Calculus

Problem 14601

Find the minimum distance of the object from the point given by $s(t)=8 t^{3}+18 t^{2}-24 t+10$ for $0 \leq t \leq 10$ .

See Solution

Problem 14602

When does the bacterial population $N(t)=500\left[1+1.2 e^{-0.03 t}\right]^{-1}$ reach about $75\%$ of its capacity?

See Solution

Problem 14603

Find the derivative of $f(x) = x^2 + 3x$ at $x=4$ using limits, and verify $f(4) = 11$ .

See Solution

Problem 14604

Derivatives Quiz: Given $f(x)=\left\{\begin{array}{lll}\frac{4 x^{2}-4}{x-1}, & x \neq 1 \\ 4, & x=1\end{array}\right.$ , which statements are true? a. none b. I only c. I and II only d. I, II, and III

See Solution

Problem 14605

Find the asymptotes of the function $f(x)=\frac{1+\ln (x-2)}{x}$ .

See Solution

Problem 14606

Find the limit: $\lim _{x \rightarrow 2+} \frac{1+\ln (x-2)}{x}$ .

See Solution

Problem 14607

Find the antiderivative $F(t)$ of $f(t)=7 \sec ^{2}(t)-8 t^{2}$ with $F(0)=0$ . What is $F(t)$ ?

See Solution

Problem 14608

Find the antiderivative $F(x)$ of $f(x)=6 x^{3}-8 x^{2}+6 x-5$ in the form $F(x)=A x^{4}+B x^{3}+C x^{2}+D x$ .

See Solution

Problem 14609

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

See Solution

Problem 14610

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

See Solution

Problem 14611

Find the additional profit when production increases from 10 to 11, given $P(q)=300 q-10 q^{2}-200$ .

See Solution

Problem 14612

A plane at 1.5 miles altitude flies at $650 \mathrm{mi} / \mathrm{h}$ . Find the distance rate increase when it's 2.5 miles from a radar.

See Solution

Problem 14613

Find $f^{\prime}(x)$ given $f^{\prime \prime}(x)=2x+5$ , $f^{\prime}(-3)=3$ , and $f(-3)=4$ . Also, find $f(2)$ .

See Solution

Problem 14614

Find the average change in profit when production increases from 10 to 14, given $P(q)=300 q-10 q^{2}-200$ .

See Solution

Problem 14615

Find $f\left(\frac{\pi}{2}\right)$ given $f^{\prime \prime}(x)=-4 \sin (2 x)$ , $f^{\prime}(0)=3$ , and $f(0)=4$ .

See Solution

Problem 14616

A student is 50 m from a track. A train moves at 120 km/h. Find the rate at which distance decreases when the train is 60 m away.

See Solution

Problem 14617

Find $f\left(\frac{\pi}{2}\right)$ given $f^{\prime \prime}(x)=-4 \sin (2 x)$ , $f^{\prime}(0)=3$ , $f(0)=4$ .

See Solution

Problem 14618

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

See Solution

Problem 14619

Find the derivative of the function: $\frac{7-\sqrt{6 x}}{\sqrt{3 x}}$ .

See Solution

Problem 14620

Find $f\left(\frac{\pi}{6}\right)$ given $f^{\prime \prime}(x)=-36 \sin (6 x)$ , $f^{\prime}(0)=-2$ , and $f(0)=-6$ .

See Solution

Problem 14621

Find $f(5)$ for the function with $f^{\prime \prime}(x)=3 x+4 \sin (x)$ , given $f(0)=3$ and $f^{\prime}(0)=4$ .

See Solution

Problem 14622

A balloon loses air at $2 \mathrm{~cm}^{3} / \mathrm{sec}$ . Find the radius decrease rate when the diameter is $30 \mathrm{~cm}$ .

See Solution

Problem 14623

A 6 ft person walks away from a $12 \mathrm{ft}$ light post at $4 \mathrm{ft/s}$ . Find the shadow length increase rate at $30 \mathrm{ft}$ .

See Solution

Problem 14624

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

See Solution

Problem 14625

Oblicz pochodną funkcji $f(x)=x^{2} e^{4x}$ .

See Solution

Problem 14626

Find $f(5)$ given that $f(4)=5$ and the slope of the tangent line is $4x + 2$ .

See Solution

Problem 14627

Find the antiderivative of the function $f(x)=5 x^{8}+7 x^{6}-4 x^{4}-2$ and include the constant $+\mathrm{C}$ .

See Solution

Problem 14628

Find the antiderivative of the function $f(x)=9 x^{9}+5 x^{7}-8 x^{3}-10$ .

See Solution

Problem 14629

Solve the differential equation: $y' - 2y = e^{2x}$ .

See Solution

Problem 14630

Find the first and second derivatives of the function $f(x)=2+6 x^{5}-\frac{3}{5 x^{3}}-8 \sqrt[4]{x^{3}}$ .

