Calculus

Problem 5601

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

See Solution

Problem 5602

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=-4 x^{2}$ and simplify it.

See Solution

Problem 5603

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

See Solution

Problem 5604

Find the limit as $x$ approaches 0 for the function $f(x)=\frac{2 x}{\sin 3 x}$ .

See Solution

Problem 5605

Analyze object B's motion: describe it, create a motion map, calculate displacement, average velocity, average acceleration, and instantaneous velocities.

See Solution

Problem 5606

Find $g^{\prime}(x)$ for $g(x)=\cos (x)$ and calculate $g^{\prime}(5)$ rounded to the nearest hundredth.

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Problem 5607

Find $g^{\prime}(x)$ for $g(x)=\sin (x)$ and calculate $g^{\prime}(1)$ rounded to the nearest hundredth.

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Problem 5608

Encuentra $T(t), N(t), B(t), K(t)$ para $r(t)=<5t-2, e^t, 3t>$

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Problem 5609

Find the derivative of the function $f(x)=2 \sin x+5 \cos x$ , i.e., $f^{\prime}(x)=$ .

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Problem 5610

Find $f'(x)$ for $f(x)=9x+2\sin x+6\cos x$ and calculate $f'(1)$ (round to the nearest hundredth).

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Problem 5611

Find the limits: $\lim _{h \rightarrow 0} \frac{\cos (h)-1}{h}$ and $\lim _{h \rightarrow 0} \frac{\sin (h)}{h}$ .

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Problem 5612

Find the tangent line equation to $y=2 \sin x$ at $\left(\frac{\pi}{6}, 1\right)$ in the form $y=m x+b$ . Round $m$ and $b$ to the nearest hundredth.

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Problem 5613

Simplify $f(x)=\frac{1-\cos 4 x}{x}$ and indicate if you're finding its derivative, integral, limit, or another aspect.

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Problem 5614

Solve the ODE $\frac{d^{2} y}{d x^{2}}-\omega^{2} y=0$ with $y(0)=0$ and $\left.\frac{d y}{d x}\right|_{x=\pi}=0$ .

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Problem 5615

Find the derivative $[D f]$ of the function $f\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=\left(\begin{array}{c} (1+2 x+3 y)^{-1} \\ e^{2 y-5 z} \\ (x+2)(z-4) \end{array}\right)$ .

See Solution

Problem 5616

Find the derivative $[D f]$ of the function $f(x, y, z) = \begin{pmatrix} (1+2x+3y)^{-1} \\ e^{2y-5z} \\ (x+2)(z-4) \end{pmatrix}$ , showing your work.

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Problem 5617

Find the derivative $[D f]$ of the function $f(x,y,z)$ and evaluate it at the origin. Show your work.

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Problem 5618

Find the limit as $x$ approaches 3 from the left: $\lim _{x \rightarrow 3^{-}} \ln (3-x)$ .

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Problem 5619

Find the limit: $\lim _{x \rightarrow 5^{+}} \ln (x-5)$ .

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Problem 5620

Find the derivative of $f(x)=3+\frac{3}{2} x$ using the definition of the derivative.

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Problem 5621

Find the derivatives of $f$ and $g$ at the origin for the functions $f(x,y,z)$ and $g(s,t)$ .

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Problem 5622

Find the average rate of change between points A, B, and C given: A to B is $\frac{1}{5}$ , B to C is $\frac{1}{4}$ , A to C is $\frac{1}{6}$ .

See Solution

Problem 5623

Find the limit: $\lim _{x \rightarrow-4^{+}} \frac{(x-6)^{3}(x+4)}{(x^{2}+8)(x^{2}-3 x-10)}$ .

See Solution

Problem 5624

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

See Solution

Problem 5625

A plane drops a package from $117 \mathrm{~m}$ at $42 \mathrm{~m/s}$ . Find where it lands, and its horizontal and vertical velocities before impact.

See Solution

Problem 5626

Find the tangent line equation to $y=2 \cos x$ at $x=\frac{\pi}{6}$ . Use exact values. $y=$

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Problem 5627

Find the derivative of $\frac{1}{x^{3}}-\frac{1}{x}+x^{2}$ and evaluate it at $x=-1$ .

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Problem 5628

Find $f^{\prime}(e)$ for the function $f(x)=e^{x}$ .

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Problem 5629

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt[3]{h+8}-2}{h}$ .

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Problem 5630

Find the limit: $\lim _{n \rightarrow 0} \frac{4}{\sqrt[3]{n+8}+2}$ .

See Solution

Problem 5631

Find $\lim _{x \rightarrow 3} f(x)$ given $3 x-1 \leq f(x) \leq x^{2}-3 x+8$ for $x \geq 0$ .

