Calculus

Problem 24501

A $200 \mathrm{~m}$ fence surrounds a rectangular lawn with a semi-circle.
a Explain why $y=100-x-\frac{\pi}{2} x$ . b Show $A=200 x-x^{2}\left(2+\frac{\pi}{2}\right)$ . c Use calculus to find dimensions for maximum area.

See Solution

Problem 24502

Find the sum of the infinite series: $\sum_{k=0}^{\infty} \sqrt{2}^k$ .

See Solution

Problem 24503

Find the slope of the curve defined by $x^{2}+2xy+y=4$ at the point $(1,1)$ .

See Solution

Problem 24504

Find the limit: $\lim _{x \rightarrow 5} \frac{|x-5|}{25-x^{2}}$ .

See Solution

Problem 24505

Berechne den Flächeninhalt $A$ zwischen $f(x)=x^{2}+1$ , der $x$ -Achse und den Linien $x=1$ und $x=2$ : $A = \int_{1}^{2} (x^{2} + 1) \, dx$

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Problem 24506

Find the distance $s$ (in cm) moved by a cam-shaft follower in 4 seconds, given $s=\int_{0}^{4} t \sqrt{4+9 t^{2}} dt$ .

See Solution

Problem 24507

Calculate the sum of the series: $\sum_{m=0}^{\infty} 5^{-m}$ .

See Solution

Problem 24508

Calculate the area under the curve $y=5-2x^{2}$ between $x=0$ and $x=1$ .

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Problem 24509

Evaluate the integral using the substitution $u=4 x^{2}+5 x$ :
$\int(8 x+5) \sqrt{4 x^{2}+5 x} d x=\int(\sqrt{u}) d u$
Find the result:
$\int(8 x+5) \sqrt{4 x^{2}+5 x} d x=\square$

See Solution

Problem 24510

Bestimmen Sie das unbestimmte Integral: $\int(3-5 x)^{7} d x=$

See Solution

Problem 24511

Find the area between the $x$ -axis and $f(x)=3 e^{x}-4$ over the interval $[-3,2]$ . Check for $x$ -axis crossings.

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Problem 24512

Find the limits: h) $\lim _{x \rightarrow 5} \frac{|x-5|}{25-x^{2}}$ i) $\lim _{z \rightarrow 1} \frac{\frac{1}{z}-1}{\frac{1}{z^{2}}-1}$

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Problem 24513

Find the limit as $\alpha$ approaches $\pi$ from the right: $\lim _{\alpha \rightarrow \pi^{+}} \tan \frac{3 \pi}{2}-\alpha$ .

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Problem 24514

Find the limit: $\lim _{x \rightarrow \infty} \frac{2 e^{-x}}{7+8 e^{-x}}$ .

See Solution

Problem 24515

Find the area between the $x$ -axis and $f(x)=e^{x}-2$ over the interval $[-1,3]$ . Does the graph cross the $x$ -axis?

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Problem 24516

Find the length of the curve given by $y^{2}=4(x+3)^{3}$ for $0 \leq x \leq 1$ , $y>0$ .

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Problem 24517

Find the first and second derivatives of the function $y=7x+1$ .

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Problem 24518

Find the limit: $\lim_{x \rightarrow 2} (-3x + 4)$ . Choose the correct rewriting option from A, B, C, or D.

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Problem 24519

Calculate the length of the curve given by $y=3+8 x^{3/2}$ for $0 \leq x \leq 1$ .

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Problem 24520

Chewing frequency $c$ is related to body mass $M$ by $c=kM^{-0.128}$ . Given $M(t)=1+2\sqrt{t}$ , find $\frac{d c}{d t}$ . Also, $L=rM^{0.312}$ ; find $\frac{d c}{d L}$ .

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Problem 24521

Differentiate $y=9 x e^{x}-9 e^{x}$ .

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Problem 24522

Differentiate the equation $y^{8}-5 x^{3}=5 x$ to find $\frac{d y}{d x}$ . Choose the correct answer.

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Problem 24523

Calculate the integral of the function: $\int\left(\frac{x^{2}}{4}-4\right) d x$ .

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Problem 24524

Find the limit: $\lim _{x \rightarrow-\infty} \frac{4 x^{2}+x}{x+1}$ .

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Problem 24525

Evaluate the infinite series: $\sum_{k=0}^{\infty} \frac{k-1}{k !}$ .

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Problem 24526

Find the exact length of the polar curve $r=\theta^{2}$ for $0 \leq \theta \leq 8\pi$ .

