Calculus

Problem 12401

Find the derivative $h^{\prime}(x)$ of the function $h(x)=\frac{(3-x)^{5}}{\sqrt{x^{2}-5}}$ .

See Solution

Problem 12402

Find the differential $d y$ for the function $y=8 x^{3/4}$ . Use " $d x$ " for $d x$ .

See Solution

Problem 12403

Find the center of mass coordinates $(\bar{x}, \bar{y})$ for the region bounded by $x=2$ , $y=x+2$ , and axes with density $\rho=3x$ .

See Solution

Problem 12404

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

See Solution

Problem 12405

Evaluate and compare $\Delta y$ and $d y$ for $y=x^{3}$ at $x=2$ with $\Delta x=d x=0.1$ . Round to three decimal places.

See Solution

Problem 12406

Can we apply the mean value theorem to find a solution for $h^{\prime}(x)=2$ in the interval $11<x<19$ ?

See Solution

Problem 12407

Find the tangent line $T$ to $f(x)=x^{2}$ at the point (6,36) and use it for the table. Round answers to four decimals.

See Solution

Problem 12408

Find the tangent line equation $T(x)$ for $f(x) = \frac{20}{x^{2}}$ at point (5, $\frac{4}{5}$ ).

See Solution

Problem 12409

Find the derivative of the function $y=\left(\frac{5 x^{3}}{6 x-3}\right)^{3}$ with respect to $x$ .

See Solution

Problem 12410

Can we apply the mean value theorem to find if $f^{\prime}(x)=-9$ has a solution for $0<x<5$ ? Choose: (A) No, (B) No, (C) Yes.

See Solution

Problem 12411

Find the value of $c$ that satisfies the Mean Value Theorem for $h(x)=\sqrt{4x-8}$ on $[3,11]$ . Choices: (A) 3.5 (B) 6 (C) 8 (D) 9.5.

See Solution

Problem 12412

Find the number $c$ that satisfies the Mean Value Theorem for $f(x)=x^{3}-6 x^{2}+12 x$ on $[0,3]$ . What is $c$ ? (A) 0 (B) 1 (C) 2 (D) 3

See Solution

Problem 12413

Find the tangent line $T$ to $f(x)=\frac{20}{x^{2}}$ at $(5, \frac{4}{5})$ and use it for the table. Round to four decimals.

See Solution

Problem 12414

Find the derivative of the function $y=\frac{x}{5-6 x^{4}}$ in simplified form.

See Solution

Problem 12415

Given values of the function $h$ : $h(-3)=-6$ , $h(-2)=-1$ , $h(-1)=-4$ , $h(0)=-5$ . Is Finn's justification for $h^{\prime}(x)=-2$ in $(-2,0)$ complete? Why?

See Solution

Problem 12416

Can we apply the mean value theorem to find $f^{\prime}(x)=-9$ for $0<x<5$ given $f(0)=30$ ?
Choose 1 answer: (A) No, we don't know if the function is differentiable on that interval. (B) No, the average rate of change isn't -9. (C) Yes, both conditions for the theorem are satisfied.

See Solution

Problem 12417

Find the derivative $\frac{d y}{d x}$ of the function $y=\sqrt{\frac{5 x^{2}}{3 x^{3}-3}}$ .

See Solution

Problem 12418

Is there a solution to $h'(x)=0$ for $-1<x<1$ using the mean value theorem for $h(x)=|x|$ ? Choose A, B, or C.

See Solution

Problem 12419

Find the local maximum of the function $f(x)=x^{3}-9x^{2}-48x+52$ .

See Solution

Problem 12420

Find the derivative $\frac{d y}{d x}$ for the function $y=\sqrt[3]{\frac{4+6 x^{2}}{5+2 x}}$ .

See Solution

Problem 12421

Is there a solution to $f^{\prime}(x)=40$ for $-3<x<0$ using the mean value theorem? Choose: (A) No (B) No (C) Yes.

See Solution

Problem 12422

Given $f(2)=5$ , $f^{\prime}(2)=-2$ , and $f^{\prime}(3)=1$ , determine true statements about local extrema between specified intervals.

See Solution

Problem 12423

Is Rafael's justification complete for the equation $f^{\prime}(x)=\frac{1}{4}$ in the interval $-2<x<-1$ ?
Choose 1 answer: (A) Yes, complete. (B) No, he didn't show the average rate of change is $\frac{1}{4}$ . (C) No, he didn't show $f$ is differentiable.

See Solution

Problem 12424

What method finds the dose $a$ that maximizes blood pressure $b$ ? Choose one: a. $\frac{\mathrm{d} b}{\mathrm{~d} a}=0$ b. $\frac{\mathrm{d} b}{\mathrm{~d} a}$ at $a=0$ c. $\frac{\mathrm{d} a}{\mathrm{~d} b}=0$ d. $\frac{\mathrm{d} a}{\mathrm{~d} b}$ at $b=0$ .

See Solution

Problem 12425

Is Amrita's justification for the mean value theorem correct based on her calculations for $g$ over $[-1,4]$ ?
Choose 1 answer: (A) Yes, it's complete. (B) No, she didn't show the average rate of change equals 0.4. (C) No, she didn't show $g$ is differentiable.

