding local extrema and intervals?

See Solution

Problem 7613

Describe the concavity of $f(x)=\frac{x}{x+4}$ and find any points of inflection.

See Solution

Problem 7614

Select a suitable $u$ for integration by parts without evaluating the integral: $\int e^{3 x} \cos (5 x) d x$

See Solution

Problem 7615

Calculate the integral of $\ln(4x+1)$ with respect to $x$ : $\int \ln(4x+1) \, dx = +C$

See Solution

Problem 7616

Determine the concavity and points of inflection for $f(x)=\frac{1}{4} x^{4}-6 x^{2}$ .

See Solution

Problem 7617

Find the average rate of change of $f$ from $x_{1}$ to $x_{2}$ : A. $\frac{f\left(x_{2}\right)+f\left(x_{1}\right)}{x_{2}+x_{1}}$ , B. $\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}+x_{1}}$ , C. $\frac{f\left(x_{2}\right)+f\left(x_{1}\right)}{x_{2}-x_{1}}$ , D. $\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}}$ .

See Solution

Problem 7618

Graph the linear approximation of $f(x) = 2$ at $a=2$ using $\tilde{f}(x)=f(a)+f^{\prime}(a)(x-a)$ for $x=3$ .

See Solution

Problem 7619

Find the x-coordinates of all points of inflection for the function $f(x)$ given $f^{\prime \prime}(x)=\frac{x^{3}-2 x^{2}}{(x-5)^{3}(x-6)^{3}}$ .

See Solution

Problem 7620

Estimate the change in area of a circular oil spill as the radius increases from 6 ft to 6.023 ft. Calculate new area too.

See Solution

Problem 7621

Find the linear approximation $L(x)$ of $f(x)=\sin (x)$ at $a=\frac{11 \pi}{6}$ . Enter the exact answer.

See Solution

Problem 7622

Find the derivative of $\frac{\pi}{3}(24 r^{2}-r^{3})$ .

See Solution

Problem 7623

Find the base radius $r$ and height $h$ of a cone (with $r+h=24$ cm) that maximize volume, then calculate max volume.

See Solution

Problem 7624

What should we set $d v$ to when using integration by parts to evaluate $\int x \ln (x) d x$ ?

See Solution

Problem 7625

Estimate $\sqrt{25.05}$ using linear approximation. Round your answer to three decimal places.

See Solution

Problem 7626

Find the volume of the solid formed by rotating the region $R$ (bounded by $y=0$ , $y=\sqrt{x}$ , and $x=4$ ) around the $x$ -axis using the disk method.

See Solution

Problem 7627

Given the function $y=2 x^{3}+3 x^{2}-36 x+1$ , find local max/min using the second derivative test and determine concavity intervals.

See Solution

Problem 7628

Find the first and second derivatives of $h(x)=b^{\log _{b}(x)}$ for $b > 1$ .

See Solution

Problem 7629

Find the derivative of $y=\tan \left(\frac{\pi}{4}-\frac{x}{2}\right)$ with respect to $x$ .

See Solution

Problem 7630

Find the volume of the solid formed by rotating the region $R$ (bounded by $y=2$ , $y=\sqrt{x}$ , $x=0$ ) around the $y$ -axis.

See Solution

Problem 7631

Find the $x$ -coordinate of the inflection point for the curve $y=f(x)=\frac{5}{3} x^{3}+25 x^{2}-48 x-196$ .

See Solution

Problem 7632

Choose $u$ from $x^3$ , $\cos(x)$ , $x^6$ , $x^3\sin(x)$ , $\sin(x)$ , or $x$ to evaluate $\int x^{6} \sin (x) dx$ using integration by parts.

See Solution

Problem 7633

Differentiate: $y=\tan \left(\frac{\pi}{4}-\frac{x}{2}\right)$

See Solution

Problem 7634

Find the first and second derivatives of $\operatorname{TR}(Q)=Q \cdot\left(\frac{15-Q}{3}\right)^{2}$ .

See Solution

Problem 7635

Find the first and second derivatives of $TR(Q)=-3 \sqrt{P}+15 \cdot\left(\frac{15-Q}{3}\right)^{2}$ .

See Solution

Problem 7636

Estimate $f(2.02)$ for $y=f(x)=5 x^{2}-2 x+3$ using $f^{\prime}(x)$ : $\frac{1}{0.02} \times[f(2.02)-$

See Solution

Problem 7637

Estimate $\Delta y$ for $y=\sin(5x)$ with $\Delta x=0.3$ at $x=0$ using linear approximation. Find the percentage error.

See Solution

Problem 7638

Estimate the paint needed in cm³ for a 0.07 cm thick coat on a hemispherical dome with a diameter of 7500 cm. Use $V=\frac{2}{3} \pi r^{3}$ .