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Problem 24527

Find the average rate of change of $y = x^{2} - 2$ from $x = 0$ to $x = 1$ .

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Problem 24528

Fish interactions can be modeled by the Morse potential $V(r)=e^{-r}-A e^{-a r}$ . For $A=1$ , $a=\frac{1}{2}$ , find $\lim_{r \to 0} V(r)$ .

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Problem 24529

Find the average rate of change of $y = x^2 - 2$ from $x = 0$ to $x = 1$ .

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Problem 24530

Evaluate the integral: $\int \frac{(\ln x)^{8}}{x} d x$ , assuming $u>0$ for $\ln u$ .

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Problem 24531

Approximate the area under $f(x)=e^{x}+6$ from $x=-2$ to $x=2$ using $n=4$ rectangles: left, right, average, and midpoints.

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Problem 24532

Find the limit: $\lim _{h \rightarrow 0} \frac{\sin (x+h) \cos (x+h)-\sin x \cos x}{h}$ .

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Problem 24533

Find the average rate of change of $y=2x^{2}-2x+2$ over the interval $[0,1]$ .

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Problem 24534

Find the area under $f(x)=(10-x)$ from $x=0$ to $x=10$ using 10 rectangles and midpoints. Area is approximately $\square$ square units.

See Solution

Problem 24535

Find the indefinite integral and verify by differentiating:
$\int\left(\sec ^{2} x-2\right) d x=\square$

See Solution

Problem 24536

A substance grows at $14\%$ per day. If it starts at 46 grams, find its mass after 4 days, exact and rounded to the nearest tenth.

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Problem 24537

Estimate the age of cave paintings with $6\%$ carbon-14 remaining using the decay model $A = A_0 e^{-kt}$ .

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Problem 24538

Find the area between the $x$ -axis and $f(x)=2 e^{x}-1$ over the interval [-2,3]. Check for $x$ -axis crossings.

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Problem 24539

Calculate the area between the curves $y=2,916 \sqrt{x}$ and $y=108 x^{2}$ . The area is $\square$ .

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Problem 24540

Calculez $f^{\prime}(x)$ pour $f(x)=x^{2}-4x$ , $f(x)=\sqrt{x+1}-4$ , $f(x)=\frac{7}{1-2x}$ , et $f(x)=\frac{1}{\sqrt{x-3}}$ . Esquissez le graphique de la dérivée pour deux fonctions.

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Problem 24541

Differentiate the function: $u=\sqrt[5]{t}+6 \sqrt{t^{5}}$ .

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Problem 24542

Differentiate the function $u=\sqrt[5]{t}+4 \sqrt{t^{5}}$ .

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Problem 24543

Calculate the area in the first quadrant between $y=10$ and $y=10 \sin x$ from $x=0$ to $x=\frac{\pi}{2}$ . Area: $\square$ .

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Problem 24544

How long for a tranquilizer to decay to $95\%$ of its original dosage if its half-life is 40 hours? Use $A=A_{0} e^{kt}$ .

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Problem 24545

Find critical points and use the first derivative test for $f(x)=x \sqrt{4-x^{2}}$ on $[-2,2]$ to determine max/min values.

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Problem 24546

Find the critical points of $f(x)=x \sqrt{36-x^{2}}$ on $[-6,6]$ and determine local/absolute max/min values.

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Problem 24547

Find the absolute extrema of $f(x)=\frac{1}{3} x^{3}-\frac{5}{2} x^{2}-6 x-2$ on $[-2,7]$ .

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Problem 24548

Find the area between $f(x)=x$ and $g(x)=x^{1/n}$ for $x \geq 0$ , expressed in terms of positive integer $n \geq 2$ .

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Problem 24549

Find the absolute extrema of $f(x)=\frac{1}{3} x^{3}-\frac{5}{2} x^{2}-6 x-2$ on $[-2,7]$ .

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Problem 24550

Find critical points of $f(x)=x \sqrt{4-x^{2}}$ on $[-2,2]$ and use the first derivative test for extrema.

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Problem 24551

Differentiate the function $y=e^{x+4}+1$ .

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Problem 24552

Differentiate the function $y=e^{x+4}+1$ and find $y'=\square$ .

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Problem 24553

Differentiate the function $S(R)=2 \pi R^{2}$ and find $S^{\prime}(R)$ .

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Problem 24554

Differentiate the function $g(u) = \sqrt{7} u + \sqrt{10u}$ . Find $g'(u) =$ .

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Problem 24555

Find critical points and use the first derivative test for $f(x)=x \sqrt{36-x^{2}}$ on $[-6,6]$ to determine local extrema.

