Calculus

Problem 5501

Zeigen Sie, dass die Gerade $g$ mit $y=f^{\prime}(a)(x-a)+f(a)$ eine Tangente an $f$ im Punkt $P(a \mid f(a))$ ist, indem Sie (1) P auf $g$ und (2) die Steigung $f^{\prime}(a)$ überprüfen.

See Solution

Problem 5502

Find the coordinates where $f(x)=x^{3}-12x$ has a horizontal tangent line (zero slope) using differentiation shortcuts.

See Solution

Problem 5503

A stone is thrown up at $20 \mathrm{~ms}^{-1}$ . Find the max height, time to max height, and distance in 3 seconds.

See Solution

Problem 5504

An investor bought two buildings with growth rates $r_{1}(t)=0.062(1.043^{t})$ and $r_{2}(t)=0.046(1.039^{t})$ . Find the area between their graphs from $0 \leq t \leq 10$ . What does this area represent?

See Solution

Problem 5505

1) Show that $f'(x)=\frac{(x-3)(x+3)}{x^{2}}$ for $f(x)=\frac{(x+1)(x+9)}{x}$ . 2) Find $f'(x)$ for $f(x)=\frac{9x^{2}+3}{2\sqrt[3]{x}}$ . 3) Find the derivative of $y=\frac{(2\sqrt{x}-3)^{2}}{\sqrt{x^{3}}}$ .

See Solution

Problem 5506

Solve the differential equation $2 \frac{d y}{d x}=6 x$ to find $y$ .

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Problem 5507

Calculate the average rate of change of $f(x)=-1 x^{2}+3 x-8$ from $x=-4$ to $x=0$ . Average rate of change $=$

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Problem 5508

Find the tangent plane equation to the surface $x^{2}+y^{2}-z^{2}-59=0$ at points $(8,2,3)$ and $(-2,-8,3)$ .

See Solution

Problem 5509

Find the derivative of $y=\frac{2 x \sqrt{3-3 x}}{e^{4 x}\left(\sin ^{2}(x)\right)}$ using logarithmic differentiation. $y'=$

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Problem 5510

Find the derivative of $d(t)=16 t^{2}$ , its units, and compute the height for $t=4.9$ s and speed at impact in mph.

See Solution

Problem 5511

Find the derivative of the piecewise function:
$f(x)=\begin{cases} x^{2}+1 & \text{if } x \leq 0 \\ 2x^{2}+x+1 & \text{if } x > 0 \end{cases}$

See Solution

Problem 5512

Gegeben ist $f_{t}(x)=(x-t) \cdot e^{x}$ . Bestimme Schnittpunkt mit $y$ -Achse, Nullstellen, Extrem- und Wendepunkte für $t=-2, -1, 0, 1, 2$ .

See Solution

Problem 5513

Evaluate the integral $\int_{0}^{\frac{1}{3}} \frac{d x}{1+9 x^{2}}$ . Options: A. $\frac{\pi}{12}$ B. $\frac{\pi}{6}$ C. $\frac{\ln (2)}{18}$ D. $\frac{\pi}{4}$ E. $\ln (2)$

See Solution

Problem 5514

1. Find $f^{\prime}(x)$ and $f^{\prime}(2)$ for $f(x)=3 x 10^{x}$ .
2. Find $f^{\prime}(x)$ and $f^{\prime}(9)$ for $f(x)=\frac{9 x}{5 x+3}$ .

See Solution

Problem 5515

Find the tangent line equation for $f(x)=-2 x \ln (x+7)$ at $x=1$ . Answer: $y=$

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Problem 5516

Find the population growth rate in thousands at $t=6$ for $P(t)=(0.8 t-7)(0.4 t+6)+45$ . Round to two decimal places.

See Solution

Problem 5517

Find the derivative of $f(x)=\frac{8 \sqrt{x}}{5 x+6}$ using the Product or Quotient Rule.

See Solution

Problem 5518

Find the derivative of $f(t)=\left(4+\frac{1}{t}\right)\left(t^{4}-\frac{1}{4}\right)$ using Product or Quotient Rule.

See Solution

Problem 5519

Find the derivative of $g(t)=\left(9-\frac{8}{t^{2}}\right)\left(t^{4}+2 t^{2}-7\right)$ using the Product or Quotient Rule.

See Solution

Problem 5520

Find the derivative of $g(x)=\frac{-7 x}{x-6}$ using the Product or Quotient Rule.

See Solution

Problem 5521

Find the derivative of $f(x)=x^{2}(4 x^{5}+4)$ using the Product or Quotient Rule.

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Problem 5522

Find the derivative of $f(x)=(-2 x^{-2}-7)(-x-3)$ using the Product or Quotient Rule.