See Solution

Problem 5632

Find the derivative of the function $y=3 \log _{3}(x^{3}-5)$ .

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Problem 5633

Find the derivative of $y=\log_{19} x$ . Use logarithm properties to simplify before calculating $f^{\prime}(x)$ .

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Problem 5634

Find the horizontal asymptotes of the function $f(x)=\frac{x+e^{x}}{3+4 x^{3}}$ .

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Problem 5635

Find the horizontal asymptotes of the function $f(x)=\frac{5 x^{2}+e^{x}}{x^{3}-2}$ .

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Problem 5636

Find the speed of a searchlight beam along the shoreline when it makes a $45^{\circ}$ angle, turning at 5 rev/min.

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Problem 5637

Find the general solution of $y'' + 4y' + 4y = 0$ and show $y=e^{-2x}$ and $y=xe^{-2x}$ are independent using the Wronskian.

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Problem 5638

Find a relation between angle $\theta$ and side length $c$ in $\triangle ABC$ . Also, find the rate of increase of $c$ when $\frac{d \theta}{d t}=5 \text{ radians/sec}$ and $\cos \theta=\frac{2}{3}$ .

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Problem 5639

Bestimme den Differenzquotienten für $f(x)=2 x^{2}+5$ im Intervall [-2, 4].

See Solution

Problem 5640

Bestimme die mittlere Änderungsrate der Funktion $f(x)=2 x^{2}+5$ im Intervall $[-2, 4]$ .

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Problem 5641

How fast is the height of a conical gravel pile increasing when it is 7 ft high, given a rate of $25 \mathrm{ft}^{3}/\mathrm{min}$ ?

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Problem 5642

Finde das Extremum (Maxima oder Minima) der Funktion $f(x)=2x^{2}+5$ im Intervall $[-2, 4]$ .

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Problem 5643

Find the horizontal asymptotes of the function $f(x)=5x^{5}+x^{3}+x^{4}$ .

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Problem 5644

Find if the limit is $\infty$ , $-\infty$ , or a specific value: $\lim_{x \rightarrow-\infty} \frac{2x^{2}+4}{6x^{2}-e^{x}}$

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Problem 5645

What is the limit as $x$ approaches infinity for $\frac{3 x^{8}}{-4(3)^{x}+x^{8}}$ ? Options: 0, does not exist, or exists and not 0.

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Problem 5646

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{3}-41}}{10 x+9+3 x^{2}}$ . If infinite, state DNE.

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Problem 5647

Find $x$ such that $f(x)=\frac{x}{x+4}$ and $f^{\prime}(x)=3$ . Provide exact values for $x$ .

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Problem 5648

Find the second derivative of the function $h(u)=-6 e^{u} u^{4}$ . What is $h^{\prime \prime}(u)=?$

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Problem 5649

Calculate the average rate of change for $f(x)=x^{2}-1$ over the interval $[1, 1.1]$ .

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Problem 5650

Find the derivative $f'(x)$ of $f(x)=\frac{\sqrt{x}-2}{\sqrt{x}+2}$ and calculate $f'(2)$ rounded to the nearest hundredth.

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Problem 5651

Find the vertical and horizontal asymptotes for the function $p(x)=\frac{4}{x^{2}+1}$ .

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Problem 5652

Find $f'(x)$ for $f(x)=\frac{5 \sin x}{2 \sin x+4 \cos x}$ . Determine the tangent line at $a=\frac{\pi}{6}$ : $y=mx+b$ , round $m$ and $b$ to the nearest hundredth.

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Problem 5653

Verify if the derivative of a product $(f g)^{\prime}(x)$ equals the product of the derivatives $f^{\prime}(x) g^{\prime}(x)$ for $f(x)=7x+8, g(x)=3x+10$ .

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Problem 5654

Berechnen Sie die mittlere Änderungsrate von $f(x)=0,5x^{2}-2x+2$ im Intervall $[0, 4]$ .

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Problem 5655

Find the correct product rule for derivatives and use it to find the derivative of $f(x)=(x^{6}+10) \sqrt{x}$ .

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Problem 5656

If $h(x)=f(x) \cdot g(x)$ , find $h^{\prime}(1)$ . If $h(x)=\frac{f(x)}{g(x)}$ , find $h^{\prime}(1)$ .

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Problem 5657

Find the derivative of $g(x)=(5x^2-3x-3)e^x$ . What is $g'(x)=?$

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Problem 5658

Find the derivative of $\left(-4 x^{8}-3 x^{6}\right)\left(8 e^{x}-1\right)$ using the product rule.

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Problem 5659

Find the first and second derivatives of the function $f(x)=\frac{-3 x+4}{x+3}$ . Simplify your answers.