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Problem 24556

Find the absolute extrema of $f(x)=3 x^{4}-150 x^{2}+1$ on $[-6,6]$ and their corresponding $x$ values.

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Problem 24557

Find critical points and use the first derivative test for $f(x)=x \sqrt{4-x^{2}}$ on $[-2,2]$ . Identify extrema.

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Problem 24558

Find the velocity $v(1)$ and acceleration $a(1)$ at $t=1$ for $s(t)=t^{2}-4t$ . $v(1)=\square$

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Problem 24559

Find the absolute extrema of $f(x)=3 x^{4}-150 x^{2}+1$ on $[-6,6]$ and the $x$ values where they occur.

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Problem 24560

Find the first and second derivative of the function $G(r)=\sqrt{r}+\sqrt[6]{r}$ . What are $G^{\prime}(r)$ and $G^{\prime \prime}(r)$ ?

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Problem 24561

Verify if the function $f(x)=-4 x^{2}+4 x-7$ has a local extremum and find the absolute extrema.

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Problem 24562

Find $f^{\prime}(1)$ given that $f(x)+x^{2}[f(x)]^{3}=10$ and $f(1)=2$ .

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Problem 24563

Find the absolute extrema of $f(x)=\frac{1-x}{5+x}$ on $[0,5]$ . What are the max value and $x$ where it occurs?

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Problem 24564

Find the first and second derivatives of the function $f(x)=12 x^{12}+9 x^{9}-x$ . What are $f'(x)$ and $f''(x)$ ?

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Problem 24565

Find the tangent line equation at (3,4) for the curve $x^{2}+2xy-y^{2}+x=20$ using implicit differentiation.

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Problem 24566

Verify if the function $A(r)=\frac{16}{r}+2 \pi r^{2}$ has an absolute extremum for $r>0$ , then find its location and value.

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Problem 24567

Find the absolute extrema of $f(x)=\frac{6+x}{7-x}$ on $[-7,0]$ . What are the values and where do they occur?

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Problem 24568

Find the absolute extrema of $f(x)=3 x^{3}-24 x^{2}+48 x+3$ on $[0,7]$ . What is the absolute minimum and where does it occur?

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Problem 24569

Bestimme die Ableitung der Funktionen und klammere aus: a) $f(x)=x \cdot e^{x}$ b) $f(x)=(x-3) \cdot e^{x}$ c) $f(x)=x^{2} \cdot e^{x}$ d) $f(x)=20 \cdot e^{0,1 x}+x+5$ e) $f(x)=\frac{1}{3} x^{3}-2 \cdot e^{-0,25 x}$ f) $f(x)=2 x^{2}-e^{-x+3}+x^{3}$ g) $f(x)=\sin (x) \cdot e^{x}$ h) $f(x)=\left(x^{2}-5\right) \cdot e^{x}$ i) $f(x)=x^{4} \cdot e^{x}-7$ j) $f(x)=5 x-x \cdot e^{x}$

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Problem 24570

Differentiate the function: $y = 5 e^{x} + \frac{8}{\sqrt[3]{x}}$ .

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Problem 24571

Find the absolute extrema of $f(x)=3 x^{3}+18 x^{2}+27 x+6$ on $[-6,0]$ . What is the max and where does it occur?

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Problem 24572

Verify if the function $A(r)=\frac{20}{r}+2 \pi r^{2}, r>0$ has an absolute extremum. Find its location and value.

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Problem 24573

Given the motion $s=t^{3}-3t$ , find velocity $v(t)$ and acceleration $a(t)$ . Also, find $a(5)$ and $a(v=0)$ .

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Problem 24574

Determine if the function $f(x)$ is continuous at $x=-2$ using $f(-2)$ and $\lim_{x \to -2} f(x)$ .

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Problem 24575

Find the absolute extrema of $f(x)=3x^{3}+18x^{2}+27x+6$ on $[-6,0]$ and their $x$ values.

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Problem 24576

The function $f(x) = \frac{|x|}{x}$ is not continuous on $[-1,1]$ . True or False?

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Problem 24577

Find the absolute extremum of the function $A(r) = \frac{20}{r} + 2 \pi r^{2}$ for $r > 0$ .

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Problem 24578

Find the derivative of $g(x) = \frac{\arcsin(3x)}{x}$ .

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Problem 24579

Find the absolute extreme values of $f(x)=4 \sin ^{2} x$ on $[0, \pi]$ . Select A, B, C, or D and fill in the blanks.