See Solution

Problem 14631

a. Calculate the derivative of $y=10 \cdot 16^{x}-x^{6}$ . What is $\frac{d y}{d x}=$ ?
b. Determine the derivative of $f(x)=\frac{x^{10}}{e^{9}}+10^{x}$ . What is $f^{\prime}(x)=$ ?

See Solution

Problem 14632

Evaluate the expression $\left(\frac{e^{5 x}}{5}-\frac{e^{7 x}}{7}\right)$ from $x=0$ to $x=1$ .

See Solution

Problem 14633

Find the second derivative of $f(x)=(x^{2}-8) \cdot e^{x}$ .

See Solution

Problem 14634

Find the position of a particle at time $t=11$ given $a(t)=18t+4$ , $s(0)=7$ , and $v(0)=14$ .

See Solution

Problem 14635

Find the derivative of the function: $\frac{7-\sqrt{6 x}}{\sqrt{3 x}}$ .

See Solution

Problem 14636

Calculate the integral of the constant function -42 from 0 to $2\pi$ .

See Solution

Problem 14637

Calculate the integral $\int_{0}^{2 \pi}(64) d t$ .

See Solution

Problem 14638

1. Given $P(t)=\frac{t^{2}-t+1}{t^{2}+1}$ , find when $P(t)$ is least and when $P'(t)$ is maximized for $t>0$ .

See Solution

Problem 14639

A cone-shaped tank (base radius $2 \mathrm{~m}$ , height $6 \mathrm{~m}$ ) fills at $1.5 \mathrm{~m}^{3}/\mathrm{min}$ . Find the water rise rate when depth is $3 \mathrm{~m}$ .

See Solution

Problem 14640

Find $\frac{d y}{d x}$ for the equation $x^{2}+y^{2}=x y+39$ at $x=7$ , $y=2$ .

See Solution

Problem 14641

Find the absolute extreme values of $f(x)=x^{\frac{1}{3}}(x-4)$ on the interval $[-64,64]$ .

See Solution

Problem 14642

Given the function $r(x)=\frac{4 x+1}{x-7}$ , complete the tables and analyze the asymptotes.
(a) Fill in the values for $r(x)$ at specified $x$ values. (b) Describe $r(x)$ as $x$ approaches 7 from both sides. (c) Find the horizontal asymptote.

See Solution

Problem 14643

Differentiate the equation $x^{6} y = 3$ implicitly to find $\frac{d y}{d x}$ .

See Solution

Problem 14644

Ein Schauspieler springt von einer $43 \mathrm{~m}$ hohen Brücke.
1) Wie lange dauert der Fall? 2) Welche Geschwindigkeit hat er beim Aufprall?
Nutze $s(t)=5 t^{2}$ .

See Solution

Problem 14645

Find the derivative of $f(x) = x(x-10)^{2}$ .

See Solution

Problem 14646

Bestimmen Sie die Steigung von $\mathrm{f}$ an $x_{0}$ durch eine Näherungstabelle für: a) $f(x)=x^{2}, x_{0}=1$ b) $f(x)=\frac{1}{4} x^{3}, x_{0}=2$ c) $f(x)=\frac{1}{x}, x_{0}=2$ d) $f(x)=\sqrt{x}, x_{0}=1$

See Solution

Problem 14647

Leiten Sie folgende Funktionen ab: a) $f(x)=x e^{x}$ , b) $f(x)=\frac{e^{x}}{x}$ , c) $f(x)=\frac{x}{e^{x}}$ , d) $f(x)=(x+1) e^{x}$ , e) $f(x)=\frac{x}{e^{-0,5 x}}$ , f) $f(x)=\frac{e^{x}+1}{x}$ , g) $f(x)=\frac{e^{x}}{x-1}$ , h) $f(x)=\frac{e^{3 x}}{x+2}$ , i) $f(x)=x^{2}+x e^{0,1 x}$ , j) $f(x)=x \cdot e^{-2 x+1}$ , k) $f(x)=x^{2} \cdot e^{a x}$ , l) $f(x)=x \cdot e^{2 x^{2}+1}$ .

See Solution

Problem 14648

Use implicit differentiation on the equation $x^{6} y=3$ to find $\frac{d y}{d x}$ .

See Solution

Problem 14649

Find the tangent line equation for $f(x)=(2x+1)^{1/4}$ at $x=1$ . Tangent line: $y=$

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Problem 14650

Use implicit differentiation on the equation $xy - 5x = 8$ to find $\frac{dy}{dx}$ .

See Solution

Problem 14651

Find the integral of $\cos^{17} \theta \sin \theta$ using the substitution $u=\cos \theta$ .

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Problem 14652

Find the price for max profit from $f(p)=-80 p^{2}+2880 p-25,600$ . Also, find max profit and break-even prices.

