Calculus

Problem 27501

A- 1) Prove that $g(x)=x-1+e^{x}$ is strictly increasing and create its variation table. 2) Find $g(0)$ and analyze the sign of $g(x)$ for different $x$ .
B 1) For $f(x)=\frac{(x-2)e^{x}}{1+e^{x}}$ , find $\lim_{x \rightarrow -\infty} f(x)$ and identify an asymptote. 2) Analyze the positions of the curve $(C)$ and line $(\Delta): y=x-2$ . 3) Calculate $\lim_{x \rightarrow +\infty} f(x)$ and show $(\Delta)$ is an asymptote. 4) Show $f'(x)=\frac{e^{x} g(x)}{(1+e^{x})^2}$ and create the variation table for $f$ . 5) Plot $(\Delta)$ and $(C)$ .

See Solution

Problem 27502

Find the derivative of $y=3 \ln (5 x-2)$ with respect to $x$ : $\frac{\mathrm{d} y}{\mathrm{~d} x}$ .

See Solution

Problem 27503

Find the derivative of the function $y=5 e^{2 x+1}$ with respect to $x$ : $\frac{\mathrm{d} y}{\mathrm{~d} x}$ .

See Solution

Problem 27504

Find the derivative of $y=\left(x^{2}+7\right)^{\frac{1}{2}}$ with respect to $x$ : $\frac{d y}{d x}$ .

See Solution

Problem 27505

Differentiate $y = \{x+\ln (2 x)\}^{3}$ with respect to $x$ .

See Solution

Problem 27506

Berechnen Sie $f^{\prime}(x)$ für $6 f(x)=\frac{6}{x}+3 \sqrt{x}-a x^{2}$ oder $f(x)=-\frac{3}{8} x^{-4}+a b^{2} x-\pi^{2}$ .

See Solution

Problem 27507

Untersuchen Sie die Funktion $f(x)=\frac{x^{2}}{x^{2}-4}$ vollständig und zeichnen Sie ihren Graphen in einem geeigneten Intervall.

See Solution

Problem 27508

Find the normal line equation at point $P$ on the curve $y=\frac{3}{(5-3x)^{2}}$ where $x=2$ . Format: $ax+by+c=0$ .

See Solution

Problem 27509

Find the tangent line equation at point $P(2, \sqrt{9})$ on the curve $y=\sqrt{4x+1}$ in the form $ax + by + c = 0$ .

See Solution

Problem 27510

Find the following limits:
1. $\lim_{x \rightarrow 2} (2x + 6)$
2. $2 \cdot \lim_{x \rightarrow 3} ((x + 3)(2x - 1))$
3. $\lim_{x \rightarrow 3} \frac{x - 3}{x^2 - 9}$
4. $\lim_{x \rightarrow 2} \frac{x + 2}{x - 1}$
5. $\lim_{x \rightarrow 0} \frac{3}{\sqrt{3x + 1} + 1}$
6. $\lim_{x \rightarrow 0} \frac{\sqrt{3x + 1} - 1}{x}$

See Solution

Problem 27511

Wie ändert sich $f(x, y)=x^{3} y^{2}$ bei $(2,5)$ , wenn $x$ um 0.7 steigt und $y$ um 0.3 sinkt? Berechnen Sie das totale Differential! Ergebnis als Dezimalzahl, gerundet auf zwei Nachkommastellen.

See Solution

Problem 27512

Finde den stationären Punkt der Funktion $f(x, y)=y(3x^{2}-y^{2})$ . Was ist der stationäre Punkt an ..-- bitte auswählen ---. $\hat{v}$ ?

See Solution

Problem 27513

Gegeben ist die Funktion $Y(A, K)=25 A^{0,7} K^{0,3}$ mit $A=\sqrt{t}+20$ und $K=t^{2}+8$ . Welche Ableitung ist richtig?

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Problem 27514

Berechne die partielle Elastizität von $k=f(t, w, z)=w t^{5} \frac{z^{3}}{a}+\frac{3}{t w z}$ bezüglich $t$ .

See Solution

Problem 27515

Bestimme die Extremstellen der Funktion $f(x, y)=x^{3}-2 x^{2}-7 x+y^{3}-2 y^{2}-15 y-10$ und die Anzahl der Sattelpunkte.

See Solution

Problem 27516

Gegeben ist die Funktion $f(x, y, z)=x^{2}-3 y+4 \sqrt{z}$ mit $y=7 x^{5}$ und $z=x^{2}+5$ . Welche Ableitungen sind korrekt?

See Solution

Problem 27517

Bestimme die Veränderung von $f(x, y)=\ln(x^{9})+4y^{2}$ bei $(0.4, 3.2)$ für $\Delta x=0.49$ , $\Delta y=-0.32$ .

See Solution

Problem 27518

Bestimme die Elastizität der Funktion $f\left(x_{1}, x_{2}\right)=4 x_{1}^{3}+0.3 x_{2}^{3}$ bezüglich $x_{1}$ bei $(2.1 ; 3.7)$ . Ergebnis auf vier Dezimalstellen.

See Solution

Problem 27519

Gegeben ist die Nachfragefunktion $x(p)=-\ln \left(0.04 p^{5}\right)$ . Berechne die Änderung $\Delta x$ bei $p=3.4$ und einer Erhöhung um 0.7.