See Solution

Problem 7639

Estimate $f(2.02)$ for $y=f(x)=5 x^{2}-2 x+3$ using $f^{\prime}(x)$ : $\frac{1}{0.02} \times[f(2.02)-\square] \approx \square$

See Solution

Problem 7640

A spherical ball bearing has a diameter of $5 \mathrm{~mm}$ and a possible error of $0.1 \mathrm{~mm}$ . Find: a. Maximum error in volume using differentials. b. Relative error in volume using differentials.

See Solution

Problem 7641

Estimate the paint needed for a hemispherical dome (diameter 4000 cm, thickness 0.09 cm). Use $V=\frac{2}{3} \pi r^{3}$ .

See Solution

Problem 7642

Find the max error in volume of a ball bearing with diameter $6.8 \mathrm{~mm}$ and error $0.09 \mathrm{~mm}$ using differentials.

See Solution

Problem 7643

A ball bearing's diameter is $6.8 \mathrm{~mm}$ with a $0.09 \mathrm{~mm}$ error. Find max error and relative error in volume.

See Solution

Problem 7644

Given $f(x)=7-1 x+18 x^{2}$ , find $f(a)$ , $f(a+h)$ , and $\frac{f(a+h)-f(a)}{h}$ for $h \neq 0$ .

See Solution

Problem 7645

Given $y=2 x^{2}$ , calculate $\Delta y$ for $x=4$ , $\Delta x=0.4$ and find $d y$ for $d x=0.4$ .

See Solution

Problem 7646

Find $d y$ for $y=\tan(3x+4)$ at $x=1$ for $d x=0.3$ and $d x=0.6$ .

See Solution

Problem 7647

A ball bearing has a diameter of $5.8 \mathrm{~mm}$ and a possible error of $0.1 \mathrm{~mm}$ . Find the max and relative error in volume.

See Solution

Problem 7648

Bestimmen Sie die Ableitung von $f(x)=3 x^{5}+0,5 x^{3}+x-12$ .

See Solution

Problem 7649

Bestimmen Sie die Ableitung von $f(x)=-2 x^{4}+\frac{1}{3} x^{3}-0,2 x+c$ .

See Solution

Problem 7650

Analysiere die Funktion $f(x)=x^{3}-2 x^{2}-3 x$ : Finde Nullstellen, zeichne sie, und bestimme Ableitung und Integral.

See Solution

Problem 7651

How fast is the water level in Jon's $16 \mathrm{ft}^{2}$ bathtub rising if filled at $0.5 \mathrm{ft}^{3}/\mathrm{min}$ ?

See Solution

Problem 7652

The oil slick's radius grows at $3 \mathrm{~m/min}$ . Find the area increase rate when the radius is $59 \mathrm{~m}$ . $\frac{d A}{d t} =$

See Solution

Problem 7653

An oil slick's radius grows at $3 \mathrm{~m/min}$ . Find area growth rate when radius is $59 \mathrm{~m}$ and after $4$ min.

See Solution

Problem 7654

A ladder of length 8.9 m slides down a wall at $1.1 \mathrm{~m/s}$ . Find $\frac{d x}{d t}$ when $h=3.8$ .

See Solution

Problem 7655

Gegeben ist die Funktion $f(x) = \frac{1}{x} \cdot \cos(x)$ . Bestimmen Sie:
a) Definitionsmenge, b) Nullstellen, d) Grenzwert für $x \rightarrow +\infty$ , e) $x$ für Werte < 0,01 vom Grenzwert. Zeichnen Sie auch $f$ und $g(x) = \frac{1}{x}$ sowie $h(x) = \frac{1}{x} + \cos(x)$ .

See Solution

Problem 7656

A car travels past a farmhouse 1.6 km away at 86 km/h. Find how fast the distance to the farmhouse increases when it's 3.4 km past the intersection.

See Solution

Problem 7657

Bestimme den Steigungswinkel der Funktion $f(x)=3 x^{2}+4 x-3$ an der Stelle $x_{0}=2$ .

See Solution

Problem 7658

A police car moves south at $160 \mathrm{~km/h}$ and a truck east at $140 \mathrm{~km/h}$ . Find the distance change rate at $t=5$ min.

See Solution

Problem 7659

Untersuchen Sie das Verhalten von $f(x)=\frac{6 x-x^{2}}{x^{3}}$ für $x \to +\infty$ und $x \to -\infty$ .

See Solution

Problem 7660

Find the derivative of $f(x)=\sqrt{x}-\frac{4}{x^{3}}$ at $x=1$ . Also, clarify the issue with $f(x)=x^{2}-8x+7$ .

See Solution

Problem 7661

Die Kochdauer $t$ (in min) für ein Ei hängt vom Durchmesser $d$ (in $\mathrm{mm}$ ) ab: $t(d)=0,00262 d^{2}$ . Berechne die mittleren Änderungsraten für [40; 50], [40; 45], [40; 41] und die lokale Änderungsrate bei 40 mm.

See Solution

Problem 7662

Berechnen Sie die mittlere Änderungsrate von $f$ in den Intervallen: a) $f(x)=2x$ , $I=[0;1]$ ; b) $f(x)=0,5x^{2}$ , $I=[1;4]$ ; c) $f(x)=1-x^{2}$ .