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Problem 24580

Find the tangent line to $y=x \sqrt{x}$ that is parallel to $y=6+9x$ . What is the equation? $y=$

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Problem 24581

Find the area under $f(x)=(10-x)$ from $x=0$ to $x=10$ using 10 rectangles with midpoints. Area is $\square$ square units.

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Problem 24582

Find the linear approximation of $g(x)=\sqrt[3]{1+x}$ at $a=0$ . What is $g(x) \approx$ ?

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Problem 24583

Find an antiderivative of $\sin 23 x$ and verify by differentiating. What is $\int \sin 23 x \, dx = \square$ ?

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Problem 24584

A ball on a 2 m string moves in a circle at $4 \mathrm{~m/s}$ . Find its centripetal acceleration.

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Problem 24585

Evaluate the integral $\int \sqrt{7 x+1} \, dx$ using the substitution $u = ax + b$ . Result: $\int \sqrt{7 x+1} \, dx = \square$ .

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Problem 24586

Use $u$ -substitution to evaluate $\int_{2}^{4}(2 x-3)^{4} d x$ and find $u$ , $d u$ , $a$ , $b$ , and $f(u)$ .

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Problem 24587

Find local maxima and minima for $y=x^{3}-27x+5$ . Identify intervals of increase and decrease.

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Problem 24588

Find where the function $f(x)=3 x^{4}-2 x^{3}+6$ is concave up or down and identify inflection points.

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Problem 24589

Find the area under $f(x)=(10-x)$ from $x=0$ to $x=10$ using midpoints of 10 rectangles. Also, calculate the integral $\int_{0}^{10}(10-x) dx$ .

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Problem 24590

Estimate $e^{-0.01}$ using linear approximation or differentials.

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Problem 24591

Find the linear approximation of $f(x)=\sqrt{4-x}$ at $a=0$ . What is $L(x)=?$

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Problem 24592

Approximate $\sqrt{3.9}$ and $\sqrt{3.99}$ using $L(x)$ . Round answers to four decimal places.

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Problem 24593

Find where the tangent line is horizontal for the curve $y=2 x^{3}+3 x^{2}-12 x+3$ . Identify the points $(x, y)$ .

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Problem 24594

Find where the function $f(x)=x^{4}-2 x^{3}+3$ is concave up/down and identify inflection points.

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Problem 24595

Evaluate the integral: $\int \sec 5 w \tan 5 w \, d w = \square$ using a variable change or integration table.

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Problem 24596

Determine if the piecewise function $f(x)$ is continuous at $x=1$ for a given $c$ . Choose the correct option about continuity.

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Problem 24597

Evaluate the integral: $\int \frac{2 x^{4}}{\sqrt{1-6 x^{5}}} d x$ using a change of variables or a table.

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Problem 24598

Find the indefinite integral using a variable change and verify by differentiation: $\int \frac{d x}{(9 x-2)^{2}}=\square$

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Problem 24599

Evaluate the integral: $\int x^{5}\left(x^{6}+14\right)^{8} d x$ using a change of variables.

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Problem 24600

Find the interval where the function $f$ is increasing, given its derivative $f^{\prime}(x)=(x+2)^{2}(x-5)^{5}(x-6)^{4}$ .

See Solution

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Calculus

Problem 24501

A 200 m200 \mathrm{~m}200 m fence surrounds a rectangular lawn with a semi-circle. a Explain why y=100−x−π2xy=100-x-\frac{\pi}{2} xy=100−x−2π​x. b Show A=200x−x2(2+π2)A=200 x-x^{2}\left(2+\frac{\pi}{2}\right)A=200x−x2(2+2π​). c Use calculus to find dimensions for maximum area.

Problem 24502

Find the sum of the infinite series: ∑k=0∞2k\sum_{k=0}^{\infty} \sqrt{2}^kk=0∑∞​2​k.

Problem 24503

Find the slope of the curve defined by x2+2xy+y=4x^{2}+2xy+y=4x2+2xy+y=4 at the point (1,1)(1,1)(1,1).

Problem 24504

Find the limit: lim⁡x→5∣x−5∣25−x2\lim _{x \rightarrow 5} \frac{|x-5|}{25-x^{2}}limx→5​25−x2∣x−5∣​.

Problem 24505

Berechne den Flächeninhalt AAA zwischen f(x)=x2+1f(x)=x^{2}+1f(x)=x2+1, der xxx-Achse und den Linien x=1x=1x=1 und x=2x=2x=2: A=∫12(x2+1) dxA = \int_{1}^{2} (x^{2} + 1) \, dxA=∫12​(x2+1)dx

Problem 24506

Find the distance sss (in cm) moved by a cam-shaft follower in 4 seconds, given s=∫04t4+9t2dts=\int_{0}^{4} t \sqrt{4+9 t^{2}} dts=∫04​t4+9t2​dt.

