Calculus

Problem 8601

Berechnen Sie die Werte der folgenden Integrale: a) $\int_{0}^{4} \frac{2 x}{x^{2}+1} d x$ , b) $\int_{0}^{\frac{\pi}{\pi}} \frac{-\sin x}{\cos x} d x$ , c) $\int_{1}^{5} \frac{3 x^{2}}{x^{3}+8} d x$ , d) $\int_{e}^{1} \frac{\frac{1}{x}}{\ln x} d x$ , e) $\int_{-2}^{0} \frac{e^{x}}{e^{x}+e} d x$ , f) $\int_{\frac{\pi}{6}}^{\frac{\pi}{6}} \frac{\cos x}{\sin x} d x$ , g) $\int_{0}^{5} \frac{e^{x}}{e} d x$ , h) $\int_{1}^{2} \frac{2 x+4 x^{3}}{x^{2}\left(1+x^{2}\right)} d x$ , i) $\int_{0}^{-2} \sin (2 x) d x$ , j) $\int_{-2}^{6} e^{-x} d x$ , k) $\int_{-1}^{1} e^{2 x+3} d x$ , l) $\int_{-\frac{1}{2}}^{0} \cos (\pi x) d x$ .

See Solution

Problem 8602

Calculate the average rate of change of $f(x)=\sin \left(\frac{x}{2}\right)$ from 0 to $\frac{\pi}{2}$ .

See Solution

Problem 8603

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

See Solution

Problem 8604

How many years ago was a wooden artifact made if it has 25% carbon-14 left? (Half-life of carbon-14 is 5730 years.)

See Solution

Problem 8605

Bestimme die Änderungsrate der Blutgeschwindigkeit $v=C \cdot (R^{2}-r^{2})$ bezüglich $r$ und erkläre die negative Änderungsrate. Wie beeinflusst die Erweiterung der Blutgefäße die Geschwindigkeit?

See Solution

Problem 8606

Ein Wasserbecken wird durch $x=4$ , $y=4$ und $f(x)=-10 x \cdot e^{-x-1}$ begrenzt. Bestimmen Sie die Länge der Beckenränder und die Flächeninhalte.

See Solution

Problem 8607

Ein Ölfleck hat anfangs eine Fläche von $30 \mathrm{~cm}^{2}$ und wächst um $6 \mathrm{~cm}^{2}$ pro Minute.
a) Finde $A(t)$ für die Fläche in Abhängigkeit von $t$ in Minuten. b) Bestimme $r(t)$ für den Radius und berechne $r^{\prime}(12)$ .

See Solution

Problem 8608

How many years will \$4000 grow to \$50000 at a 4% annual interest rate, compounded continuously? Round to the nearest tenth.

See Solution

Problem 8609

Find the derivative of $y=\ln \left(\mathrm{e}^{-9 x}+9 x \mathrm{e}^{-9 x}\right)$ .

See Solution

Problem 8610

Find $f^{\prime}(1)$ for $f(x)=\frac{\ln (x)}{x^{4}}$ . Options: 1, 0, $\frac{1}{\ln (4)}$ , 5.

See Solution

Problem 8611

What is the rate of temperature change if water goes from $18^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ in 6 minutes?

See Solution

Problem 8612

Differentiate $y=3^{5x^3 + 5x + 2}$ . Find $\frac{dy}{dx}$ .

See Solution

Problem 8613

Evaluate the derivative of $\sin \left(\cot \frac{\pi}{6}\right)$ with respect to $x$ .

See Solution

Problem 8614

Find the derivative of $y=-3 e^{x}$ . What is $\frac{d y}{d x}=?$

See Solution

Problem 8615

Differentiate the function $y=4^{4 x^{3}+4}$ .

See Solution

Problem 8616

Differentiate $k(x)=\mathrm{e}^{\left(-1-4 x^{4}\right)^{4}}$ and find $k^{\prime}(x)$ . No simplification needed.

See Solution

Problem 8617

Differentiate $g(x)=\frac{8 x^{6}}{e^{x^{4}}}$ . Find $g^{\prime}(x)$ .

See Solution

Problem 8618

Find the derivative of $f(x)=\ln(x^{21})$ with respect to $x$ .