See Solution

Problem 7663

A baseball diamond is a square with sides of $90 \mathrm{ft}$ . A player runs to first base at $23 \mathrm{ft/s}$ . Find the rate of change of distance to second base when halfway to first. Give answer to two decimal places. $\frac{d h}{d t} =$

See Solution

Problem 7664

Calculate $\int_{-2}^{2} x^{2} dx + \int_{3}^{5} x^{2} dx + \int_{2}^{3} x^{2} dx$ .

See Solution

Problem 7665

Differentiate the function by expanding: $f(x)=(4x+3)(3x^{2}+1)$ .

See Solution

Problem 7666

Find the derivative of the function $f(w)=\frac{7 w^{3}-3 w}{8 w}$ after simplifying it.

See Solution

Problem 7667

Garten Teich: Bestimmen Sie die Symmetrie von $h(x)$ , das Verhalten für $x \rightarrow+\infty$ , und den Flächeninhalt eines Dreiecks. Analysieren Sie lokale Extrempunkte und Tangenten auf $G_{2}$ . Finden Sie den Parameter für einen Sattelpunkt und berechnen Sie die Seitenlängen einer Plane. Bestimmen Sie die Fläche im Schatten der Brücke und die Gleichung einer symmetrischen Parabel.

See Solution

Problem 7668

Differentiate the function by expanding: $f(x)=(5 x-6)^{2}$ .

See Solution

Problem 7669

Calculate the integrals: $\int_{-2}^{2} x^{2} dx + \int_{3}^{5} x^{2} dx + \int_{2}^{3} x^{2} dx$ .

See Solution

Problem 7670

Find the derivative of the function $f(w) = 3w^{4} + 6w^{2} + 5w$ .

See Solution

Problem 7671

Find the derivative of $g(w)=\frac{6 e^{2 w}+7 e^{w}}{e^{w}}$ after simplifying the expression.

See Solution

Problem 7672

Find values of $K$ for a local minimum at the origin for $f(x, y)=K x^{2}+8 x y+2 K y^{2}$ . Options: $|K|<2\sqrt{2}$ , $|K|>\sqrt{2}$ , $K>\sqrt{2}$ , $K>2\sqrt{2}$ , $K<\sqrt{2}$ , None, $K<-2$ .

See Solution

Problem 7673

Find the global minimum of $f(x, y)=e^{3 x-5 y+2 z}$ for $-1 \leq x, y, z \leq 1$ .

See Solution

Problem 7674

Find the tangent line equation at $x=\ln 13$ for the curve $y=e^{x}$ and graph both.

See Solution

Problem 7675

Find the derivative of $y=\operatorname{sech} x(1+\ln \operatorname{sech} x)$ .

See Solution

Problem 7676

Find $x$ where the slope of $y=f(x)=2x^2-12x-1$ is 0 and where it is 4.

See Solution

Problem 7677

Optimize the cost function $f(x, y)=x y$ with constraint $a x+b y=c$ . Find the Lagrange equations.

See Solution

Problem 7678

Find points on the graph of $f(x)=50 \sqrt{x}-5 x$ where the tangent line is horizontal or has slope $-\frac{5}{2}$ .

See Solution

Problem 7679

Find the first, second, and third derivatives of the function $f(x) = 6 e^{x}$ .

See Solution

Problem 7680

Bestimmen Sie die Ableitungen für die folgenden Funktionen: a) $f(x)=x^{3}+\sqrt{x}$ b) $f(x)=3 x^{17}-5 x^{12}+4 x^{3}-6$ c) $f(x)=x^{n}+x-1+\frac{1}{x}$ d) $f(x)=-(x-1)^{2}+4$ e) $f(x)=\sqrt{4 x+7}$ f) $f(x)=\frac{2}{3 x-6}$ g) $f(x)=(x^{2}+1)(x^{2}-1)$ h) $f(x)=\frac{x}{x+5}$

See Solution

Problem 7681

Find points on the graph of $f(x)=50 \sqrt{x}-5 x$ where the tangent line is horizontal or has slope $-\frac{5}{2}$ .

See Solution

Problem 7682

Find the first three derivatives $f^{\prime}(x)$ , $f^{\prime \prime}(x)$ , and $f^{(3)}(x)$ for $f(x)=\frac{x^{2}-2x-15}{x+3}$ .

See Solution

Problem 7683

Find the derivative of $y=e^{2 x} \csc (1-2 x)$ with respect to $x$ .

See Solution

Problem 7684

Momentangeschwindigkeit: Ein Snowboarder hat die Bewegung $s(t)=1,5 t^2$ . Berechne: a) Weg nach 1s und 5s, b) mittlere Geschwindigkeit in 5s, c) Momentangeschwindigkeit nach 5s.

See Solution

Problem 7685

Find a function $f$ and a number $a$ such that $\lim _{x \rightarrow 0} \frac{e^{x}-1}{x} = f^{\prime}(a)$ . Then, compute the limit.

