Calculus

Problem 10701

Find the derivatives of these functions: (a) $f(x)=\frac{\sin x+\cos x}{\tan x-1}$ (b) $f(x)=e^{x} x^{e}-e^{2} \ln (x^{2})$ (c) $f(x)=(1+x^{2})^{x^{3}+x}$ (use logarithmic differentiation) (d) $f(x)=\cos (\sin^{3} x+x^{4} \cos x)$

See Solution

Problem 10702

Find the second derivative of $f(x)=x^{2} e^{2 x}(x^{-2}+e^{-2 x})$ and calculate $f^{\prime \prime}(1)$ . Simplify $f$ first.

See Solution

Problem 10703

Bestimmen Sie die Ableitung der Funktion $g(x)=a \cdot x^{2}+c$ an der Stelle $x_0$ .

See Solution

Problem 10704

A toy rocket is launched from a 94 ft building with an initial velocity of 250 ft/s. Find the height function, max height/time, when above 575 ft, and time to hit the ground.

See Solution

Problem 10705

Bacteria growth is given by $n(t)=985 e^{0.25 t}$ . Find the growth rate, initial population, and $n(5)$ .

See Solution

Problem 10706

Bestimmen Sie die Ableitung von $g(x) = 2 a x$ an $x_{0}=2$ direkt und mit der Definition. Vervollständigen Sie die Berechnung.

See Solution

Problem 10707

A toy rocket is launched from a 99 ft building at 202 ft/s. Find the height function, max height/time, and when it hits the ground.

See Solution

Problem 10708

A toy rocket launches from a 58 ft building at 150 ft/s. Find: a) height function $h(t)$ , b) max height & time, c) time above 273 ft, d) time to hit ground.

See Solution

Problem 10709

A toy rocket is launched from a 58 ft tall building with an initial velocity of 150 ft/s. Find the height function, max height/time, time above 273 ft, and time to hit the ground.

See Solution

Problem 10710

A toy rocket is launched from a 139 ft building at 203 ft/s. Find the height function, max height/time, time > 395 ft, and ground hit time.

See Solution

Problem 10711

Find $\int_{0}^{2} f(x) \mathrm{d} x$ given: $3\int_{0}^{1} f(x) \mathrm{d} x + 2\int_{1}^{2} f(x) \mathrm{d} x = 7$ and $\int_{0}^{2} f(x) \mathrm{d} x + \int_{1}^{2} f(x) \mathrm{d} x = 1$ . Options: (a) -1 (b) 0 (c) $\frac{1}{2}$ (d) 2.

See Solution

Problem 10712

Evaluate the limits: (a) $\lim _{x \rightarrow-3} \frac{x|x+3|}{x^{2}+x-6}$ and (b) $\lim _{x \rightarrow \pi / 2} \frac{\sin \left(\cos ^{2} x\right)}{2 \cos (x)}$ .

See Solution

Problem 10713

Find $h+k$ if $\lim _{x \rightarrow-1} \frac{x^{2}-2 x-h}{k x+2}=-2$ .

See Solution

Problem 10714

Untersuchen Sie die Funktion $f(x)=x^{5}-15 x^{3}-30$ auf Extrem- und Wendepunkte.

See Solution

Problem 10715

Untersuchen Sie die Extrem- und Sattelstellen der Funktion $f(x)=x^{5}-15 x^{3}-30$ .

See Solution

Problem 10716

Find the limit: $\lim _{x \rightarrow+\infty}\left(\sqrt[3]{x^{3}+3 x}-\sqrt{x^{2}+x-1}\right)$ .

See Solution

Problem 10717

Untersuchen Sie die Extrem- und Sattelstellen der Funktionen: a) $f(x)=x^{2}-4$ b) $f(x)=\frac{1}{3} x^{3}-2 x^{2}+4 x$ c) $f(x)=x^{3}+3 x^{2}+1$ d) $f(x)=\frac{1}{9} x^{3}-x^{2}+3 x$ e) $f(t)=t^{4}-2 t^{2}$ f) $f(a)=-a^{3}+2,5 a^{2}+3,5$ g) $f(x)=x^{5}-15 x^{3}-30$ h) $f(x)=-3 x^{4}+4 x^{3}+8$

See Solution

Problem 10718

Determine the horizontal asymptote of $f(x)=\frac{4 x^{3}-8 x-8}{9 x^{3}-9 x+8}$ .

