Calculus

Problem 4501

Find the derivative of the function $f(x)=-(x+1)^{2}$ using the limit of the difference quotient and compute $f^{\prime}(1)$ .

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Problem 4502

A ball is dropped from a building 480 m high. Use $d(t)=4.9 t^{2}$ to find average velocity for $6 \leq t \leq 7, 6.1, 6.01, 6.001$ . What is the instantaneous velocity at $t=6$ s?

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Problem 4503

Find the derivative of $f(x)=5 \tan (3 x)+\cos (x^{2})$ .

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Problem 4504

Evaluate the limits or explain their non-existence: (a) $\lim _{x \rightarrow-1} \frac{x^{3}+1}{x^{3}-5 x^{2}-6 x}$ ; (b) $\lim _{x \rightarrow 3} \frac{2}{x^{2}-7 x+12}$ .

See Solution

Problem 4505

Find the derivative of $f(x)=5 x^{2} \sin (9 x)$ .

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Problem 4506

Find values of $c$ for continuity of $f(x)$ at $x=2$ and check if $f(x)$ is continuous at $x=1$ .

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Problem 4507

Give an example of a function $f$ that is continuous at $a=2$ but not differentiable there, or explain why.

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Problem 4508

Find a function $f$ that is continuous at $a=2$ but not differentiable at that point.

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Problem 4509

Find a function $g$ that is differentiable at $a=3$ but lacks a limit at $a=3$ .

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Problem 4510

Find the limits or state if they don't exist: (a) $\lim _{x \rightarrow-1} \frac{x^{3}+1}{x^{3}-5 x^{2}-6 x}$ ; (b) $\lim _{x \rightarrow 3} \frac{2}{x^{2}-7 x+12}$ .

See Solution

Problem 4511

Find a function $h$ that has a limit at $a=-2$ , is defined at $a=-2$ , but is not continuous at $a=-2$ .

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Problem 4512

Aufgabe 1: Bestimmen Sie den Schnittpunkt der Tangenten an $f(x)=-\frac{1}{2} x^{2}-2 x$ in $A(-1, 1.5)$ und $B(1, 2.5)$ . Aufgabe 2: Finden Sie Stammfunktionen für $f(x)=x^{3}$ , $f(x)=\frac{1}{x^{4}}$ , $f(x)=3 x^{4}-\frac{1}{3} x^{2}+12$ , $f(x)=\frac{1}{\sqrt{x}}$ . Aufgabe 3: Bestimmen Sie Wendepunkte von $f(x)=0,5 x^{4}-x^{3}-6 x^{2}+2$ und deren Informationen. Aufgabe 4: Berechnen Sie den Gesamtinhalt $A$ zwischen $f(x)=x^{4}-5 x^{2}+4$ und der $x$ -Achse, Nullstellen bei $x_{1}=-2, x_{2}=-1, x_{3}=1, x_{4}=2$ . Aufgabe 5: Finde Werte für $k \in R$ , sodass $\int_{k}^{k+1}(k^{2}+2 x) dx=9$ .

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Problem 4513

Finde für jede Funktion $f$ eine Stammfunktion $F$ : a) $f(x)=x^{3}$ b) $f(x)=\frac{1}{x^{4}}$ c) $f(x)=3 x^{4}-\frac{1}{3} x^{2}+12$ d) $f(x)=\frac{1}{\sqrt{x}}$

See Solution

Problem 4514

Find $f^{\prime}(6)$ if $f(x)=\frac{1}{3} g(x)+\frac{1}{5} h(x)$ , given $g^{\prime}(6)=5$ and $h^{\prime}(6)=15$ .

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Problem 4515

Find the derivative of $y=\frac{6 \ln (x+8)}{x^{2}}$ . Which differentiation rule helps most: A. chain, B. power, C. sum, D. quotient?

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Problem 4516

Find the limit as $x$ approaches 3 for the expression $\frac{2 x^{2}-18}{x+3}$ .

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Problem 4517

Find the slope of the tangent line for $y=3 x^{1/2}+x^{3/2}$ at $x=4$ .

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Problem 4518

Find the limit: $\lim _{x \rightarrow 3} \frac{2 x^{2}-18}{x+3}$ .

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Problem 4519

Estimate the population in 2030 using $p(t)=37.75(1.021)^{t}$ and find the rate of change at $t=30$ .

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Problem 4520

Find the slope of the tangent line for $y=x^{4}-6 x^{3}+5$ at $x=2$ and its equation. How to find the slope? A. Substitute $x$ into the equation and solve for $y$ . B. Set the derivative to zero and solve for $x$ . C. Substitute 2 into the derivative and evaluate. D. Substitute $y$ into the equation and solve for $x$ .

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Problem 4521

Find the limit of the difference quotient for the derivative of $f(x)=-(x+4)^{2}$ .

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Problem 4522

Find the slope of the tangent line for $y=x^{4}-6 x^{3}+5$ at $x=2$ . Choose the correct method from the options.

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Problem 4523

Gegeben ist die Änderungsrate $f$ eines Tankinhalts.
a) Zeichne den Graphen der Integralfunktion $I_{-3}(x)=\int_{-3}^{x} f(t) dt$ .
b) Erkläre die Bedeutung der Integralfunktion.
c) Was lässt sich über den Tankinhalt am Tag 7 sagen?