See Solution

Problem 7686

Freier Fall auf dem Mond:
a) Fallstrecke in der ersten Sekunde? b) Durchschnittsgeschwindigkeit in der ersten Sekunde? c) Momentangeschwindigkeit nach 1s und 10s? d) Aufprallgeschwindigkeit aus 40 m Höhe? Weg-Zeit-Gesetz: $s(t)=0,8 t^{2}$ .

See Solution

Problem 7687

Finde die Stellen, an denen der Graph von $f(x)=2 x^{2}+2$ die Steigung $m=4$ hat und die gleiche Steigung wie $g(x)=x^{3}-4 x-1$ .

See Solution

Problem 7688

Estimate the distance a cyclist traveled using 3 equal-width strips under the curve from (0,0) to (9,25).

See Solution

Problem 7689

Gegeben sind die Funktionen $f(x)=2 x^{2}-6 x+6,5$ , $g(x)=x$ , $h(x)=x^{4}-5 x^{2}+4$ . Finde die Ableitungen und Extrempunkte.

See Solution

Problem 7690

Berechnen Sie die Ableitungen $\mathrm{f}^{\prime}$ für die Funktionen a) bis f): a) $f(x)=\frac{1}{4} x^{4}-2 x^{2}$ , b) $f(x)=-3 x^{2}+4$ , c) $f(x)=3(x-2)^{2}+x$ , d) $f(x)=a x^{3}+b x^{2}+c x+d$ , e) $f(x)=2 \sqrt{x}$ , f) $f(x)=\frac{4}{x}+1$ .

See Solution

Problem 7691

Bestimmen Sie die Ableitung von $f(t)=\sqrt{\frac{2}{t}}$ .

See Solution

Problem 7692

Berechnen Sie die Ableitungen $\mathrm{f}^{\prime}$ für die Funktionen: a) $f(x)=\frac{1}{4} x^{4}-2 x^{2}$ , b) $f(x)=-3 x^{2}+4$ , c) $f(x)=3(x-2)^{2}+x$ , d) $f(x)=a x^{3}+b x^{2}+c x+d$ , e) $f(x)=2 \sqrt{x}$ , f) $f(x)=\frac{4}{x}+1$ .

See Solution

Problem 7693

Gegeben ist die Funktion $f: x \mapsto \frac{1}{x} \cdot \cos (x)$ . Bestimmen Sie die Definitionsmenge, Nullstellen, Grenzwert und zeichnen Sie die Graphen.

See Solution

Problem 7694

Find the derivative of $f(x)=e^{4 x}+x$ at the point $A(0, f(0))$ .

See Solution

Problem 7695

Differentiate and expand:
1. Find $\frac{d}{d x}\left[e^{x}\right]$ .
2. For $f(x)=3 x(x^{2}-1)(x^{3}+2 x+3)$ , expand and differentiate.

See Solution

Problem 7696

Untersuchen Sie die Funktion $f_{t}(x)=\left(x+\frac{1}{t}\right) \cdot e^{-t x}$ auf Nullstellen, Extrempunkte und Wendepunkte.

See Solution

Problem 7697

Find the tangent line equation for $f(x)=e^{-2 x}+0.25^{x}$ at point $A(-1, f(-1))$ using its derivative.

See Solution

Problem 7698

Untersuchen Sie die Funktion $f_{1}(x)=(x+\frac{1}{t}) e^{-t x}$ auf Nullstellen, Extrempunkte und Wendepunkte. Finden Sie eine Gleichung für die Wendepunkte.

See Solution

Problem 7699

Berechnen Sie die Ableitungen $f_{a}^{\prime}(x)$ und $f_{a}^{(21)}(x)$ für die Funktionen: a) $f_{0}(x)=\cos (a x)$ , b) $f_{a}(x)=(a x+5)^{21}$ , c) $f_{a}(x)=\sin \left(-a^{2} x\right)$ .

See Solution

Problem 7700

Gegeben ist die Funktion $y=f_{1}(x)=\left(x+\frac{1}{t}\right) e^{-t x}$ .
a) Untersuchen Sie Nullstellen und Extrempunkte. b) Geben Sie die Gleichung der Wendepunkte an. c) Zeichnen Sie $f_{-1}$ und $f_{0,5}$ . d) Berechnen Sie den Flächeninhalt $A(k)$ und $\lim_{k \to \infty} A(k)$ . e) Bestimmen Sie das Volumen des Rotationskörpers über $0 \leq x \leq 1$ . f) Für $W_{1}\left(\frac{1}{1} ; \frac{2}{1 e}\right)$ : Finden Sie $t$ für Flächeninhalt 2 in $\triangle P_{1} Q_{1} W_{1}$ . g) Bestimmen Sie den Wert für $I$ , so dass $g_{1}$ die Funktion $f_{1}$ in einem Punkt schneidet.