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Problem 5523

Find the derivative of $f(x)=(-2 x^{-2}+1)(-5 x+9)$ using the Product or Quotient Rule.

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Problem 5524

Find the derivative of the function using the Quotient Rule: $f(x)=\left(-x^{-2}-2\right)\left(-2 x^{-3}+7\right)$ .

See Solution

Problem 5525

Find the derivative of $f(x)=\frac{8 x-5}{2 x^{2}+2}$ using the Product or Quotient Rule.

See Solution

Problem 5526

Find the marginal revenue for the demand function $D(x)=\frac{168}{3x+3}$ at $x=7$ , rounded to the nearest cent.

See Solution

Problem 5527

Find the population growth rate in thousands for the city in 21 years, where $P(t)=(0.7 t-8)(0.2 t+4)+85$ .

See Solution

Problem 5528

Find the marginal revenue for the demand function $D(x)=\frac{171}{4 x+3}$ at $x=8$ . Round to the nearest cent.

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Problem 5529

Find the marginal revenue for the demand function $D(x)=\frac{195}{4x+2}$ at $x=8$ . Round to the nearest cent.

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Problem 5530

City planners use $A(t)=-\frac{1}{40} t^{2}+2 t+30$ for city size. Find $A'(t)$ , its units, and what it measures.

See Solution

Problem 5531

Find the derivative of $h(x)=\frac{2 \sqrt{x}}{2 x+7}$ using the Product or Quotient Rule.

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Problem 5532

Find the absolute extrema of the function $f(x)=-x^{3}-6$ on the interval $[-5,7]$ . Provide your answer as $(x, f(x))$ .

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Problem 5533

Geben Sie Beispiele für Folgen an: a) $a_n \to 3$ , monoton steigend; b) $b_n \to g$ , monoton fallend; c) $c_n$ divergent; d) $d_n$ konvergent mit unendlich vielen positiven und negativen Gliedern.

See Solution

Problem 5534

Gegeben ist die Funktion $f(x)=4+5 e^{-2 x}-4 e^{-0,5 x}$ .
a) Bestimme den Anfangsbestand, erstelle eine Tabelle für 10 Jahre und plotte die Funktion. b) Finde Extremal- und Wendepunkt von $f$ und erkläre deren Bedeutung. c) Berechne die Fläche $A$ unter $f$ im Intervall $I = [0 ; 10]$ . d) Was bedeutet der Term $\frac{1}{10} \int_{0}^{10} f(x)$ ?

See Solution

Problem 5535

Find the absolute extrema of $f(x)=-7 x^{2}-42 x-6$ on $[-7,7]$ . Present your answer as $(x, f(x))$ .

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Problem 5536

Find the absolute extrema of $f(x)=-2 x^{3}+24 x$ on $[-5,5]$ . Provide your answer as $(x, f(x))$ pairs.

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Problem 5537

Berechne die verbleibende Menge an Cs 137 nach 2, 5, 10, 33 und n Jahren, beginnend mit 500 g und 2.1\% Zerfall pro Jahr.

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Problem 5538

What will \$ 4,000,000 grow to in 25 years at a continuous 5% interest rate? A. \$ 13,853,617.10 B. \$ 13,961,371.83 C. \$ 13,925,161.81 D. \$ 13,748,434.88

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Problem 5539

Finde die Stelle im Intervall $[1 ; 2]$ , wo die Steigung der Tangente gleich der Sekantensteigung ist. Zeichne beide ein.

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Problem 5540

The nth derivative $\frac{d^{n}}{d x^{n}}\left(5 x^{3}+2 x+5\right)$ is 0 for $n \geq 3$ . True or False?

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Problem 5541

Find the tangent lines at $x=4$ for: a. $y=f(x)+g(x)$ , b. $y=f(x)-4g(x)$ , c. $y=2f(x)$ .

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Problem 5542

Find possible values for the limit $\lim _{x \rightarrow 0}\left(\frac{f(x)}{x}-\frac{g(x)}{x}\right)$ given $f(0)=1$ and $g(0)=1$ . Options: A 0, B 1, C 17, D DNE.

See Solution

Problem 5543

Find the slope of the tangent line for $f(x)=9 x^{2}-11 x-5$ at $x=5$ .

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Problem 5544

(a) What are the units of $f^{\prime}(x)$ ? (Write each word in its entirety, in lower case). Units: (b) Estimate the cost of producing one more sheet after 1660 sheets if $f^{\prime}(1660)=1400$ . Cost:

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Problem 5545

Find the function $f(x)$ and the number $a$ such that $\lim _{h \rightarrow 0} \frac{(2+h)^{3}-8}{h} = f^{\prime}(a)$ . $f(x)=$ $a=$

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Problem 5546

Find the slope of the tangent line for $f(x)=5 x^{2}+2 x+2$ at $x=12$ .