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Problem 4524

Find the slope of the tangent line for $y=x^{4}-6 x^{3}+5$ at $x=2$ . Choose the correct method: A, B, C, or D.

See Solution

Problem 4525

Find the derivative: $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{-(x+h+4)^{2}+(x+4)^{2}}{h}$ .

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Problem 4526

Bestimme die Stellen, an denen die Funktion $f(x)=x^{2}$ die Steigung hat: a) 6, b) -2, c) 0, d) -4.

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Problem 4527

Find the bullet's acceleration when fired straight down from a cliff, ignoring air resistance. Use $9.8 \, m/s^2$ for gravity.

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Problem 4528

Finde die x-Werte, an denen die Funktionen $f(x)=\frac{2}{3} x^{3}-10 x$ und $g(x)=2 x^{2}+6 x-4$ die gleiche Steigung haben.

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Problem 4529

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

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Problem 4530

Ein Pkw beschleunigt. a) Skizzieren Sie den Geschwindigkeitsgraphen v. b) Untersuchen Sie v'. c) Deuten Sie $v(2)=3$ und $v'(12)=1$ im Kontext.

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Problem 4531

Find the rate of change of sales for the function $S(t)=102-100 e^{-0.4 t}$ at (a) 1 year, (b) 5 years. What happens to the rate over time? Does it ever equal zero?

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Problem 4532

Is the derivative of $f(t)=\ln(t)$ equal to $f^{\prime}(t)=1/t$ true or false? Explain.

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Problem 4533

Is the derivative of $f(t)=\ln (t)$ equal to $f^{\prime}(t)=1 / t$ true or false? Explain your reasoning.

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Problem 4534

Find the derivative of $u(x)=C(x)$ where $v(x)=x$ . What is $u^{\prime}(x)$ ? A. $C^{\prime}(x)$ B. $\frac{1}{C^{\prime}(x)}$ C. $C(x) \cdot C^{\prime}(x)$

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Problem 4535

What is the quotient rule for the derivative of $f(x)=\frac{u(x)}{v(x)}$ ? A, B, or C?

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Problem 4536

Find the derivative of $u(x)=C(x)$ and $v(x)=x$ . What are $u^{\prime}(x)$ and $v^{\prime}(x)$ ?

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Problem 4537

Beurteilen Sie, ob der Grenzwert von $f(x) = 0,001 x^{4}-0,11 x^{2}+3$ im Unendlichen 0 ist.

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Problem 4538

Bestimmen Sie die Ableitungen der Funktionen: a) $f(x)=x^{-4}$ , b) $f(x)=x^{-1}$ , c) $g(x)=x^{-2}$ , d) $h(x)=x^{\frac{1}{2}}$ , e) $g(x)=x^{\frac{1}{3}}$ .

See Solution

Problem 4539

Leiten Sie folgende Funktionen ab: a) $f(x)=x^{-4}$ , b) $f(x)=x^{-1}$ , c) $g(x)=x^{-2}$ , d) $h(x)=x^{\frac{1}{2}}$ , e) $g(x)=x^{\frac{1}{3}}$ .

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Problem 4540

Find the marginal average cost function $\bar{C}^{\prime}(x)$ and verify using the quotient rule.

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Problem 4541

Untersuchen Sie das Krümmungsverhalten der Funktionen: a) $f(x)=-x^{2}+2x+4$ , b) $f(x)=x^{3}-x$ , c) $f(x)=x^{3}-3x^{2}-9x-5$ , d) $f(x)=x^{4}+x^{2}$ , e) $f(x)=x^{4}-6x^{2}$ , f) $f(x)=\frac{1}{4}x^{4}+3x^{2}-2$ , g) $f(x)=\frac{1}{3}x^{6}-20x^{2}$ , h) $f(x)=\frac{1}{20}x^{5}+\frac{1}{2}x^{4}+\frac{3}{2}x^{3}$ , i) $f(x)=(x+2)^{2}(x-1)^{2}-3$ .

See Solution

Problem 4542

Find the derivative of $f(x) = (5x + 2)^3$ .

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Problem 4543

Untersuchen Sie die Krümmung der Funktion $f(x)=x^{4}+x^{2}$ .

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Problem 4544

Schreibe die Funktion als Potenz mit Basis x und finde die Ableitung. a) $f(x)=\sqrt{x}$ b) $f(x)=\sqrt[3]{x}$ c) $f(x)=\sqrt[4]{x}$ d) $f(x)=\sqrt[5]{x^{2}}$ e) $f(x)=\frac{1}{\sqrt{x^{3}}}$

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Problem 4545

For a 134-lb female, find the height's BMI change rate by deriving $f(h)=\frac{703(134)}{h^{2}}$ . Calculate $f^{\prime}(64)$ . Round final answer to two decimal places.

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Problem 4546

Find the marginal revenue for the demand function $p=150+\frac{80}{\ln x}$ , and revenue for $x=8$ .

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Problem 4547

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

See Solution

Problem 4548

Bestimmen Sie die Ableitung von $f$ und formen Sie die Funktion um: a) $f(x)=x \cdot(x-3)$ , b) $f(x)=\frac{5}{x^{3}}-\frac{x^{3}}{5}$ , c) $f(x)=(x+4)^{2}$ , d) $f(x)=\sqrt[4]{x}+\frac{1}{x^{2}}$ , e) $f(x)=x \cdot\left(x+\frac{1}{x}\right)$ , f) $f(x)=\frac{2}{\sqrt{x}}-\frac{3}{x^{5}}$ .