See Solution

Problem 10719

Bestimme die Ableitung der Funktionen: a) $f(x)=e^{7 x}$ , b) $f(x)=e^{-0,2 x}$ , c) $f(x)=3 \cdot e^{2 x}$ , d) $f(x)=0,5 \cdot e^{-4 x}$ , e) $f(x)=e^{x^{2}}$ , f) $f(x)=4 \cdot e^{2 x-3}$ , g) $f(x)=\frac{1}{e^{x}}$ , h) $f(x)=e^{-\frac{2}{3} x^{3}}$ , i) $f(x)=0,2 \cdot e^{1-5 x}$ , j) $f(x)=-\frac{1}{10 \cdot e^{5 x+1}}$ , k) $f(x)=e^{-4 x+2 x^{2}}$ , l) $f(x)=\frac{5}{e^{3 x-x^{3}}}$ .

See Solution

Problem 10720

Approximate $\int_{0}^{1} e^{t^{2}} dt$ using the degree 3 Taylor polynomial of $f(x)=\int_{0}^{x} e^{t^{2}} dt$ .

See Solution

Problem 10721

Find the difference quotient for $f(x) = x^{3}$ using $\frac{f(x+\Delta x)-f(x)}{\Delta x}$ .

See Solution

Problem 10722

If a function is a power series, is its Taylor polynomial identical to the series?

See Solution

Problem 10723

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{8 x}-1-8 x}{x^{2}}$ . Choose A (exact answer) or B (limit does not exist).

See Solution

Problem 10724

Evaluate the integral $\iiint_{\square} f(x, y, z) d V$ where $f(x, y, z)=x z^{2}$ and $\square = [0,2] \times[3,10] \times[4,9]$ .

See Solution

Problem 10725

Bestimmen Sie die Wendepunkte der Funktion $f_{t}(x)=x^{3}-3 t x^{2}$ für $t>0$ .

See Solution

Problem 10726

Find the slope of the tangent line to the curve $x^{2}-x y+y^{2}=19$ at the point $(2,5)$ .

See Solution

Problem 10727

Find the slope of the relation $y^{2}(4-x)=x^{2}$ at the point where $x=3$ and $y=-3$ .

See Solution

Problem 10728

Find the rate of change of $\sin y + x = \frac{7}{2}$ at the point $\left(3, \frac{\pi}{6}\right)$ .

See Solution

Problem 10729

Find the derivative $\frac{d y}{d x}$ for the equation $3 \sqrt[3]{x}-12 \sqrt[3]{y^{4}}=9$ .

See Solution

Problem 10730

Find the derivative $\frac{d y}{d x}$ for the equation $\tan (x y) = x + y$ .

See Solution

Problem 10731

Find $\frac{d x}{d t}$ for the circle $x^{2}+y^{2}=16$ at $(2,y)$ when $\frac{d y}{d t}=-3$ .

See Solution

Problem 10732

The function $f(t)=\frac{108,000}{1+5000 e^{-t}}$ models flu cases. Find: a) initial cases, b) cases after 4 weeks, c) max cases.

See Solution

Problem 10733

Zeige, dass $F(x)=(x+2) \cdot e^{x}$ eine Stammfunktion von $f(x)=(x+3) \cdot e^{x}$ ist.

See Solution

Problem 10734

Zeige, dass $F(x)=-3 x^{3} \cdot e^{x}$ eine Stammfunktion von $f(x)=\left(-9 x^{2}-3 x^{3}\right) \cdot e^{x}$ ist.

See Solution

Problem 10735

Differentiate $\sin(x^{6}+4)$ with respect to $x$ .

See Solution

Problem 10736

Find the derivative of $y=\sin^{6} x$ . What is $\frac{dy}{dx}=?$

See Solution

Problem 10737

Find the derivative of $y=\sin^{6} x$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 10738

Find the derivative of $y=\frac{3 \cos x}{9-8 \cos x}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 10739

Evaluate the integral: $\int e^{-x^{2}} d x$

See Solution

Problem 10740

Find the derivative of $y=(4x^4-4x^2+3)^4$ . What is $\frac{dy}{dx}$ ?

See Solution

Problem 10741

Find the critical numbers of the function $g(x) = x^{1/7} - x^{-6/7}$ .

See Solution

Problem 10742

Find the absolute extrema of $f(x)=x^{3}-4$ on $[-5,7]$ . Provide your answer as an ordered pair $(x, f(x))$ .

See Solution

Problem 10743

Find the derivative of $y=-2(7x^{2}+3)^{-6}$ . What is $\frac{dy}{dx}$ ?

See Solution

Problem 10744

Find the absolute extrema of $f(x)=-6 x^{3}+9 x^{2}+36 x$ on $[-3,4]$ . Provide $(x, f(x))$ as an ordered pair.