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Problem 5547

Find the tangent line equation for the function $f(x)=-9 x^{2}-3 x-10$ at $x=6$ .

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Problem 5548

Find the slope of the tangent line for $f(x)=x^{2}+7$ at $x=1$ .

See Solution

Problem 5549

Find the equation of the tangent line for $f(x)=-2 x^{2}-11 x+6$ at $x=-1$ .

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Problem 5550

Find the derivative of the function $f(x)=2 x^{3}+x$ using the limit definition. No simplification needed.

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Problem 5551

Find the tangent line equation for $f(x)=x^{2}+12$ at $x=10$ .

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Problem 5552

Find the tangent line equation for the function $f(x)=7 x^{2}+10 x-1$ at $x=-7$ .

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Problem 5553

Find the derivative of $f(x)=\sqrt{x+5}$ at $x=3$ using the limit method provided. No need to simplify.

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Problem 5554

Find the instantaneous rate of change of $f(x)=\ln (x+4)$ at $x=4$ using the limit definition. No simplification needed.

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Problem 5555

Find the slope of the tangent line for $f(x)=5x-3$ at $x=-11$ .

See Solution

Problem 5556

Find the slope of the tangent line for $f(x)=3 x^{3}+x^{5}$ at $x=4$ using the limit method. No need to simplify.

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Problem 5557

Find the slope of the tangent line for $f(x)=3 x^{3}+x^{5}$ at $x=4$ using the given limit. No need to simplify.

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Problem 5558

Find the instantaneous rate of change of $f(x)=\cos x$ at $x=\pi$ using the limit definition. No need to simplify.

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Problem 5559

Find the instantaneous rate of change of $f(x)=\sqrt{x+4}$ at $x=-3$ using the limit definition.

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Problem 5560

Finde den Fehler in den Ableitungen der Funktionen:
a) $f(x)=2 x^{3}-4 x^{2}+5$ , $f^{\prime}(x)=6 x^{2}+8 x+5$
b) $f(x)=4 x^{2}+\frac{1}{x}+2$ , $f^{\prime}(x)=8 x+\frac{1}{x^{2}}$
c) $f(x)=x^{2}+c^{3}+2 x+3$ , $f^{\prime}(x)=2 x+3 c^{2}+2$
d) $f(x)=\sqrt{x}+x^{3}+\frac{1}{x}$ , $f^{\prime}(x)=\frac{1}{\sqrt{x}}+3 x^{2}+\frac{1}{x^{2}}$

See Solution

Problem 5561

Find the instantaneous rate of change of $f(x)=3x-x^5$ at $x=-8$ using the limit definition.

See Solution

Problem 5562

Find the instantaneous rate of change of $f(x)=\sqrt{x+4}$ at $x=-3$ using the limit definition. No simplification needed.

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Problem 5563

Find the derivative of the function $f(x)=x^{5}$ using the limit definition of a derivative. No need to simplify.

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Problem 5564

Find the slope of the tangent line for $f(x)=e^{3x}$ at $x=-5$ using the limit definition of the derivative.

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Problem 5565

Find the derivative of the function $f(x)=5 x^{4}+x^{2}$ using the limit definition, without simplifying.

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Problem 5566

Find the derivative of $f(x)=4 x^{3}-x$ at $x=-2$ using the limit definition. No need to simplify.

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Problem 5567

Find the derivative of the function $f(x)=\frac{5}{x+3}$ using the limit definition, without simplifying.

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Problem 5568

Find the derivative of $f(x)=e^{x}$ at $x=7$ using the limit definition. No simplification needed.

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Problem 5569

Find the derivative of the function $f(x)=2 x^{5}-2 x$ using the limit definition without simplifying.

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Problem 5570

Find the derivative of $f(x)=\sqrt{x+4}$ .

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Problem 5571

Given $f(-7)=3$ and $f(0)=6$ , use the Intermediate Value Theorem on $(-7,0)$ to find possible values of $f(x)$ .

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Problem 5572

Find the derivative of $f(x)=e^{x-5}$ at $x=-1$ using the limit definition. No need to simplify your answer.

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Problem 5573

Find the derivative of $f(x)=e^{x-5}$ at $x=-1$ using the limit definition. No need to simplify.

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Problem 5574

Given $f(-6)=6$ and $f(-1)=4$ , use the Intermediate Value Theorem on $(-6,-1)$ to draw conclusions about $f(x)$ .

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Problem 5575

Find the slope of the tangent line of $f(x)=\sqrt{3 x}$ at $x=6$ using the limit definition. No need to simplify.

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Problem 5576

Find where the tangent line $l_{T}$ is horizontal for $y=x^{1/3}(4-x)^{2/3}$ .