See Solution

Problem 4549

Find the marginal revenue for the demand function $p=150+\frac{80}{\ln x}$ and revenue for $x=8$ .

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Problem 4550

Untersuchen Sie das Krümmungsverhalten der Funktion $f(x)=x^{3}-3 x^{2}-9 x-5$ .

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Problem 4551

Bestimme den Wert von a, für den die Funktion $f_{a}(x)=x^{4}-(a+1) x^{2}+a$ an $x=1$ einen Extrempunkt hat.

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Problem 4552

Schreibe die Funktionstermine um, um einen konstanten Faktor zu erhalten, und leite sie dann ab: a) $f(x)=\frac{x^{4}}{2}$ b) $f(x)=\frac{7}{x}$ c) $f(x)=-\frac{3}{x^{2}}$ d) $f(x)=\frac{5}{3 x^{3}}$ e) $f(x)=(3 x)^{3}$ f) $f(x)=\sqrt{9 x}$

See Solution

Problem 4553

Show that the derivative of $y = kn^p$ is $\frac{kp}{n^{1-p}}$ and describe its behavior as $n$ increases.

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Problem 4554

Schreibe die Funktionstermine um, um konstante Faktoren zu erhalten, und leite sie dann ab: a) $f(x)=\frac{x^{4}}{2}$ , b) $f(x)=\frac{7}{x}$ , c) $f(x)=-\frac{3}{x^{2}}$ , d) $f(x)=\frac{5}{3 x^{3}}$ , e) $f(x)=(3 x)^{3}$ , f) $f(x)=\sqrt{9 x}$ .

See Solution

Problem 4555

Bestimmen Sie die Ableitungen der folgenden Funktionen: a) $f(x)=a x^{2}$ , b) $g(y)=-2 y^{3}$ , c) $h(t)=v t$ , d) $f(a)=5 a b$ , e) $f(x)=k^{2}$ , f) $s(t)=\frac{3}{a \sqrt{t}}$ .

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Problem 4556

Given $Y=kn^p$ with $0<p<1$ , show $\frac{d Y}{d n}=\frac{k p}{n^{1-p}}$ and explain diminishing returns.

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Problem 4557

Find the derivative of $y=(4 x^{2}+5)(3 x-4)$ using the product rule. Which option is correct?

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Problem 4558

Find the derivative of $y=(13 x^{2}-26 x+26)e^{-8 x}$ . What rule do you use first?

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Problem 4559

Fehlerbeschreibung und -korrektur:
a) $f(x)=x^{2}-6$ , $f^{\prime}(x)=2x-6$
b) $f(x)=a^{2}+x$ , $f^{\prime}(x)=1$
c) $f(x)=x^{2}(3x-1)$ , $f^{\prime}(x)=2x(3x-1)+x^{2} \cdot 3$

See Solution

Problem 4560

Zeigen Sie, dass die 1. Ableitungen von $f(x)=0,25 x^{4}-x^{3}$ und $g(x)=0,2 x^{5}-0,75 x^{4}$ die gleichen Nullstellen haben.

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Problem 4561

Find the derivative of $y=(13x^2-26x+26)e^{-8x}$ . Identify the first rule to apply. $\frac{dy}{dx} =$

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Problem 4562

Find the derivative using the product rule for the function with derivative:
$\frac{d y}{d x}=\left(13 x^{2}-26 x+26\right) \frac{d}{d x}\left(e^{-8 x}\right)+e^{-8 x} \frac{d}{d x}\left(13 x^{2}-26 x+26\right)$
Choose the correct expression from the options.

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Problem 4563

Find the derivative of $y=(13 x^{2}-26 x+26) e^{-8 x}$ .

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Problem 4564

Find the derivative of $y=e^{12 x}$ . Which rule applies: A, B, C, or D?

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Problem 4565

Gegeben ist die Funktion $f(t)=5 t^{3}-30 t^{2}+60 t$ für Niederschlag in $\mathrm{mm}$ .
a) Berechne $f(1)$ , $f(2)$ , $f(3)$ . b) Bestimme $f^{\prime}(1)$ und erkläre seine Bedeutung. c) Zeige, dass es am Ende des 2. Tages eine Regenpause gab. d) Finde den Zeitpunkt $t$ mit maximalem Regen und berechne die Regenstärke.

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Problem 4566

Is it true or false that a constant function's derivative is always 0? Explain your reasoning.