See Solution

Problem 27520

Leite die Funktion $f(g, x)=4g-x^{2}$ mit $g=3x^{2}+x+1$ ab. Gib die Ableitung ohne Leerzeichen an: $\frac{d f}{d x}=\square$

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Problem 27521

Was ist die erste Ableitung von $f(x, y)=x^{2} y+8 y^{6}$ nach $x$ bei $(x^{}=2.2, y^{}=1)$ ? Ergebnis als Dezimalzahl.

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Problem 27522

Given the function $f(x)=(1-x)e^{x}+2$ , find limits, inflection points, and analyze the curve's relation to $y=2x$ .

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Problem 27523

A soccer ball's height (m) at times (s) is given. Estimate the rate of change at $t=2.0$ s using two methods. Which do you prefer?

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Problem 27524

Estimate the instantaneous rate of change of height $h(x)=-5 x^{2}+3 x+65$ at $x=3$ seconds.

See Solution

Problem 27525

A raccoon population is modeled by $P(t)=100+30t+4t^{2}$ . Find $P(2.5)$ , average rate from 0 to 2.5, and rate at 2.5.

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Problem 27526

Consider the function $f(x)=(1-x)e^{x}+2$ .
1) a- Find $\lim_{x \to -\infty} f(x)$ and an asymptote (d) to (C). b- Find $\lim_{x \to +\infty} f(x)$ , then calculate $f(1)$ and $f(2)$ .
2) a- Show $f'(x)=-x e^{x}$ and create the variation table for $f$ . b- Prove the curve (C) has an inflection point $I$ and find its coordinates.
3) Sketch (d) and (C).
4) Let $(\Delta)$ be the line $y=2x$ . a- Show $f(x)-2x=(e^{x}+2)(1-x)$ and analyze positions of (C) and $(\Delta)$ . b- Find an antiderivative $F$ of $f$ . c- Draw $(\Delta)$ and calculate the area bounded by (C), the $y$ -axis, and $(\Delta)$ .
5) Let $g(x)=\ln[f(x)-2]$ . a- Show the domain of $g$ is $(-\infty, 1)$ . b- Is there a point on $(G)$ where the tangent is parallel to $(\Delta)$ ? Justify.

See Solution

Problem 27527

Graph the function $f(x)=x^{3}-2 x^{2}+x$ to find its zeros and estimate where the rate of change is positive, negative, or zero.

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Problem 27528

Given the function $f(x)=3(x-2)^{2}-2$ , find average rates of change on specified intervals and discuss positivity and negativity of rates.

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Problem 27529

A construction worker drops a bolt from $320 \mathrm{~m}$ .
a) Find the average velocity from $t=3$ to $t=8$ . b) Find velocity at $t=2$ .

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Problem 27530

How many years until $94\%$ of a radioactive isotope remains if it decays as $y(t)=y_{0} e^{-0.0002 t}$ ?

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Problem 27531

Estimate the instantaneous rate of change of the car's value, given $V(t)=18999(0.93)^{t}$ , at $t=5$ . What does it mean?

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Problem 27532

Find the average population change from 2000 to 2024 using $P(t)=-1.5 t^{2}+36 t+6$ . Discuss results for 2000-2012 and 2012-2024. When is the rate of change 0?

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Problem 27533

Given $f(x)=3 x^{2}-4 x-1$ , find the tangent line at $x=1$ : a) slope, b) $y$ -coordinate, c) equation.

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Problem 27534

A watermelon drops from $10 \mathrm{~m}$ . What is its speed upon hitting the ground?

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Problem 27535

For 500 g of a substance with a half-life of 5.2 h, use $M(t)=500(0.5)^{\frac{t}{5.2}}$ to find:
a) Amount after 1 day.
b) Instantaneous rate of change at 1 day.

See Solution

Problem 27536

A boat's distance from the dock is given by $d(t)=\left(\frac{1}{200}\right) t^{2}(t-8)^{2}$ .
a) Find when the rate of change is positive, negative, and zero. b) Explain the significance of zero and negative rates of change.

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Problem 27537

Determine the end behavior of $f(x)=2^{-x}+4$ as $x \to \infty$ .

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Problem 27538

Find the end behavior of $f(x)=2^{-x}+4$ as $x$ approaches infinity. What does $f(x)$ approach?

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Problem 27539

Find the end behavior of $f(x)=2^{-x}+4$ as $x \to \infty$ . What does $f(x)$ approach?

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Problem 27540

Graph $s(t)=\frac{t}{t^{2}+5}$ for $0 \leq t \leq 16$ . Find when the particle reverses direction, speeds up/slows down, and max/min velocities.

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Problem 27541

Find the limiting value of cumulative sales $S(t)=\frac{64}{1+8 e^{-0.47 t}}$ after a product launch.

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Problem 27542

Find the derivative of the function $f(x) = 3x^2 + 2x - 2$ . What is $f'(x)$ ?

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Problem 27543

Bestimme die Asymptote von $x^{-3}$ .

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Problem 27544

Sia $\{a_n\}$ convergente con $a_n > 0$ . Studia la convergenza di $\{b_n\}, \{c_n\}, \{d_n\}$ definiti come sopra.