See Solution

Problem 12426

Analyze the quadratic function $f(x)=2 x^{2}-12 x+17$ . Does it have a minimum or maximum? Where does it occur, and what is its value?

See Solution

Problem 12427

Find the tangent line equation for the curve $2 y^{3}+x y-y=54 x^{4}$ at $x=1$ .

See Solution

Problem 12428

Analyze the function $f(x)=(\ln (x))^{2}$ for $x \in (e, e^{2})$ : 1.1 Is it increasing or decreasing? 1.2 Is it concave or convex?

See Solution

Problem 12429

Find the first and second derivatives of the function $f(x)=3 x^{3}-9 x+1$ using the limit definition.

See Solution

Problem 12430

Find the limit as $h$ approaches 0 for $f(t)=\sqrt{t}$ and $x=8$ : $\lim _{h \rightarrow 0}\left(\frac{f(8+h)-f(8)}{h}\right) =$

See Solution

Problem 12431

Find the derivative of $f(x)=6+\frac{2}{x}$ and the slope of the tangent line at $x=3$ .

See Solution

Problem 12432

Given $g(x)$ with $g(\pi)=-2$ , $g^{\prime}(\pi)=0$ , $g^{\prime \prime}(\pi)=-3$ , determine the nature of $f(x)=2 \cos (x)+g(x)$ at $x=\pi$ .

See Solution

Problem 12433

Find the slope and intercept of the tangent line to $f(x)=a x^{2}+b x+c$ at $x=d$ for various values.

See Solution

Problem 12434

Given a twice differentiable function $g(x)$ , determine which statement about $f(x)=g(x)+x^{2}$ is TRUE.

See Solution

Problem 12435

Find the derivative of $f(t)=\frac{3}{t^{2}}$ using $f^{\prime}(t)=\lim _{h \rightarrow 0}\left(\frac{f(t+h)-f(t)}{h}\right)$ . Then compute $f^{\prime}(-5)$ , $f^{\prime}(2)$ , and $f^{\prime}(\sqrt{5})$ .

See Solution

Problem 12436

Find the slope of the tangent line to $f(x)=13-x^{2}$ at $x=2$ . Then, write the tangent line equation at $(2,9)$ .

See Solution

Problem 12437

Bestimme den Grenzwert von $f_{2}(x)=\frac{x-2}{x^{2}-4}$ für $x \neq 2$ und $f_{2}(2)=1$ , wenn $x \to 2$ .

See Solution

Problem 12438

Berechnen Sie den Grenzwert von $f_{3}(x)$ für $x \to 2$ , wobei $f_{3}(x) = \begin{cases} x-1 & \text{für } x \leq 2 \\ -(x-2)^{2} & \text{für } x > 2 \end{cases}$ .

See Solution

Problem 12439

Un ballon de basket rebondit à $70\%$ de sa hauteur précédente. Défends l'idée que le ballon ne s'arrête jamais de rebondir.

See Solution

Problem 12440

Find the slope of the tangent line to $f(x)=\frac{9}{x}$ at $(-2,-\frac{9}{2})$ using the derivative limit.

See Solution

Problem 12441

Approximate the area under $f(x)=\cos x$ from $x=0$ to $x=\frac{\pi}{2}$ using subintervals and area calculations.

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Problem 12442

Estimate the area under $f(x)=18-x^{2}$ on $[0,4]$ using 4 equal subintervals and right endpoints.

See Solution

Problem 12443

Find the slope of the secant line $PQ$ for $x=25.1$ , $25.01$ , $24.9$ , and $24.99$ on $y=\sqrt{x}+3$ . Estimate the tangent slope at $P(25,8)$ .

See Solution

Problem 12444

Find the tangent and normal lines to $f(x)=x^{2}$ at $x=-3$ . What are their equations?

See Solution

Problem 12445

Express the limit as a definite integral: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(4 x_{i}^{} \sin x_{i}^{}\right) \Delta x$ for $n$ subintervals in $[2,6]$ .

See Solution

Problem 12446

Find the slope of the tangent line for $f(x)=\frac{x^{2}}{x-5}$ at the point $\left(-1,-\frac{1}{6}\right)$ .

See Solution

Problem 12447

Find the height of the helicopter after 8 seconds using the velocity function $v(t)=2 t+\frac{252}{(t+6)^{2}}$ .

See Solution

Problem 12448

An ice cube with side length $6 \mathrm{~cm}$ melts at $5 \mathrm{~cm}^{3}$ per minute. Find the rate of side length decrease.

See Solution

Problem 12449

Find the height increase rate of water in a triangular trough (7 ft wide, 7 ft deep, 22 ft long) when height is 3 ft, with water input at 3 ft³/min.

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Problem 12450

Given $f(x)=(e^{a x}+b e^{-c x})^{2}$ , find $\theta_1$ , $\theta_2$ , $\theta_3$ , their product, and the sum when $a=b=c=1$ .

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Problem 12451

Find the value of $I=\int_{0}^{5} f(x) d x$ given $\int_{0}^{7} f(x) d x=11$ and $\int_{5}^{7} f(x) d x=5$ .

See Solution

Problem 12452

A firm uses $x$ kgs of apples to produce juice: $f(x)=\left(e^{2 x}\right)^{0.5}-1$ . Find $x^$ , $\lambda^$ , max juice, and identify an incorrect statement. Can we find max juice without Lagrangian?