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Problem 4567

Find the derivative of $y=e^{12 x}$ . Which rule applies? A. $\frac{d}{d x}\left[e^{g(x)}\right]=e^{g(x)} g^{\prime}(x)$ B. $\frac{\mathrm{d}}{\mathrm{dx}}[e]=0$ C. $\frac{d}{d x}\left[a^{x}\right]=(\ln a) a^{x}$ D. $\frac{\mathrm{d}}{\mathrm{dx}}\left[e^{\mathrm{x}}\right]=e^{\mathrm{x}}$

See Solution

Problem 4568

Differentiate $y=e^{12 x}$ . Which rule applies? A. $\frac{d}{d x}\left[e^{g(x)}\right]=e^{g(x)} g^{\prime}(x)$ B. $\frac{\mathrm{d}}{\mathrm{dx}}[e]=0$ C. $\frac{d}{d x}\left[a^{x}\right]=(\ln a) a^{x}$ D. $\frac{d}{d x}\left[e^{x}\right]=e^{x}$

See Solution

Problem 4569

Gegeben sind die Funktionen $f(x)=0,25 x^{4}-x^{3}$ und $g(x)=0,2 x^{5}-0,75 x^{4}$ . Zeigen Sie, dass die 1. Ableitungen von $f$ und $g$ die gleichen Nullstellen haben. Untersuchen Sie mit der 2. Ableitung, ob Hoch- oder Tiefpunkte existieren und skizzieren Sie die Graphen.

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Problem 4570

What is the instantaneous rate of change at the vertex of an upright parabola? (a) -1 (b) 0 (c) 1 (d) $\infty$

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Problem 4571

Leite $f(x)=\frac{e^{x}}{x}$ zweimal ab.

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Problem 4572

Finde die Wendepunkte der Funktion $f(x)=x^{3}+6 x^{2}$ .

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Problem 4573

A diver on a 10m platform dives, height at $t$ seconds is $h(t)=10 \div 2 t-4.9 t^{3}$ . Find: (a) height at $t=1$ ; (b) avg. rate change in first second; (c) instant. rate change at $t=1$ .

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Problem 4574

A diver on a 10m platform dives. The height after $t$ seconds is $h(t)=10 - 4.9 t^{2}$ . Find:
(a) Height at $t=1$ second. (b) Average rate of change in the first second. (c) Instantaneous rate of change at $t=1$ second.

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Problem 4575

Bestimmen Sie den Wendepunkt und die Wendetangente für $f(x)=0,5 x^{3}-3 x^{2}+5 x$ und die anderen Funktionen.

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Problem 4576

Finde den Wendepunkt und die Wendetangente für $f(x)=x^{3}+3x^{2}+x+2$ und $f(x)=-x^{3}-3x^{2}+4x+4$ .

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Problem 4577

Find the average rate of change of the height $h(t)=18t-0.8t^{2}$ for $10 \leq t \leq 15$ on the Moon.

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Problem 4578

Determine the inflection points of the function $f(x) = -2x^3 + 6x$ .

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Problem 4579

Find the change in gravity $g$ when moving a distance $\Delta r = x$ from Earth's center: $\Delta g \approx g \times\left(-\frac{2 x}{r}\right)$ . Is $\Delta g$ positive or negative? Calculate the percentage change in $g$ from sea level to Mount Elbert (4.35 km) with Earth's radius 6400 km.

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Problem 4580

Given the function $s(t)=t^{4}-18 t^{2}+81$ , find:
(a) Velocity $v(t)$ and acceleration $a(t)$ functions. (b) Position, velocity, speed, and acceleration at $t=2$ . (c) When is the particle stopped? (d) When is the particle speeding up or slowing down?

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Problem 4581

Bestimmen Sie die Stammfunktion von $5x$ .

See Solution

Problem 4582

Bestimmen Sie die Ableitung von $f(x) = e^{x^{2}}$ .

See Solution

Problem 4583

An element has a half-life of 1.5 billion years. Find the age of rocks with $5\%$ and $60\%$ of the original element remaining.

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Problem 4584

A diver on a $10 \mathrm{~m}$ platform has height $h(t)=10+2t-4.9t^{2}$ . Find when they hit the water and their rate of height change.

See Solution

Problem 4585

A rocket launches at 128 ft/s from a 65-ft structure. What is its maximum height?

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Problem 4586

Find the average rate of change of area with respect to radius for circles from $0 \mathrm{~cm}$ to $100 \mathrm{~cm}$ and at $120 \mathrm{~cm}$ .

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Problem 4587

Find the time $t$ when the pilot reaches maximum altitude using $h(t)=-16 t^{2}+90 t+10000$ . Relate max altitude to tangent slope.

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Problem 4588

A particle moves on the circle $x^{2}+y^{2}=25$ . At $(3,4)$ with $\frac{d x}{d t}=6$ , find $\frac{d y}{d t}$ .

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Problem 4589

Find the average rate of change of the ship's height $h(t)=\sin \frac{\pi}{5} t$ over 5 seconds and the instantaneous rate at $t=6$ .

See Solution

Problem 4590

A conical tank (diameter 10 ft, height 15 ft) has water leaking at 12 ft³/hr. Volume is $27 \pi$ ft³. Find the height change rate.

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Problem 4591

What condition ensures the tangent line at $x=5$ overestimates $f(5.25)$ ? A) $f$ decreasing, B) $f$ increasing, C) concave down, D) concave up.

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Problem 4592

Find the slope of the tangent line to $y=x^{2}+8x$ at $(-3,-15)$ . Then, find the tangent line equation and graph both.

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Problem 4593

Water flows into a tub at $F(t)=\arctan \left(\frac{\pi}{2}-\frac{t}{10}\right)$ and leaks at $L(t)=0.03(20t-t^2-75)$ . At $t=3$ , $W(3)=2.5$ . Find the linear approximation of $W$ at $t=3$ to estimate $W(3.5)$ .