See Solution

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Calculus

Problem 7601

Find the acceleration of a car at t=2t=2t=2 seconds, where its position is given by s(t)=13(t2+5)3/2s(t)=\frac{1}{3}(t^{2}+5)^{3/2}s(t)=31​(t2+5)3/2.

Problem 7602

Find the rate of water draining from a 2000-gallon tank after 10 minutes, using V=2000(1−t40)2V=2000\left(1-\frac{t}{40}\right)^{2}V=2000(1−40t​)2.

Problem 7603

Analyze the concavity of f(x)=3(x−2)5/3f(x)=3(x-2)^{5/3}f(x)=3(x−2)5/3 and identify any points of inflection.

Problem 7604

Given the polynomial f(x)=x3+Ax2+Bxf(x)=x^{3}+A x^{2}+B xf(x)=x3+Ax2+Bx, if x=2x=2x=2 is a critical point and x=−3x=-3x=−3 is an inflection point, find f(1)f(1)f(1). Options: −38-38−38, −36-36−36, −57-57−57, −35-35−35, None.

Problem 7605

Find the first and second derivatives of u(x)=ex+5u(x)=e^{x}+5u(x)=ex+5.

Problem 7606

Analyze the concavity of f(x)=12x4−12x2f(x)=\frac{1}{2} x^{4}-12 x^{2}f(x)=21​x4−12x2 and identify any points of inflection.

Problem 7607

Find x-coordinates of inflection points for f(x)f(x)f(x) given f′′(x)=x2−25(x−6)3(x−10)3f''(x)=\frac{x^{2}-25}{(x-6)^{3}(x-10)^{3}}f′′(x)=(x−6)3(x−10)3x2−25​.

Problem 7608

Find the xxx-coordinates of points of inflection for the function with f′′(x)=x2−25(x−6)3(x−10)3f^{\prime \prime}(x)=\frac{x^{2}-25}{(x-6)^{3}(x-10)^{3}}f′′(x)=(x−6)3(x−10)3x2−25​.

Problem 7609

Determine the concavity of f(x)=xx−3f(x)=\frac{x}{x-3}f(x)=x−3x​ and find any points of inflection.

Problem 7610

Find ccc such that the function f(x)=cx2+x−2f(x)=c x^{2}+x^{-2}f(x)=cx2+x−2 has an inflection point at (2,f(2))(2, f(2))(2,f(2)).

Problem 7611

Calculate the integral: ∫8x2cos⁡(x) dx=\int 8 x^{2} \cos (x) \, dx =∫8x2cos(x)dx=

Problem 7612

Which statements about the graph of f(x)=6x3−12x2+6x+4f(x)=6 x^{3}-12 x^{2}+6 x+4f(x)=6x3−12x2+6x+4 are true regarding local extrema and intervals?

Problem 7613

Describe the concavity of f(x)=xx+4f(x)=\frac{x}{x+4}f(x)=x+4x​ and find any points of inflection.

Problem 7614

Select a suitable uuu for integration by parts without evaluating the integral: ∫e3xcos⁡(5x)dx\int e^{3 x} \cos (5 x) d x∫e3xcos(5x)dx

Problem 7615

Calculate the integral of ln⁡(4x+1)\ln(4x+1)ln(4x+1) with respect to xxx: ∫ln⁡(4x+1) dx=+C\int \ln(4x+1) \, dx = +C∫ln(4x+1)dx=+C

Problem 7616

Determine the concavity and points of inflection for f(x)=14x4−6x2f(x)=\frac{1}{4} x^{4}-6 x^{2}f(x)=41​x4−6x2.

Problem 7617

Problem 7618

Graph the linear approximation of f(x)=2f(x) = 2f(x)=2 at a=2a=2a=2 using f~(x)=f(a)+f′(a)(x−a)\tilde{f}(x)=f(a)+f^{\prime}(a)(x-a)f~​(x)=f(a)+f′(a)(x−a) for x=3x=3x=3.

Problem 7619

Find the x-coordinates of all points of inflection for the function f(x)f(x)f(x) given f′′(x)=x3−2x2(x−5)3(x−6)3f^{\prime \prime}(x)=\frac{x^{3}-2 x^{2}}{(x-5)^{3}(x-6)^{3}}f′′(x)=(x−5)3(x−6)3x3−2x2​.

Problem 7620

Estimate the change in area of a circular oil spill as the radius increases from 6 ft to 6.023 ft. Calculate new area too.

Problem 7621

Find the linear approximation L(x)L(x)L(x) of f(x)=sin⁡(x)f(x)=\sin (x)f(x)=sin(x) at a=11π6a=\frac{11 \pi}{6}a=611π​. Enter the exact answer.

Problem 7622

Find the derivative of π3(24r2−r3)\frac{\pi}{3}(24 r^{2}-r^{3})3π​(24r2−r3).

Problem 7623

Find the base radius rrr and height hhh of a cone (with r+h=24r+h=24r+h=24 cm) that maximize volume, then calculate max volume.