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Problem 5577

Find the derivative of $f(x)=e^{x-5}$ at $x=-1$ using the limit definition. No need to simplify.

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Problem 5578

Determine conclusions from the Intermediate Value Theorem for $f(x)$ on $(-1,5)$ given $f(-1)=2$ and $f(5)=4$ .

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Problem 5579

Find the derivative of $f(\theta) = 4\sin\theta - \theta$ at the point $(0,0)$ .

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Problem 5580

Find the slope of the tangent line for $f(x)=4 \ln (x+1)$ at $x=8$ using the limit method.

See Solution

Problem 5581

Find the slope of the tangent line for $f(x)=\frac{5}{x-1}$ at $x=-3$ using the limit definition of the derivative.

See Solution

Problem 5582

Find the derivative of $f(x)=4 \ln (x+5)$ at $x=-4$ using the given limit. No need to simplify your answer.

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Problem 5583

Find the slope of the tangent line for $f(x)=\frac{5}{x-1}$ at $x=-3$ using the limit definition of the derivative.

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Problem 5584

Find the derivative of the function $f(x)=2 x^{3}+x^{4}$ using the limit definition. No simplification needed.

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Problem 5585

Determine the horizontal asymptote of the function $f(x)=(2x^{2}-4x)e^{x}$ .

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Problem 5586

Find the slope of the tangent line for $f(x)=x^{4}+x^{3}$ at $x=-8$ using the limit definition.

See Solution

Problem 5587

Find where the tangent line to $y=x-2 \cos x$ is horizontal.

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Problem 5588

Use the Intermediate Value Theorem for $f(x)$ on $(-2,1)$ given $f(-2)=2$ and $f(1)=-6$ . What can you conclude?

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Problem 5589

Find the derivative of the function $f(x)=5 \cos x$ using the limit definition without simplifying.

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Problem 5590

Find the derivative of $y=(3x+1)^{4}(x^{2}+1)^{3}(3x^{4}-1)^{2}$ .

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Problem 5591

Find the derivative of $f(x)=3 x^{3}$ at $x=-2$ using the limit definition. No need to simplify your answer.

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Problem 5592

Find the derivative of the function $f(x)=3 x^{2}+x^{5}$ using the limit definition. No need to simplify.

See Solution

Problem 5593

Find the limit: $\lim _{x \rightarrow-\infty} \frac{\sqrt{8 x^{2}-4 x}}{x+2}$ .

See Solution

Problem 5594

Find the derivative of $f(x)=\sqrt[3]{x}(\sqrt{x}+3)$ .

See Solution

Problem 5595

Find the derivative of $y=x^{4} \sin \left(\frac{1}{x}\right)$ .

See Solution

Problem 5596

Find the derivative of $f(x)=x^{4}-2 x^{5}$ at $x=-7$ using the limit definition of the derivative.

See Solution

Problem 5597

Find the tangent line equation for the function $f(x)=x^{2}+11$ at the point where $x=5$ .

See Solution

Problem 5598

Find the slope of the tangent line for the function $f(x)=-3 x^{2}-9 x+7$ at $x=-9$ .

See Solution

Problem 5599

Determine conclusions from the Intermediate Value Theorem for $f(x)$ on $(-5,2)$ , given $f(-5)=-3$ and $f(2)=7$ .

See Solution

Problem 5600

Find the slope of the secant line for $f(x)=x^{2}+6$ between $x=-3$ and $x=3$ .

See Solution

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Calculus

Problem 5501

Zeigen Sie, dass die Gerade ggg mit y=f′(a)(x−a)+f(a)y=f^{\prime}(a)(x-a)+f(a)y=f′(a)(x−a)+f(a) eine Tangente an fff im Punkt P(a∣f(a))P(a \mid f(a))P(a∣f(a)) ist, indem Sie (1) P auf ggg und (2) die Steigung f′(a)f^{\prime}(a)f′(a) überprüfen.

Problem 5502

Find the coordinates where f(x)=x3−12xf(x)=x^{3}-12xf(x)=x3−12x has a horizontal tangent line (zero slope) using differentiation shortcuts.

Problem 5503

A stone is thrown up at 20 ms−120 \mathrm{~ms}^{-1}20 ms−1. Find the max height, time to max height, and distance in 3 seconds.

Problem 5504

An investor bought two buildings with growth rates r1(t)=0.062(1.043t)r_{1}(t)=0.062(1.043^{t})r1​(t)=0.062(1.043t) and r2(t)=0.046(1.039t)r_{2}(t)=0.046(1.039^{t})r2​(t)=0.046(1.039t). Find the area between their graphs from 0≤t≤100 \leq t \leq 100≤t≤10. What does this area represent?