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Problem 5660

Find the derivative of $f(x) = x^{8} e^{x}$ . What is $f^{\prime}(x) =$ ?

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Problem 5661

Bestimme die Steigung an den Schnittpunkten von $f(x)=-x^{2}+4x$ mit der $x$ -Achse und vergleiche mit $f^{\prime}(2)$ .

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Problem 5662

Find the second derivative of the function $g(x)=-5 e^{x} x^{5}$ . What is $\frac{d^{2} g}{d x^{2}}$ ?

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Problem 5663

Find the derivative of the function $f(x)=\frac{5 \sin x}{2 \sin x+6 \cos x}$ . What is $f^{\prime}(x)$ ?

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Problem 5664

Find the derivative $f'(x)$ for $f(x)=\frac{5 \sin x}{2 \sin x+6 \cos x}$ and the tangent line at $a=\frac{\pi}{6}$ in $y=mx+b$ form. Round $m$ and $b$ to the nearest hundredth.

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Problem 5665

A pilot's flight distance is modeled by $d(t)=t^{3}+30t+100$ .
a) Find average velocity from 1 to 4 hours. b) Find instantaneous rate of change at 3 hours. c) Explain the meanings of a and b.

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Problem 5666

Find the tangent line to $y=3 x \cos x$ at $(\pi,-3 \pi)$ . Express it as $y=m x+b$ with $m=$ and $b=$ .

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Problem 5667

Verify if the derivative of a quotient is the quotient of the derivatives for $f(x)=4x+5$ and $g(x)=9x$ .

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Problem 5668

You buy a car for \ $40000. Its value after$ t $years is$ V(t)=40000(0.55)^{t} $. Find$ V(3) $and$ V^{\prime}(3) $. Interpret$ V(7)=609 $and$ V^{\prime}(7)=-7234 $. How does$ V^{\prime}(1)=-13152$ affect your decision?

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Problem 5669

Verify if the derivative of a quotient is the quotient of the derivatives for $f(x)=4x+5$ and $g(x)=9x$ . Calculate $f'(x)$ , $g'(x)$ , and check the claims.

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Problem 5670

Find the derivative of $g(x)=\frac{e^{x}}{1-5 x}$ . What is $g^{\prime}(x)=?$

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Problem 5671

Find the derivative of the function $f(x)=\frac{7 \sin x}{2 \sin x+4 \cos x}$ . What is $f^{\prime}(x)$ ?

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Problem 5672

Verify if the derivative of a product is the product of the derivatives for $f(x)=5x+4$ and $g(x)=7x+9$ .

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Problem 5673

Find the derivative of $f(x)=5 x(\sin x+\cos x)$ and calculate $f^{\prime}(4)$ rounded to the nearest hundredth.

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Problem 5674

Calculate the derivative $(fg)'(x)$ using the product rule for $f(x)=7x+5$ and $g(x)=3x+10$ .

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Problem 5675

Check if the derivative of a product $(f g)^{\prime}(x)$ equals the product of derivatives $f^{\prime}(x) g^{\prime}(x)$ for $f(x)=7x+5, g(x)=3x+10$ .

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Problem 5676

Find $f^{\prime}(-1)$ for $f(x)=x^{7} h(x)$ given $h(-1)=4$ and $h^{\prime}(-1)=7$ .

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Problem 5677

Find the derivative $f'(x)$ for $f(x)=\frac{7 \sin x}{2 \sin x + 4 \cos x}$ . Determine the tangent line at $a=\frac{\pi}{3}$ : $y=mx+b$ with $m$ and $b$ rounded to the nearest hundredth.

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Problem 5678

Find the derivative of $(9 x^{3}+2 x^{8})(5 e^{x}+2)$ using the product rule without expanding.

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Problem 5679

Bestimme die Ableitung von $f$ , finde die Extremstellen mit $f^{\prime}(x)=0$ und prüfe mit dem Vorzeichenwechselkriterium. a) $f(x)=x^{2}-8 x+15$ b) $f(x)=2 x^{3}-3 x^{2}-12 x+4$

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Problem 5680

Simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=2x-3$ .

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Problem 5681

Find the correct product rule for derivatives: $(ab)' = ?$ Then use it to find $f'(x)$ for $f(x)=(x^6+4)\sqrt{x}$ .

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Problem 5682

Find the tangent line to $y=2 x \cos x$ at $(\pi,-2 \pi)$ in the form $y=m x+b$ where $m=$ and $b=$ .

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Problem 5683

Find the derivative $f'(x)$ of $f(x)=5x(\sin x+\cos x)$ , then compute $f'(2)$ rounded to the nearest hundredth.