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Problem 12453

Find the derivative $y'$ of $y=\ln(x^{2}+y^{2})$ at the point $\left(-\sqrt{e^{3}-9}, 3\right)$ .

See Solution

Problem 12454

A cylindrical water heater with radius $1 \mathrm{ft}$ and height $4 \mathrm{ft}$ drains. Find the drainage rate in $\mathrm{ft}^{3} / \mathrm{min}$ if the water level drops at $6 \mathrm{in} / \mathrm{min}$ .

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Problem 12455

Find the rate of change of the area $A = \pi r^2$ of a circle when the radius $r$ is 3 feet.

See Solution

Problem 12456

Untersuchen Sie das Verhalten von $f(x)=10^{10} x^{6}-0,1 x^{7}+250 x$ für $x \to +\infty$ .

See Solution

Problem 12457

Find the antiderivative $F(x)$ of $f(x)=\frac{9}{x^{2}}-\frac{2}{x^{6}}$ with $F(1)=0$ .

See Solution

Problem 12458

Find the derivative of $(3 x^{2}+5) e^{-x}$ and evaluate it at $x=-1$ .

See Solution

Problem 12459

Finde die Maße eines offenen Kartons mit quadratischer Grundfläche, um bei $100 \mathrm{~cm}^{2}$ Oberfläche das Volumen zu maximieren.

See Solution

Problem 12460

Bestimme den Wert von $x$ , der das Volumen einer Schachtel maximiert, wenn an einem $16 \mathrm{~cm} \times 10 \mathrm{~cm}$ Rechteck Ecken mit $x$ ausgeschnitten werden.

See Solution

Problem 12461

Find the antiderivative $F(x)$ of $f(x)=\frac{10}{x^{3}}-\frac{9}{x^{5}}$ with $F(1)=0$ . What is $F(x)$ ?

See Solution

Problem 12462

Find $f(4)$ given that $f^{\prime \prime}(x)=8 x+10 \sin (x)$ , $f(0)=3$ , and $f^{\prime}(0)=2$ .

See Solution

Problem 12463

Calculate $h^{\prime}(3)$ for $h(x)=g(f(x))$ using the given values for $f(x)$ and $g(x)$ .

See Solution

Problem 12464

Eine Pflanze hat Blattläuse. $f(t)=\frac{300}{0,1 t+1}$ beschreibt die Anzahl der Läuse.
a) Wann sind die Läuse nach 15 Tagen ausgestorben? b) Wann müssen Marienkäfer eingesetzt werden, damit die Läuse nach 30 Tagen ausgestorben sind?

See Solution

Problem 12465

Find the first year when Sydney's population exceeds 5.5 million, given $P = P_{0} e^{0.012 t}$ and $P_{0} = 4.9$ million.

See Solution

Problem 12466

Berechne für die Funktion $f(x)=-0,0029 x^{4}+0,306 x^{3}-10,28 x^{2}+109,1 x$ den Weg und die Entfernung zwischen Start- und Landeplatz.

See Solution

Problem 12467

Bestimmen Sie die 1. Ableitung für die Funktionen: a) $f(x)=x^{t-3}$ , b) $f(x)=\frac{1}{3} x^{3}+x-1$ , c) $f(b)=4 a b$ , d) $f(z)=\frac{4}{z}+\sqrt{z}$ , e) $f(x)=-4 x^{2}+0,5 x+1$ .

See Solution

Problem 12468

Bestimme die Ableitung von $f(x) = \sqrt{x^{3}}$ .

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Problem 12469

Zeichnen Sie die Entwicklung von Zufluss und Abfluss in einem Regenrückhaltebecken. Was bedeutet das Integral in diesem Zusammenhang?

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Problem 12470

Berechne die Flächeninhalte der Funktionen: a) $f(x)=x^{2}-x+1$ und c) $f(x)=x^{3}-x$ über $[0; 2]$ .

See Solution

Problem 12471

Berechne den Flächeninhalt A unter der Funktion $F(x)=-0,5x^2+2x$ .

See Solution

Problem 12472

Gezeitenkraftwerk:
a) Was bedeutet 1 FE unter dem Graphen von $d$ ?
b) Wann steigt die Wassermenge im Speicher am schnellsten und wann ist sie maximal/minimal?
c) Bei Springflut fließen $25 \%$ mehr Wasser. Wie ändert sich die Fläche zwischen $d$ und der x-Achse?

See Solution

Problem 12473

Find when the object is at rest for the position equation $s(t)=t^{3}-3 t^{2}$ . Options: A. $t=0,3$ B. $t=0,2$ C. $t=0$ D. $t=1$ E. $t=2$ .

See Solution

Problem 12474

Find $f^{\prime}(1)$ for $f(x)=\sqrt{x+3}$ . Which limit option is correct? A, B, C, D, or E?

See Solution

Problem 12475

Which function has a nonexistent $f^{\prime}(0)$ ? I. $f(x)=|x|$ II. $f(x)=x^{3}$ III. $f(x)=\sqrt[3]{x}$ . Choices: A, B, C, D, E.