Problem 5505

Problem 5506

Solve the differential equation 2dydx=6x2 \frac{d y}{d x}=6 x2dxdy​=6x to find yyy.

Problem 5507

Calculate the average rate of change of f(x)=−1x2+3x−8f(x)=-1 x^{2}+3 x-8f(x)=−1x2+3x−8 from x=−4x=-4x=−4 to x=0x=0x=0. Average rate of change ===

Problem 5508

Find the tangent plane equation to the surface x2+y2−z2−59=0x^{2}+y^{2}-z^{2}-59=0x2+y2−z2−59=0 at points (8,2,3)(8,2,3)(8,2,3) and (−2,−8,3)(-2,-8,3)(−2,−8,3).

Problem 5509

Find the derivative of y=2x3−3xe4x(sin⁡2(x))y=\frac{2 x \sqrt{3-3 x}}{e^{4 x}\left(\sin ^{2}(x)\right)}y=e4x(sin2(x))2x3−3x​​ using logarithmic differentiation. y′=y'=y′=

Problem 5510

Find the derivative of d(t)=16t2d(t)=16 t^{2}d(t)=16t2, its units, and compute the height for t=4.9t=4.9t=4.9 s and speed at impact in mph.

Problem 5511

Find the derivative of the piecewise function: f(x)={x2+1if x≤02x2+x+1if x>0f(x)=\begin{cases} x^{2}+1 & \text{if } x \leq 0 \\ 2x^{2}+x+1 & \text{if } x > 0 \end{cases}f(x)={x2+12x2+x+1​if x≤0if x>0​

Problem 5512

Gegeben ist ft(x)=(x−t)⋅exf_{t}(x)=(x-t) \cdot e^{x}ft​(x)=(x−t)⋅ex. Bestimme Schnittpunkt mit yyy-Achse, Nullstellen, Extrem- und Wendepunkte für t=−2,−1,0,1,2t=-2, -1, 0, 1, 2t=−2,−1,0,1,2.

Problem 5513

Evaluate the integral ∫013dx1+9x2\int_{0}^{\frac{1}{3}} \frac{d x}{1+9 x^{2}}∫031​​1+9x2dx​. Options: A. π12\frac{\pi}{12}12π​ B. π6\frac{\pi}{6}6π​ C. ln⁡(2)18\frac{\ln (2)}{18}18ln(2)​ D. π4\frac{\pi}{4}4π​ E. ln⁡(2)\ln (2)ln(2)

Problem 5514

1. Find f′(x)f^{\prime}(x)f′(x) and f′(2)f^{\prime}(2)f′(2) for f(x)=3x10xf(x)=3 x 10^{x}f(x)=3x10x. 2. Find f′(x)f^{\prime}(x)f′(x) and f′(9)f^{\prime}(9)f′(9) for f(x)=9x5x+3f(x)=\frac{9 x}{5 x+3}f(x)=5x+39x​.

Problem 5515

Find the tangent line equation for f(x)=−2xln⁡(x+7)f(x)=-2 x \ln (x+7)f(x)=−2xln(x+7) at x=1x=1x=1. Answer: y=y=y=

Problem 5516

Find the population growth rate in thousands at t=6t=6t=6 for P(t)=(0.8t−7)(0.4t+6)+45P(t)=(0.8 t-7)(0.4 t+6)+45P(t)=(0.8t−7)(0.4t+6)+45. Round to two decimal places.

Problem 5517

Find the derivative of f(x)=8x5x+6f(x)=\frac{8 \sqrt{x}}{5 x+6}f(x)=5x+68x​​ using the Product or Quotient Rule.

Problem 5518

Find the derivative of f(t)=(4+1t)(t4−14)f(t)=\left(4+\frac{1}{t}\right)\left(t^{4}-\frac{1}{4}\right)f(t)=(4+t1​)(t4−41​) using Product or Quotient Rule.

Problem 5519

Find the derivative of g(t)=(9−8t2)(t4+2t2−7)g(t)=\left(9-\frac{8}{t^{2}}\right)\left(t^{4}+2 t^{2}-7\right)g(t)=(9−t28​)(t4+2t2−7) using the Product or Quotient Rule.

Problem 5520

Find the derivative of g(x)=−7xx−6g(x)=\frac{-7 x}{x-6}g(x)=x−6−7x​ using the Product or Quotient Rule.

Problem 5521

Find the derivative of f(x)=x2(4x5+4)f(x)=x^{2}(4 x^{5}+4)f(x)=x2(4x5+4) using the Product or Quotient Rule.

Problem 5522

Find the derivative of f(x)=(−2x−2−7)(−x−3)f(x)=(-2 x^{-2}-7)(-x-3)f(x)=(−2x−2−7)(−x−3) using the Product or Quotient Rule.