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Problem 27545

Calcula $\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ para $f(x)=x^{3}$ y $f(x)=\sqrt{x}$ , $x>0$ .

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Problem 27546

Encuentra $\lim _{x \rightarrow 1} f(x)$ donde $f(x)=\frac{\sqrt[3]{x}-1}{x-1}$ si $x \neq 1$ y $f(1)=1$ .

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Problem 27547

Dans un repère orthonormé, pour la fonction $f(x)=2xe^{-x+1}+1$ , faites les tâches suivantes : 1) Trouvez $\lim_{x \to -\infty} f(x)$ . 2) Montrez que $\lim_{x \to +\infty} f(x)=1$ et déduisez-en une asymptote. 3) Montrez que $f'(x)=2(1-x)e^{-x+1}$ .

See Solution

Problem 27548

Find the limit: $\lim _{x \rightarrow \infty} \frac{2 x^{2}-3 x}{x^{2}-1}$ .

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Problem 27549

Find the derivative of the function $f(x)=\frac{2x-5}{2-x}$ .

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Problem 27550

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

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Problem 27551

Find the third derivative of $P(x) = (3x + 2)^3$ .

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Problem 27552

Find the decay constant for Pu-239 (half-life 24,110 years), remaining amount after 5,000 years, and time to decay from 20g to 1g.

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Problem 27553

Find the relationship between the perimeter $P$ and the side length $a$ of an equilateral triangle over time.

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Problem 27554

A fossil has 70% of its original carbon-14. Using the half-life of 5730 years, find when the creature died.

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Problem 27555

Explain the graph behavior of $f(x)$ near its vertical asymptote for:
1) $f(x)=\frac{3}{x-9}$ . 2) $f(x)=\frac{-5}{(x-7)^{2}}$ .

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Problem 27556

Determine the horizontal asymptote of the function $f(x)=\frac{17 x}{5 x^{2}+4}$ .

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Problem 27557

Find the tangent line approximation for $f(2.1)$ given $f(2)=3$ and $f^{\prime}(x)=\cos \left(\frac{1}{x^{2}}+x\right)$ . Is $f(2.1)=2.12$ an overestimate or underestimate?

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Problem 27558

Find the derivative of $f(x)=6x+2$ using the limit definition: $f'(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ .

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Problem 27559

A skier descends from $100 \mathrm{~m}$ . What is his velocity at this height, assuming no friction or air resistance?

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Problem 27560

Berechnen Sie den Grenzwert der Reihe $s_{n} = \sum_{i=1}^{n} \frac{3^{i}-2^{2(i-1)}}{5^{i}}$ für $n \rightarrow \infty$ .

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Problem 27561

Find the derivative $\frac{d y}{d x}$ of the function $y=2 \sqrt{x}$ using the first principle of calculus.

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Problem 27562

Berechne die Grenzwerte:
1. $\lim _{n \rightarrow \infty} U_{n} = \frac{1}{3}$
2. $\lim _{n \rightarrow \infty} O_{n} = \frac{1}{3}$

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Problem 27563

Bestimmen Sie die Tangentengleichung der Exponentialfunktion bei $A(2, e^2)$ und finden Sie die Achsenschnittpunkte.

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Problem 27564

An object falls with position $s(t)=30-5 t^{2}$ . Find average velocities and limits as $h$ approaches 0.

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Problem 27565

Find the slope of the tangent to $f(x)=\sqrt{x}$ at $x=16$ using points near $P(16,4)$ , then find the tangent line's equation. $y=$

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Problem 27566

Solve for $y(x)$ in the equation $y' - 7y = 21x^2 - 34x - 10$ with initial condition $y(0) = 7$ .

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Problem 27567

Evaluate the limit: $\lim _{x \rightarrow 0^{+}} \frac{x^{2}-1}{x}$ . If it doesn't exist, write DNE.

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Problem 27568

Find the limit: $\lim _{x \rightarrow \pi / 2} \frac{6 x}{\tan (x)}$ . If it doesn't exist, write DNE.

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Problem 27569

Find the limit as $x$ approaches 0: $\lim _{x \rightarrow 0} \frac{5+e^{x}}{1-e^{x^{2}}}$ .

See Solution

Problem 27570

Find the average rate of change of $f(x)=x^{2}-x-8$ from $x=-1$ to $x=5$ .

See Solution

Problem 27571

Find the average rate of change of the function $h(x)=-x^{2}+6x+11$ from $x=-3$ to $x=6$ .

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Problem 27572

Find the average rate of change of $h(x)=-x^{2}+5x+12$ from $x=2$ to $x=8$ .

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Problem 27573

Find the average rate of change of $g(x)=x^{2}+6x+1$ from $x=-4$ to $x=0$ .

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Problem 27574

Find the average rate of change of $g(x)=-x^{2}-8x+18$ on the interval $-9 \leq x \leq -1$ .

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Problem 27575

Find the rate of change of the cylinder's volume with base radius $\sqrt{t+2}$ and height $\frac{1}{2} \sqrt{t}$ .

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Problem 27576

Find the speed of a $1.00-\mathrm{kg}$ cart at $0.340 \mathrm{~m}$ height, starting at $2.35 \mathrm{~m/s}$ from $0.125 \mathrm{~m}$ .