See Solution

Problem 12476

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

See Solution

Problem 12477

Find the instantaneous rate of change of $f(x)=4^{x-2}$ at $x=3.4$ using a calculator. Options: A. 9.655 B. 6.964 C. 2.438 D. -0.0002 E. 154.475

See Solution

Problem 12478

Find $\frac{d y}{d x}$ for $y=x^{2} f(x)$ . Choose from: A. $f^{\prime}(x)+2 x$ , B. $2 x f^{\prime}(x)$ , C. $2 x f(x)$ , D. $2 x f(1)$ , E. $x^{2} f^{\prime}(x)+2 x f(x)$ .

See Solution

Problem 12479

Find $h'(2)$ for $h(x) = f(x) \cdot g(x)$ given $f(2) = -2$ , $g(2) = 2$ , $f'(2) = -2$ , $g'(2) = 2$ .

See Solution

Problem 12480

Find the limit: $\lim _{x \rightarrow \infty} \frac{e^{-x}}{\sin x}+3$ .

See Solution

Problem 12481

Calculate the average rate of change of $f(x)=\frac{3}{x+1}$ from $x=-4$ to $x=5$ .

See Solution

Problem 12482

Find the derivative of $f(x)=x \sqrt{x}$ : $f^{\prime}(x)=$ A. $\frac{1}{2 \sqrt{x}}$ B. $\frac{3}{2} \sqrt{x}$ C. $\frac{1}{2} \sqrt{x}-\sqrt{x}$ D. $\frac{3}{2 \sqrt{x}}$ L. $\sqrt{x}$

See Solution

Problem 12483

Find an equation guaranteed to have a solution for $4<x<17$ given $h(4)=2$ and $h(17)=9$ .

See Solution

Problem 12484

Find the values of $x$ where the function $f(x)=x^{4}-4 x^{3}+4 x^{2}-10$ has horizontal tangents.

See Solution

Problem 12485

Find the derivative of $y=x^{2} \ln x$ . What is $\frac{d y}{d x}=?$ A. 2 B. $x^{2} \ln x+2 x \ln x$ C. $1+2 x \ln x$ D. $x+2 x \ln x$ E. $x+2$

See Solution

Problem 12486

Find the tangent line to $f(x)=\ln(x^{2}+1)-3\sin(e^{x})$ at $x=2$ .
Point: (_, _) Slope: Equation:

See Solution

Problem 12487

Find the derivative of the function $y=\frac{3 x^{5}}{5}+\frac{2}{x^{2}}-\sqrt[3]{x^{5}}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 12488

Find the average rate of change of $f(x)$ on the interval $[-2,0]$ given $f(-2)=3$ , $f(0)=-2$ .

See Solution

Problem 12489

Gegeben ist die Funktion $f(x)=-2 x^{3}+5 x^{2}-2 x$ . Finde Nullpunkte, Hoch- und Tiefpunkte, Tangentengleichung und skizziere den Graphen.

See Solution

Problem 12490

Which limits are true? I. $\lim _{x \rightarrow x} e^{x}=\infty$ II. $\lim _{x \rightarrow-\infty} e^{x}=0$ III. $\lim _{x \rightarrow 0} e^{x}=0$ A. I only B. II only C. I and II D. II and III E. I, II, and III

See Solution

Problem 12491

Një drejtëz është tangjente me $y=x^{2}+3 x-1$ në $(1, k)$ . Gjeni $k$ , koeficientin këndor dhe ekuacionin e tangjentes.

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Problem 12492

Find the tangent line equation to $y=2 \ln x$ at $(1,0)$ . Options: A. $y=x-1$ , B. $y=0$ , C. $y=2(x-1)$ , D. $y=x+1$ , E. $y=2x+1$ .

See Solution

Problem 12493

Evaluate the limits as derivatives:
(A) $\lim _{x \rightarrow 3} \frac{2 \ln x-2 \ln 3}{x-3}$
(B) $\lim _{h \rightarrow 0} \frac{\sqrt{9+h}-\sqrt{9}}{h}$
Identify $f(x)$ and $f'(x)$ for each.

See Solution

Problem 12494

Find the derivative of the function $y=e^{\pi}-\frac{4}{x}+\frac{x}{3}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 12495

Find the slope of the tangent to $y=4-\cot x$ at $\left(\frac{\pi}{4}, 3\right)$ . Options: A. $-\sqrt{2}$ B. $4+\sqrt{2}$ C. $\frac{\sqrt{2}}{2}$ D. 2 E. $\sqrt{2}$

See Solution

Problem 12496

Differentiate the following functions and find the tangent line at given points:
1) $y=\frac{\sin x}{\ln x}$ , find $\frac{dy}{dx}$ . 2) $y=3 e^{x} \cos x$ , find $\frac{dy}{dx}$ . 3) $y=\frac{3 x^{5}}{5}+\frac{2}{x^{2}}-\sqrt[3]{x^{5}}$ , find $\frac{dy}{dx}$ . 4) $y=e^{\pi}-\frac{4}{x}+\frac{x}{3}$ , find $\frac{dy}{dx}$ . 5) $f(x)=\ln(x^{2}+1)-3\sin(e^{x})$ , find the tangent line at $x=2$ .