Problem 5523

Find the derivative of f(x)=(−2x−2+1)(−5x+9)f(x)=(-2 x^{-2}+1)(-5 x+9)f(x)=(−2x−2+1)(−5x+9) using the Product or Quotient Rule.

Problem 5524

Find the derivative of the function using the Quotient Rule: f(x)=(−x−2−2)(−2x−3+7)f(x)=\left(-x^{-2}-2\right)\left(-2 x^{-3}+7\right)f(x)=(−x−2−2)(−2x−3+7).

Problem 5525

Find the derivative of f(x)=8x−52x2+2f(x)=\frac{8 x-5}{2 x^{2}+2}f(x)=2x2+28x−5​ using the Product or Quotient Rule.

Problem 5526

Find the marginal revenue for the demand function D(x)=1683x+3D(x)=\frac{168}{3x+3}D(x)=3x+3168​ at x=7x=7x=7, rounded to the nearest cent.

Problem 5527

Find the population growth rate in thousands for the city in 21 years, where P(t)=(0.7t−8)(0.2t+4)+85P(t)=(0.7 t-8)(0.2 t+4)+85P(t)=(0.7t−8)(0.2t+4)+85.

Problem 5528

Find the marginal revenue for the demand function D(x)=1714x+3D(x)=\frac{171}{4 x+3}D(x)=4x+3171​ at x=8x=8x=8. Round to the nearest cent.

Problem 5529

Find the marginal revenue for the demand function D(x)=1954x+2D(x)=\frac{195}{4x+2}D(x)=4x+2195​ at x=8x=8x=8. Round to the nearest cent.

Problem 5530

City planners use A(t)=−140t2+2t+30A(t)=-\frac{1}{40} t^{2}+2 t+30A(t)=−401​t2+2t+30 for city size. Find A′(t)A'(t)A′(t), its units, and what it measures.

Problem 5531

Find the derivative of h(x)=2x2x+7h(x)=\frac{2 \sqrt{x}}{2 x+7}h(x)=2x+72x​​ using the Product or Quotient Rule.

Problem 5532

Find the absolute extrema of the function f(x)=−x3−6f(x)=-x^{3}-6f(x)=−x3−6 on the interval [−5,7][-5,7][−5,7]. Provide your answer as (x,f(x))(x, f(x))(x,f(x)).

Problem 5533

Geben Sie Beispiele für Folgen an: a) an→3a_n \to 3an​→3, monoton steigend; b) bn→gb_n \to gbn​→g, monoton fallend; c) cnc_ncn​ divergent; d) dnd_ndn​ konvergent mit unendlich vielen positiven und negativen Gliedern.

Problem 5534

Problem 5535

Find the absolute extrema of f(x)=−7x2−42x−6f(x)=-7 x^{2}-42 x-6f(x)=−7x2−42x−6 on [−7,7][-7,7][−7,7]. Present your answer as (x,f(x))(x, f(x))(x,f(x)).

Problem 5536

Find the absolute extrema of f(x)=−2x3+24xf(x)=-2 x^{3}+24 xf(x)=−2x3+24x on [−5,5][-5,5][−5,5]. Provide your answer as (x,f(x))(x, f(x))(x,f(x)) pairs.

Problem 5537

Berechne die verbleibende Menge an Cs 137 nach 2, 5, 10, 33 und n Jahren, beginnend mit 500 g und 2.1\% Zerfall pro Jahr.

Problem 5538

What will \$ 4,000,000 grow to in 25 years at a continuous 5% interest rate? A. \$ 13,853,617.10 B. \$ 13,961,371.83 C. \$ 13,925,161.81 D. \$ 13,748,434.88

Problem 5539

Finde die Stelle im Intervall [1;2][1 ; 2][1;2], wo die Steigung der Tangente gleich der Sekantensteigung ist. Zeichne beide ein.

Problem 5540

The nth derivative dndxn(5x3+2x+5)\frac{d^{n}}{d x^{n}}\left(5 x^{3}+2 x+5\right)dxndn​(5x3+2x+5) is 0 for n≥3n \geq 3n≥3. True or False?

Problem 5541

Zeigen Sie, dass die Gerade $g$ mit $y=f^{\prime}(a)(x-a)+f(a)$ eine Tangente an $f$ im Punkt $P(a \mid f(a))$ ist, indem Sie (1) P auf $g$ und (2) die Steigung $f^{\prime}(a)$ überprüfen.

Find the coordinates where $f(x)=x^{3}-12x$ has a horizontal tangent line (zero slope) using differentiation shortcuts.

A stone is thrown up at $20 \mathrm{~ms}^{-1}$ . Find the max height, time to max height, and distance in 3 seconds.