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Problem 5684

Bestimme den Funktionstyp von $f$ und ob $f$ für $x \rightarrow+\infty$ konvergiert oder divergiert: a) $f(x) = 3x^{3} - x^{2} - 2x + 5$ b) $f(x) = -\frac{4}{x} - 7$ c) $f(x) = 0,01 \cdot \sin(x) + \frac{1}{10}$

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Problem 5685

Find the tangent line to $y=3 x \cos x$ at $(\pi,-3 \pi)$ . Write it as $y=m x+b$ with $m=$ and $b=$ .

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Problem 5686

Find the derivative of $\frac{10 e^{x}+8}{-3 x^{5}+2 x^{7}}$ using the quotient rule. No need to expand.

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Problem 5687

Find the derivative $f'(x)$ of $f(x)=\frac{5 \sin x}{1+\cos x}$ and compute $f'(4)$ (rounded to the nearest hundredth).

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Problem 5688

Eine Pflanze hat Blattläuse. Die Funktion $f(t)=\frac{300}{0,1 t+1}$ beschreibt die Anzahl der Blattläuse.
a) Finde den Zeitpunkt, wann die Blattläuse nach 15 Tagen mit Marienkäfern ausgestorben sind. b) Bestimme, wann Marienkäfer eingesetzt werden müssen, damit die Blattläuse nach 30 Tagen aussterben.

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Problem 5689

Approximate $f(0.82)$ using the tangent line of $f(x)=x \sin x$ at $x=\frac{\pi}{2}$ with slope 1.

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Problem 5690

Approximate sugar production when the price is 28 cents using $f'(30)=50$ and $f(30)=6700$ .

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Problem 5691

What does $f^{\prime}(3)=2$ mean for rainfall at 3:00 am? It indicates a rate of 2 cm/h and predicts 2 cm more rain.

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Problem 5692

Ergänzen Sie die Tabelle mit dem Verhalten der Funktionen für $x \rightarrow +\infty$ und $x \rightarrow -\infty$ . Skizzieren Sie eine Funktion zu den gegebenen Asymptoten.

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Problem 5693

Untersuchen Sie die Stetigkeit und Differenzierbarkeit von $f$ . Bestimmen Sie eine knickfreie Fortsetzung und eine neue Funktion $g(x)$ .
1. $f(x)=\begin{cases} x^{2}-8 & \text{für } x \leq 2 \\ 0.5 x^{2}+2 x-10 & \text{für } x > 2 \end{cases}$
2. $f(x)=0.25 x^{2}-1$ für $x \in [-3, 3]$ .

See Solution

Problem 5694

Approximate $f(0.82)$ using the tangent line of $f(x)=x \sin x$ at $x=\frac{\pi}{2}$ , which rises $\frac{\pi}{2}$ per $\frac{\pi}{2}$ .

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Problem 5695

Gegeben ist $f(x)=3 x^{4}-6 x^{3}$ . Untersuchen Sie Symmetrie, Nullstellen, Extrempunkte, Wendepunkte und die Ortskurve für $k \neq 0$ .

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Problem 5696

Untersuchen Sie die Stetigkeit und Differenzierbarkeit der Funktion $f$ definiert durch: $f(x)=\left\{\begin{array}{ll} x^{2}-8 & \text { für } x \leq 2 \\ 0.5 x^{2}+2 x-10 & \text { für } x>2 \end{array}\right.$

See Solution

Problem 5697

Determine the features of $f(x)=\frac{1}{2} e^{2 x}(x^{2}-2)$ : y-intercept, endpoint behavior, symmetry, extrema, inflection points, and asymptotes.

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Problem 5698

Find the first and second derivatives of $f(x)=\frac{1}{2} e^{2 x}(x^{2}-2)$ , then determine extreme points, inflection points, limits as $x \to \pm \infty$ , and roots.

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Problem 5699

Untersuchen Sie die Funktion $f_{k}(x)=3 x^{4}-k x^{3}$ für $k \neq 0$ . Hat der Graph immer eine Tiefstelle? Bestimmen Sie die Ortskurve der Wendepunkte und den Wert von $k$ , wenn die Tangente im Wendepunkt einen Winkel von $\alpha=-21^{\circ}$ zur $x$ -Achse bildet.

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Problem 5700

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

See Solution

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Calculus

Problem 5601

Calculate the average rate of change of g(x)=3x3+4g(x)=3 x^{3}+4g(x)=3x3+4 between x=−4x=-4x=−4 and x=4x=4x=4.

Problem 5602

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=−4x2f(x)=-4 x^{2}f(x)=−4x2 and simplify it.