See Solution

Problem 10745

Find the derivative of $y=4 e^{2x+2}$ . What is $\frac{dy}{dx}=?$

See Solution

Problem 10746

Find the absolute extrema of $f(x)=-4 x^{3}-12 x^{2}+288 x$ on $[-8,6]$ . Provide $(x, f(x))$ as the answer.

See Solution

Problem 10747

Find the derivative of the function $f(x)=8 \sqrt{7 x^{2}+3}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 10748

Find the derivative of the function $r(t)=\frac{(7 t-4)^{7}}{4 t^{2}+9}$ . What is $r^{\prime}(t)$ ?

See Solution

Problem 10749

Find the derivative of the function $y=x^{10} e^{x}$ . What is $y^{\prime}=?$

See Solution

Problem 10750

Differentiate the function $f(x)=-4 e^{3 x}$ . What is $\frac{d}{d x}\left(-4 e^{3 x}\right)$ ?

See Solution

Problem 10751

Find the tangent line equation for $f(x)=\sqrt{x^{2}+11}$ at $x=5$ : $y=\square$ (use $x$ ).

See Solution

Problem 10752

Find the area between $y=\cos ^{2} x$ and $y=\frac{8 x^{2}}{\pi^{2}}$ from $x=-\frac{\pi}{2}$ to $\frac{\pi}{2}$ .

See Solution

Problem 10753

Find the area between the curves $y=\cos^{2} x$ and $y=\frac{8 x^{2}}{\pi^{2}}$ using their intersection points.

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Problem 10754

Calculate the integral $\int_{-4}^{4}\left(t^{2}-3\right) dt$ .

See Solution

Problem 10755

Find the derivative of $y$ where $y=(4x^{2}-8x+8)e^{-6x}$ . What is $y'$ ?

See Solution

Problem 10756

Find coordinates of point $P$ where the tangent to $f(x)=x^{2} e^{-x}$ is horizontal.

See Solution

Problem 10757

Find the derivative of the function $y=e^{x \sqrt{9 x+1}}$ . What is $\frac{d y}{d x}=?$

See Solution

Problem 10758

Evaluate the integral $\int_{0}^{2}(2-t) \sqrt{t} \, dt$ .

See Solution

Problem 10759

Find all values of $c$ in $[3,6]$ for $f(x)=x-\frac{1}{x}$ that satisfy the Mean-Value Theorem.

See Solution

Problem 10760

Find all values of $c$ in $[3,6]$ for $f(x)=x-\frac{1}{x}$ that satisfy the Mean-Value Theorem.

See Solution

Problem 10761

A book of mass 1.68 kg slides down a table at an angle of 39.7° with kinetic friction $\mu_{k}=0.360$ . Find its speed at the bottom of a 1.26 m table.

See Solution

Problem 10762

Exercice 2(8 pt) : Soit $h(x)=\operatorname{Arctan}\left(\frac{x}{\sqrt{1+x^{2}}+1}\right)$ 1) Montrer que $h$ est impaire, et calculer $\lim _{x \rightarrow+\infty} h(x)$ et $\lim _{x \rightarrow 0} \frac{h(x)}{x}$ 2) Montrer que $\frac{x}{\sqrt{x^{2}+1}+1}=\tan \left(\frac{\theta}{2}\right)$ avec $\theta=\operatorname{Arctan}(x)$ .

See Solution

Problem 10763

Find the limit as $x$ approaches 1 from the left of $\arcsin^{\prime}(x)$ . What does this indicate about the graph of $y=\arcsin x$ ?

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Problem 10764

Set up an integral using the Disk or Washer Method to find the volume of the solid formed by rotating the region bounded by $y=x^{2}+1$ , $y=19-x^{2}$ , and $x=0$ about the $x$ -axis. Do not evaluate the integral.

See Solution

Problem 10765

Soit $h(x)=\operatorname{Arctan}\left(\frac{x}{\sqrt{1+x^{2}}+1}\right)$ . Montrez que $h$ est impaire et calculez les limites.

See Solution

Problem 10766

Find the tangent line equation for $f(x)=\frac{1}{1+x^{2}}$ at $x=3$ with slope $-\frac{3}{50}$ .

See Solution

Problem 10767

Find the first-order partial derivatives of these functions: (a) $f(x, y)=\cos (2 x) \sin (2 x)$ (b) $f(x, y)=\ln (2 x+y^{2}+x y)$ (c) $f(x, y)=\frac{x y}{x+y}$

See Solution

Problem 10768

Find the $3 \times 3$ Hessian matrix for $f(x, y, z)=x^{2} y^{3} z-3 x y^{2} z^{3}+x^{2} z$ .

See Solution

Problem 10769

Find the tangent line equation for $f(x)=\frac{1}{3+x^{2}}$ at $x=3$ with slope $-\frac{1}{24}$ .