See Solution

Problem 8619

Find the derivative using the limit definition for the function $f(x) = \frac{1}{x^{2}}$ .

See Solution

Problem 8620

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\log _{5}(x^{3}-2 x^{2}+10)$ .

See Solution

Problem 8621

Find $f^{\prime}(x)$ for $f(x)=\log _{4}(x^{3}-4 x^{2}+9)$ .

See Solution

Problem 8622

Find the derivative of $\frac{1}{x^{2}}$ .

See Solution

Problem 8623

Differentiate $f(x)=e^{\sqrt{x}} x^{15}$ without simplifying. Find $f'(x)=$

See Solution

Problem 8624

Differentiate the function $y=7^{5 x^{3}+2 x^{2}-3}$ .

See Solution

Problem 8625

Find the derivative $\frac{d y}{d x}$ for the function $y=\ln \left(x^{15} e^{-3 x}\right)$ .

See Solution

Problem 8626

Given $f(x)=\ln \left(\frac{\left(-3 x^{3}+1\right)^{6}}{\left(3 x^{2}-5\right)^{7}}\right)$ , find an equivalent form and $f^{\prime}(x)$ .

See Solution

Problem 8627

Find the limit: $\lim_{x \rightarrow 0} \frac{1-\sqrt{1-x^{2}}}{x}$

See Solution

Problem 8628

Compute the integral: $\int \tan^{7}(x) \sec^{4}(x) \, dx = \square + C$

See Solution

Problem 8629

Evaluate the integral using $u$ -substitution: $\int \cos^{5}(6x) \sin(6x) \, dx = +C$

See Solution

Problem 8630

Find $\frac{d x}{d t}$ at $x=-1$ given $y=-5 x^{2}-3$ and $\frac{d y}{d t}=4$ .

See Solution

Problem 8631

Find $\frac{d x}{d t}$ at $x=1$ given $y=2 x^{2}+5$ and $\frac{d y}{d t}=4$ .

See Solution

Problem 8632

Evaluate the integral using a double-angle formula: $\int \cos^{2}(11x) \, dx = \square + C$

See Solution

Problem 8633

Identify the improper integrals from the following list:
1. $\int_{0=\pi / 4}^{3 \pi / 4} \sec \theta d \theta$
2. $\int_{x=2}^{2} \sqrt{2-x} d x$
3. $\int_{x=0}^{1} \frac{d x}{\sqrt{e^{x}}-10}$
4. $\int_{x=0}^{1} \frac{d x}{x-1}$
5. $\int_{-\infty}^{\infty} e^{-\pi / 2} d x$
6. $\int_{0 \rightarrow-1 / 4}^{x / 4} \sec \theta d \theta$
7. $\int_{z \rightarrow-1}^{1} \ln \left(1+x^{2}\right) d x$
8. $\int_{t=-1,000,000}^{1200000} e^{-2 / 2} d x$
9. $\int_{a=0}^{1} \frac{d x}{x^{2}}$
10. $\int_{t=0}^{\infty} t^{-1} e^{-1} d t$

See Solution

Problem 8634

Explain why $\lim _{x \rightarrow 0} \frac{\sin |x|}{x}$ does not exist, given that $\lim _{x \rightarrow 0} \frac{\sin (x)}{x}=1$ .

See Solution

Problem 8635

Evaluate the integral from -2π to -π: $\int_{-2 \pi}^{-\pi} \sin(2x) \cos(x) \, dx =$

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Problem 8636

Finde Gegenbeispiele für die falschen Aussagen: a) $f^{\prime}$ monoton $\Rightarrow f$ monoton, b) Rechtskurve $\Rightarrow f^{\prime \prime}<0$ , c) $f^{\prime}(x_{0})=0 \Rightarrow f^{\prime \prime}(x_{0}) \neq 0$ .

See Solution

Problem 8637

Check if the intermediate value theorem shows a zero for $g(x)=2 x^{3}-13 x^{2}+18 x+5$ in intervals: a. $[1,2]$ , b. $[2,3]$ , c. $[3,4]$ , d. $[4,5]$ .

See Solution

Problem 8638

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sqrt{11+9 x^{2}}}{8+9 x}$ .