See Solution

Problem 12497

Find the tangent line to $f(x)=\ln(x^{2}+1)-3\sin(e^{x})$ at $x=2$ . Point: (_, _), Slope: _, Equation: _

See Solution

Problem 12498

Find the derivatives: (5) $y=\frac{\sin x}{\ln x}$ , (6) $y=3 e^{x} \cdot \cos x$ .

See Solution

Problem 12499

You have \ $15,000 to invest. Which is better in 5 years:$ 8\% $daily or$ 7.94\%$ continuously?

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Problem 12500

Differentiate the function $e^{x} \cos x$ with respect to $x$ .

See Solution

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Calculus

Problem 12401

Find the derivative h′(x)h^{\prime}(x)h′(x) of the function h(x)=(3−x)5x2−5h(x)=\frac{(3-x)^{5}}{\sqrt{x^{2}-5}}h(x)=x2−5​(3−x)5​.

Problem 12402

Find the differential dyd ydy for the function y=8x3/4y=8 x^{3/4}y=8x3/4. Use "dxd xdx" for dxd xdx.

Problem 12403

Find the center of mass coordinates (xˉ,yˉ)(\bar{x}, \bar{y})(xˉ,yˉ​) for the region bounded by x=2x=2x=2, y=x+2y=x+2y=x+2, and axes with density ρ=3x\rho=3xρ=3x.

Problem 12404

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=(2x+2)42+3xy=\frac{(2 x+2)^{4}}{\sqrt{2+3 x}}y=2+3x​(2x+2)4​.

Problem 12405

Evaluate and compare Δy\Delta yΔy and dyd ydy for y=x3y=x^{3}y=x3 at x=2x=2x=2 with Δx=dx=0.1\Delta x=d x=0.1Δx=dx=0.1. Round to three decimal places.

Problem 12406

Can we apply the mean value theorem to find a solution for h′(x)=2h^{\prime}(x)=2h′(x)=2 in the interval 11<x<1911<x<1911<x<19?

Problem 12407

Find the tangent line TTT to f(x)=x2f(x)=x^{2}f(x)=x2 at the point (6,36) and use it for the table. Round answers to four decimals.

Problem 12408

Find the tangent line equation T(x)T(x)T(x) for f(x)=20x2f(x) = \frac{20}{x^{2}}f(x)=x220​ at point (5, 45\frac{4}{5}54​).

Problem 12409

Find the derivative of the function y=(5x36x−3)3y=\left(\frac{5 x^{3}}{6 x-3}\right)^{3}y=(6x−35x3​)3 with respect to xxx.

Problem 12410

Can we apply the mean value theorem to find if f′(x)=−9f^{\prime}(x)=-9f′(x)=−9 has a solution for 0<x<50<x<50<x<5? Choose: (A) No, (B) No, (C) Yes.

Problem 12411

Find the value of ccc that satisfies the Mean Value Theorem for h(x)=4x−8h(x)=\sqrt{4x-8}h(x)=4x−8​ on [3,11][3,11][3,11]. Choices: (A) 3.5 (B) 6 (C) 8 (D) 9.5.

Problem 12412

Find the number ccc that satisfies the Mean Value Theorem for f(x)=x3−6x2+12xf(x)=x^{3}-6 x^{2}+12 xf(x)=x3−6x2+12x on [0,3][0,3][0,3]. What is ccc? (A) 0 (B) 1 (C) 2 (D) 3

Problem 12413

Find the tangent line TTT to f(x)=20x2f(x)=\frac{20}{x^{2}}f(x)=x220​ at (5,45)(5, \frac{4}{5})(5,54​) and use it for the table. Round to four decimals.

Problem 12414

Find the derivative of the function y=x5−6x4y=\frac{x}{5-6 x^{4}}y=5−6x4x​ in simplified form.

Problem 12415

Given values of the function hhh: h(−3)=−6h(-3)=-6h(−3)=−6, h(−2)=−1h(-2)=-1h(−2)=−1, h(−1)=−4h(-1)=-4h(−1)=−4, h(0)=−5h(0)=-5h(0)=−5. Is Finn's justification for h′(x)=−2h^{\prime}(x)=-2h′(x)=−2 in (−2,0)(-2,0)(−2,0) complete? Why?

Problem 12416

Problem 12417

Find the derivative dydx\frac{d y}{d x}dxdy​ of the function y=5x23x3−3y=\sqrt{\frac{5 x^{2}}{3 x^{3}-3}}y=3x3−35x2​​.

Problem 12418

Is there a solution to h′(x)=0h'(x)=0h′(x)=0 for −1<x<1-1<x<1−1<x<1 using the mean value theorem for h(x)=∣x∣h(x)=|x|h(x)=∣x∣? Choose A, B, or C.

Problem 12419

Find the local maximum of the function f(x)=x3−9x2−48x+52f(x)=x^{3}-9x^{2}-48x+52f(x)=x3−9x2−48x+52.

Problem 12420

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=4+6x25+2x3y=\sqrt[3]{\frac{4+6 x^{2}}{5+2 x}}y=35+2x4+6x2​​.

Problem 12421

Is there a solution to f′(x)=40f^{\prime}(x)=40f′(x)=40 for −3<x<0-3<x<0−3<x<0 using the mean value theorem? Choose: (A) No (B) No (C) Yes.