See Solution

Problem 27577

Matthew slides down an $8.45 \mathrm{~m}$ hill at a $32^\circ$ angle. With a mass of $27.5 \mathrm{~kg}$ and friction $0.128$ , find the stopping distance.

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Problem 27578

Determine the speed of a $1.00-\mathrm{kg}$ cart at $0.340 \mathrm{~m}$ height if it starts at $2.35 \mathrm{~m/s}$ and $0.125 \mathrm{~m}$ .

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Problem 27579

Connor $(m=76.0 \mathrm{~kg})$ dives from $3.00 \mathrm{~m}$ with $5.94 \mathrm{~m/s}$ . Find his speed on impact and average water resistance force at $2.15 \mathrm{~m}$ .

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Problem 27580

Bestimmen Sie die Extremstellen der Funktionen, indem Sie $f^{\prime}(x)=0$ lösen: a) $f(x)=x^{2}-4x+4$ , b) $f(x)=7x^{2}-42x+35$ , c) $f(x)=\frac{1}{3}x^{3}+3x^{2}+8x$ , d) $f(x)=\frac{1}{4}x^{4}-8x$ .

See Solution

Problem 27581

Given that $f(x)$ is continuous on $[-2,5]$ with a critical point at $x=1$ and $f$ is negative, what is true about $f$ at $x=1$ ?

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Problem 27582

Ein Stein sinkt mit der Geschwindigkeit $v(t)=2,5 \cdot\left(1-e^{-0,1 t}\right)$ .
a) Wann erreicht er $2 \frac{\mathrm{m}}{\mathrm{s}}$ ? b) Zeigen Sie, dass $v(t)$ stetig wächst. c) Was bedeutet $v(t+5)-v(t)=0,05$ ? d) Wie löst man: „Wann steigt $v$ in der nächsten Sekunde um 0,13 $\frac{\mathrm{m}}{\mathrm{s}}$ ?“ e) Wann ist die Beschleunigung $2,6 \frac{\mathrm{cm}}{\mathrm{s}^{2}}$ ?

See Solution

Problem 27583

Find the derivative of $f(x)$ and simplify $f'(x) = \frac{x}{\sqrt{x} - 1}$ .

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Problem 27584

Find the second derivative of $f(x)$ if $f'(x) = \frac{x}{\sqrt{x} - 1}$ .

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Problem 27585

Check if the graph of $f(x)=7+(9-x)^{3/7}$ has a vertical tangent or cusp at $x=9$ .

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Problem 27586

Check if the graph of $f(x)=4-(4-x)^{3/7}$ has a vertical tangent or cusp at $x=4$ .

See Solution

Problem 27587

Check if the graph of $f(x)=-3 x \sqrt[3]{x-8}$ has a vertical tangent or cusp at $x=8$ .

See Solution

Problem 27588

Check if the graph of $f(x)=4(x-4)^{4/5}$ has a vertical tangent or cusp at $x=4$ .

See Solution

Problem 27589

If 0 is a root of multiplicity 4 of a DE, then any polynomial of degree 3 or less is a solution. True or False?

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Problem 27590

Find the cell reaction and calculate the standard Gibbs energy, enthalpy, and entropy at 298.15 K for the cell:
$E^{\circ} = 0.23659 - (4.8564 \times 10^{-4}) t - (3.4205 \times 10^{-6}) t^2 + (5.869 \times 10^{-9}) t^3$ for $0^{\circ}C \leq t \leq 50^{\circ}C$ .

See Solution

Problem 27591

Given the rational function $f(x)=\frac{1}{x^{2}-4}$ , which statement about $f(x)$ is FALSE?

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Problem 27592

Which statements about the graph of $f(x)=64 x^{2}+\frac{128}{x}+3$ are true?

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Problem 27593

Consider the function $f(x)=(4x^2-x)^{\frac{1}{3}}$ . Which statement about $f(x)$ is FALSE?

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Problem 27594

Identify the FALSE statement about the function $f(x)=\frac{6 x}{(x-1)^{2}}$ and its derivatives.

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Problem 27595

Determine the vertical and horizontal asymptotes of the function $f(x)=\frac{x}{\sqrt{x^{2}+5}}$ .

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Problem 27596

Find the limit: $\lim _{x \rightarrow-\infty} \frac{\sqrt{4 x^{2}+3}}{2 x-1}$ . Choose from (A) -2, (B) -1, (C) 1, (D) Does not exist.

See Solution

Problem 27597

Find the tangent line to the curve $2x^{2}-y^{4}=1$ at $(-1,1)$ . Choose from options (A) to (D).

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Problem 27598

Определете вида на уравнението $x^{2} u_{xx}+2xy u_{xy}+y^{2} u_{yy}=0$ и намерете решението с трансформации $\xi=\frac{y}{x}, \eta=x$ .

See Solution

Problem 27599

Дадено е ЧДУ: $x^{2} u_{x x}+2 x y u_{x y}+y^{2} u_{y y}=0$ . Определете вида и намерете решението с $\xi=\frac{y}{x}$ , $\eta=x$ .