An investor bought two buildings with growth rates $r_{1}(t)=0.062(1.043^{t})$ and $r_{2}(t)=0.046(1.039^{t})$ . Find the area between their graphs from $0 \leq t \leq 10$ . What does this area represent?

Solve the differential equation $2 \frac{d y}{d x}=6 x$ to find $y$ .

Calculate the average rate of change of $f(x)=-1 x^{2}+3 x-8$ from $x=-4$ to $x=0$ . Average rate of change $=$

Find the tangent plane equation to the surface $x^{2}+y^{2}-z^{2}-59=0$ at points $(8,2,3)$ and $(-2,-8,3)$ .

Find the derivative of $y=\frac{2 x \sqrt{3-3 x}}{e^{4 x}\left(\sin ^{2}(x)\right)}$ using logarithmic differentiation. $y'=$

Find the derivative of $d(t)=16 t^{2}$ , its units, and compute the height for $t=4.9$ s and speed at impact in mph.

Find the derivative of the piecewise function:
$f(x)=\begin{cases} x^{2}+1 & \text{if } x \leq 0 \\ 2x^{2}+x+1 & \text{if } x > 0 \end{cases}$

Gegeben ist $f_{t}(x)=(x-t) \cdot e^{x}$ . Bestimme Schnittpunkt mit $y$ -Achse, Nullstellen, Extrem- und Wendepunkte für $t=-2, -1, 0, 1, 2$ .

Evaluate the integral $\int_{0}^{\frac{1}{3}} \frac{d x}{1+9 x^{2}}$ . Options: A. $\frac{\pi}{12}$ B. $\frac{\pi}{6}$ C. $\frac{\ln (2)}{18}$ D. $\frac{\pi}{4}$ E. $\ln (2)$

1. Find $f^{\prime}(x)$ and $f^{\prime}(2)$ for $f(x)=3 x 10^{x}$ .
2. Find $f^{\prime}(x)$ and $f^{\prime}(9)$ for $f(x)=\frac{9 x}{5 x+3}$ .

Find the tangent line equation for $f(x)=-2 x \ln (x+7)$ at $x=1$ . Answer: $y=$

Find the population growth rate in thousands at $t=6$ for $P(t)=(0.8 t-7)(0.4 t+6)+45$ . Round to two decimal places.

Find the derivative of $f(x)=\frac{8 \sqrt{x}}{5 x+6}$ using the Product or Quotient Rule.

Find the derivative of $f(t)=\left(4+\frac{1}{t}\right)\left(t^{4}-\frac{1}{4}\right)$ using Product or Quotient Rule.

Find the derivative of $g(t)=\left(9-\frac{8}{t^{2}}\right)\left(t^{4}+2 t^{2}-7\right)$ using the Product or Quotient Rule.

Find the derivative of $g(x)=\frac{-7 x}{x-6}$ using the Product or Quotient Rule.

Find the derivative of $f(x)=x^{2}(4 x^{5}+4)$ using the Product or Quotient Rule.

Find the derivative of $f(x)=(-2 x^{-2}-7)(-x-3)$ using the Product or Quotient Rule.

Find the derivative of $f(x)=(-2 x^{-2}+1)(-5 x+9)$ using the Product or Quotient Rule.

Find the derivative of the function using the Quotient Rule: $f(x)=\left(-x^{-2}-2\right)\left(-2 x^{-3}+7\right)$ .

Find the derivative of $f(x)=\frac{8 x-5}{2 x^{2}+2}$ using the Product or Quotient Rule.

Find the marginal revenue for the demand function $D(x)=\frac{168}{3x+3}$ at $x=7$ , rounded to the nearest cent.

Find the population growth rate in thousands for the city in 21 years, where $P(t)=(0.7 t-8)(0.2 t+4)+85$ .

Find the marginal revenue for the demand function $D(x)=\frac{171}{4 x+3}$ at $x=8$ . Round to the nearest cent.

Find the marginal revenue for the demand function $D(x)=\frac{195}{4x+2}$ at $x=8$ . Round to the nearest cent.

City planners use $A(t)=-\frac{1}{40} t^{2}+2 t+30$ for city size. Find $A'(t)$ , its units, and what it measures.

Find the derivative of $h(x)=\frac{2 \sqrt{x}}{2 x+7}$ using the Product or Quotient Rule.

Find the absolute extrema of the function $f(x)=-x^{3}-6$ on the interval $[-5,7]$ . Provide your answer as $(x, f(x))$ .

Geben Sie Beispiele für Folgen an: a) $a_n \to 3$ , monoton steigend; b) $b_n \to g$ , monoton fallend; c) $c_n$ divergent; d) $d_n$ konvergent mit unendlich vielen positiven und negativen Gliedern.