Problem 12422

Given f(2)=5f(2)=5f(2)=5, f′(2)=−2f^{\prime}(2)=-2f′(2)=−2, and f′(3)=1f^{\prime}(3)=1f′(3)=1, determine true statements about local extrema between specified intervals.

Problem 12423

Problem 12424

Problem 12425

Is Amrita's justification for the mean value theorem correct based on her calculations for ggg over [−1,4][-1,4][−1,4]? Choose 1 answer: (A) Yes, it's complete. (B) No, she didn't show the average rate of change equals 0.4. (C) No, she didn't show ggg is differentiable.

Problem 12426

Analyze the quadratic function f(x)=2x2−12x+17f(x)=2 x^{2}-12 x+17f(x)=2x2−12x+17. Does it have a minimum or maximum? Where does it occur, and what is its value?

Problem 12427

Find the tangent line equation for the curve 2y3+xy−y=54x42 y^{3}+x y-y=54 x^{4}2y3+xy−y=54x4 at x=1x=1x=1.

Problem 12428

Analyze the function f(x)=(ln⁡(x))2f(x)=(\ln (x))^{2}f(x)=(ln(x))2 for x∈(e,e2)x \in (e, e^{2})x∈(e,e2): 1.1 Is it increasing or decreasing? 1.2 Is it concave or convex?

Problem 12429

Find the first and second derivatives of the function f(x)=3x3−9x+1f(x)=3 x^{3}-9 x+1f(x)=3x3−9x+1 using the limit definition.

Problem 12430

Find the limit as hhh approaches 0 for f(t)=tf(t)=\sqrt{t}f(t)=t​ and x=8x=8x=8: lim⁡h→0(f(8+h)−f(8)h)=\lim _{h \rightarrow 0}\left(\frac{f(8+h)-f(8)}{h}\right) = h→0lim​(hf(8+h)−f(8)​)=

Problem 12431

Find the derivative of f(x)=6+2xf(x)=6+\frac{2}{x}f(x)=6+x2​ and the slope of the tangent line at x=3x=3x=3.

Problem 12432

Given g(x)g(x)g(x) with g(π)=−2g(\pi)=-2g(π)=−2, g′(π)=0g^{\prime}(\pi)=0g′(π)=0, g′′(π)=−3g^{\prime \prime}(\pi)=-3g′′(π)=−3, determine the nature of f(x)=2cos⁡(x)+g(x)f(x)=2 \cos (x)+g(x)f(x)=2cos(x)+g(x) at x=πx=\pix=π.

Problem 12433

Find the slope and intercept of the tangent line to f(x)=ax2+bx+cf(x)=a x^{2}+b x+cf(x)=ax2+bx+c at x=dx=dx=d for various values.

Problem 12434

Given a twice differentiable function g(x)g(x)g(x), determine which statement about f(x)=g(x)+x2f(x)=g(x)+x^{2}f(x)=g(x)+x2 is TRUE.

Problem 12435

Problem 12436

Find the slope of the tangent line to f(x)=13−x2f(x)=13-x^{2}f(x)=13−x2 at x=2x=2x=2. Then, write the tangent line equation at (2,9)(2,9)(2,9).

Problem 12437

Bestimme den Grenzwert von f2(x)=x−2x2−4f_{2}(x)=\frac{x-2}{x^{2}-4}f2​(x)=x2−4x−2​ für x≠2x \neq 2x=2 und f2(2)=1f_{2}(2)=1f2​(2)=1, wenn x→2x \to 2x→2.

Problem 12438

Berechnen Sie den Grenzwert von f3(x)f_{3}(x)f3​(x) für x→2x \to 2x→2, wobei f3(x)={x−1fu¨r x≤2−(x−2)2fu¨r x>2f_{3}(x) = \begin{cases} x-1 & \text{für } x \leq 2 \\ -(x-2)^{2} & \text{für } x > 2 \end{cases}f3​(x)={x−1−(x−2)2​fu¨r x≤2fu¨r x>2​.

Problem 12439

Un ballon de basket rebondit à 70%70\%70% de sa hauteur précédente. Défends l'idée que le ballon ne s'arrête jamais de rebondir.

Problem 12440

Find the slope of the tangent line to f(x)=9xf(x)=\frac{9}{x}f(x)=x9​ at (−2,−92)(-2,-\frac{9}{2})(−2,−29​) using the derivative limit.

Problem 12441

Approximate the area under f(x)=cos⁡xf(x)=\cos xf(x)=cosx from x=0x=0x=0 to x=π2x=\frac{\pi}{2}x=2π​ using subintervals and area calculations.

Problem 12442

Find the derivative $h^{\prime}(x)$ of the function $h(x)=\frac{(3-x)^{5}}{\sqrt{x^{2}-5}}$ .

Find the differential $d y$ for the function $y=8 x^{3/4}$ . Use " $d x$ " for $d x$ .

Find the center of mass coordinates $(\bar{x}, \bar{y})$ for the region bounded by $x=2$ , $y=x+2$ , and axes with density $\rho=3x$ .

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

Evaluate and compare $\Delta y$ and $d y$ for $y=x^{3}$ at $x=2$ with $\Delta x=d x=0.1$ . Round to three decimal places.