See Solution

Problem 4594

Find the slope of the tangent line to $y=x^{2}+8x$ at $(-3,-15)$ using $m=\lim _{x \rightarrow -3} \frac{f(x)-f(-3)}{x+3}$ .

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Problem 4595

Find the area between the curves $y=12-x^{2}$ and $y=x^{2}-6$ .

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Problem 4596

Find the slope $m$ of the tangent line to the curve $y=3x^{2}-11x+1$ at the point $(4,5)$ and its equation.

See Solution

Problem 4597

Calculate the average rate of change of $g(x)=7 x^{4}+\frac{4}{x^{3}}$ from $x=-2$ to $x=4$ .

See Solution

Problem 4598

Evaluate and simplify the expression for the function $f(x)=6 x^{2}+3 x$ : $\frac{f(x+h)-f(x)}{h}$

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Problem 4599

Evaluate and simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=6x^{2}+5x$ .

See Solution

Problem 4600

Calculate the average rate of change of $g(x)=-3 x^{3}-1$ between $x=-4$ and $x=1$ .

See Solution

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Calculus

Problem 4501

Find the derivative of the function f(x)=−(x+1)2f(x)=-(x+1)^{2}f(x)=−(x+1)2 using the limit of the difference quotient and compute f′(1)f^{\prime}(1)f′(1).

Problem 4502

A ball is dropped from a building 480 m high. Use d(t)=4.9t2d(t)=4.9 t^{2}d(t)=4.9t2 to find average velocity for 6≤t≤7,6.1,6.01,6.0016 \leq t \leq 7, 6.1, 6.01, 6.0016≤t≤7,6.1,6.01,6.001. What is the instantaneous velocity at t=6t=6t=6 s?

Problem 4503

Find the derivative of f(x)=5tan⁡(3x)+cos⁡(x2)f(x)=5 \tan (3 x)+\cos (x^{2})f(x)=5tan(3x)+cos(x2).

Problem 4504

Evaluate the limits or explain their non-existence: (a) lim⁡x→−1x3+1x3−5x2−6x\lim _{x \rightarrow-1} \frac{x^{3}+1}{x^{3}-5 x^{2}-6 x}limx→−1​x3−5x2−6xx3+1​; (b) lim⁡x→32x2−7x+12\lim _{x \rightarrow 3} \frac{2}{x^{2}-7 x+12}limx→3​x2−7x+122​.

Problem 4505

Find the derivative of f(x)=5x2sin⁡(9x)f(x)=5 x^{2} \sin (9 x)f(x)=5x2sin(9x).

Problem 4506

Find values of ccc for continuity of f(x)f(x)f(x) at x=2x=2x=2 and check if f(x)f(x)f(x) is continuous at x=1x=1x=1.

Problem 4507

Give an example of a function fff that is continuous at a=2a=2a=2 but not differentiable there, or explain why.

Problem 4508

Find a function fff that is continuous at a=2a=2a=2 but not differentiable at that point.

Problem 4509

Find a function ggg that is differentiable at a=3a=3a=3 but lacks a limit at a=3a=3a=3.

Problem 4510

Find the limits or state if they don't exist: (a) lim⁡x→−1x3+1x3−5x2−6x\lim _{x \rightarrow-1} \frac{x^{3}+1}{x^{3}-5 x^{2}-6 x}limx→−1​x3−5x2−6xx3+1​; (b) lim⁡x→32x2−7x+12\lim _{x \rightarrow 3} \frac{2}{x^{2}-7 x+12}limx→3​x2−7x+122​.

Problem 4511

Find a function hhh that has a limit at a=−2a=-2a=−2, is defined at a=−2a=-2a=−2, but is not continuous at a=−2a=-2a=−2.

Problem 4512

Problem 4513

Finde für jede Funktion fff eine Stammfunktion FFF: a) f(x)=x3f(x)=x^{3}f(x)=x3 b) f(x)=1x4f(x)=\frac{1}{x^{4}}f(x)=x41​ c) f(x)=3x4−13x2+12f(x)=3 x^{4}-\frac{1}{3} x^{2}+12f(x)=3x4−31​x2+12 d) f(x)=1xf(x)=\frac{1}{\sqrt{x}}f(x)=x​1​

Problem 4514

Find f′(6)f^{\prime}(6)f′(6) if f(x)=13g(x)+15h(x)f(x)=\frac{1}{3} g(x)+\frac{1}{5} h(x)f(x)=31​g(x)+51​h(x), given g′(6)=5g^{\prime}(6)=5g′(6)=5 and h′(6)=15h^{\prime}(6)=15h′(6)=15.

Problem 4515

Find the derivative of y=6ln⁡(x+8)x2y=\frac{6 \ln (x+8)}{x^{2}}y=x26ln(x+8)​. Which differentiation rule helps most: A. chain, B. power, C. sum, D. quotient?

Problem 4516

Find the limit as xxx approaches 3 for the expression 2x2−18x+3\frac{2 x^{2}-18}{x+3}x+32x2−18​.

Problem 4517

Find the slope of the tangent line for y=3x1/2+x3/2y=3 x^{1/2}+x^{3/2}y=3x1/2+x3/2 at x=4x=4x=4.