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Problem 14653

Differentiate the function $f(x)=\frac{2}{(2-3 x^{2})^{5}}$ . Find $f^{\prime}(x)=\square$ .

See Solution

Problem 14654

Leiten Sie die folgenden Funktionen ab: a) $f(x)=x e^{x}$ , b) $f(x)=\frac{e^{x}}{x}$ , c) $f(x)=\frac{x}{e^{x}}$ , d) $f(x)=(x+1) e^{x}$ , e) $f(x)=\frac{x}{e^{-0,5 x}}$ , f) $f(x)=\frac{e^{x}+1}{x}$ , g) $f(x)=\frac{e^{x}}{x-1}$ , h) $f(x)=\frac{e^{3 x}}{x+2}$ , i) $f(x)=x^{2}+x e^{0,1 x}$ , j) $f(x)=x \cdot e^{-2 x+1}$ , k) $f(x)=x^{2} \cdot e^{a x}$ , l) $f(x)=x \cdot e^{2 x^{2}+1}$ .

See Solution

Problem 14655

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{\sin \left(\frac{\pi}{6}+\Delta x\right)-\frac{1}{2}}{\Delta x}$ .

See Solution

Problem 14656

Druzinsky (1993) found that chewing frequency $c=kM^{-0.128}$ . Given $M(t)=1+2\sqrt{t}$ , find $\frac{dc}{dt}$ .

See Solution

Problem 14657

Calculate $\frac{d c}{d t}$ for $c=kM^{-0.128}$ where $M(t)=1+2\sqrt{t}$ . Use the chain rule.

See Solution

Problem 14658

Given mass $M(t)=1+2\sqrt{t}$ , find $\frac{d c}{d t}$ using $c=kM^{-0.128}$ . Calculate $\frac{d c}{d t}=\square$ .

See Solution

Problem 14659

Find the derivative of $\sin^{2}(\pi x)$ .

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Problem 14660

Calculate the work done by the force field $\vec{F}=(3 x^{2}+2 y) \hat{i}+(4 y+2 x) \hat{j}$ along the path $y=x^{2}$ from $x=-1$ to $x=1$ .

See Solution

Problem 14661

Find the limit: $\lim _{x \rightarrow 21} \frac{x^{2}-42 x+441}{\sin ^{2} \pi x}$ .

See Solution

Problem 14662

Chewing frequency $c$ relates to body mass $M$ as $c=kM^{-0.128}$ . Given $M(t)=1+2\sqrt{t}$ , find $\frac{d c}{d t}$ and $\frac{d c}{d L}$ .

See Solution

Problem 14663

Graph the functions: $y=x^{2/3}+(x-1)^{1/3}$ and $y=x^{2/3}+(x-1)^{2/3}$ . Analyze their first derivatives.

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Problem 14664

Differentiate $f(x)$ and $g(x)$ , then evaluate at $x=1$ and $x=2$ . Find $d/dx(f+g)$ , $d/dx((f-g)^2)$ , and $d/dx(\sqrt{f+g})$ .

See Solution

Problem 14665

Solve the differential equation $y' - 2y = e^{2x}$ with the function $u(x) = x e^{2x}$ .

See Solution

Problem 14666

Besucherzahlen im Freizeitpark: $f(x)=100(x-10)e^{-0,05x}+10000$ . Analysiere Verlauf, Maximum, Abnahme und Rentabilität.

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Problem 14667

Compute $R_{6}$ , $L_{6}$ , and $M_{3}$ to estimate distance over $[0,3]$ using given velocity data. Provide exact answers.

See Solution

Problem 14668

Solve the differential equation $y' - 2y = e^{2x}$ with the initial condition $y(0) = 2$ .

See Solution

Problem 14669

Find the area represented by the limit: $\lim _{N \rightarrow \infty} \frac{1}{N} \sum_{j=1}^{N}\left(\frac{j}{N}\right)^{7}$ .

See Solution

Problem 14670

Find the limit: $\lim _{N \rightarrow \infty} \frac{4}{N} \sum_{j=0}^{N-1} e^{-2+\frac{4 j}{N}}$ . What area does it represent?

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Problem 14671

Express the area under $f(x)=8 \sin (x)$ from $0$ to $\frac{\pi}{3}$ as a limit using $R_{N}$ approximation.

See Solution

Problem 14672

Find the derivative $\frac{d y}{d x}$ using implicit differentiation for the equation $4 x^{4} y^{5}-x+y^{2}=16$ .

See Solution

Problem 14673

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x^{3}+8 y^{4}=\ln y$ .

See Solution

Problem 14674

Evaluate $A(t)=0.4\left(\frac{1}{2}\right)^{\frac{t}{138.4}}$ for (a) $t=138.4$ and (b) $t=276.8$ . Interpret results.

See Solution

Problem 14675

Differentiate implicitly: $\ln(5xy) = e^{xy}$ , find $\frac{dy}{dx}$ , where $y \neq 0$ .