Problem 24507

Calculate the sum of the series: ∑m=0∞5−m\sum_{m=0}^{\infty} 5^{-m}∑m=0∞​5−m.

Problem 24508

Calculate the area under the curve y=5−2x2y=5-2x^{2}y=5−2x2 between x=0x=0x=0 and x=1x=1x=1.

Problem 24509

Evaluate the integral using the substitution u=4x2+5xu=4 x^{2}+5 xu=4x2+5x: ∫(8x+5)4x2+5xdx=∫(u)du\int(8 x+5) \sqrt{4 x^{2}+5 x} d x=\int(\sqrt{u}) d u∫(8x+5)4x2+5x​dx=∫(u​)du Find the result: ∫(8x+5)4x2+5xdx=□\int(8 x+5) \sqrt{4 x^{2}+5 x} d x=\square∫(8x+5)4x2+5x​dx=□

Problem 24510

Bestimmen Sie das unbestimmte Integral: ∫(3−5x)7dx=\int(3-5 x)^{7} d x=∫(3−5x)7dx=

Problem 24511

Find the area between the xxx-axis and f(x)=3ex−4f(x)=3 e^{x}-4f(x)=3ex−4 over the interval [−3,2][-3,2][−3,2]. Check for xxx-axis crossings.

Problem 24512

Find the limits: h) lim⁡x→5∣x−5∣25−x2\lim _{x \rightarrow 5} \frac{|x-5|}{25-x^{2}}limx→5​25−x2∣x−5∣​ i) lim⁡z→11z−11z2−1\lim _{z \rightarrow 1} \frac{\frac{1}{z}-1}{\frac{1}{z^{2}}-1}limz→1​z21​−1z1​−1​

Problem 24513

Find the limit as α\alphaα approaches π\piπ from the right: lim⁡α→π+tan⁡3π2−α\lim _{\alpha \rightarrow \pi^{+}} \tan \frac{3 \pi}{2}-\alphalimα→π+​tan23π​−α.

Problem 24514

Find the limit: lim⁡x→∞2e−x7+8e−x\lim _{x \rightarrow \infty} \frac{2 e^{-x}}{7+8 e^{-x}}limx→∞​7+8e−x2e−x​.

Problem 24515

Find the area between the xxx-axis and f(x)=ex−2f(x)=e^{x}-2f(x)=ex−2 over the interval [−1,3][-1,3][−1,3]. Does the graph cross the xxx-axis?

Problem 24516

Find the length of the curve given by y2=4(x+3)3y^{2}=4(x+3)^{3}y2=4(x+3)3 for 0≤x≤10 \leq x \leq 10≤x≤1, y>0y>0y>0.

Problem 24517

Find the first and second derivatives of the function y=7x+1y=7x+1y=7x+1.

Problem 24518

Find the limit: lim⁡x→2(−3x+4)\lim_{x \rightarrow 2} (-3x + 4)x→2lim​(−3x+4). Choose the correct rewriting option from A, B, C, or D.

Problem 24519

Calculate the length of the curve given by y=3+8x3/2y=3+8 x^{3/2}y=3+8x3/2 for 0≤x≤10 \leq x \leq 10≤x≤1.

Problem 24520

Chewing frequency ccc is related to body mass MMM by c=kM−0.128c=kM^{-0.128}c=kM−0.128. Given M(t)=1+2tM(t)=1+2\sqrt{t}M(t)=1+2t​, find dcdt\frac{d c}{d t}dtdc​. Also, L=rM0.312L=rM^{0.312}L=rM0.312; find dcdL\frac{d c}{d L}dLdc​.

Problem 24521

Differentiate y=9xex−9exy=9 x e^{x}-9 e^{x}y=9xex−9ex.

Problem 24522

Differentiate the equation y8−5x3=5xy^{8}-5 x^{3}=5 xy8−5x3=5x to find dydx\frac{d y}{d x}dxdy​. Choose the correct answer.

Problem 24523

Calculate the integral of the function: ∫(x24−4)dx\int\left(\frac{x^{2}}{4}-4\right) d x∫(4x2​−4)dx.

Problem 24524

Find the limit: lim⁡x→−∞4x2+xx+1\lim _{x \rightarrow-\infty} \frac{4 x^{2}+x}{x+1}limx→−∞​x+14x2+x​.