Problem 4518

Find the limit: lim⁡x→32x2−18x+3\lim _{x \rightarrow 3} \frac{2 x^{2}-18}{x+3}limx→3​x+32x2−18​.

Problem 4519

Estimate the population in 2030 using p(t)=37.75(1.021)tp(t)=37.75(1.021)^{t}p(t)=37.75(1.021)t and find the rate of change at t=30t=30t=30.

Problem 4520

Problem 4521

Find the limit of the difference quotient for the derivative of f(x)=−(x+4)2f(x)=-(x+4)^{2}f(x)=−(x+4)2.

Problem 4522

Find the slope of the tangent line for y=x4−6x3+5y=x^{4}-6 x^{3}+5y=x4−6x3+5 at x=2x=2x=2. Choose the correct method from the options.

Problem 4523

Gegeben ist die Änderungsrate fff eines Tankinhalts.a) Zeichne den Graphen der Integralfunktion I−3(x)=∫−3xf(t)dtI_{-3}(x)=\int_{-3}^{x} f(t) dtI−3​(x)=∫−3x​f(t)dt.b) Erkläre die Bedeutung der Integralfunktion.c) Was lässt sich über den Tankinhalt am Tag 7 sagen?

Problem 4524

Find the slope of the tangent line for y=x4−6x3+5y=x^{4}-6 x^{3}+5y=x4−6x3+5 at x=2x=2x=2. Choose the correct method: A, B, C, or D.

Problem 4525

Find the derivative: f′(x)=lim⁡h→0−(x+h+4)2+(x+4)2hf^{\prime}(x)=\lim _{h \rightarrow 0} \frac{-(x+h+4)^{2}+(x+4)^{2}}{h}f′(x)=limh→0​h−(x+h+4)2+(x+4)2​.

Problem 4526

Bestimme die Stellen, an denen die Funktion f(x)=x2f(x)=x^{2}f(x)=x2 die Steigung hat: a) 6, b) -2, c) 0, d) -4.

Problem 4527

Find the bullet's acceleration when fired straight down from a cliff, ignoring air resistance. Use 9.8 m/s29.8 \, m/s^29.8m/s2 for gravity.

Problem 4528

Finde die x-Werte, an denen die Funktionen f(x)=23x3−10xf(x)=\frac{2}{3} x^{3}-10 xf(x)=32​x3−10x und g(x)=2x2+6x−4g(x)=2 x^{2}+6 x-4g(x)=2x2+6x−4 die gleiche Steigung haben.

Problem 4529

Bestimme die Stellen, an denen der Graph von f(x)=2x2+2f(x)=2 x^{2}+2f(x)=2x2+2 die Steigung m=4m=4m=4 hat und dieselbe Steigung wie g(x)=x3−4x−1g(x)=x^{3}-4 x-1g(x)=x3−4x−1.

Problem 4530

Ein Pkw beschleunigt. a) Skizzieren Sie den Geschwindigkeitsgraphen v. b) Untersuchen Sie v'. c) Deuten Sie v(2)=3v(2)=3v(2)=3 und v′(12)=1v'(12)=1v′(12)=1 im Kontext.

Problem 4531

Find the rate of change of sales for the function S(t)=102−100e−0.4tS(t)=102-100 e^{-0.4 t}S(t)=102−100e−0.4t at (a) 1 year, (b) 5 years. What happens to the rate over time? Does it ever equal zero?

Problem 4532

Is the derivative of f(t)=ln⁡(t)f(t)=\ln(t)f(t)=ln(t) equal to f′(t)=1/tf^{\prime}(t)=1/tf′(t)=1/t true or false? Explain.

Problem 4533

Is the derivative of f(t)=ln⁡(t)f(t)=\ln (t)f(t)=ln(t) equal to f′(t)=1/tf^{\prime}(t)=1 / tf′(t)=1/t true or false? Explain your reasoning.

Problem 4534

Find the derivative of u(x)=C(x)u(x)=C(x)u(x)=C(x) where v(x)=xv(x)=xv(x)=x. What is u′(x)u^{\prime}(x)u′(x)? A. C′(x)C^{\prime}(x)C′(x) B. 1C′(x)\frac{1}{C^{\prime}(x)}C′(x)1​ C. C(x)⋅C′(x)C(x) \cdot C^{\prime}(x)C(x)⋅C′(x)

Problem 4535

What is the quotient rule for the derivative of f(x)=u(x)v(x)f(x)=\frac{u(x)}{v(x)}f(x)=v(x)u(x)​? A, B, or C?

Problem 4536

Find the derivative of u(x)=C(x)u(x)=C(x)u(x)=C(x) and v(x)=xv(x)=xv(x)=x. What are u′(x)u^{\prime}(x)u′(x) and v′(x)v^{\prime}(x)v′(x)?

Problem 4537

Beurteilen Sie, ob der Grenzwert von f(x)=0,001x4−0,11x2+3f(x) = 0,001 x^{4}-0,11 x^{2}+3f(x)=0,001x4−0,11x2+3 im Unendlichen 0 ist.

Problem 4538

Bestimmen Sie die Ableitungen der Funktionen: a) f(x)=x−4f(x)=x^{-4}f(x)=x−4, b) f(x)=x−1f(x)=x^{-1}f(x)=x−1, c) g(x)=x−2g(x)=x^{-2}g(x)=x−2, d) h(x)=x12h(x)=x^{\frac{1}{2}}h(x)=x21​, e) g(x)=x13g(x)=x^{\frac{1}{3}}g(x)=x31​.