See Solution

Problem 14676

Berechnen Sie die Ableitungen der folgenden Funktionen: a) $f(x)=\sqrt{x}+\sqrt[6]{x}$ b) $g(t)=-\frac{2}{t}+\frac{1}{\sqrt{t}}$ c) $h(x)=(3 x-2)^{2}$

See Solution

Problem 14677

Evaluate $A(138.4)=0.4\left(\frac{1}{2}\right)^{\frac{138.4}{138.4}}$ to find the amount of ${ }^{210} \mathrm{Po}$ after 138.4 years.

See Solution

Problem 14678

Deux gaz réagissent pour former l'ammoniac. Trouvez $Q(t)$ , sa variation, et évaluez des limites et dérivées.

See Solution

Problem 14679

Find the derivative $\frac{d y}{d x}$ using implicit differentiation for $(1+e^{3 x})^{2}=3+\ln (3 x+y)$ , where $y \neq -3 x$ .

See Solution

Problem 14680

Differentiate implicitly: $(1+e^{3 x})^{2}=4+\ln (x+y)$ , find $\frac{d y}{d x}$ , where $y \neq -x$ .

See Solution

Problem 14681

Find the limit as $x$ approaches infinity for the expression $-x + \frac{4}{x+1}$ .

See Solution

Problem 14682

Find the rate of change of $q$ with respect to $p$ for the demand equation $p=\frac{10}{(q+6)^{2}}$ .

See Solution

Problem 14683

Differentiate the function using logarithmic differentiation: $f(x)=\frac{(x+1)(6 x+1)(9 x+1)}{\sqrt{8 x+1}}$ .

See Solution

Problem 14684

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

See Solution

Problem 14685

Evaluate the integral: $\int \frac{\cos \left(\pi / x^{5}\right)}{x^{6}} d x$

See Solution

Problem 14686

Find the derivative of $y$ with respect to $x$ using logarithmic differentiation, where $y=\frac{x \sqrt{x^{2}+6}}{(x+9)^{2/3}}$ .

See Solution

Problem 14687

Find the derivative $y^{\prime}$ for the function $y=7 x^{\sqrt{x}}$ .

See Solution

Problem 14688

A girl throws a baseball up at $20.66 \mathrm{~m/s}$ . What is the maximum height (in $\mathrm{m}$ ) it reaches?

See Solution

Problem 14689

Calculate the integral $\int_{0}^{3} f(x) dx$ where $f(x)=\left\{\begin{array}{ll}\frac{x}{2}-1, & x \leq 2 \\ x^{2}-6 x+8, & x>2\end{array}\right.$

See Solution

Problem 14690

Finde die Extrem- und Wendepunkte für die folgenden Funktionen: a) $f(x)=x \cdot(x+3)^{2}$ , b) $f(t)=(3 t-1)^{2}$ , c) $g(x)=(2 x-5)^{3}-150 x$ .

See Solution

Problem 14691

Find the derivative using logarithmic differentiation: $\frac{d}{d x}\left(1+\frac{3}{x}\right)^{x}$ .

See Solution

Problem 14692

Find the derivative of $y$ with respect to $x$ for $y=(4x+7)^{x}$ .

See Solution

Problem 14693

Find the tangent line equation for $y=(x+1)(x+2)^{2}(x+3)^{2}$ at $x=1$ .

See Solution

Problem 14694

Find the tangent line equation for $y=x^{21 x}$ at $x=e$ .

See Solution

Problem 14695

Find the tangent line equation for $y=x^{14 x}$ at $x=e$ .

See Solution

Problem 14696

Find the tangent line equation for $y=x^{11 x}$ at $x=e$ .

See Solution

Problem 14697

Find the tangent line equation for $y=x^{20 x}$ at $x=e$ .

See Solution

Problem 14698

Find the domain and range of $f(x)=e^{x-2}+2$ . Options: domain $=(-\infty, \infty)$ , range $(2, \infty)$ ; etc.

See Solution

Problem 14699

Find the limit as $x$ approaches $3$ from the right for a function with a horizontal asymptote at $y=-1$ : $\lim_{{x \to 3^+}} f(x)$ .

See Solution

Problem 14700

Find the limit of the function as $x \rightarrow 3^{+}$ for a rational function with $y=-1$ (horizontal) and $x=-4$ , $x=3$ (vertical) asymptotes.

See Solution

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Calculus

Problem 14601

Find the minimum distance of the object from the point given by s(t)=8t3+18t2−24t+10s(t)=8 t^{3}+18 t^{2}-24 t+10s(t)=8t3+18t2−24t+10 for 0≤t≤100 \leq t \leq 100≤t≤10.