See Solution

Problem 10770

Find the tangent line equation for $f(x)=\frac{1}{7+x^{2}}$ at $x=3$ with slope $-\frac{3}{128}$ .

See Solution

Problem 10771

Das Profil eines Vulkans wird durch $f(x)=\frac{25}{x^{2}}$ beschrieben. Wie hoch ist der Vulkan bei $x=2$ und $x=6$ ?

See Solution

Problem 10772

How long for insulin to reduce to half if it breaks down by $5\%$ each minute from a 10-unit dose?

See Solution

Problem 10773

Verify that $A(y)=\pi(2+0.5y)^{2}$ matches the table values, then find the most accurate volume using $A(y)$ .

See Solution

Problem 10774

Use Newton's method to find a root of $f(x)=3 x^{2}-\sqrt{x}-3$ starting from $x_{0}=1$ . Create a table of approximations.

See Solution

Problem 10775

Approximate a root of the function $f(x)=7 x^{2}-\sqrt{x}-2$ using Newton's method with $x_{0}=1$ . Show your work in a table.

See Solution

Problem 10776

Find the critical numbers of the function $h(t)=t^{3/4}-6t^{1/4}$ .

See Solution

Problem 10777

Estimate the volume of water in cubic inches when the depth is 6 inches using the area function $A(y)=\pi(2+0.5 y)^{2}$ .

See Solution

Problem 10778

Calculate the relative rate of change of $f(x)=6 x^{2}-\ln x$ at $x=3$ . Round your answer to three decimal places.

See Solution

Problem 10779

Bestimmen Sie die marginal-benefit-Funktion aus der Total-Benefit-Funktion $T B(Q)=108 \cdot \sqrt{Q}$ .

See Solution

Problem 10780

Find the limit: $\lim _{h \rightarrow 0} \frac{\sin ^{2}(3 x+3 h)-\sin ^{2}(3 x)}{h}$ .

See Solution

Problem 10781

Evaluate the integral $\int_{0}^{2}(x^{3}-24 x^{2}+144 x) d x$ .

See Solution

Problem 10782

Finde die Funktion $f$ für die Zuflussgeschwindigkeit nach einem Gewitter und beantworte Fragen zu Zunahme, Höchstwert und Zeitraum.

See Solution

Problem 10783

Find the derivative of the function $\frac{x}{\sqrt{x^{2}+9}}$ .

See Solution

Problem 10784

Bestimmen Sie die Funktion $f$ für die Zuflussgeschwindigkeit nach einem Gewitter und beantworten Sie Fragen zu Zunahme und Zeitintervallen.

See Solution

Problem 10785

Fruit fly growth model: $N(t)=\frac{400}{1+19 e^{-0.2 t}}$ . Answer: A. Initial count? B. Limiting count? C. Max growth time? D. Growth rate after 2 days?

See Solution

Problem 10786

Find the $3 \times 3$ Hessian matrix for the function $f(x, y, z)=x^{2} y^{3} z-3 x y^{2} z^{3}+x^{2} z$ .

See Solution

Problem 10787

Find the critical numbers of the function $g(x)=x^{1/9}-x^{-8/9}$ .

See Solution

Problem 10788

Determine the rates of change for $f(x) = x^3 - 18x^2 + 33x - 10$ at $x=0.7$ and $x=2$ .

See Solution

Problem 10789

Find the rate of change of the volume of a cone with height $10 \mathrm{~cm}$ and radius $20 \mathrm{~cm}$ , given rates of $3 \mathrm{~cm/min}$ and $2 \mathrm{~cm/min}$ .

See Solution

Problem 10790

Estimate the fox population in 2008 given a continuous growth rate of 8% and a 2000 population of 14900. Use $P(t) = P_0 e^{rt}$ .

See Solution

Problem 10791

Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ for $z = y z^{2} + x \ln y$ using implicit differentiation.

See Solution

Problem 10792

Evaluate the integral $\int_{1}^{3} \frac{1}{(x-2)^{2}} dx$ by splitting at the singularity $x=2$ . Fill in the missing expressions.

See Solution

Problem 10793

A company's revenue is $R(q)=-q^{3}+280 q^{2}$ and cost is $C(q)=330+16 q$ .
A) Find the marginal profit function $M P(q)=$ . B) Determine the quantity (in hundreds) for maximum profit (round to two decimal places).
Answer: hundred units must be sold.

See Solution

Problem 10794

Find the tangent line equation for $f(x)=\frac{2}{\sqrt[4]{x^{3}}}$ at point $(1,2)$ .

See Solution

Problem 10795

Evaluate the integral $\int_{1}^{3} \frac{1}{(x-2)^{2}} d x$ using limits as $s \rightarrow 2$ and $t \rightarrow 2$ .