Problem 24525

Evaluate the infinite series: ∑k=0∞k−1k!\sum_{k=0}^{\infty} \frac{k-1}{k !}∑k=0∞​k!k−1​.

Problem 24526

Find the exact length of the polar curve r=θ2r=\theta^{2}r=θ2 for 0≤θ≤8π0 \leq \theta \leq 8\pi0≤θ≤8π.

Problem 24527

Find the average rate of change of y=x2−2y = x^{2} - 2y=x2−2 from x=0x = 0x=0 to x=1x = 1x=1.

Problem 24528

Fish interactions can be modeled by the Morse potential V(r)=e−r−Ae−arV(r)=e^{-r}-A e^{-a r}V(r)=e−r−Ae−ar. For A=1A=1A=1, a=12a=\frac{1}{2}a=21​, find lim⁡r→0V(r)\lim_{r \to 0} V(r)limr→0​V(r).

Problem 24529

Find the average rate of change of y=x2−2y = x^2 - 2y=x2−2 from x=0x = 0x=0 to x=1x = 1x=1.

Problem 24530

Evaluate the integral: ∫(ln⁡x)8xdx\int \frac{(\ln x)^{8}}{x} d x∫x(lnx)8​dx, assuming u>0u>0u>0 for ln⁡u\ln ulnu.

Problem 24531

Approximate the area under f(x)=ex+6f(x)=e^{x}+6f(x)=ex+6 from x=−2x=-2x=−2 to x=2x=2x=2 using n=4n=4n=4 rectangles: left, right, average, and midpoints.

Problem 24532

Find the limit: lim⁡h→0sin⁡(x+h)cos⁡(x+h)−sin⁡xcos⁡xh\lim _{h \rightarrow 0} \frac{\sin (x+h) \cos (x+h)-\sin x \cos x}{h}limh→0​hsin(x+h)cos(x+h)−sinxcosx​.

Problem 24533

Find the average rate of change of y=2x2−2x+2y=2x^{2}-2x+2y=2x2−2x+2 over the interval [0,1][0,1][0,1].

Problem 24534

Find the area under f(x)=(10−x)f(x)=(10-x)f(x)=(10−x) from x=0x=0x=0 to x=10x=10x=10 using 10 rectangles and midpoints. Area is approximately □\square□ square units.

Problem 24535

Find the indefinite integral and verify by differentiating: ∫(sec⁡2x−2)dx=□ \int\left(\sec ^{2} x-2\right) d x=\square ∫(sec2x−2)dx=□

Problem 24536

A substance grows at 14%14\%14% per day. If it starts at 46 grams, find its mass after 4 days, exact and rounded to the nearest tenth.

Problem 24537

Estimate the age of cave paintings with 6%6\%6% carbon-14 remaining using the decay model A=A0e−ktA = A_0 e^{-kt}A=A0​e−kt.

Problem 24538

Find the area between the xxx-axis and f(x)=2ex−1f(x)=2 e^{x}-1f(x)=2ex−1 over the interval [-2,3]. Check for xxx-axis crossings.

Problem 24539

Calculate the area between the curves y=2,916xy=2,916 \sqrt{x}y=2,916x​ and y=108x2y=108 x^{2}y=108x2. The area is □\square□.

Problem 24540

A $200 \mathrm{~m}$ fence surrounds a rectangular lawn with a semi-circle.
a Explain why $y=100-x-\frac{\pi}{2} x$ . b Show $A=200 x-x^{2}\left(2+\frac{\pi}{2}\right)$ . c Use calculus to find dimensions for maximum area.

Find the sum of the infinite series: $\sum_{k=0}^{\infty} \sqrt{2}^k$ .

Find the slope of the curve defined by $x^{2}+2xy+y=4$ at the point $(1,1)$ .

Find the limit: $\lim _{x \rightarrow 5} \frac{|x-5|}{25-x^{2}}$ .

Berechne den Flächeninhalt $A$ zwischen $f(x)=x^{2}+1$ , der $x$ -Achse und den Linien $x=1$ und $x=2$ : $A = \int_{1}^{2} (x^{2} + 1) \, dx$

Find the distance $s$ (in cm) moved by a cam-shaft follower in 4 seconds, given $s=\int_{0}^{4} t \sqrt{4+9 t^{2}} dt$ .

Calculate the sum of the series: $\sum_{m=0}^{\infty} 5^{-m}$ .

Calculate the area under the curve $y=5-2x^{2}$ between $x=0$ and $x=1$ .