See Solution

Problem 8639

Find the limits: (a) $\lim _{x \rightarrow \infty} \frac{\sqrt{11+2 x^{2}}}{6+7 x}$ and (b) $\lim _{x \rightarrow-\infty} \frac{\sqrt{11+2 x^{2}}}{6+7 x}$ .

See Solution

Problem 8640

Find the horizontal asymptotes by calculating these limits:
1. $\lim _{x \rightarrow \infty} \frac{-3 x}{11+2 x}=$
2. $\lim _{x \rightarrow-\infty} \frac{14 x-8}{x^{3}+8 x-7}=$
3. $\lim _{x \rightarrow \infty} \frac{x^{2}-3 x-15}{13-14 x^{2}}=$
4. $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+10 x}}{10-10 x}=$
5. $\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}+10 x}}{10-10 x}=$

See Solution

Problem 8641

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{3+11 x}{11-3 x}$ .

See Solution

Problem 8642

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{(4-x)(7+8 x)}{(3-3 x)(2+7 x)}$ .

See Solution

Problem 8643

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sqrt{7+9 x^{2}}}{(2+11 x)}$ .

See Solution

Problem 8644

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{3 x^{3}-3 x^{2}-6 x}{11-4 x-9 x^{3}}$ .

See Solution

Problem 8645

Find the limits: (a) $\lim _{x \rightarrow \infty} \frac{(4-x)(7+2 x)}{(3-3 x)(10+6 x)}$ and (b) $\lim _{x \rightarrow-\infty} \frac{(4-x)(7+2 x)}{(3-3 x)(10+6 x)}$ .

See Solution

Problem 8646

Find the limits: (a) $\lim _{x \rightarrow \infty} \frac{2+11 x}{6-3 x}$ and (b) $\lim _{x \rightarrow-\infty} \frac{2+11 x}{6-3 x}$ .

See Solution

Problem 8647

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sqrt{9+4 x^{2}}}{4+10 x}$ .

See Solution

Problem 8648

Find the limits: (a) $\lim _{x \rightarrow \infty} \frac{6 x^{3}-6 x^{2}-7 x}{4-9 x-2 x^{3}}$ and (b) $\lim _{x \rightarrow-\infty} \frac{6 x^{3}-6 x^{2}-7 x}{4-9 x-2 x^{3}}$ .

See Solution

Problem 8649

Find the horizontal asymptote of $r(x)=\frac{-4 x^{2}+5 x-1}{x^{2}+2}$ and where it crosses, if applicable.

See Solution

Problem 8650

Find the displacement at time $t$ and total distance traveled for $v(t)=4-2t$ from $t=0$ to $t=2$ .

See Solution

Problem 8651

If \$8000 is invested at 8% interest compounded continuously, find when future value is \$12000 and complete the table.

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Problem 8652

Evaluate the limit as $x$ approaches 2 for $\frac{2 x^{3}-16}{x-2}$ .

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Problem 8653

Determine the truth of these limits for $f(x)=\frac{x^{2}-1}{x^{2}}$ : I. $\lim_{x \to 0} f(x)=-\infty$ II. $\lim_{x \to \infty} f(x)=-1$ III. $\lim_{x \to \infty} f(x)=1$ .

See Solution

Problem 8654

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation $x^{2}+x y+y^{2}=3$ .

See Solution

Problem 8655

Find the second derivative $y^{\prime \prime}$ of the function $y=\frac{\csc x}{2}$ .

See Solution

Problem 8656

Find the limit as $x$ approaches 2 for the piecewise function $f(x)$ defined as: $(x+1)^{2}$ for $x \leq 2$ and $x+7$ for $x > 2$ .

See Solution

Problem 8657

Which statement about the function $f$ is always true if defined for all real numbers? A) If $\lim _{x \rightarrow 3} f(x)=4$ , then $f(3)=4$ . B) If $\lim _{x \rightarrow 3} f(x)=4$ , then 4 is in the range of $f$ . C) If $\lim _{x \rightarrow 3^{+}} f(x)=\lim _{x \rightarrow 3^{-}} f(x)$ , then $f(3)=\lim _{x \rightarrow 3^{+}} f(x)$ .