Problem 5603

Calculate the average rate of change of g(x)=3x3+4g(x)=3x^{3}+4g(x)=3x3+4 from x=−4x=-4x=−4 to x=4x=4x=4.

Problem 5604

Find the limit as xxx approaches 0 for the function f(x)=2xsin⁡3xf(x)=\frac{2 x}{\sin 3 x}f(x)=sin3x2x​.

Problem 5605

Analyze object B's motion: describe it, create a motion map, calculate displacement, average velocity, average acceleration, and instantaneous velocities.

Problem 5606

Find g′(x)g^{\prime}(x)g′(x) for g(x)=cos⁡(x)g(x)=\cos (x)g(x)=cos(x) and calculate g′(5)g^{\prime}(5)g′(5) rounded to the nearest hundredth.

Problem 5607

Find g′(x)g^{\prime}(x)g′(x) for g(x)=sin⁡(x)g(x)=\sin (x)g(x)=sin(x) and calculate g′(1)g^{\prime}(1)g′(1) rounded to the nearest hundredth.

Problem 5608

Encuentra T(t),N(t),B(t),K(t)T(t), N(t), B(t), K(t)T(t),N(t),B(t),K(t) para r(t)=<5t−2,et,3t>r(t)=<5t-2, e^t, 3t>r(t)=<5t−2,et,3t>

Problem 5609

Find the derivative of the function f(x)=2sin⁡x+5cos⁡xf(x)=2 \sin x+5 \cos xf(x)=2sinx+5cosx, i.e., f′(x)=f^{\prime}(x)=f′(x)=.

Problem 5610

Find f′(x)f'(x)f′(x) for f(x)=9x+2sin⁡x+6cos⁡xf(x)=9x+2\sin x+6\cos xf(x)=9x+2sinx+6cosx and calculate f′(1)f'(1)f′(1) (round to the nearest hundredth).

Problem 5611

Find the limits: lim⁡h→0cos⁡(h)−1h\lim _{h \rightarrow 0} \frac{\cos (h)-1}{h}limh→0​hcos(h)−1​ and lim⁡h→0sin⁡(h)h\lim _{h \rightarrow 0} \frac{\sin (h)}{h}limh→0​hsin(h)​.

Problem 5612

Find the tangent line equation to y=2sin⁡xy=2 \sin xy=2sinx at (π6,1)\left(\frac{\pi}{6}, 1\right)(6π​,1) in the form y=mx+by=m x+by=mx+b. Round mmm and bbb to the nearest hundredth.

Problem 5613

Simplify f(x)=1−cos⁡4xxf(x)=\frac{1-\cos 4 x}{x}f(x)=x1−cos4x​ and indicate if you're finding its derivative, integral, limit, or another aspect.

Problem 5614

Solve the ODE d2ydx2−ω2y=0\frac{d^{2} y}{d x^{2}}-\omega^{2} y=0dx2d2y​−ω2y=0 with y(0)=0y(0)=0y(0)=0 and dydx∣x=π=0\left.\frac{d y}{d x}\right|_{x=\pi}=0dxdy​∣∣​x=π​=0.

Problem 5615

Problem 5616

Find the derivative [Df][D f][Df] of the function f(x,y,z)=((1+2x+3y)−1e2y−5z(x+2)(z−4))f(x, y, z) = \begin{pmatrix} (1+2x+3y)^{-1} \\ e^{2y-5z} \\ (x+2)(z-4) \end{pmatrix}f(x,y,z)=⎝⎛​(1+2x+3y)−1e2y−5z(x+2)(z−4)​⎠⎞​, showing your work.

Problem 5617

Find the derivative [Df][D f][Df] of the function f(x,y,z)f(x,y,z)f(x,y,z) and evaluate it at the origin. Show your work.

Problem 5618

Find the limit as xxx approaches 3 from the left: lim⁡x→3−ln⁡(3−x)\lim _{x \rightarrow 3^{-}} \ln (3-x)limx→3−​ln(3−x).

Problem 5619

Find the limit: lim⁡x→5+ln⁡(x−5)\lim _{x \rightarrow 5^{+}} \ln (x-5)limx→5+​ln(x−5).

Problem 5620

Find the derivative of f(x)=3+32xf(x)=3+\frac{3}{2} xf(x)=3+23​x using the definition of the derivative.

Problem 5621

Find the derivatives of fff and ggg at the origin for the functions f(x,y,z)f(x,y,z)f(x,y,z) and g(s,t)g(s,t)g(s,t).

Problem 5622

Find the average rate of change between points A, B, and C given: A to B is 15\frac{1}{5}51​, B to C is 14\frac{1}{4}41​, A to C is 16\frac{1}{6}61​.