Problem 14602

When does the bacterial population N(t)=500[1+1.2e−0.03t]−1N(t)=500\left[1+1.2 e^{-0.03 t}\right]^{-1}N(t)=500[1+1.2e−0.03t]−1 reach about 75%75\%75% of its capacity?

Problem 14603

Find the derivative of f(x)=x2+3xf(x) = x^2 + 3xf(x)=x2+3x at x=4x=4x=4 using limits, and verify f(4)=11f(4) = 11f(4)=11.

Problem 14604

Derivatives Quiz: Given f(x)={4x2−4x−1,x≠14,x=1f(x)=\left\{\begin{array}{lll}\frac{4 x^{2}-4}{x-1}, & x \neq 1 \\ 4, & x=1\end{array}\right.f(x)={x−14x2−4​,4,​x=1x=1​, which statements are true? a. none b. I only c. I and II only d. I, II, and III

Problem 14605

Find the asymptotes of the function f(x)=1+ln⁡(x−2)xf(x)=\frac{1+\ln (x-2)}{x}f(x)=x1+ln(x−2)​.

Problem 14606

Find the limit: lim⁡x→2+1+ln⁡(x−2)x\lim _{x \rightarrow 2+} \frac{1+\ln (x-2)}{x}limx→2+​x1+ln(x−2)​.

Problem 14607

Find the antiderivative F(t)F(t)F(t) of f(t)=7sec⁡2(t)−8t2f(t)=7 \sec ^{2}(t)-8 t^{2}f(t)=7sec2(t)−8t2 with F(0)=0F(0)=0F(0)=0. What is F(t)F(t)F(t)?

Problem 14608

Find the antiderivative F(x)F(x)F(x) of f(x)=6x3−8x2+6x−5f(x)=6 x^{3}-8 x^{2}+6 x-5f(x)=6x3−8x2+6x−5 in the form F(x)=Ax4+Bx3+Cx2+DxF(x)=A x^{4}+B x^{3}+C x^{2}+D xF(x)=Ax4+Bx3+Cx2+Dx.

Problem 14609

Find the limit: lim⁡x→2+1+ln⁡(x−2)x\lim _{x \rightarrow 2^{+}} \frac{1+\ln (x-2)}{x}limx→2+​x1+ln(x−2)​.

Problem 14610

Find the limit: lim⁡x→2−1+ln⁡(x−2)x\lim _{x \rightarrow 2^{-}} \frac{1+\ln (x-2)}{x}limx→2−​x1+ln(x−2)​.

Problem 14611

Find the additional profit when production increases from 10 to 11, given P(q)=300q−10q2−200P(q)=300 q-10 q^{2}-200P(q)=300q−10q2−200.

Problem 14612

A plane at 1.5 miles altitude flies at 650mi/h650 \mathrm{mi} / \mathrm{h}650mi/h. Find the distance rate increase when it's 2.5 miles from a radar.

Problem 14613

Find f′(x)f^{\prime}(x)f′(x) given f′′(x)=2x+5f^{\prime \prime}(x)=2x+5f′′(x)=2x+5, f′(−3)=3f^{\prime}(-3)=3f′(−3)=3, and f(−3)=4f(-3)=4f(−3)=4. Also, find f(2)f(2)f(2).

Problem 14614

Find the average change in profit when production increases from 10 to 14, given P(q)=300q−10q2−200P(q)=300 q-10 q^{2}-200P(q)=300q−10q2−200.

Problem 14615

Find f(π2)f\left(\frac{\pi}{2}\right)f(2π​) given f′′(x)=−4sin⁡(2x)f^{\prime \prime}(x)=-4 \sin (2 x)f′′(x)=−4sin(2x), f′(0)=3f^{\prime}(0)=3f′(0)=3, and f(0)=4f(0)=4f(0)=4.

Problem 14616

A student is 50 m from a track. A train moves at 120 km/h. Find the rate at which distance decreases when the train is 60 m away.

Problem 14617

Find f(π2)f\left(\frac{\pi}{2}\right)f(2π​) given f′′(x)=−4sin⁡(2x)f^{\prime \prime}(x)=-4 \sin (2 x)f′′(x)=−4sin(2x), f′(0)=3f^{\prime}(0)=3f′(0)=3, f(0)=4f(0)=4f(0)=4.

Problem 14618

Find the derivative of the function f(x)=1x2f(x)=\frac{1}{x^{2}}f(x)=x21​.

Problem 14619

Find the derivative of the function: 7−6x3x\frac{7-\sqrt{6 x}}{\sqrt{3 x}}3x​7−6x​​.

Problem 14620

Find f(π6)f\left(\frac{\pi}{6}\right)f(6π​) given f′′(x)=−36sin⁡(6x)f^{\prime \prime}(x)=-36 \sin (6 x)f′′(x)=−36sin(6x), f′(0)=−2f^{\prime}(0)=-2f′(0)=−2, and f(0)=−6f(0)=-6f(0)=−6.