Problem 4539

Leiten Sie folgende Funktionen ab: a) f(x)=x−4f(x)=x^{-4}f(x)=x−4, b) f(x)=x−1f(x)=x^{-1}f(x)=x−1, c) g(x)=x−2g(x)=x^{-2}g(x)=x−2, d) h(x)=x12h(x)=x^{\frac{1}{2}}h(x)=x21​, e) g(x)=x13g(x)=x^{\frac{1}{3}}g(x)=x31​.

Problem 4540

Find the marginal average cost function Cˉ′(x)\bar{C}^{\prime}(x)Cˉ′(x) and verify using the quotient rule.

Problem 4541

Find the derivative of the function $f(x)=-(x+1)^{2}$ using the limit of the difference quotient and compute $f^{\prime}(1)$ .

A ball is dropped from a building 480 m high. Use $d(t)=4.9 t^{2}$ to find average velocity for $6 \leq t \leq 7, 6.1, 6.01, 6.001$ . What is the instantaneous velocity at $t=6$ s?

Find the derivative of $f(x)=5 \tan (3 x)+\cos (x^{2})$ .

Evaluate the limits or explain their non-existence: (a) $\lim _{x \rightarrow-1} \frac{x^{3}+1}{x^{3}-5 x^{2}-6 x}$ ; (b) $\lim _{x \rightarrow 3} \frac{2}{x^{2}-7 x+12}$ .

Find the derivative of $f(x)=5 x^{2} \sin (9 x)$ .

Find values of $c$ for continuity of $f(x)$ at $x=2$ and check if $f(x)$ is continuous at $x=1$ .

Give an example of a function $f$ that is continuous at $a=2$ but not differentiable there, or explain why.

Find a function $f$ that is continuous at $a=2$ but not differentiable at that point.

Find a function $g$ that is differentiable at $a=3$ but lacks a limit at $a=3$ .

Find the limits or state if they don't exist: (a) $\lim _{x \rightarrow-1} \frac{x^{3}+1}{x^{3}-5 x^{2}-6 x}$ ; (b) $\lim _{x \rightarrow 3} \frac{2}{x^{2}-7 x+12}$ .

Find a function $h$ that has a limit at $a=-2$ , is defined at $a=-2$ , but is not continuous at $a=-2$ .

Finde für jede Funktion $f$ eine Stammfunktion $F$ : a) $f(x)=x^{3}$ b) $f(x)=\frac{1}{x^{4}}$ c) $f(x)=3 x^{4}-\frac{1}{3} x^{2}+12$ d) $f(x)=\frac{1}{\sqrt{x}}$

Find $f^{\prime}(6)$ if $f(x)=\frac{1}{3} g(x)+\frac{1}{5} h(x)$ , given $g^{\prime}(6)=5$ and $h^{\prime}(6)=15$ .

Find the derivative of $y=\frac{6 \ln (x+8)}{x^{2}}$ . Which differentiation rule helps most: A. chain, B. power, C. sum, D. quotient?

Find the limit as $x$ approaches 3 for the expression $\frac{2 x^{2}-18}{x+3}$ .

Find the slope of the tangent line for $y=3 x^{1/2}+x^{3/2}$ at $x=4$ .

Find the limit: $\lim _{x \rightarrow 3} \frac{2 x^{2}-18}{x+3}$ .

Estimate the population in 2030 using $p(t)=37.75(1.021)^{t}$ and find the rate of change at $t=30$ .

Find the limit of the difference quotient for the derivative of $f(x)=-(x+4)^{2}$ .

Find the slope of the tangent line for $y=x^{4}-6 x^{3}+5$ at $x=2$ . Choose the correct method from the options.

Gegeben ist die Änderungsrate $f$ eines Tankinhalts.
a) Zeichne den Graphen der Integralfunktion $I_{-3}(x)=\int_{-3}^{x} f(t) dt$ .
b) Erkläre die Bedeutung der Integralfunktion.
c) Was lässt sich über den Tankinhalt am Tag 7 sagen?

Find the slope of the tangent line for $y=x^{4}-6 x^{3}+5$ at $x=2$ . Choose the correct method: A, B, C, or D.

Find the derivative: $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{-(x+h+4)^{2}+(x+4)^{2}}{h}$ .

Bestimme die Stellen, an denen die Funktion $f(x)=x^{2}$ die Steigung hat: a) 6, b) -2, c) 0, d) -4.

Find the bullet's acceleration when fired straight down from a cliff, ignoring air resistance. Use $9.8 \, m/s^2$ for gravity.

Finde die x-Werte, an denen die Funktionen $f(x)=\frac{2}{3} x^{3}-10 x$ und $g(x)=2 x^{2}+6 x-4$ die gleiche Steigung haben.

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

Ein Pkw beschleunigt. a) Skizzieren Sie den Geschwindigkeitsgraphen v. b) Untersuchen Sie v'. c) Deuten Sie $v(2)=3$ und $v'(12)=1$ im Kontext.