See Solution

Problem 10796

Find the production level $x$ that minimizes the marginal cost $C(x)=x^{2}-160x+7200$ and its minimum value.

See Solution

Problem 10797

Find $\int f(x) dx$ for $f(x)=\frac{3x+1}{x+2}$ . Also, find the derivative of $y=x^2(4x+3)$ at $x=2$ .

See Solution

Problem 10798

Find the maximum height of a 1 kg ball launched at 40 m/s as $k \to 0$ in $M(k)=\frac{40 k-9.8 \ln \left(\frac{200 k}{49}+1\right)}{k^{2}}$ . (2 decimal places)

See Solution

Problem 10799

Given the profit function $P(q)=-0.03 q^{2}+4 q-31$ , find the marginal profit $M P(q)$ , the quantity $q$ for max profit, and max profit value.

See Solution

Problem 10800

Evaluate $\lim_{x \rightarrow 1} n f(x)$ for $f(x)=\frac{\sqrt{x}-1}{x-1}$ . Options: a. $7x-1$ , b. 32.

See Solution

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Calculus

Problem 10701

Problem 10702

Find the second derivative of f(x)=x2e2x(x−2+e−2x)f(x)=x^{2} e^{2 x}(x^{-2}+e^{-2 x})f(x)=x2e2x(x−2+e−2x) and calculate f′′(1)f^{\prime \prime}(1)f′′(1). Simplify fff first.

Problem 10703

Bestimmen Sie die Ableitung der Funktion g(x)=a⋅x2+cg(x)=a \cdot x^{2}+cg(x)=a⋅x2+c an der Stelle x0x_0x0​.

Problem 10704

A toy rocket is launched from a 94 ft building with an initial velocity of 250 ft/s. Find the height function, max height/time, when above 575 ft, and time to hit the ground.

Problem 10705

Bacteria growth is given by n(t)=985e0.25tn(t)=985 e^{0.25 t}n(t)=985e0.25t. Find the growth rate, initial population, and n(5)n(5)n(5).

Problem 10706

Bestimmen Sie die Ableitung von g(x)=2axg(x) = 2 a xg(x)=2ax an x0=2x_{0}=2x0​=2 direkt und mit der Definition. Vervollständigen Sie die Berechnung.

Problem 10707

A toy rocket is launched from a 99 ft building at 202 ft/s. Find the height function, max height/time, and when it hits the ground.

Problem 10708

A toy rocket launches from a 58 ft building at 150 ft/s. Find: a) height function h(t)h(t)h(t), b) max height & time, c) time above 273 ft, d) time to hit ground.

Problem 10709

A toy rocket is launched from a 58 ft tall building with an initial velocity of 150 ft/s. Find the height function, max height/time, time above 273 ft, and time to hit the ground.

Problem 10710

A toy rocket is launched from a 139 ft building at 203 ft/s. Find the height function, max height/time, time > 395 ft, and ground hit time.

Problem 10711

Problem 10712

Problem 10713

Find h+kh+kh+k if lim⁡x→−1x2−2x−hkx+2=−2\lim _{x \rightarrow-1} \frac{x^{2}-2 x-h}{k x+2}=-2limx→−1​kx+2x2−2x−h​=−2.

Problem 10714

Untersuchen Sie die Funktion f(x)=x5−15x3−30f(x)=x^{5}-15 x^{3}-30f(x)=x5−15x3−30 auf Extrem- und Wendepunkte.

Problem 10715

Untersuchen Sie die Extrem- und Sattelstellen der Funktion f(x)=x5−15x3−30f(x)=x^{5}-15 x^{3}-30f(x)=x5−15x3−30.

Problem 10716

Find the limit: lim⁡x→+∞(x3+3x3−x2+x−1)\lim _{x \rightarrow+\infty}\left(\sqrt[3]{x^{3}+3 x}-\sqrt{x^{2}+x-1}\right)limx→+∞​(3x3+3x​−x2+x−1​).

Problem 10717

Problem 10718

Determine the horizontal asymptote of f(x)=4x3−8x−89x3−9x+8f(x)=\frac{4 x^{3}-8 x-8}{9 x^{3}-9 x+8}f(x)=9x3−9x+84x3−8x−8​.

Problem 10719

Problem 10720

Approximate ∫01et2dt\int_{0}^{1} e^{t^{2}} dt∫01​et2dt using the degree 3 Taylor polynomial of f(x)=∫0xet2dtf(x)=\int_{0}^{x} e^{t^{2}} dtf(x)=∫0x​et2dt.