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Problem 27600

Find the acceleration of a particle with velocity $v(t)=5+e^{t / 3}$ at time $t=4$ . Options: (A) 0.422 (B) 0.698 (C) 1.265 (D) 8.794

See Solution

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Calculus

Problem 27501

Problem 27502

Find the derivative of y=3ln⁡(5x−2)y=3 \ln (5 x-2)y=3ln(5x−2) with respect to xxx: dy dx\frac{\mathrm{d} y}{\mathrm{~d} x} dxdy​.

Problem 27503

Find the derivative of the function y=5e2x+1y=5 e^{2 x+1}y=5e2x+1 with respect to xxx: dy dx\frac{\mathrm{d} y}{\mathrm{~d} x} dxdy​.

Problem 27504

Find the derivative of y=(x2+7)12y=\left(x^{2}+7\right)^{\frac{1}{2}}y=(x2+7)21​ with respect to xxx: dydx\frac{d y}{d x}dxdy​.

Problem 27505

Differentiate y={x+ln⁡(2x)}3y = \{x+\ln (2 x)\}^{3}y={x+ln(2x)}3 with respect to xxx.

Problem 27506

Berechnen Sie f′(x)f^{\prime}(x)f′(x) für 6f(x)=6x+3x−ax26 f(x)=\frac{6}{x}+3 \sqrt{x}-a x^{2}6f(x)=x6​+3x​−ax2 oder f(x)=−38x−4+ab2x−π2f(x)=-\frac{3}{8} x^{-4}+a b^{2} x-\pi^{2}f(x)=−83​x−4+ab2x−π2.

Problem 27507

Untersuchen Sie die Funktion f(x)=x2x2−4f(x)=\frac{x^{2}}{x^{2}-4}f(x)=x2−4x2​ vollständig und zeichnen Sie ihren Graphen in einem geeigneten Intervall.

Problem 27508

Find the normal line equation at point PPP on the curve y=3(5−3x)2y=\frac{3}{(5-3x)^{2}}y=(5−3x)23​ where x=2x=2x=2. Format: ax+by+c=0ax+by+c=0ax+by+c=0.

Problem 27509

Find the tangent line equation at point P(2,9)P(2, \sqrt{9})P(2,9​) on the curve y=4x+1y=\sqrt{4x+1}y=4x+1​ in the form ax+by+c=0ax + by + c = 0ax+by+c=0.

Problem 27510

Problem 27511

Wie ändert sich f(x,y)=x3y2f(x, y)=x^{3} y^{2}f(x,y)=x3y2 bei (2,5)(2,5)(2,5), wenn xxx um 0.7 steigt und yyy um 0.3 sinkt? Berechnen Sie das totale Differential! Ergebnis als Dezimalzahl, gerundet auf zwei Nachkommastellen.

Problem 27512

Finde den stationären Punkt der Funktion f(x,y)=y(3x2−y2)f(x, y)=y(3x^{2}-y^{2})f(x,y)=y(3x2−y2). Was ist der stationäre Punkt an ..-- bitte auswählen ---. v^\hat{v}v^?

Problem 27513

Gegeben ist die Funktion Y(A,K)=25A0,7K0,3Y(A, K)=25 A^{0,7} K^{0,3}Y(A,K)=25A0,7K0,3 mit A=t+20A=\sqrt{t}+20A=t​+20 und K=t2+8K=t^{2}+8K=t2+8. Welche Ableitung ist richtig?

Problem 27514

Berechne die partielle Elastizität von k=f(t,w,z)=wt5z3a+3twzk=f(t, w, z)=w t^{5} \frac{z^{3}}{a}+\frac{3}{t w z}k=f(t,w,z)=wt5az3​+twz3​ bezüglich ttt.

Problem 27515

Bestimme die Extremstellen der Funktion f(x,y)=x3−2x2−7x+y3−2y2−15y−10f(x, y)=x^{3}-2 x^{2}-7 x+y^{3}-2 y^{2}-15 y-10f(x,y)=x3−2x2−7x+y3−2y2−15y−10 und die Anzahl der Sattelpunkte.

Problem 27516

Gegeben ist die Funktion f(x,y,z)=x2−3y+4zf(x, y, z)=x^{2}-3 y+4 \sqrt{z}f(x,y,z)=x2−3y+4z​ mit y=7x5y=7 x^{5}y=7x5 und z=x2+5z=x^{2}+5z=x2+5. Welche Ableitungen sind korrekt?

Problem 27517

Bestimme die Veränderung von f(x,y)=ln⁡(x9)+4y2f(x, y)=\ln(x^{9})+4y^{2}f(x,y)=ln(x9)+4y2 bei (0.4,3.2)(0.4, 3.2)(0.4,3.2) für Δx=0.49\Delta x=0.49Δx=0.49, Δy=−0.32\Delta y=-0.32Δy=−0.32.

Problem 27518

Bestimme die Elastizität der Funktion f(x1,x2)=4x13+0.3x23f\left(x_{1}, x_{2}\right)=4 x_{1}^{3}+0.3 x_{2}^{3}f(x1​,x2​)=4x13​+0.3x23​ bezüglich x1x_{1}x1​ bei (2.1;3.7)(2.1 ; 3.7)(2.1;3.7). Ergebnis auf vier Dezimalstellen.