Find the rate of change of sales for the function $S(t)=102-100 e^{-0.4 t}$ at (a) 1 year, (b) 5 years. What happens to the rate over time? Does it ever equal zero?

Is the derivative of $f(t)=\ln(t)$ equal to $f^{\prime}(t)=1/t$ true or false? Explain.

Is the derivative of $f(t)=\ln (t)$ equal to $f^{\prime}(t)=1 / t$ true or false? Explain your reasoning.

Find the derivative of $u(x)=C(x)$ where $v(x)=x$ . What is $u^{\prime}(x)$ ? A. $C^{\prime}(x)$ B. $\frac{1}{C^{\prime}(x)}$ C. $C(x) \cdot C^{\prime}(x)$

What is the quotient rule for the derivative of $f(x)=\frac{u(x)}{v(x)}$ ? A, B, or C?

Find the derivative of $u(x)=C(x)$ and $v(x)=x$ . What are $u^{\prime}(x)$ and $v^{\prime}(x)$ ?

Beurteilen Sie, ob der Grenzwert von $f(x) = 0,001 x^{4}-0,11 x^{2}+3$ im Unendlichen 0 ist.

Bestimmen Sie die Ableitungen der Funktionen: a) $f(x)=x^{-4}$ , b) $f(x)=x^{-1}$ , c) $g(x)=x^{-2}$ , d) $h(x)=x^{\frac{1}{2}}$ , e) $g(x)=x^{\frac{1}{3}}$ .

Leiten Sie folgende Funktionen ab: a) $f(x)=x^{-4}$ , b) $f(x)=x^{-1}$ , c) $g(x)=x^{-2}$ , d) $h(x)=x^{\frac{1}{2}}$ , e) $g(x)=x^{\frac{1}{3}}$ .

Find the marginal average cost function $\bar{C}^{\prime}(x)$ and verify using the quotient rule.

Find the derivative of $f(x) = (5x + 2)^3$ .

Untersuchen Sie die Krümmung der Funktion $f(x)=x^{4}+x^{2}$ .

Schreibe die Funktion als Potenz mit Basis x und finde die Ableitung. a) $f(x)=\sqrt{x}$ b) $f(x)=\sqrt[3]{x}$ c) $f(x)=\sqrt[4]{x}$ d) $f(x)=\sqrt[5]{x^{2}}$ e) $f(x)=\frac{1}{\sqrt{x^{3}}}$

For a 134-lb female, find the height's BMI change rate by deriving $f(h)=\frac{703(134)}{h^{2}}$ . Calculate $f^{\prime}(64)$ . Round final answer to two decimal places.

Find the marginal revenue for the demand function $p=150+\frac{80}{\ln x}$ , and revenue for $x=8$ .

Find the marginal revenue for the demand function $p=150+\frac{80}{\ln x}$ and revenue for $x=8$ .

Untersuchen Sie das Krümmungsverhalten der Funktion $f(x)=x^{3}-3 x^{2}-9 x-5$ .

Bestimme den Wert von a, für den die Funktion $f_{a}(x)=x^{4}-(a+1) x^{2}+a$ an $x=1$ einen Extrempunkt hat.

Show that the derivative of $y = kn^p$ is $\frac{kp}{n^{1-p}}$ and describe its behavior as $n$ increases.

Bestimmen Sie die Ableitungen der folgenden Funktionen: a) $f(x)=a x^{2}$ , b) $g(y)=-2 y^{3}$ , c) $h(t)=v t$ , d) $f(a)=5 a b$ , e) $f(x)=k^{2}$ , f) $s(t)=\frac{3}{a \sqrt{t}}$ .

Given $Y=kn^p$ with $0<p<1$ , show $\frac{d Y}{d n}=\frac{k p}{n^{1-p}}$ and explain diminishing returns.

Find the derivative of $y=(4 x^{2}+5)(3 x-4)$ using the product rule. Which option is correct?

Find the derivative of $y=(13 x^{2}-26 x+26)e^{-8 x}$ . What rule do you use first?

Zeigen Sie, dass die 1. Ableitungen von $f(x)=0,25 x^{4}-x^{3}$ und $g(x)=0,2 x^{5}-0,75 x^{4}$ die gleichen Nullstellen haben.

Find the derivative of $y=(13x^2-26x+26)e^{-8x}$ . Identify the first rule to apply. $\frac{dy}{dx} =$

Find the derivative of $y=(13 x^{2}-26 x+26) e^{-8 x}$ .

Find the derivative of $y=e^{12 x}$ . Which rule applies: A, B, C, or D?

What is the instantaneous rate of change at the vertex of an upright parabola? (a) -1 (b) 0 (c) 1 (d) $\infty$

Leite $f(x)=\frac{e^{x}}{x}$ zweimal ab.

Finde die Wendepunkte der Funktion $f(x)=x^{3}+6 x^{2}$ .

A diver on a 10m platform dives, height at $t$ seconds is $h(t)=10 \div 2 t-4.9 t^{3}$ . Find: (a) height at $t=1$ ; (b) avg. rate change in first second; (c) instant. rate change at $t=1$ .

A diver on a 10m platform dives. The height after $t$ seconds is $h(t)=10 - 4.9 t^{2}$ . Find:
(a) Height at $t=1$ second. (b) Average rate of change in the first second. (c) Instantaneous rate of change at $t=1$ second.