Can we apply the mean value theorem to find a solution for $h^{\prime}(x)=2$ in the interval $11<x<19$ ?

Find the tangent line $T$ to $f(x)=x^{2}$ at the point (6,36) and use it for the table. Round answers to four decimals.

Find the tangent line equation $T(x)$ for $f(x) = \frac{20}{x^{2}}$ at point (5, $\frac{4}{5}$ ).

Find the derivative of the function $y=\left(\frac{5 x^{3}}{6 x-3}\right)^{3}$ with respect to $x$ .

Can we apply the mean value theorem to find if $f^{\prime}(x)=-9$ has a solution for $0<x<5$ ? Choose: (A) No, (B) No, (C) Yes.

Find the value of $c$ that satisfies the Mean Value Theorem for $h(x)=\sqrt{4x-8}$ on $[3,11]$ . Choices: (A) 3.5 (B) 6 (C) 8 (D) 9.5.

Find the number $c$ that satisfies the Mean Value Theorem for $f(x)=x^{3}-6 x^{2}+12 x$ on $[0,3]$ . What is $c$ ? (A) 0 (B) 1 (C) 2 (D) 3

Find the tangent line $T$ to $f(x)=\frac{20}{x^{2}}$ at $(5, \frac{4}{5})$ and use it for the table. Round to four decimals.

Find the derivative of the function $y=\frac{x}{5-6 x^{4}}$ in simplified form.

Given values of the function $h$ : $h(-3)=-6$ , $h(-2)=-1$ , $h(-1)=-4$ , $h(0)=-5$ . Is Finn's justification for $h^{\prime}(x)=-2$ in $(-2,0)$ complete? Why?

Find the derivative $\frac{d y}{d x}$ of the function $y=\sqrt{\frac{5 x^{2}}{3 x^{3}-3}}$ .

Is there a solution to $h'(x)=0$ for $-1<x<1$ using the mean value theorem for $h(x)=|x|$ ? Choose A, B, or C.

Find the local maximum of the function $f(x)=x^{3}-9x^{2}-48x+52$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=\sqrt[3]{\frac{4+6 x^{2}}{5+2 x}}$ .

Is there a solution to $f^{\prime}(x)=40$ for $-3<x<0$ using the mean value theorem? Choose: (A) No (B) No (C) Yes.

Given $f(2)=5$ , $f^{\prime}(2)=-2$ , and $f^{\prime}(3)=1$ , determine true statements about local extrema between specified intervals.

Is Amrita's justification for the mean value theorem correct based on her calculations for $g$ over $[-1,4]$ ?
Choose 1 answer: (A) Yes, it's complete. (B) No, she didn't show the average rate of change equals 0.4. (C) No, she didn't show $g$ is differentiable.

Analyze the quadratic function $f(x)=2 x^{2}-12 x+17$ . Does it have a minimum or maximum? Where does it occur, and what is its value?

Find the tangent line equation for the curve $2 y^{3}+x y-y=54 x^{4}$ at $x=1$ .

Analyze the function $f(x)=(\ln (x))^{2}$ for $x \in (e, e^{2})$ : 1.1 Is it increasing or decreasing? 1.2 Is it concave or convex?

Find the first and second derivatives of the function $f(x)=3 x^{3}-9 x+1$ using the limit definition.

Find the limit as $h$ approaches 0 for $f(t)=\sqrt{t}$ and $x=8$ : $\lim _{h \rightarrow 0}\left(\frac{f(8+h)-f(8)}{h}\right) =$

Find the derivative of $f(x)=6+\frac{2}{x}$ and the slope of the tangent line at $x=3$ .

Given $g(x)$ with $g(\pi)=-2$ , $g^{\prime}(\pi)=0$ , $g^{\prime \prime}(\pi)=-3$ , determine the nature of $f(x)=2 \cos (x)+g(x)$ at $x=\pi$ .

Find the slope and intercept of the tangent line to $f(x)=a x^{2}+b x+c$ at $x=d$ for various values.

Given a twice differentiable function $g(x)$ , determine which statement about $f(x)=g(x)+x^{2}$ is TRUE.

Find the slope of the tangent line to $f(x)=13-x^{2}$ at $x=2$ . Then, write the tangent line equation at $(2,9)$ .

Bestimme den Grenzwert von $f_{2}(x)=\frac{x-2}{x^{2}-4}$ für $x \neq 2$ und $f_{2}(2)=1$ , wenn $x \to 2$ .

Berechnen Sie den Grenzwert von $f_{3}(x)$ für $x \to 2$ , wobei $f_{3}(x) = \begin{cases} x-1 & \text{für } x \leq 2 \\ -(x-2)^{2} & \text{für } x > 2 \end{cases}$ .

Un ballon de basket rebondit à $70\%$ de sa hauteur précédente. Défends l'idée que le ballon ne s'arrête jamais de rebondir.

Find the slope of the tangent line to $f(x)=\frac{9}{x}$ at $(-2,-\frac{9}{2})$ using the derivative limit.

Approximate the area under $f(x)=\cos x$ from $x=0$ to $x=\frac{\pi}{2}$ using subintervals and area calculations.

Estimate the area under $f(x)=18-x^{2}$ on $[0,4]$ using 4 equal subintervals and right endpoints.