Problem 7624

What should we set dvd vdv to when using integration by parts to evaluate ∫xln⁡(x)dx\int x \ln (x) d x∫xln(x)dx?

Problem 7625

Estimate 25.05\sqrt{25.05}25.05​ using linear approximation. Round your answer to three decimal places.

Problem 7626

Find the volume of the solid formed by rotating the region RRR (bounded by y=0y=0y=0, y=xy=\sqrt{x}y=x​, and x=4x=4x=4) around the xxx-axis using the disk method.

Problem 7627

Given the function y=2x3+3x2−36x+1y=2 x^{3}+3 x^{2}-36 x+1y=2x3+3x2−36x+1, find local max/min using the second derivative test and determine concavity intervals.

Problem 7628

Find the first and second derivatives of h(x)=blog⁡b(x)h(x)=b^{\log _{b}(x)}h(x)=blogb​(x) for b>1b > 1b>1.

Problem 7629

Find the derivative of y=tan⁡(π4−x2)y=\tan \left(\frac{\pi}{4}-\frac{x}{2}\right)y=tan(4π​−2x​) with respect to xxx.

Problem 7630

Find the volume of the solid formed by rotating the region RRR (bounded by y=2y=2y=2, y=xy=\sqrt{x}y=x​, x=0x=0x=0) around the yyy-axis.

Problem 7631

Find the xxx-coordinate of the inflection point for the curve y=f(x)=53x3+25x2−48x−196y=f(x)=\frac{5}{3} x^{3}+25 x^{2}-48 x-196y=f(x)=35​x3+25x2−48x−196.

Problem 7632

Choose uuu from x3x^3x3, cos⁡(x)\cos(x)cos(x), x6x^6x6, x3sin⁡(x)x^3\sin(x)x3sin(x), sin⁡(x)\sin(x)sin(x), or xxx to evaluate ∫x6sin⁡(x)dx\int x^{6} \sin (x) dx∫x6sin(x)dx using integration by parts.

Problem 7633

Differentiate: y=tan⁡(π4−x2)y=\tan \left(\frac{\pi}{4}-\frac{x}{2}\right)y=tan(4π​−2x​)

Problem 7634

Find the first and second derivatives of TR⁡(Q)=Q⋅(15−Q3)2\operatorname{TR}(Q)=Q \cdot\left(\frac{15-Q}{3}\right)^{2}TR(Q)=Q⋅(315−Q​)2.

Problem 7635

Find the first and second derivatives of TR(Q)=−3P+15⋅(15−Q3)2TR(Q)=-3 \sqrt{P}+15 \cdot\left(\frac{15-Q}{3}\right)^{2}TR(Q)=−3P​+15⋅(315−Q​)2.

Problem 7636

Estimate f(2.02)f(2.02)f(2.02) for y=f(x)=5x2−2x+3y=f(x)=5 x^{2}-2 x+3y=f(x)=5x2−2x+3 using f′(x)f^{\prime}(x)f′(x): 10.02×[f(2.02)−\frac{1}{0.02} \times[f(2.02)-0.021​×[f(2.02)−

Problem 7637

Estimate Δy\Delta yΔy for y=sin⁡(5x)y=\sin(5x)y=sin(5x) with Δx=0.3\Delta x=0.3Δx=0.3 at x=0x=0x=0 using linear approximation. Find the percentage error.

Problem 7638

Estimate the paint needed in cm³ for a 0.07 cm thick coat on a hemispherical dome with a diameter of 7500 cm. Use V=23πr3V=\frac{2}{3} \pi r^{3}V=32​πr3.

Problem 7639

Estimate f(2.02)f(2.02)f(2.02) for y=f(x)=5x2−2x+3y=f(x)=5 x^{2}-2 x+3y=f(x)=5x2−2x+3 using f′(x)f^{\prime}(x)f′(x): 10.02×[f(2.02)−□]≈□\frac{1}{0.02} \times[f(2.02)-\square] \approx \square0.021​×[f(2.02)−□]≈□

Problem 7640

A spherical ball bearing has a diameter of 5 mm5 \mathrm{~mm}5 mm and a possible error of 0.1 mm0.1 \mathrm{~mm}0.1 mm. Find: a. Maximum error in volume using differentials. b. Relative error in volume using differentials.

Find the acceleration of a car at $t=2$ seconds, where its position is given by $s(t)=\frac{1}{3}(t^{2}+5)^{3/2}$ .

Find the rate of water draining from a 2000-gallon tank after 10 minutes, using $V=2000\left(1-\frac{t}{40}\right)^{2}$ .

Analyze the concavity of $f(x)=3(x-2)^{5/3}$ and identify any points of inflection.

Given the polynomial $f(x)=x^{3}+A x^{2}+B x$ , if $x=2$ is a critical point and $x=-3$ is an inflection point, find $f(1)$ . Options: $-38$ , $-36$ , $-57$ , $-35$ , None.

Find the first and second derivatives of $u(x)=e^{x}+5$ .