Problem 5623

Find the limit: lim⁡x→−4+(x−6)3(x+4)(x2+8)(x2−3x−10)\lim _{x \rightarrow-4^{+}} \frac{(x-6)^{3}(x+4)}{(x^{2}+8)(x^{2}-3 x-10)}limx→−4+​(x2+8)(x2−3x−10)(x−6)3(x+4)​.

Problem 5624

Find the limit: lim⁡x→−3+x2(x−1)x3+2x2−2x+3\lim _{x \rightarrow-3^{+}} \frac{x^{2}(x-1)}{x^{3}+2 x^{2}-2 x+3}limx→−3+​x3+2x2−2x+3x2(x−1)​.

Problem 5625

A plane drops a package from 117 m117 \mathrm{~m}117 m at 42 m/s42 \mathrm{~m/s}42 m/s. Find where it lands, and its horizontal and vertical velocities before impact.

Problem 5626

Find the tangent line equation to y=2cos⁡xy=2 \cos xy=2cosx at x=π6x=\frac{\pi}{6}x=6π​. Use exact values. y= y= y=

Problem 5627

Find the derivative of 1x3−1x+x2\frac{1}{x^{3}}-\frac{1}{x}+x^{2}x31​−x1​+x2 and evaluate it at x=−1x=-1x=−1.

Problem 5628

Find f′(e)f^{\prime}(e)f′(e) for the function f(x)=exf(x)=e^{x}f(x)=ex.

Problem 5629

Find the limit: lim⁡h→0h+83−2h\lim _{h \rightarrow 0} \frac{\sqrt[3]{h+8}-2}{h}limh→0​h3h+8​−2​.

Problem 5630

Find the limit: lim⁡n→04n+83+2\lim _{n \rightarrow 0} \frac{4}{\sqrt[3]{n+8}+2}limn→0​3n+8​+24​.

Problem 5631

Find lim⁡x→3f(x)\lim _{x \rightarrow 3} f(x)limx→3​f(x) given 3x−1≤f(x)≤x2−3x+83 x-1 \leq f(x) \leq x^{2}-3 x+83x−1≤f(x)≤x2−3x+8 for x≥0x \geq 0x≥0.

Problem 5632

Find the derivative of the function y=3log⁡3(x3−5)y=3 \log _{3}(x^{3}-5)y=3log3​(x3−5).

Problem 5633

Find the derivative of y=log⁡19xy=\log_{19} xy=log19​x. Use logarithm properties to simplify before calculating f′(x)f^{\prime}(x)f′(x).

Problem 5634

Find the horizontal asymptotes of the function f(x)=x+ex3+4x3f(x)=\frac{x+e^{x}}{3+4 x^{3}}f(x)=3+4x3x+ex​.

Problem 5635

Find the horizontal asymptotes of the function f(x)=5x2+exx3−2f(x)=\frac{5 x^{2}+e^{x}}{x^{3}-2}f(x)=x3−25x2+ex​.

Problem 5636

Find the speed of a searchlight beam along the shoreline when it makes a 45∘45^{\circ}45∘ angle, turning at 5 rev/min.

Problem 5637

Find the general solution of y′′+4y′+4y=0y'' + 4y' + 4y = 0y′′+4y′+4y=0 and show y=e−2xy=e^{-2x}y=e−2x and y=xe−2xy=xe^{-2x}y=xe−2x are independent using the Wronskian.

Problem 5638

Find a relation between angle θ\thetaθ and side length ccc in △ABC\triangle ABC△ABC. Also, find the rate of increase of ccc when dθdt=5 radians/sec\frac{d \theta}{d t}=5 \text{ radians/sec}dtdθ​=5 radians/sec and cos⁡θ=23\cos \theta=\frac{2}{3}cosθ=32​.

Problem 5639

Bestimme den Differenzquotienten für f(x)=2x2+5f(x)=2 x^{2}+5f(x)=2x2+5 im Intervall [-2, 4].

Problem 5640

Bestimme die mittlere Änderungsrate der Funktion f(x)=2x2+5f(x)=2 x^{2}+5f(x)=2x2+5 im Intervall [−2,4][-2, 4][−2,4].

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

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=-4 x^{2}$ and simplify it.

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

Find the limit as $x$ approaches 0 for the function $f(x)=\frac{2 x}{\sin 3 x}$ .

Find $g^{\prime}(x)$ for $g(x)=\cos (x)$ and calculate $g^{\prime}(5)$ rounded to the nearest hundredth.

Find $g^{\prime}(x)$ for $g(x)=\sin (x)$ and calculate $g^{\prime}(1)$ rounded to the nearest hundredth.

Encuentra $T(t), N(t), B(t), K(t)$ para $r(t)=<5t-2, e^t, 3t>$

Find the derivative of the function $f(x)=2 \sin x+5 \cos x$ , i.e., $f^{\prime}(x)=$ .