Problem 10721

Find the difference quotient for f(x)=x3f(x) = x^{3}f(x)=x3 using f(x+Δx)−f(x)Δx\frac{f(x+\Delta x)-f(x)}{\Delta x}Δxf(x+Δx)−f(x)​.

Problem 10722

If a function is a power series, is its Taylor polynomial identical to the series?

Problem 10723

Find the limit: lim⁡x→0e8x−1−8xx2\lim _{x \rightarrow 0} \frac{e^{8 x}-1-8 x}{x^{2}}x→0lim​x2e8x−1−8x​. Choose A (exact answer) or B (limit does not exist).

Problem 10724

Evaluate the integral ∭□f(x,y,z)dV\iiint_{\square} f(x, y, z) d V∭□​f(x,y,z)dV where f(x,y,z)=xz2f(x, y, z)=x z^{2}f(x,y,z)=xz2 and □=[0,2]×[3,10]×[4,9]\square = [0,2] \times[3,10] \times[4,9]□=[0,2]×[3,10]×[4,9].

Problem 10725

Bestimmen Sie die Wendepunkte der Funktion ft(x)=x3−3tx2f_{t}(x)=x^{3}-3 t x^{2}ft​(x)=x3−3tx2 für t>0t>0t>0.

Problem 10726

Find the slope of the tangent line to the curve x2−xy+y2=19x^{2}-x y+y^{2}=19x2−xy+y2=19 at the point (2,5)(2,5)(2,5).

Problem 10727

Find the slope of the relation y2(4−x)=x2y^{2}(4-x)=x^{2}y2(4−x)=x2 at the point where x=3x=3x=3 and y=−3y=-3y=−3.

Problem 10728

Find the rate of change of sin⁡y+x=72\sin y + x = \frac{7}{2}siny+x=27​ at the point (3,π6)\left(3, \frac{\pi}{6}\right)(3,6π​).

Problem 10729

Find the derivative dydx\frac{d y}{d x}dxdy​ for the equation 3x3−12y43=93 \sqrt[3]{x}-12 \sqrt[3]{y^{4}}=933x​−123y4​=9.

Problem 10730

Find the derivative dydx\frac{d y}{d x}dxdy​ for the equation tan⁡(xy)=x+y\tan (x y) = x + ytan(xy)=x+y.

Problem 10731

Find dxdt\frac{d x}{d t}dtdx​ for the circle x2+y2=16x^{2}+y^{2}=16x2+y2=16 at (2,y)(2,y)(2,y) when dydt=−3\frac{d y}{d t}=-3dtdy​=−3.

Problem 10732

The function f(t)=108,0001+5000e−tf(t)=\frac{108,000}{1+5000 e^{-t}}f(t)=1+5000e−t108,000​ models flu cases. Find: a) initial cases, b) cases after 4 weeks, c) max cases.

Problem 10733

Zeige, dass F(x)=(x+2)⋅exF(x)=(x+2) \cdot e^{x}F(x)=(x+2)⋅ex eine Stammfunktion von f(x)=(x+3)⋅exf(x)=(x+3) \cdot e^{x}f(x)=(x+3)⋅ex ist.

Problem 10734

Zeige, dass F(x)=−3x3⋅exF(x)=-3 x^{3} \cdot e^{x}F(x)=−3x3⋅ex eine Stammfunktion von f(x)=(−9x2−3x3)⋅exf(x)=\left(-9 x^{2}-3 x^{3}\right) \cdot e^{x}f(x)=(−9x2−3x3)⋅ex ist.

Problem 10735

Differentiate sin⁡(x6+4)\sin(x^{6}+4)sin(x6+4) with respect to xxx.

Problem 10736

Find the derivative of y=sin⁡6xy=\sin^{6} xy=sin6x. What is dydx=?\frac{dy}{dx}=?dxdy​=?

Problem 10737

Find the derivative of y=sin⁡6xy=\sin^{6} xy=sin6x. What is dydx\frac{d y}{d x}dxdy​?

Problem 10738

Find the derivative of y=3cos⁡x9−8cos⁡xy=\frac{3 \cos x}{9-8 \cos x}y=9−8cosx3cosx​. What is dydx\frac{d y}{d x}dxdy​?

Problem 10739

Evaluate the integral: ∫e−x2dx\int e^{-x^{2}} d x∫e−x2dx

Problem 10740

Find the derivative of y=(4x4−4x2+3)4y=(4x^4-4x^2+3)^4y=(4x4−4x2+3)4. What is dydx\frac{dy}{dx}dxdy​?

Problem 10741

Find the critical numbers of the function g(x)=x1/7−x−6/7g(x) = x^{1/7} - x^{-6/7}g(x)=x1/7−x−6/7.