See Solution

Problem 8658

Find the derivative $h^{\prime}(x)$ for $h(x)=\ln \frac{x(x-1)^{3}}{\sqrt{x-2}}$ . No simplification needed.

See Solution

Problem 8659

Find the derivative $\frac{d y}{d x}$ using implicit differentiation for the equation $2 x^{2}-y^{2}=9$ .

See Solution

Problem 8660

Find the derivative $y^{\prime}$ for the function $y=-4 e^{\sec x}$ .

See Solution

Problem 8661

Find the differential $d G$ for $G(u, v, w) = \frac{3 u w^{2}}{v}$ and simplify $d G / G$ .

See Solution

Problem 8662

Find the tangent line equation for $f(x)=x(1-x)^{4}$ at $x=2$ .

See Solution

Problem 8663

Find the average rate of change of $f(x)=x^{2}$ from $x=6$ to $x=7$ and illustrate it graphically.

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Problem 8664

Calculate the average rate of change of $f(x)=\sqrt{2 x}$ from $x_{1}=2$ to $x_{2}=8$ . Options: A. 2 B. 7 C. $\frac{1}{3}$ D. $-\frac{3}{10}$

See Solution

Problem 8665

A. Find the height of Chicago's Sears Tower given gravity is $2.18 \times 10^{-3} \mathrm{~m/s}^{2}$ lower at the top. B. Determine the altitude for a geosynchronous satellite orbiting Earth in 2.00 hours and its gravitational strength (g). C. A scale shows $75.0 \mathrm{~kg}$ at 20.0 m above sea level. a) Does Earth's spin affect this reading? b) Will the scale read the same, less, or more at the South Pole at the same height? Explain. (Earth mass: $5.98 \times 10^{24} \mathrm{~kg}$ , radius: $6.37 \times 10^{6} \mathrm{~m}$ )

See Solution

Problem 8666

A sensitive gravimeter at the top of Sears Tower reads $2.18 \times 10^{-3} \mathrm{~m/s}^{2}$ less gravity. Find the height.

See Solution

Problem 8667

Given $f(4)=6$ , $f'(4)=7$ , $f(6)=10$ , and $f'(6)=-5$ , find $\left(f^{-1}\right)'(10)$ .

See Solution

Problem 8668

Find the derivative of the inverse function at -12 for $f(x)=\frac{1}{3} x^{3}+\frac{5}{3} x+2$ .

See Solution

Problem 8669

Find the derivative of the inverse function at -12 for $f(x)=\frac{1}{3} x^{3}+\frac{5}{3} x+2$ .

See Solution

Problem 8670

Find $f^{\prime \prime}\left(\frac{\pi}{3}\right)$ for the function $f(x)=\sec (x)$ .

See Solution

Problem 8671

Find the first and second derivatives of the function $H(\theta) = \theta \cos(\theta)$ . What are $H^{\prime}(\theta)$ and $H^{\prime \prime}(\theta)$ ?

See Solution

Problem 8672

Find the average velocity of a ball rolling down a ramp from $t_{1}=6$ to $t_{2}=6.5$ for $s(t)=11 t^{2}$ . A. $34.375 \mathrm{ft/sec}$ B. $68.75 \mathrm{ft/sec}$ C. $137.5 \mathrm{ft/sec}$ D. $464.75 \mathrm{ft/sec}$

See Solution

Problem 8673

Find $\left(f^{-1}\right)^{\prime}(-12)$ for the function $f(x)=\frac{1}{3} x^{3}+\frac{5}{3} x+2$ .

See Solution

Problem 8674

Find $d f$ for $f(x, y, z)=x^{2} y^{1/3} z^{2/3}$ and use it to approximate $(3.02)^{2}(\sqrt[3]{(1.04)(8.05)^{2}})$ .

See Solution

Problem 8675

Find the tangent line equation for $y=\sin(\sin(x))$ at the point $(3\pi, 0)$ .

See Solution

Problem 8676

Find the derivative of $G(x)=\sqrt[3]{x^{5}+6x}$ .

See Solution

Problem 8677

Find the derivative of $f(x)=\left(\frac{3x-1}{5x+2}\right)^{4}$ .