Find the absolute extrema of $f(x)=-7 x^{2}-42 x-6$ on $[-7,7]$ . Present your answer as $(x, f(x))$ .

Find the absolute extrema of $f(x)=-2 x^{3}+24 x$ on $[-5,5]$ . Provide your answer as $(x, f(x))$ pairs.

Finde die Stelle im Intervall $[1 ; 2]$ , wo die Steigung der Tangente gleich der Sekantensteigung ist. Zeichne beide ein.

The nth derivative $\frac{d^{n}}{d x^{n}}\left(5 x^{3}+2 x+5\right)$ is 0 for $n \geq 3$ . True or False?

Find the tangent lines at $x=4$ for: a. $y=f(x)+g(x)$ , b. $y=f(x)-4g(x)$ , c. $y=2f(x)$ .

Find possible values for the limit $\lim _{x \rightarrow 0}\left(\frac{f(x)}{x}-\frac{g(x)}{x}\right)$ given $f(0)=1$ and $g(0)=1$ . Options: A 0, B 1, C 17, D DNE.

Find the slope of the tangent line for $f(x)=9 x^{2}-11 x-5$ at $x=5$ .

(a) What are the units of $f^{\prime}(x)$ ? (Write each word in its entirety, in lower case). Units: (b) Estimate the cost of producing one more sheet after 1660 sheets if $f^{\prime}(1660)=1400$ . Cost:

Find the function $f(x)$ and the number $a$ such that $\lim _{h \rightarrow 0} \frac{(2+h)^{3}-8}{h} = f^{\prime}(a)$ . $f(x)=$ $a=$

Find the slope of the tangent line for $f(x)=5 x^{2}+2 x+2$ at $x=12$ .

Find the tangent line equation for the function $f(x)=-9 x^{2}-3 x-10$ at $x=6$ .

Find the slope of the tangent line for $f(x)=x^{2}+7$ at $x=1$ .

Find the equation of the tangent line for $f(x)=-2 x^{2}-11 x+6$ at $x=-1$ .

Find the derivative of the function $f(x)=2 x^{3}+x$ using the limit definition. No simplification needed.

Find the tangent line equation for $f(x)=x^{2}+12$ at $x=10$ .

Find the tangent line equation for the function $f(x)=7 x^{2}+10 x-1$ at $x=-7$ .

Find the derivative of $f(x)=\sqrt{x+5}$ at $x=3$ using the limit method provided. No need to simplify.

Find the instantaneous rate of change of $f(x)=\ln (x+4)$ at $x=4$ using the limit definition. No simplification needed.

Find the slope of the tangent line for $f(x)=5x-3$ at $x=-11$ .

Find the slope of the tangent line for $f(x)=3 x^{3}+x^{5}$ at $x=4$ using the limit method. No need to simplify.

Find the slope of the tangent line for $f(x)=3 x^{3}+x^{5}$ at $x=4$ using the given limit. No need to simplify.

Find the instantaneous rate of change of $f(x)=\cos x$ at $x=\pi$ using the limit definition. No need to simplify.

Find the instantaneous rate of change of $f(x)=\sqrt{x+4}$ at $x=-3$ using the limit definition.

Find the instantaneous rate of change of $f(x)=3x-x^5$ at $x=-8$ using the limit definition.

Find the instantaneous rate of change of $f(x)=\sqrt{x+4}$ at $x=-3$ using the limit definition. No simplification needed.

Find the derivative of the function $f(x)=x^{5}$ using the limit definition of a derivative. No need to simplify.

Find the slope of the tangent line for $f(x)=e^{3x}$ at $x=-5$ using the limit definition of the derivative.

Find the derivative of the function $f(x)=5 x^{4}+x^{2}$ using the limit definition, without simplifying.

Find the derivative of $f(x)=4 x^{3}-x$ at $x=-2$ using the limit definition. No need to simplify.

Find the derivative of the function $f(x)=\frac{5}{x+3}$ using the limit definition, without simplifying.

Find the derivative of $f(x)=e^{x}$ at $x=7$ using the limit definition. No simplification needed.

Find the derivative of the function $f(x)=2 x^{5}-2 x$ using the limit definition without simplifying.

Find the derivative of $f(x)=\sqrt{x+4}$ .

Given $f(-7)=3$ and $f(0)=6$ , use the Intermediate Value Theorem on $(-7,0)$ to find possible values of $f(x)$ .

Find the derivative of $f(x)=e^{x-5}$ at $x=-1$ using the limit definition. No need to simplify your answer.

Find the derivative of $f(x)=e^{x-5}$ at $x=-1$ using the limit definition. No need to simplify.

Given $f(-6)=6$ and $f(-1)=4$ , use the Intermediate Value Theorem on $(-6,-1)$ to draw conclusions about $f(x)$ .