Problem 10742

Find the absolute extrema of f(x)=x3−4f(x)=x^{3}-4f(x)=x3−4 on [−5,7][-5,7][−5,7]. Provide your answer as an ordered pair (x,f(x))(x, f(x))(x,f(x)).

Find the second derivative of $f(x)=x^{2} e^{2 x}(x^{-2}+e^{-2 x})$ and calculate $f^{\prime \prime}(1)$ . Simplify $f$ first.

Bestimmen Sie die Ableitung der Funktion $g(x)=a \cdot x^{2}+c$ an der Stelle $x_0$ .

Bacteria growth is given by $n(t)=985 e^{0.25 t}$ . Find the growth rate, initial population, and $n(5)$ .

Bestimmen Sie die Ableitung von $g(x) = 2 a x$ an $x_{0}=2$ direkt und mit der Definition. Vervollständigen Sie die Berechnung.

A toy rocket launches from a 58 ft building at 150 ft/s. Find: a) height function $h(t)$ , b) max height & time, c) time above 273 ft, d) time to hit ground.

Find $h+k$ if $\lim _{x \rightarrow-1} \frac{x^{2}-2 x-h}{k x+2}=-2$ .

Untersuchen Sie die Funktion $f(x)=x^{5}-15 x^{3}-30$ auf Extrem- und Wendepunkte.

Untersuchen Sie die Extrem- und Sattelstellen der Funktion $f(x)=x^{5}-15 x^{3}-30$ .

Find the limit: $\lim _{x \rightarrow+\infty}\left(\sqrt[3]{x^{3}+3 x}-\sqrt{x^{2}+x-1}\right)$ .

Determine the horizontal asymptote of $f(x)=\frac{4 x^{3}-8 x-8}{9 x^{3}-9 x+8}$ .

Approximate $\int_{0}^{1} e^{t^{2}} dt$ using the degree 3 Taylor polynomial of $f(x)=\int_{0}^{x} e^{t^{2}} dt$ .

Find the difference quotient for $f(x) = x^{3}$ using $\frac{f(x+\Delta x)-f(x)}{\Delta x}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{8 x}-1-8 x}{x^{2}}$ . Choose A (exact answer) or B (limit does not exist).

Evaluate the integral $\iiint_{\square} f(x, y, z) d V$ where $f(x, y, z)=x z^{2}$ and $\square = [0,2] \times[3,10] \times[4,9]$ .

Bestimmen Sie die Wendepunkte der Funktion $f_{t}(x)=x^{3}-3 t x^{2}$ für $t>0$ .

Find the slope of the tangent line to the curve $x^{2}-x y+y^{2}=19$ at the point $(2,5)$ .

Find the slope of the relation $y^{2}(4-x)=x^{2}$ at the point where $x=3$ and $y=-3$ .

Find the rate of change of $\sin y + x = \frac{7}{2}$ at the point $\left(3, \frac{\pi}{6}\right)$ .

Find the derivative $\frac{d y}{d x}$ for the equation $3 \sqrt[3]{x}-12 \sqrt[3]{y^{4}}=9$ .

Find the derivative $\frac{d y}{d x}$ for the equation $\tan (x y) = x + y$ .

Find $\frac{d x}{d t}$ for the circle $x^{2}+y^{2}=16$ at $(2,y)$ when $\frac{d y}{d t}=-3$ .

The function $f(t)=\frac{108,000}{1+5000 e^{-t}}$ models flu cases. Find: a) initial cases, b) cases after 4 weeks, c) max cases.

Zeige, dass $F(x)=(x+2) \cdot e^{x}$ eine Stammfunktion von $f(x)=(x+3) \cdot e^{x}$ ist.

Zeige, dass $F(x)=-3 x^{3} \cdot e^{x}$ eine Stammfunktion von $f(x)=\left(-9 x^{2}-3 x^{3}\right) \cdot e^{x}$ ist.

Differentiate $\sin(x^{6}+4)$ with respect to $x$ .

Find the derivative of $y=\sin^{6} x$ . What is $\frac{dy}{dx}=?$

Find the derivative of $y=\sin^{6} x$ . What is $\frac{d y}{d x}$ ?

Find the derivative of $y=\frac{3 \cos x}{9-8 \cos x}$ . What is $\frac{d y}{d x}$ ?

Evaluate the integral: $\int e^{-x^{2}} d x$

Find the derivative of $y=(4x^4-4x^2+3)^4$ . What is $\frac{dy}{dx}$ ?

Find the critical numbers of the function $g(x) = x^{1/7} - x^{-6/7}$ .

Find the absolute extrema of $f(x)=x^{3}-4$ on $[-5,7]$ . Provide your answer as an ordered pair $(x, f(x))$ .