Evaluate the integral using the substitution $u=4 x^{2}+5 x$ :
$\int(8 x+5) \sqrt{4 x^{2}+5 x} d x=\int(\sqrt{u}) d u$
Find the result:
$\int(8 x+5) \sqrt{4 x^{2}+5 x} d x=\square$

Bestimmen Sie das unbestimmte Integral: $\int(3-5 x)^{7} d x=$

Find the area between the $x$ -axis and $f(x)=3 e^{x}-4$ over the interval $[-3,2]$ . Check for $x$ -axis crossings.

Find the limits: h) $\lim _{x \rightarrow 5} \frac{|x-5|}{25-x^{2}}$ i) $\lim _{z \rightarrow 1} \frac{\frac{1}{z}-1}{\frac{1}{z^{2}}-1}$

Find the limit as $\alpha$ approaches $\pi$ from the right: $\lim _{\alpha \rightarrow \pi^{+}} \tan \frac{3 \pi}{2}-\alpha$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{2 e^{-x}}{7+8 e^{-x}}$ .

Find the area between the $x$ -axis and $f(x)=e^{x}-2$ over the interval $[-1,3]$ . Does the graph cross the $x$ -axis?

Find the length of the curve given by $y^{2}=4(x+3)^{3}$ for $0 \leq x \leq 1$ , $y>0$ .

Find the first and second derivatives of the function $y=7x+1$ .

Find the limit: $\lim_{x \rightarrow 2} (-3x + 4)$ . Choose the correct rewriting option from A, B, C, or D.

Calculate the length of the curve given by $y=3+8 x^{3/2}$ for $0 \leq x \leq 1$ .

Chewing frequency $c$ is related to body mass $M$ by $c=kM^{-0.128}$ . Given $M(t)=1+2\sqrt{t}$ , find $\frac{d c}{d t}$ . Also, $L=rM^{0.312}$ ; find $\frac{d c}{d L}$ .

Differentiate $y=9 x e^{x}-9 e^{x}$ .

Differentiate the equation $y^{8}-5 x^{3}=5 x$ to find $\frac{d y}{d x}$ . Choose the correct answer.

Calculate the integral of the function: $\int\left(\frac{x^{2}}{4}-4\right) d x$ .

Find the limit: $\lim _{x \rightarrow-\infty} \frac{4 x^{2}+x}{x+1}$ .

Evaluate the infinite series: $\sum_{k=0}^{\infty} \frac{k-1}{k !}$ .

Find the exact length of the polar curve $r=\theta^{2}$ for $0 \leq \theta \leq 8\pi$ .

Find the average rate of change of $y = x^{2} - 2$ from $x = 0$ to $x = 1$ .

Fish interactions can be modeled by the Morse potential $V(r)=e^{-r}-A e^{-a r}$ . For $A=1$ , $a=\frac{1}{2}$ , find $\lim_{r \to 0} V(r)$ .

Find the average rate of change of $y = x^2 - 2$ from $x = 0$ to $x = 1$ .

Evaluate the integral: $\int \frac{(\ln x)^{8}}{x} d x$ , assuming $u>0$ for $\ln u$ .

Approximate the area under $f(x)=e^{x}+6$ from $x=-2$ to $x=2$ using $n=4$ rectangles: left, right, average, and midpoints.

Find the limit: $\lim _{h \rightarrow 0} \frac{\sin (x+h) \cos (x+h)-\sin x \cos x}{h}$ .

Find the average rate of change of $y=2x^{2}-2x+2$ over the interval $[0,1]$ .

Find the area under $f(x)=(10-x)$ from $x=0$ to $x=10$ using 10 rectangles and midpoints. Area is approximately $\square$ square units.

Find the indefinite integral and verify by differentiating:
$\int\left(\sec ^{2} x-2\right) d x=\square$

A substance grows at $14\%$ per day. If it starts at 46 grams, find its mass after 4 days, exact and rounded to the nearest tenth.

Estimate the age of cave paintings with $6\%$ carbon-14 remaining using the decay model $A = A_0 e^{-kt}$ .

Find the area between the $x$ -axis and $f(x)=2 e^{x}-1$ over the interval [-2,3]. Check for $x$ -axis crossings.

Calculate the area between the curves $y=2,916 \sqrt{x}$ and $y=108 x^{2}$ . The area is $\square$ .

Calculez $f^{\prime}(x)$ pour $f(x)=x^{2}-4x$ , $f(x)=\sqrt{x+1}-4$ , $f(x)=\frac{7}{1-2x}$ , et $f(x)=\frac{1}{\sqrt{x-3}}$ . Esquissez le graphique de la dérivée pour deux fonctions.