Problem 27519

Gegeben ist die Nachfragefunktion x(p)=−ln⁡(0.04p5)x(p)=-\ln \left(0.04 p^{5}\right)x(p)=−ln(0.04p5). Berechne die Änderung Δx\Delta xΔx bei p=3.4p=3.4p=3.4 und einer Erhöhung um 0.7.

Problem 27520

Leite die Funktion f(g,x)=4g−x2f(g, x)=4g-x^{2}f(g,x)=4g−x2 mit g=3x2+x+1g=3x^{2}+x+1g=3x2+x+1 ab. Gib die Ableitung ohne Leerzeichen an: dfdx=□\frac{d f}{d x}=\squaredxdf​=□

Problem 27521

Was ist die erste Ableitung von f(x,y)=x2y+8y6f(x, y)=x^{2} y+8 y^{6}f(x,y)=x2y+8y6 nach xxx bei (x∗=2.2,y∗=1)(x^{*}=2.2, y^{*}=1)(x∗=2.2,y∗=1)? Ergebnis als Dezimalzahl.

Problem 27522

Given the function f(x)=(1−x)ex+2f(x)=(1-x)e^{x}+2f(x)=(1−x)ex+2, find limits, inflection points, and analyze the curve's relation to y=2xy=2xy=2x.

Problem 27523

A soccer ball's height (m) at times (s) is given. Estimate the rate of change at t=2.0t=2.0t=2.0 s using two methods. Which do you prefer?

Problem 27524

Estimate the instantaneous rate of change of height h(x)=−5x2+3x+65h(x)=-5 x^{2}+3 x+65h(x)=−5x2+3x+65 at x=3x=3x=3 seconds.

Problem 27525

A raccoon population is modeled by P(t)=100+30t+4t2P(t)=100+30t+4t^{2}P(t)=100+30t+4t2. Find P(2.5)P(2.5)P(2.5), average rate from 0 to 2.5, and rate at 2.5.

Problem 27526

Problem 27527

Graph the function f(x)=x3−2x2+xf(x)=x^{3}-2 x^{2}+xf(x)=x3−2x2+x to find its zeros and estimate where the rate of change is positive, negative, or zero.

Problem 27528

Given the function f(x)=3(x−2)2−2f(x)=3(x-2)^{2}-2f(x)=3(x−2)2−2, find average rates of change on specified intervals and discuss positivity and negativity of rates.

Problem 27529

A construction worker drops a bolt from 320 m320 \mathrm{~m}320 m. a) Find the average velocity from t=3t=3t=3 to t=8t=8t=8. b) Find velocity at t=2t=2t=2.

Problem 27530

How many years until 94%94\%94% of a radioactive isotope remains if it decays as y(t)=y0e−0.0002ty(t)=y_{0} e^{-0.0002 t}y(t)=y0​e−0.0002t?

Problem 27531

Estimate the instantaneous rate of change of the car's value, given V(t)=18999(0.93)tV(t)=18999(0.93)^{t}V(t)=18999(0.93)t, at t=5t=5t=5. What does it mean?

Problem 27532

Find the average population change from 2000 to 2024 using P(t)=−1.5t2+36t+6P(t)=-1.5 t^{2}+36 t+6P(t)=−1.5t2+36t+6. Discuss results for 2000-2012 and 2012-2024. When is the rate of change 0?

Problem 27533

Given f(x)=3x2−4x−1f(x)=3 x^{2}-4 x-1f(x)=3x2−4x−1, find the tangent line at x=1x=1x=1: a) slope, b) yyy-coordinate, c) equation.

Problem 27534

A watermelon drops from 10 m10 \mathrm{~m}10 m. What is its speed upon hitting the ground?

Problem 27535

For 500 g of a substance with a half-life of 5.2 h, use M(t)=500(0.5)t5.2M(t)=500(0.5)^{\frac{t}{5.2}}M(t)=500(0.5)5.2t​ to find:a) Amount after 1 day.b) Instantaneous rate of change at 1 day.

Problem 27536

A boat's distance from the dock is given by d(t)=(1200)t2(t−8)2d(t)=\left(\frac{1}{200}\right) t^{2}(t-8)^{2}d(t)=(2001​)t2(t−8)2. a) Find when the rate of change is positive, negative, and zero. b) Explain the significance of zero and negative rates of change.

Problem 27537

Determine the end behavior of f(x)=2−x+4f(x)=2^{-x}+4f(x)=2−x+4 as x→∞x \to \inftyx→∞.

Problem 27538

Find the end behavior of f(x)=2−x+4f(x)=2^{-x}+4f(x)=2−x+4 as xxx approaches infinity. What does f(x)f(x)f(x) approach?

Problem 27539

Find the end behavior of f(x)=2−x+4f(x)=2^{-x}+4f(x)=2−x+4 as x→∞x \to \inftyx→∞. What does f(x)f(x)f(x) approach?

Problem 27540

Graph s(t)=tt2+5s(t)=\frac{t}{t^{2}+5}s(t)=t2+5t​ for 0≤t≤160 \leq t \leq 160≤t≤16. Find when the particle reverses direction, speeds up/slows down, and max/min velocities.

Problem 27541

Find the limiting value of cumulative sales S(t)=641+8e−0.47tS(t)=\frac{64}{1+8 e^{-0.47 t}}S(t)=1+8e−0.47t64​ after a product launch.