Find the derivative of $y=-2(7x^{2}+3)^{-6}$ . What is $\frac{dy}{dx}$ ?

Find the absolute extrema of $f(x)=-6 x^{3}+9 x^{2}+36 x$ on $[-3,4]$ . Provide $(x, f(x))$ as an ordered pair.

Find the derivative of $y=4 e^{2x+2}$ . What is $\frac{dy}{dx}=?$

Find the absolute extrema of $f(x)=-4 x^{3}-12 x^{2}+288 x$ on $[-8,6]$ . Provide $(x, f(x))$ as the answer.

Find the derivative of the function $f(x)=8 \sqrt{7 x^{2}+3}$ . What is $f^{\prime}(x)$ ?

Find the derivative of the function $r(t)=\frac{(7 t-4)^{7}}{4 t^{2}+9}$ . What is $r^{\prime}(t)$ ?

Find the derivative of the function $y=x^{10} e^{x}$ . What is $y^{\prime}=?$

Differentiate the function $f(x)=-4 e^{3 x}$ . What is $\frac{d}{d x}\left(-4 e^{3 x}\right)$ ?

Find the tangent line equation for $f(x)=\sqrt{x^{2}+11}$ at $x=5$ : $y=\square$ (use $x$ ).

Find the area between $y=\cos ^{2} x$ and $y=\frac{8 x^{2}}{\pi^{2}}$ from $x=-\frac{\pi}{2}$ to $\frac{\pi}{2}$ .

Find the area between the curves $y=\cos^{2} x$ and $y=\frac{8 x^{2}}{\pi^{2}}$ using their intersection points.

Calculate the integral $\int_{-4}^{4}\left(t^{2}-3\right) dt$ .

Find the derivative of $y$ where $y=(4x^{2}-8x+8)e^{-6x}$ . What is $y'$ ?

Find coordinates of point $P$ where the tangent to $f(x)=x^{2} e^{-x}$ is horizontal.

Find the derivative of the function $y=e^{x \sqrt{9 x+1}}$ . What is $\frac{d y}{d x}=?$

Evaluate the integral $\int_{0}^{2}(2-t) \sqrt{t} \, dt$ .

Find all values of $c$ in $[3,6]$ for $f(x)=x-\frac{1}{x}$ that satisfy the Mean-Value Theorem.

Find all values of $c$ in $[3,6]$ for $f(x)=x-\frac{1}{x}$ that satisfy the Mean-Value Theorem.

A book of mass 1.68 kg slides down a table at an angle of 39.7° with kinetic friction $\mu_{k}=0.360$ . Find its speed at the bottom of a 1.26 m table.

Find the limit as $x$ approaches 1 from the left of $\arcsin^{\prime}(x)$ . What does this indicate about the graph of $y=\arcsin x$ ?

Set up an integral using the Disk or Washer Method to find the volume of the solid formed by rotating the region bounded by $y=x^{2}+1$ , $y=19-x^{2}$ , and $x=0$ about the $x$ -axis. Do not evaluate the integral.

Soit $h(x)=\operatorname{Arctan}\left(\frac{x}{\sqrt{1+x^{2}}+1}\right)$ . Montrez que $h$ est impaire et calculez les limites.

Find the tangent line equation for $f(x)=\frac{1}{1+x^{2}}$ at $x=3$ with slope $-\frac{3}{50}$ .

Find the first-order partial derivatives of these functions: (a) $f(x, y)=\cos (2 x) \sin (2 x)$ (b) $f(x, y)=\ln (2 x+y^{2}+x y)$ (c) $f(x, y)=\frac{x y}{x+y}$

Find the $3 \times 3$ Hessian matrix for $f(x, y, z)=x^{2} y^{3} z-3 x y^{2} z^{3}+x^{2} z$ .

Find the tangent line equation for $f(x)=\frac{1}{3+x^{2}}$ at $x=3$ with slope $-\frac{1}{24}$ .

Find the tangent line equation for $f(x)=\frac{1}{7+x^{2}}$ at $x=3$ with slope $-\frac{3}{128}$ .

Das Profil eines Vulkans wird durch $f(x)=\frac{25}{x^{2}}$ beschrieben. Wie hoch ist der Vulkan bei $x=2$ und $x=6$ ?

How long for insulin to reduce to half if it breaks down by $5\%$ each minute from a 10-unit dose?

Verify that $A(y)=\pi(2+0.5y)^{2}$ matches the table values, then find the most accurate volume using $A(y)$ .

Use Newton's method to find a root of $f(x)=3 x^{2}-\sqrt{x}-3$ starting from $x_{0}=1$ . Create a table of approximations.