Problem 14621

Find f(5)f(5)f(5) for the function with f′′(x)=3x+4sin⁡(x)f^{\prime \prime}(x)=3 x+4 \sin (x)f′′(x)=3x+4sin(x), given f(0)=3f(0)=3f(0)=3 and f′(0)=4f^{\prime}(0)=4f′(0)=4.

Problem 14622

A balloon loses air at 2 cm3/sec2 \mathrm{~cm}^{3} / \mathrm{sec}2 cm3/sec. Find the radius decrease rate when the diameter is 30 cm30 \mathrm{~cm}30 cm.

Problem 14623

A 6 ft person walks away from a 12ft12 \mathrm{ft}12ft light post at 4ft/s4 \mathrm{ft/s}4ft/s. Find the shadow length increase rate at 30ft30 \mathrm{ft}30ft.

Problem 14624

Problem 14625

Oblicz pochodną funkcji f(x)=x2e4xf(x)=x^{2} e^{4x}f(x)=x2e4x.

Problem 14626

Find f(5)f(5)f(5) given that f(4)=5f(4)=5f(4)=5 and the slope of the tangent line is 4x+24x + 24x+2.

Problem 14627

Find the antiderivative of the function f(x)=5x8+7x6−4x4−2f(x)=5 x^{8}+7 x^{6}-4 x^{4}-2f(x)=5x8+7x6−4x4−2 and include the constant +C+\mathrm{C}+C.

Problem 14628

Find the antiderivative of the function f(x)=9x9+5x7−8x3−10f(x)=9 x^{9}+5 x^{7}-8 x^{3}-10f(x)=9x9+5x7−8x3−10.

Problem 14629

Solve the differential equation: y′−2y=e2xy' - 2y = e^{2x}y′−2y=e2x.

Problem 14630

Find the first and second derivatives of the function f(x)=2+6x5−35x3−8x34f(x)=2+6 x^{5}-\frac{3}{5 x^{3}}-8 \sqrt[4]{x^{3}}f(x)=2+6x5−5x33​−84x3​.

Problem 14631

a. Calculate the derivative of y=10⋅16x−x6y=10 \cdot 16^{x}-x^{6}y=10⋅16x−x6. What is dydx=\frac{d y}{d x}=dxdy​=?b. Determine the derivative of f(x)=x10e9+10xf(x)=\frac{x^{10}}{e^{9}}+10^{x}f(x)=e9x10​+10x. What is f′(x)=f^{\prime}(x)=f′(x)=?

Problem 14632

Evaluate the expression (e5x5−e7x7)\left(\frac{e^{5 x}}{5}-\frac{e^{7 x}}{7}\right)(5e5x​−7e7x​) from x=0x=0x=0 to x=1x=1x=1.

Problem 14633

Find the second derivative of f(x)=(x2−8)⋅exf(x)=(x^{2}-8) \cdot e^{x}f(x)=(x2−8)⋅ex.

Problem 14634

Find the position of a particle at time t=11t=11t=11 given a(t)=18t+4a(t)=18t+4a(t)=18t+4, s(0)=7s(0)=7s(0)=7, and v(0)=14v(0)=14v(0)=14.

Problem 14635

Find the derivative of the function: 7−6x3x\frac{7-\sqrt{6 x}}{\sqrt{3 x}}3x​7−6x​​.

Problem 14636

Calculate the integral of the constant function -42 from 0 to 2π2\pi2π.

Problem 14637

Calculate the integral ∫02π(64)dt\int_{0}^{2 \pi}(64) d t∫02π​(64)dt.

Problem 14638

1. Given P(t)=t2−t+1t2+1P(t)=\frac{t^{2}-t+1}{t^{2}+1}P(t)=t2+1t2−t+1​, find when P(t)P(t)P(t) is least and when P′(t)P'(t)P′(t) is maximized for t>0t>0t>0.

Problem 14639

A cone-shaped tank (base radius 2 m2 \mathrm{~m}2 m, height 6 m6 \mathrm{~m}6 m) fills at 1.5 m3/min1.5 \mathrm{~m}^{3}/\mathrm{min}1.5 m3/min. Find the water rise rate when depth is 3 m3 \mathrm{~m}3 m.

Problem 14640

Find dydx\frac{d y}{d x}dxdy​ for the equation x2+y2=xy+39x^{2}+y^{2}=x y+39x2+y2=xy+39 at x=7x=7x=7, y=2y=2y=2.

Find the minimum distance of the object from the point given by $s(t)=8 t^{3}+18 t^{2}-24 t+10$ for $0 \leq t \leq 10$ .

When does the bacterial population $N(t)=500\left[1+1.2 e^{-0.03 t}\right]^{-1}$ reach about $75\%$ of its capacity?

Find the derivative of $f(x) = x^2 + 3x$ at $x=4$ using limits, and verify $f(4) = 11$ .