Find the derivative of $y=3 \ln (5 x-2)$ with respect to $x$ : $\frac{\mathrm{d} y}{\mathrm{~d} x}$ .

Find the derivative of the function $y=5 e^{2 x+1}$ with respect to $x$ : $\frac{\mathrm{d} y}{\mathrm{~d} x}$ .

Find the derivative of $y=\left(x^{2}+7\right)^{\frac{1}{2}}$ with respect to $x$ : $\frac{d y}{d x}$ .

Differentiate $y = \{x+\ln (2 x)\}^{3}$ with respect to $x$ .

Berechnen Sie $f^{\prime}(x)$ für $6 f(x)=\frac{6}{x}+3 \sqrt{x}-a x^{2}$ oder $f(x)=-\frac{3}{8} x^{-4}+a b^{2} x-\pi^{2}$ .

Untersuchen Sie die Funktion $f(x)=\frac{x^{2}}{x^{2}-4}$ vollständig und zeichnen Sie ihren Graphen in einem geeigneten Intervall.

Find the normal line equation at point $P$ on the curve $y=\frac{3}{(5-3x)^{2}}$ where $x=2$ . Format: $ax+by+c=0$ .

Find the tangent line equation at point $P(2, \sqrt{9})$ on the curve $y=\sqrt{4x+1}$ in the form $ax + by + c = 0$ .

Wie ändert sich $f(x, y)=x^{3} y^{2}$ bei $(2,5)$ , wenn $x$ um 0.7 steigt und $y$ um 0.3 sinkt? Berechnen Sie das totale Differential! Ergebnis als Dezimalzahl, gerundet auf zwei Nachkommastellen.

Finde den stationären Punkt der Funktion $f(x, y)=y(3x^{2}-y^{2})$ . Was ist der stationäre Punkt an ..-- bitte auswählen ---. $\hat{v}$ ?

Gegeben ist die Funktion $Y(A, K)=25 A^{0,7} K^{0,3}$ mit $A=\sqrt{t}+20$ und $K=t^{2}+8$ . Welche Ableitung ist richtig?

Berechne die partielle Elastizität von $k=f(t, w, z)=w t^{5} \frac{z^{3}}{a}+\frac{3}{t w z}$ bezüglich $t$ .

Bestimme die Extremstellen der Funktion $f(x, y)=x^{3}-2 x^{2}-7 x+y^{3}-2 y^{2}-15 y-10$ und die Anzahl der Sattelpunkte.

Gegeben ist die Funktion $f(x, y, z)=x^{2}-3 y+4 \sqrt{z}$ mit $y=7 x^{5}$ und $z=x^{2}+5$ . Welche Ableitungen sind korrekt?

Bestimme die Veränderung von $f(x, y)=\ln(x^{9})+4y^{2}$ bei $(0.4, 3.2)$ für $\Delta x=0.49$ , $\Delta y=-0.32$ .

Bestimme die Elastizität der Funktion $f\left(x_{1}, x_{2}\right)=4 x_{1}^{3}+0.3 x_{2}^{3}$ bezüglich $x_{1}$ bei $(2.1 ; 3.7)$ . Ergebnis auf vier Dezimalstellen.

Gegeben ist die Nachfragefunktion $x(p)=-\ln \left(0.04 p^{5}\right)$ . Berechne die Änderung $\Delta x$ bei $p=3.4$ und einer Erhöhung um 0.7.

Leite die Funktion $f(g, x)=4g-x^{2}$ mit $g=3x^{2}+x+1$ ab. Gib die Ableitung ohne Leerzeichen an: $\frac{d f}{d x}=\square$

Was ist die erste Ableitung von $f(x, y)=x^{2} y+8 y^{6}$ nach $x$ bei $(x^{}=2.2, y^{}=1)$ ? Ergebnis als Dezimalzahl.

Given the function $f(x)=(1-x)e^{x}+2$ , find limits, inflection points, and analyze the curve's relation to $y=2x$ .

A soccer ball's height (m) at times (s) is given. Estimate the rate of change at $t=2.0$ s using two methods. Which do you prefer?

Estimate the instantaneous rate of change of height $h(x)=-5 x^{2}+3 x+65$ at $x=3$ seconds.

A raccoon population is modeled by $P(t)=100+30t+4t^{2}$ . Find $P(2.5)$ , average rate from 0 to 2.5, and rate at 2.5.

Graph the function $f(x)=x^{3}-2 x^{2}+x$ to find its zeros and estimate where the rate of change is positive, negative, or zero.

Given the function $f(x)=3(x-2)^{2}-2$ , find average rates of change on specified intervals and discuss positivity and negativity of rates.

A construction worker drops a bolt from $320 \mathrm{~m}$ .
a) Find the average velocity from $t=3$ to $t=8$ . b) Find velocity at $t=2$ .

How many years until $94\%$ of a radioactive isotope remains if it decays as $y(t)=y_{0} e^{-0.0002 t}$ ?

Estimate the instantaneous rate of change of the car's value, given $V(t)=18999(0.93)^{t}$ , at $t=5$ . What does it mean?

Find the average population change from 2000 to 2024 using $P(t)=-1.5 t^{2}+36 t+6$ . Discuss results for 2000-2012 and 2012-2024. When is the rate of change 0?