See Solution

Problem 8678

Find the first and second derivatives of $y=\ln(3+\ln(x))$ . What are $y'$ and $y''$ ?

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Problem 8679

Estimate the distance traveled in 6 seconds with speeds: 110, 99.8, 90.9, 83.2, 76.4, 70.4, 65.1 ft/s. Then, use $f(t)=\frac{44000}{(t+20)^{2}}$ to find the exact distance.

See Solution

Problem 8680

Find $h^{\prime}(1)$ if $h(x)=\sqrt{4+3 f(x)}$ , with $f(1)=4$ and $f^{\prime}(1)=2$ .

See Solution

Problem 8681

Find the altitude for a geosynchronous satellite orbiting Earth in 2.00 hours and the gravitational strength $g$ there.

See Solution

Problem 8682

Verify the linear approximation $f(x) \approx 1 - 15x$ at $a=0$ and find $x$ where it's accurate within 0.1. $x \in$

See Solution

Problem 8683

Differentiate $g(x)=\sqrt{\frac{4-x}{3+x}}$ .

See Solution

Problem 8684

Find the critical numbers of the function $f(x)=8 x^{3}-12 x^{2}-144 x$ . Enter answers as a comma-separated list.

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Problem 8685

Find the differential $d F$ for $F(x, y, z)=\frac{2 x^{3} z^{2}}{\sqrt{y}}$ , and estimate output variation with $0.5\%$ input change.

See Solution

Problem 8686

Check if $f(x)=x^{3}-x^{2}-20x+6$ meets Rolle's Theorem on $[0,5]$ and find $c$ values. $c=$

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Problem 8687

Find the derivative of $f(x)=(2x^3-3x^2+4x+1)^{100}$ .

See Solution

Problem 8688

If $f''$ is continuous, and $f'(-2)=0$ and $f''(-2)=-4$ , what does this imply about $f$ at $x=-2$ ?

See Solution

Problem 8689

Find the limit: $\lim _{x \rightarrow \infty}\left(1-\frac{6}{x}\right)^{x}=$ ? Options: 6, $e^{-6}$ , 0, $e^{6}$ , $-6$ .

See Solution

Problem 8690

If $f'(4)=0$ and $f''(4)=0$ , what does this imply about the function $f$ at $x=4$ ?

See Solution

Problem 8691

Find antiderivatives for: a. $\frac{1}{y} \frac{d y}{d t}$ , b. $y^{\prime}(t) \sqrt{y(t)}$ , c. $f^{\prime}(t) \sin (f(t))$ , d. $\frac{1}{1+w^{2}} \frac{d w}{d t}$ . Use $-y, f$ , or $w$ for answers.

See Solution

Problem 8692

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

See Solution

Problem 8693

Evaluate $\lim _{n \rightarrow \infty} \arctan ^{\prime} x$ and discuss its significance for the graph of $y=\arctan x$ .

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Problem 8694

Find the vertical distance $v$ between the line $y=x+12$ and the parabola $y=x^{2}$ for $-3 \leq x \leq 4$ .
Calculate $v^{\prime}(x)$ and determine the maximum vertical distance.

See Solution

Problem 8695

Solve the differential equation $dy=(4+y^2)dx$ . What is the solution?

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Problem 8696

Find the limit as $x$ approaches -7 for $\frac{x+7}{|x+7|}$ . Choices: (A) -1 (B) 0 (C) 1 (D) nonexistent.

See Solution

Problem 8697

Find the average rate of change of $f(x)=x^2-3x-28$ over the interval $[-4,7]$ . Options: a) 0, b) 11, c) 6, d) 66.

See Solution

Problem 8698

Find the general solution of the differential equation $\frac{dy}{dx} = y - (x + 2y^3)$ . Options are: 1. $x = cy + y^2$ 2. $x = cy - y^2$ 3. $x = cy + y^3$ 4. More than one of the above.

See Solution

Problem 8699

Given $f(x) = x^{4}-50x^{2}+8$ , find local min/max, inflection points, and concavity intervals.