Derivatives Quiz: Given $f(x)=\left\{\begin{array}{lll}\frac{4 x^{2}-4}{x-1}, & x \neq 1 \\ 4, & x=1\end{array}\right.$ , which statements are true? a. none b. I only c. I and II only d. I, II, and III

Find the asymptotes of the function $f(x)=\frac{1+\ln (x-2)}{x}$ .

Find the limit: $\lim _{x \rightarrow 2+} \frac{1+\ln (x-2)}{x}$ .

Find the antiderivative $F(t)$ of $f(t)=7 \sec ^{2}(t)-8 t^{2}$ with $F(0)=0$ . What is $F(t)$ ?

Find the antiderivative $F(x)$ of $f(x)=6 x^{3}-8 x^{2}+6 x-5$ in the form $F(x)=A x^{4}+B x^{3}+C x^{2}+D x$ .

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

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

Find the additional profit when production increases from 10 to 11, given $P(q)=300 q-10 q^{2}-200$ .

A plane at 1.5 miles altitude flies at $650 \mathrm{mi} / \mathrm{h}$ . Find the distance rate increase when it's 2.5 miles from a radar.

Find $f^{\prime}(x)$ given $f^{\prime \prime}(x)=2x+5$ , $f^{\prime}(-3)=3$ , and $f(-3)=4$ . Also, find $f(2)$ .

Find the average change in profit when production increases from 10 to 14, given $P(q)=300 q-10 q^{2}-200$ .

Find $f\left(\frac{\pi}{2}\right)$ given $f^{\prime \prime}(x)=-4 \sin (2 x)$ , $f^{\prime}(0)=3$ , and $f(0)=4$ .

Find $f\left(\frac{\pi}{2}\right)$ given $f^{\prime \prime}(x)=-4 \sin (2 x)$ , $f^{\prime}(0)=3$ , $f(0)=4$ .

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

Find the derivative of the function: $\frac{7-\sqrt{6 x}}{\sqrt{3 x}}$ .

Find $f\left(\frac{\pi}{6}\right)$ given $f^{\prime \prime}(x)=-36 \sin (6 x)$ , $f^{\prime}(0)=-2$ , and $f(0)=-6$ .

Find $f(5)$ for the function with $f^{\prime \prime}(x)=3 x+4 \sin (x)$ , given $f(0)=3$ and $f^{\prime}(0)=4$ .

A balloon loses air at $2 \mathrm{~cm}^{3} / \mathrm{sec}$ . Find the radius decrease rate when the diameter is $30 \mathrm{~cm}$ .

A 6 ft person walks away from a $12 \mathrm{ft}$ light post at $4 \mathrm{ft/s}$ . Find the shadow length increase rate at $30 \mathrm{ft}$ .

Oblicz pochodną funkcji $f(x)=x^{2} e^{4x}$ .

Find $f(5)$ given that $f(4)=5$ and the slope of the tangent line is $4x + 2$ .

Find the antiderivative of the function $f(x)=5 x^{8}+7 x^{6}-4 x^{4}-2$ and include the constant $+\mathrm{C}$ .

Find the antiderivative of the function $f(x)=9 x^{9}+5 x^{7}-8 x^{3}-10$ .

Solve the differential equation: $y' - 2y = e^{2x}$ .

Find the first and second derivatives of the function $f(x)=2+6 x^{5}-\frac{3}{5 x^{3}}-8 \sqrt[4]{x^{3}}$ .

a. Calculate the derivative of $y=10 \cdot 16^{x}-x^{6}$ . What is $\frac{d y}{d x}=$ ?
b. Determine the derivative of $f(x)=\frac{x^{10}}{e^{9}}+10^{x}$ . What is $f^{\prime}(x)=$ ?

Evaluate the expression $\left(\frac{e^{5 x}}{5}-\frac{e^{7 x}}{7}\right)$ from $x=0$ to $x=1$ .

Find the second derivative of $f(x)=(x^{2}-8) \cdot e^{x}$ .

Find the position of a particle at time $t=11$ given $a(t)=18t+4$ , $s(0)=7$ , and $v(0)=14$ .

Find the derivative of the function: $\frac{7-\sqrt{6 x}}{\sqrt{3 x}}$ .

Calculate the integral of the constant function -42 from 0 to $2\pi$ .

Calculate the integral $\int_{0}^{2 \pi}(64) d t$ .

1. Given $P(t)=\frac{t^{2}-t+1}{t^{2}+1}$ , find when $P(t)$ is least and when $P'(t)$ is maximized for $t>0$ .

A cone-shaped tank (base radius $2 \mathrm{~m}$ , height $6 \mathrm{~m}$ ) fills at $1.5 \mathrm{~m}^{3}/\mathrm{min}$ . Find the water rise rate when depth is $3 \mathrm{~m}$ .