Find the slope of the secant line $PQ$ for $x=25.1$ , $25.01$ , $24.9$ , and $24.99$ on $y=\sqrt{x}+3$ . Estimate the tangent slope at $P(25,8)$ .

Find the tangent and normal lines to $f(x)=x^{2}$ at $x=-3$ . What are their equations?

Express the limit as a definite integral: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(4 x_{i}^{} \sin x_{i}^{}\right) \Delta x$ for $n$ subintervals in $[2,6]$ .

Find the slope of the tangent line for $f(x)=\frac{x^{2}}{x-5}$ at the point $\left(-1,-\frac{1}{6}\right)$ .

Find the height of the helicopter after 8 seconds using the velocity function $v(t)=2 t+\frac{252}{(t+6)^{2}}$ .

An ice cube with side length $6 \mathrm{~cm}$ melts at $5 \mathrm{~cm}^{3}$ per minute. Find the rate of side length decrease.

Given $f(x)=(e^{a x}+b e^{-c x})^{2}$ , find $\theta_1$ , $\theta_2$ , $\theta_3$ , their product, and the sum when $a=b=c=1$ .

Find the value of $I=\int_{0}^{5} f(x) d x$ given $\int_{0}^{7} f(x) d x=11$ and $\int_{5}^{7} f(x) d x=5$ .

A firm uses $x$ kgs of apples to produce juice: $f(x)=\left(e^{2 x}\right)^{0.5}-1$ . Find $x^$ , $\lambda^$ , max juice, and identify an incorrect statement. Can we find max juice without Lagrangian?

Find the derivative $y'$ of $y=\ln(x^{2}+y^{2})$ at the point $\left(-\sqrt{e^{3}-9}, 3\right)$ .

A cylindrical water heater with radius $1 \mathrm{ft}$ and height $4 \mathrm{ft}$ drains. Find the drainage rate in $\mathrm{ft}^{3} / \mathrm{min}$ if the water level drops at $6 \mathrm{in} / \mathrm{min}$ .

Find the rate of change of the area $A = \pi r^2$ of a circle when the radius $r$ is 3 feet.

Untersuchen Sie das Verhalten von $f(x)=10^{10} x^{6}-0,1 x^{7}+250 x$ für $x \to +\infty$ .

Find the antiderivative $F(x)$ of $f(x)=\frac{9}{x^{2}}-\frac{2}{x^{6}}$ with $F(1)=0$ .

Find the derivative of $(3 x^{2}+5) e^{-x}$ and evaluate it at $x=-1$ .

Finde die Maße eines offenen Kartons mit quadratischer Grundfläche, um bei $100 \mathrm{~cm}^{2}$ Oberfläche das Volumen zu maximieren.

Bestimme den Wert von $x$ , der das Volumen einer Schachtel maximiert, wenn an einem $16 \mathrm{~cm} \times 10 \mathrm{~cm}$ Rechteck Ecken mit $x$ ausgeschnitten werden.

Find the antiderivative $F(x)$ of $f(x)=\frac{10}{x^{3}}-\frac{9}{x^{5}}$ with $F(1)=0$ . What is $F(x)$ ?

Find $f(4)$ given that $f^{\prime \prime}(x)=8 x+10 \sin (x)$ , $f(0)=3$ , and $f^{\prime}(0)=2$ .

Calculate $h^{\prime}(3)$ for $h(x)=g(f(x))$ using the given values for $f(x)$ and $g(x)$ .

Eine Pflanze hat Blattläuse. $f(t)=\frac{300}{0,1 t+1}$ beschreibt die Anzahl der Läuse.
a) Wann sind die Läuse nach 15 Tagen ausgestorben? b) Wann müssen Marienkäfer eingesetzt werden, damit die Läuse nach 30 Tagen ausgestorben sind?

Find the first year when Sydney's population exceeds 5.5 million, given $P = P_{0} e^{0.012 t}$ and $P_{0} = 4.9$ million.

Berechne für die Funktion $f(x)=-0,0029 x^{4}+0,306 x^{3}-10,28 x^{2}+109,1 x$ den Weg und die Entfernung zwischen Start- und Landeplatz.

Bestimme die Ableitung von $f(x) = \sqrt{x^{3}}$ .

Berechne die Flächeninhalte der Funktionen: a) $f(x)=x^{2}-x+1$ und c) $f(x)=x^{3}-x$ über $[0; 2]$ .

Berechne den Flächeninhalt A unter der Funktion $F(x)=-0,5x^2+2x$ .

Gezeitenkraftwerk:
a) Was bedeutet 1 FE unter dem Graphen von $d$ ?
b) Wann steigt die Wassermenge im Speicher am schnellsten und wann ist sie maximal/minimal?
c) Bei Springflut fließen $25 \%$ mehr Wasser. Wie ändert sich die Fläche zwischen $d$ und der x-Achse?

Find when the object is at rest for the position equation $s(t)=t^{3}-3 t^{2}$ . Options: A. $t=0,3$ B. $t=0,2$ C. $t=0$ D. $t=1$ E. $t=2$ .

Find $f^{\prime}(1)$ for $f(x)=\sqrt{x+3}$ . Which limit option is correct? A, B, C, D, or E?