Analyze the concavity of $f(x)=\frac{1}{2} x^{4}-12 x^{2}$ and identify any points of inflection.

Find x-coordinates of inflection points for $f(x)$ given $f''(x)=\frac{x^{2}-25}{(x-6)^{3}(x-10)^{3}}$ .

Find the $x$ -coordinates of points of inflection for the function with $f^{\prime \prime}(x)=\frac{x^{2}-25}{(x-6)^{3}(x-10)^{3}}$ .

Determine the concavity of $f(x)=\frac{x}{x-3}$ and find any points of inflection.

Find $c$ such that the function $f(x)=c x^{2}+x^{-2}$ has an inflection point at $(2, f(2))$ .

Calculate the integral: $\int 8 x^{2} \cos (x) \, dx =$

Which statements about the graph of $f(x)=6 x^{3}-12 x^{2}+6 x+4$ are true regarding local extrema and intervals?

Describe the concavity of $f(x)=\frac{x}{x+4}$ and find any points of inflection.

Select a suitable $u$ for integration by parts without evaluating the integral: $\int e^{3 x} \cos (5 x) d x$

Calculate the integral of $\ln(4x+1)$ with respect to $x$ : $\int \ln(4x+1) \, dx = +C$

Determine the concavity and points of inflection for $f(x)=\frac{1}{4} x^{4}-6 x^{2}$ .

Graph the linear approximation of $f(x) = 2$ at $a=2$ using $\tilde{f}(x)=f(a)+f^{\prime}(a)(x-a)$ for $x=3$ .

Find the x-coordinates of all points of inflection for the function $f(x)$ given $f^{\prime \prime}(x)=\frac{x^{3}-2 x^{2}}{(x-5)^{3}(x-6)^{3}}$ .

Find the linear approximation $L(x)$ of $f(x)=\sin (x)$ at $a=\frac{11 \pi}{6}$ . Enter the exact answer.

Find the derivative of $\frac{\pi}{3}(24 r^{2}-r^{3})$ .

Find the base radius $r$ and height $h$ of a cone (with $r+h=24$ cm) that maximize volume, then calculate max volume.

What should we set $d v$ to when using integration by parts to evaluate $\int x \ln (x) d x$ ?

Estimate $\sqrt{25.05}$ using linear approximation. Round your answer to three decimal places.

Find the volume of the solid formed by rotating the region $R$ (bounded by $y=0$ , $y=\sqrt{x}$ , and $x=4$ ) around the $x$ -axis using the disk method.

Given the function $y=2 x^{3}+3 x^{2}-36 x+1$ , find local max/min using the second derivative test and determine concavity intervals.

Find the first and second derivatives of $h(x)=b^{\log _{b}(x)}$ for $b > 1$ .

Find the derivative of $y=\tan \left(\frac{\pi}{4}-\frac{x}{2}\right)$ with respect to $x$ .

Find the volume of the solid formed by rotating the region $R$ (bounded by $y=2$ , $y=\sqrt{x}$ , $x=0$ ) around the $y$ -axis.

Find the $x$ -coordinate of the inflection point for the curve $y=f(x)=\frac{5}{3} x^{3}+25 x^{2}-48 x-196$ .

Choose $u$ from $x^3$ , $\cos(x)$ , $x^6$ , $x^3\sin(x)$ , $\sin(x)$ , or $x$ to evaluate $\int x^{6} \sin (x) dx$ using integration by parts.

Differentiate: $y=\tan \left(\frac{\pi}{4}-\frac{x}{2}\right)$

Find the first and second derivatives of $\operatorname{TR}(Q)=Q \cdot\left(\frac{15-Q}{3}\right)^{2}$ .

Find the first and second derivatives of $TR(Q)=-3 \sqrt{P}+15 \cdot\left(\frac{15-Q}{3}\right)^{2}$ .

Estimate $f(2.02)$ for $y=f(x)=5 x^{2}-2 x+3$ using $f^{\prime}(x)$ : $\frac{1}{0.02} \times[f(2.02)-$

Estimate $\Delta y$ for $y=\sin(5x)$ with $\Delta x=0.3$ at $x=0$ using linear approximation. Find the percentage error.

Estimate the paint needed in cm³ for a 0.07 cm thick coat on a hemispherical dome with a diameter of 7500 cm. Use $V=\frac{2}{3} \pi r^{3}$ .

Estimate $f(2.02)$ for $y=f(x)=5 x^{2}-2 x+3$ using $f^{\prime}(x)$ : $\frac{1}{0.02} \times[f(2.02)-\square] \approx \square$

A spherical ball bearing has a diameter of $5 \mathrm{~mm}$ and a possible error of $0.1 \mathrm{~mm}$ . Find: a. Maximum error in volume using differentials. b. Relative error in volume using differentials.

Estimate the paint needed for a hemispherical dome (diameter 4000 cm, thickness 0.09 cm). Use $V=\frac{2}{3} \pi r^{3}$ .