Find $f'(x)$ for $f(x)=9x+2\sin x+6\cos x$ and calculate $f'(1)$ (round to the nearest hundredth).

Find the limits: $\lim _{h \rightarrow 0} \frac{\cos (h)-1}{h}$ and $\lim _{h \rightarrow 0} \frac{\sin (h)}{h}$ .

Find the tangent line equation to $y=2 \sin x$ at $\left(\frac{\pi}{6}, 1\right)$ in the form $y=m x+b$ . Round $m$ and $b$ to the nearest hundredth.

Simplify $f(x)=\frac{1-\cos 4 x}{x}$ and indicate if you're finding its derivative, integral, limit, or another aspect.

Solve the ODE $\frac{d^{2} y}{d x^{2}}-\omega^{2} y=0$ with $y(0)=0$ and $\left.\frac{d y}{d x}\right|_{x=\pi}=0$ .

Find the derivative $[D f]$ of the function $f(x, y, z) = \begin{pmatrix} (1+2x+3y)^{-1} \\ e^{2y-5z} \\ (x+2)(z-4) \end{pmatrix}$ , showing your work.

Find the derivative $[D f]$ of the function $f(x,y,z)$ and evaluate it at the origin. Show your work.

Find the limit as $x$ approaches 3 from the left: $\lim _{x \rightarrow 3^{-}} \ln (3-x)$ .

Find the limit: $\lim _{x \rightarrow 5^{+}} \ln (x-5)$ .

Find the derivative of $f(x)=3+\frac{3}{2} x$ using the definition of the derivative.

Find the derivatives of $f$ and $g$ at the origin for the functions $f(x,y,z)$ and $g(s,t)$ .

Find the average rate of change between points A, B, and C given: A to B is $\frac{1}{5}$ , B to C is $\frac{1}{4}$ , A to C is $\frac{1}{6}$ .

Find the limit: $\lim _{x \rightarrow-4^{+}} \frac{(x-6)^{3}(x+4)}{(x^{2}+8)(x^{2}-3 x-10)}$ .

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

A plane drops a package from $117 \mathrm{~m}$ at $42 \mathrm{~m/s}$ . Find where it lands, and its horizontal and vertical velocities before impact.

Find the tangent line equation to $y=2 \cos x$ at $x=\frac{\pi}{6}$ . Use exact values. $y=$

Find the derivative of $\frac{1}{x^{3}}-\frac{1}{x}+x^{2}$ and evaluate it at $x=-1$ .

Find $f^{\prime}(e)$ for the function $f(x)=e^{x}$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt[3]{h+8}-2}{h}$ .

Find the limit: $\lim _{n \rightarrow 0} \frac{4}{\sqrt[3]{n+8}+2}$ .

Find $\lim _{x \rightarrow 3} f(x)$ given $3 x-1 \leq f(x) \leq x^{2}-3 x+8$ for $x \geq 0$ .

Find the derivative of the function $y=3 \log _{3}(x^{3}-5)$ .

Find the derivative of $y=\log_{19} x$ . Use logarithm properties to simplify before calculating $f^{\prime}(x)$ .

Find the horizontal asymptotes of the function $f(x)=\frac{x+e^{x}}{3+4 x^{3}}$ .

Find the horizontal asymptotes of the function $f(x)=\frac{5 x^{2}+e^{x}}{x^{3}-2}$ .

Find the speed of a searchlight beam along the shoreline when it makes a $45^{\circ}$ angle, turning at 5 rev/min.

Find the general solution of $y'' + 4y' + 4y = 0$ and show $y=e^{-2x}$ and $y=xe^{-2x}$ are independent using the Wronskian.

Find a relation between angle $\theta$ and side length $c$ in $\triangle ABC$ . Also, find the rate of increase of $c$ when $\frac{d \theta}{d t}=5 \text{ radians/sec}$ and $\cos \theta=\frac{2}{3}$ .

Bestimme den Differenzquotienten für $f(x)=2 x^{2}+5$ im Intervall [-2, 4].

Bestimme die mittlere Änderungsrate der Funktion $f(x)=2 x^{2}+5$ im Intervall $[-2, 4]$ .

How fast is the height of a conical gravel pile increasing when it is 7 ft high, given a rate of $25 \mathrm{ft}^{3}/\mathrm{min}$ ?

Finde das Extremum (Maxima oder Minima) der Funktion $f(x)=2x^{2}+5$ im Intervall $[-2, 4]$ .

Find the horizontal asymptotes of the function $f(x)=5x^{5}+x^{3}+x^{4}$ .