Given $f(x)=3 x^{2}-4 x-1$ , find the tangent line at $x=1$ : a) slope, b) $y$ -coordinate, c) equation.

A watermelon drops from $10 \mathrm{~m}$ . What is its speed upon hitting the ground?

For 500 g of a substance with a half-life of 5.2 h, use $M(t)=500(0.5)^{\frac{t}{5.2}}$ to find:
a) Amount after 1 day.
b) Instantaneous rate of change at 1 day.

A boat's distance from the dock is given by $d(t)=\left(\frac{1}{200}\right) t^{2}(t-8)^{2}$ .
a) Find when the rate of change is positive, negative, and zero. b) Explain the significance of zero and negative rates of change.

Determine the end behavior of $f(x)=2^{-x}+4$ as $x \to \infty$ .

Find the end behavior of $f(x)=2^{-x}+4$ as $x$ approaches infinity. What does $f(x)$ approach?

Find the end behavior of $f(x)=2^{-x}+4$ as $x \to \infty$ . What does $f(x)$ approach?

Graph $s(t)=\frac{t}{t^{2}+5}$ for $0 \leq t \leq 16$ . Find when the particle reverses direction, speeds up/slows down, and max/min velocities.

Find the limiting value of cumulative sales $S(t)=\frac{64}{1+8 e^{-0.47 t}}$ after a product launch.

Find the derivative of the function $f(x) = 3x^2 + 2x - 2$ . What is $f'(x)$ ?

Bestimme die Asymptote von $x^{-3}$ .

Sia $\{a_n\}$ convergente con $a_n > 0$ . Studia la convergenza di $\{b_n\}, \{c_n\}, \{d_n\}$ definiti come sopra.

Calcula $\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ para $f(x)=x^{3}$ y $f(x)=\sqrt{x}$ , $x>0$ .

Encuentra $\lim _{x \rightarrow 1} f(x)$ donde $f(x)=\frac{\sqrt[3]{x}-1}{x-1}$ si $x \neq 1$ y $f(1)=1$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{2 x^{2}-3 x}{x^{2}-1}$ .

Find the derivative of the function $f(x)=\frac{2x-5}{2-x}$ .

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

Find the third derivative of $P(x) = (3x + 2)^3$ .

Find the relationship between the perimeter $P$ and the side length $a$ of an equilateral triangle over time.

Explain the graph behavior of $f(x)$ near its vertical asymptote for:
1) $f(x)=\frac{3}{x-9}$ . 2) $f(x)=\frac{-5}{(x-7)^{2}}$ .

Determine the horizontal asymptote of the function $f(x)=\frac{17 x}{5 x^{2}+4}$ .

Find the tangent line approximation for $f(2.1)$ given $f(2)=3$ and $f^{\prime}(x)=\cos \left(\frac{1}{x^{2}}+x\right)$ . Is $f(2.1)=2.12$ an overestimate or underestimate?

Find the derivative of $f(x)=6x+2$ using the limit definition: $f'(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ .

A skier descends from $100 \mathrm{~m}$ . What is his velocity at this height, assuming no friction or air resistance?

Berechnen Sie den Grenzwert der Reihe $s_{n} = \sum_{i=1}^{n} \frac{3^{i}-2^{2(i-1)}}{5^{i}}$ für $n \rightarrow \infty$ .

Find the derivative $\frac{d y}{d x}$ of the function $y=2 \sqrt{x}$ using the first principle of calculus.

Berechne die Grenzwerte:
1. $\lim _{n \rightarrow \infty} U_{n} = \frac{1}{3}$
2. $\lim _{n \rightarrow \infty} O_{n} = \frac{1}{3}$

Bestimmen Sie die Tangentengleichung der Exponentialfunktion bei $A(2, e^2)$ und finden Sie die Achsenschnittpunkte.

An object falls with position $s(t)=30-5 t^{2}$ . Find average velocities and limits as $h$ approaches 0.

Find the slope of the tangent to $f(x)=\sqrt{x}$ at $x=16$ using points near $P(16,4)$ , then find the tangent line's equation. $y=$

Solve for $y(x)$ in the equation $y' - 7y = 21x^2 - 34x - 10$ with initial condition $y(0) = 7$ .

Evaluate the limit: $\lim _{x \rightarrow 0^{+}} \frac{x^{2}-1}{x}$ . If it doesn't exist, write DNE.

Find the limit: $\lim _{x \rightarrow \pi / 2} \frac{6 x}{\tan (x)}$ . If it doesn't exist, write DNE.

Find the limit as $x$ approaches 0: $\lim _{x \rightarrow 0} \frac{5+e^{x}}{1-e^{x^{2}}}$ .

Find the average rate of change of $f(x)=x^{2}-x-8$ from $x=-1$ to $x=5$ .

Find the average rate of change of the function $h(x)=-x^{2}+6x+11$ from $x=-3$ to $x=6$ .

Find the average rate of change of $h(x)=-x^{2}+5x+12$ from $x=2$ to $x=8$ .

Find the average rate of change of $g(x)=x^{2}+6x+1$ from $x=-4$ to $x=0$ .

Find the average rate of change of $g(x)=-x^{2}-8x+18$ on the interval $-9 \leq x \leq -1$ .