Find $\frac{d y}{d x}$ for the equation $x^{2}+y^{2}=x y+39$ at $x=7$ , $y=2$ .

Find the absolute extreme values of $f(x)=x^{\frac{1}{3}}(x-4)$ on the interval $[-64,64]$ .

Given the function $r(x)=\frac{4 x+1}{x-7}$ , complete the tables and analyze the asymptotes.
(a) Fill in the values for $r(x)$ at specified $x$ values. (b) Describe $r(x)$ as $x$ approaches 7 from both sides. (c) Find the horizontal asymptote.

Differentiate the equation $x^{6} y = 3$ implicitly to find $\frac{d y}{d x}$ .

Ein Schauspieler springt von einer $43 \mathrm{~m}$ hohen Brücke.
1) Wie lange dauert der Fall? 2) Welche Geschwindigkeit hat er beim Aufprall?
Nutze $s(t)=5 t^{2}$ .

Find the derivative of $f(x) = x(x-10)^{2}$ .

Use implicit differentiation on the equation $x^{6} y=3$ to find $\frac{d y}{d x}$ .

Find the tangent line equation for $f(x)=(2x+1)^{1/4}$ at $x=1$ . Tangent line: $y=$

Use implicit differentiation on the equation $xy - 5x = 8$ to find $\frac{dy}{dx}$ .

Find the integral of $\cos^{17} \theta \sin \theta$ using the substitution $u=\cos \theta$ .

Find the price for max profit from $f(p)=-80 p^{2}+2880 p-25,600$ . Also, find max profit and break-even prices.

Differentiate the function $f(x)=\frac{2}{(2-3 x^{2})^{5}}$ . Find $f^{\prime}(x)=\square$ .

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{\sin \left(\frac{\pi}{6}+\Delta x\right)-\frac{1}{2}}{\Delta x}$ .

Druzinsky (1993) found that chewing frequency $c=kM^{-0.128}$ . Given $M(t)=1+2\sqrt{t}$ , find $\frac{dc}{dt}$ .

Calculate $\frac{d c}{d t}$ for $c=kM^{-0.128}$ where $M(t)=1+2\sqrt{t}$ . Use the chain rule.

Given mass $M(t)=1+2\sqrt{t}$ , find $\frac{d c}{d t}$ using $c=kM^{-0.128}$ . Calculate $\frac{d c}{d t}=\square$ .

Find the derivative of $\sin^{2}(\pi x)$ .

Calculate the work done by the force field $\vec{F}=(3 x^{2}+2 y) \hat{i}+(4 y+2 x) \hat{j}$ along the path $y=x^{2}$ from $x=-1$ to $x=1$ .

Find the limit: $\lim _{x \rightarrow 21} \frac{x^{2}-42 x+441}{\sin ^{2} \pi x}$ .

Chewing frequency $c$ relates to body mass $M$ as $c=kM^{-0.128}$ . Given $M(t)=1+2\sqrt{t}$ , find $\frac{d c}{d t}$ and $\frac{d c}{d L}$ .

Graph the functions: $y=x^{2/3}+(x-1)^{1/3}$ and $y=x^{2/3}+(x-1)^{2/3}$ . Analyze their first derivatives.

Differentiate $f(x)$ and $g(x)$ , then evaluate at $x=1$ and $x=2$ . Find $d/dx(f+g)$ , $d/dx((f-g)^2)$ , and $d/dx(\sqrt{f+g})$ .

Solve the differential equation $y' - 2y = e^{2x}$ with the function $u(x) = x e^{2x}$ .

Besucherzahlen im Freizeitpark: $f(x)=100(x-10)e^{-0,05x}+10000$ . Analysiere Verlauf, Maximum, Abnahme und Rentabilität.

Compute $R_{6}$ , $L_{6}$ , and $M_{3}$ to estimate distance over $[0,3]$ using given velocity data. Provide exact answers.

Solve the differential equation $y' - 2y = e^{2x}$ with the initial condition $y(0) = 2$ .

Find the area represented by the limit: $\lim _{N \rightarrow \infty} \frac{1}{N} \sum_{j=1}^{N}\left(\frac{j}{N}\right)^{7}$ .

Find the limit: $\lim _{N \rightarrow \infty} \frac{4}{N} \sum_{j=0}^{N-1} e^{-2+\frac{4 j}{N}}$ . What area does it represent?

Express the area under $f(x)=8 \sin (x)$ from $0$ to $\frac{\pi}{3}$ as a limit using $R_{N}$ approximation.

Find the derivative $\frac{d y}{d x}$ using implicit differentiation for the equation $4 x^{4} y^{5}-x+y^{2}=16$ .

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x^{3}+8 y^{4}=\ln y$ .

Evaluate $A(t)=0.4\left(\frac{1}{2}\right)^{\frac{t}{138.4}}$ for (a) $t=138.4$ and (b) $t=276.8$ . Interpret results.