Differentiate the function: $u=\sqrt[5]{t}+6 \sqrt{t^{5}}$ .

Differentiate the function $u=\sqrt[5]{t}+4 \sqrt{t^{5}}$ .

Calculate the area in the first quadrant between $y=10$ and $y=10 \sin x$ from $x=0$ to $x=\frac{\pi}{2}$ . Area: $\square$ .

How long for a tranquilizer to decay to $95\%$ of its original dosage if its half-life is 40 hours? Use $A=A_{0} e^{kt}$ .

Find critical points and use the first derivative test for $f(x)=x \sqrt{4-x^{2}}$ on $[-2,2]$ to determine max/min values.

Find the critical points of $f(x)=x \sqrt{36-x^{2}}$ on $[-6,6]$ and determine local/absolute max/min values.

Find the absolute extrema of $f(x)=\frac{1}{3} x^{3}-\frac{5}{2} x^{2}-6 x-2$ on $[-2,7]$ .

Find the area between $f(x)=x$ and $g(x)=x^{1/n}$ for $x \geq 0$ , expressed in terms of positive integer $n \geq 2$ .

Find the absolute extrema of $f(x)=\frac{1}{3} x^{3}-\frac{5}{2} x^{2}-6 x-2$ on $[-2,7]$ .

Find critical points of $f(x)=x \sqrt{4-x^{2}}$ on $[-2,2]$ and use the first derivative test for extrema.

Differentiate the function $y=e^{x+4}+1$ .

Differentiate the function $y=e^{x+4}+1$ and find $y'=\square$ .

Differentiate the function $S(R)=2 \pi R^{2}$ and find $S^{\prime}(R)$ .

Differentiate the function $g(u) = \sqrt{7} u + \sqrt{10u}$ . Find $g'(u) =$ .

Find critical points and use the first derivative test for $f(x)=x \sqrt{36-x^{2}}$ on $[-6,6]$ to determine local extrema.

Find the absolute extrema of $f(x)=3 x^{4}-150 x^{2}+1$ on $[-6,6]$ and their corresponding $x$ values.

Find critical points and use the first derivative test for $f(x)=x \sqrt{4-x^{2}}$ on $[-2,2]$ . Identify extrema.

Find the velocity $v(1)$ and acceleration $a(1)$ at $t=1$ for $s(t)=t^{2}-4t$ . $v(1)=\square$

Find the absolute extrema of $f(x)=3 x^{4}-150 x^{2}+1$ on $[-6,6]$ and the $x$ values where they occur.

Find the first and second derivative of the function $G(r)=\sqrt{r}+\sqrt[6]{r}$ . What are $G^{\prime}(r)$ and $G^{\prime \prime}(r)$ ?

Verify if the function $f(x)=-4 x^{2}+4 x-7$ has a local extremum and find the absolute extrema.

Find $f^{\prime}(1)$ given that $f(x)+x^{2}[f(x)]^{3}=10$ and $f(1)=2$ .

Find the absolute extrema of $f(x)=\frac{1-x}{5+x}$ on $[0,5]$ . What are the max value and $x$ where it occurs?

Find the first and second derivatives of the function $f(x)=12 x^{12}+9 x^{9}-x$ . What are $f'(x)$ and $f''(x)$ ?

Find the tangent line equation at (3,4) for the curve $x^{2}+2xy-y^{2}+x=20$ using implicit differentiation.

Verify if the function $A(r)=\frac{16}{r}+2 \pi r^{2}$ has an absolute extremum for $r>0$ , then find its location and value.

Find the absolute extrema of $f(x)=\frac{6+x}{7-x}$ on $[-7,0]$ . What are the values and where do they occur?

Find the absolute extrema of $f(x)=3 x^{3}-24 x^{2}+48 x+3$ on $[0,7]$ . What is the absolute minimum and where does it occur?

Differentiate the function: $y = 5 e^{x} + \frac{8}{\sqrt[3]{x}}$ .

Find the absolute extrema of $f(x)=3 x^{3}+18 x^{2}+27 x+6$ on $[-6,0]$ . What is the max and where does it occur?

Verify if the function $A(r)=\frac{20}{r}+2 \pi r^{2}, r>0$ has an absolute extremum. Find its location and value.

Given the motion $s=t^{3}-3t$ , find velocity $v(t)$ and acceleration $a(t)$ . Also, find $a(5)$ and $a(v=0)$ .

Determine if the function $f(x)$ is continuous at $x=-2$ using $f(-2)$ and $\lim_{x \to -2} f(x)$ .