Find if the limit is $\infty$ , $-\infty$ , or a specific value: $\lim_{x \rightarrow-\infty} \frac{2x^{2}+4}{6x^{2}-e^{x}}$

What is the limit as $x$ approaches infinity for $\frac{3 x^{8}}{-4(3)^{x}+x^{8}}$ ? Options: 0, does not exist, or exists and not 0.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{3}-41}}{10 x+9+3 x^{2}}$ . If infinite, state DNE.

Find $x$ such that $f(x)=\frac{x}{x+4}$ and $f^{\prime}(x)=3$ . Provide exact values for $x$ .

Find the second derivative of the function $h(u)=-6 e^{u} u^{4}$ . What is $h^{\prime \prime}(u)=?$

Calculate the average rate of change for $f(x)=x^{2}-1$ over the interval $[1, 1.1]$ .

Find the derivative $f'(x)$ of $f(x)=\frac{\sqrt{x}-2}{\sqrt{x}+2}$ and calculate $f'(2)$ rounded to the nearest hundredth.

Find the vertical and horizontal asymptotes for the function $p(x)=\frac{4}{x^{2}+1}$ .

Find $f'(x)$ for $f(x)=\frac{5 \sin x}{2 \sin x+4 \cos x}$ . Determine the tangent line at $a=\frac{\pi}{6}$ : $y=mx+b$ , round $m$ and $b$ to the nearest hundredth.

Verify if the derivative of a product $(f g)^{\prime}(x)$ equals the product of the derivatives $f^{\prime}(x) g^{\prime}(x)$ for $f(x)=7x+8, g(x)=3x+10$ .

Berechnen Sie die mittlere Änderungsrate von $f(x)=0,5x^{2}-2x+2$ im Intervall $[0, 4]$ .

Find the correct product rule for derivatives and use it to find the derivative of $f(x)=(x^{6}+10) \sqrt{x}$ .

If $h(x)=f(x) \cdot g(x)$ , find $h^{\prime}(1)$ . If $h(x)=\frac{f(x)}{g(x)}$ , find $h^{\prime}(1)$ .

Find the derivative of $g(x)=(5x^2-3x-3)e^x$ . What is $g'(x)=?$

Find the derivative of $\left(-4 x^{8}-3 x^{6}\right)\left(8 e^{x}-1\right)$ using the product rule.

Find the first and second derivatives of the function $f(x)=\frac{-3 x+4}{x+3}$ . Simplify your answers.

Find the derivative of $f(x) = x^{8} e^{x}$ . What is $f^{\prime}(x) =$ ?

Bestimme die Steigung an den Schnittpunkten von $f(x)=-x^{2}+4x$ mit der $x$ -Achse und vergleiche mit $f^{\prime}(2)$ .

Find the second derivative of the function $g(x)=-5 e^{x} x^{5}$ . What is $\frac{d^{2} g}{d x^{2}}$ ?

Find the derivative of the function $f(x)=\frac{5 \sin x}{2 \sin x+6 \cos x}$ . What is $f^{\prime}(x)$ ?

Find the derivative $f'(x)$ for $f(x)=\frac{5 \sin x}{2 \sin x+6 \cos x}$ and the tangent line at $a=\frac{\pi}{6}$ in $y=mx+b$ form. Round $m$ and $b$ to the nearest hundredth.

A pilot's flight distance is modeled by $d(t)=t^{3}+30t+100$ .
a) Find average velocity from 1 to 4 hours. b) Find instantaneous rate of change at 3 hours. c) Explain the meanings of a and b.

Find the tangent line to $y=3 x \cos x$ at $(\pi,-3 \pi)$ . Express it as $y=m x+b$ with $m=$ and $b=$ .

Verify if the derivative of a quotient is the quotient of the derivatives for $f(x)=4x+5$ and $g(x)=9x$ .

Verify if the derivative of a quotient is the quotient of the derivatives for $f(x)=4x+5$ and $g(x)=9x$ . Calculate $f'(x)$ , $g'(x)$ , and check the claims.

Find the derivative of $g(x)=\frac{e^{x}}{1-5 x}$ . What is $g^{\prime}(x)=?$

Find the derivative of the function $f(x)=\frac{7 \sin x}{2 \sin x+4 \cos x}$ . What is $f^{\prime}(x)$ ?

Verify if the derivative of a product is the product of the derivatives for $f(x)=5x+4$ and $g(x)=7x+9$ .

Find the derivative of $f(x)=5 x(\sin x+\cos x)$ and calculate $f^{\prime}(4)$ rounded to the nearest hundredth.

Calculate the derivative $(fg)'(x)$ using the product rule for $f(x)=7x+5$ and $g(x)=3x+10$ .