Find the max error in volume of a ball bearing with diameter $6.8 \mathrm{~mm}$ and error $0.09 \mathrm{~mm}$ using differentials.

A ball bearing's diameter is $6.8 \mathrm{~mm}$ with a $0.09 \mathrm{~mm}$ error. Find max error and relative error in volume.

Given $f(x)=7-1 x+18 x^{2}$ , find $f(a)$ , $f(a+h)$ , and $\frac{f(a+h)-f(a)}{h}$ for $h \neq 0$ .

Given $y=2 x^{2}$ , calculate $\Delta y$ for $x=4$ , $\Delta x=0.4$ and find $d y$ for $d x=0.4$ .

Find $d y$ for $y=\tan(3x+4)$ at $x=1$ for $d x=0.3$ and $d x=0.6$ .

A ball bearing has a diameter of $5.8 \mathrm{~mm}$ and a possible error of $0.1 \mathrm{~mm}$ . Find the max and relative error in volume.

Bestimmen Sie die Ableitung von $f(x)=3 x^{5}+0,5 x^{3}+x-12$ .

Bestimmen Sie die Ableitung von $f(x)=-2 x^{4}+\frac{1}{3} x^{3}-0,2 x+c$ .

Analysiere die Funktion $f(x)=x^{3}-2 x^{2}-3 x$ : Finde Nullstellen, zeichne sie, und bestimme Ableitung und Integral.

How fast is the water level in Jon's $16 \mathrm{ft}^{2}$ bathtub rising if filled at $0.5 \mathrm{ft}^{3}/\mathrm{min}$ ?

The oil slick's radius grows at $3 \mathrm{~m/min}$ . Find the area increase rate when the radius is $59 \mathrm{~m}$ . $\frac{d A}{d t} =$

An oil slick's radius grows at $3 \mathrm{~m/min}$ . Find area growth rate when radius is $59 \mathrm{~m}$ and after $4$ min.

A ladder of length 8.9 m slides down a wall at $1.1 \mathrm{~m/s}$ . Find $\frac{d x}{d t}$ when $h=3.8$ .

Bestimme den Steigungswinkel der Funktion $f(x)=3 x^{2}+4 x-3$ an der Stelle $x_{0}=2$ .

A police car moves south at $160 \mathrm{~km/h}$ and a truck east at $140 \mathrm{~km/h}$ . Find the distance change rate at $t=5$ min.

Untersuchen Sie das Verhalten von $f(x)=\frac{6 x-x^{2}}{x^{3}}$ für $x \to +\infty$ und $x \to -\infty$ .

Find the derivative of $f(x)=\sqrt{x}-\frac{4}{x^{3}}$ at $x=1$ . Also, clarify the issue with $f(x)=x^{2}-8x+7$ .

Die Kochdauer $t$ (in min) für ein Ei hängt vom Durchmesser $d$ (in $\mathrm{mm}$ ) ab: $t(d)=0,00262 d^{2}$ . Berechne die mittleren Änderungsraten für [40; 50], [40; 45], [40; 41] und die lokale Änderungsrate bei 40 mm.

Berechnen Sie die mittlere Änderungsrate von $f$ in den Intervallen: a) $f(x)=2x$ , $I=[0;1]$ ; b) $f(x)=0,5x^{2}$ , $I=[1;4]$ ; c) $f(x)=1-x^{2}$ .

A baseball diamond is a square with sides of $90 \mathrm{ft}$ . A player runs to first base at $23 \mathrm{ft/s}$ . Find the rate of change of distance to second base when halfway to first. Give answer to two decimal places. $\frac{d h}{d t} =$

Calculate $\int_{-2}^{2} x^{2} dx + \int_{3}^{5} x^{2} dx + \int_{2}^{3} x^{2} dx$ .

Differentiate the function by expanding: $f(x)=(4x+3)(3x^{2}+1)$ .

Find the derivative of the function $f(w)=\frac{7 w^{3}-3 w}{8 w}$ after simplifying it.

Differentiate the function by expanding: $f(x)=(5 x-6)^{2}$ .

Calculate the integrals: $\int_{-2}^{2} x^{2} dx + \int_{3}^{5} x^{2} dx + \int_{2}^{3} x^{2} dx$ .

Find the derivative of the function $f(w) = 3w^{4} + 6w^{2} + 5w$ .

Find the derivative of $g(w)=\frac{6 e^{2 w}+7 e^{w}}{e^{w}}$ after simplifying the expression.

Find values of $K$ for a local minimum at the origin for $f(x, y)=K x^{2}+8 x y+2 K y^{2}$ . Options: $|K|<2\sqrt{2}$ , $|K|>\sqrt{2}$ , $K>\sqrt{2}$ , $K>2\sqrt{2}$ , $K<\sqrt{2}$ , None, $K<-2$ .

Find the global minimum of $f(x, y)=e^{3 x-5 y+2 z}$ for $-1 \leq x, y, z \leq 1$ .

Find the tangent line equation at $x=\ln 13$ for the curve $y=e^{x}$ and graph both.