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Problem 8700

Find the approximation of $f(8.1)$ using the tangent line at $x=8$ for $| \frac{dy}{dx} = \frac{x}{y}$ and $f(8)=2$ .

See Solution

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Calculus

Problem 8601

Problem 8602

Calculate the average rate of change of f(x)=sin⁡(x2)f(x)=\sin \left(\frac{x}{2}\right)f(x)=sin(2x​) from 0 to π2\frac{\pi}{2}2π​.

Problem 8603

Find the derivative of h(x)=32x−4h(x)=3^{2x-4}h(x)=32x−4.

Problem 8604

How many years ago was a wooden artifact made if it has 25% carbon-14 left? (Half-life of carbon-14 is 5730 years.)

Problem 8605

Bestimme die Änderungsrate der Blutgeschwindigkeit v=C⋅(R2−r2)v=C \cdot (R^{2}-r^{2})v=C⋅(R2−r2) bezüglich rrr und erkläre die negative Änderungsrate. Wie beeinflusst die Erweiterung der Blutgefäße die Geschwindigkeit?

Problem 8606

Ein Wasserbecken wird durch x=4x=4x=4, y=4y=4y=4 und f(x)=−10x⋅e−x−1f(x)=-10 x \cdot e^{-x-1}f(x)=−10x⋅e−x−1 begrenzt. Bestimmen Sie die Länge der Beckenränder und die Flächeninhalte.

Problem 8607

Ein Ölfleck hat anfangs eine Fläche von 30 cm230 \mathrm{~cm}^{2}30 cm2 und wächst um 6 cm26 \mathrm{~cm}^{2}6 cm2 pro Minute. a) Finde A(t)A(t)A(t) für die Fläche in Abhängigkeit von ttt in Minuten. b) Bestimme r(t)r(t)r(t) für den Radius und berechne r′(12)r^{\prime}(12)r′(12).

Problem 8608

How many years will \$4000 grow to \$50000 at a 4% annual interest rate, compounded continuously? Round to the nearest tenth.

Problem 8609

Find the derivative of y=ln⁡(e−9x+9xe−9x)y=\ln \left(\mathrm{e}^{-9 x}+9 x \mathrm{e}^{-9 x}\right)y=ln(e−9x+9xe−9x).

Problem 8610

Find f′(1)f^{\prime}(1)f′(1) for f(x)=ln⁡(x)x4f(x)=\frac{\ln (x)}{x^{4}}f(x)=x4ln(x)​. Options: 1, 0, 1ln⁡(4)\frac{1}{\ln (4)}ln(4)1​, 5.

Problem 8611

What is the rate of temperature change if water goes from 18∘C18^{\circ} \mathrm{C}18∘C to 100∘C100^{\circ} \mathrm{C}100∘C in 6 minutes?

Problem 8612

Differentiate y=35x3+5x+2y=3^{5x^3 + 5x + 2}y=35x3+5x+2. Find dydx\frac{dy}{dx}dxdy​.

Problem 8613

Evaluate the derivative of sin⁡(cot⁡π6)\sin \left(\cot \frac{\pi}{6}\right)sin(cot6π​) with respect to xxx.

Problem 8614

Find the derivative of y=−3exy=-3 e^{x}y=−3ex. What is dydx=?\frac{d y}{d x}=?dxdy​=?

Problem 8615

Differentiate the function y=44x3+4y=4^{4 x^{3}+4}y=44x3+4.

Problem 8616

Differentiate k(x)=e(−1−4x4)4k(x)=\mathrm{e}^{\left(-1-4 x^{4}\right)^{4}}k(x)=e(−1−4x4)4 and find k′(x)k^{\prime}(x)k′(x). No simplification needed.

Problem 8617

Differentiate g(x)=8x6ex4g(x)=\frac{8 x^{6}}{e^{x^{4}}}g(x)=ex48x6​. Find g′(x)g^{\prime}(x)g′(x).

Problem 8618

Find the derivative of f(x)=ln⁡(x21)f(x)=\ln(x^{21})f(x)=ln(x21) with respect to xxx.

Problem 8619

Find the derivative using the limit definition for the function f(x)=1x2f(x) = \frac{1}{x^{2}}f(x)=x21​.

Problem 8620

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=log⁡5(x3−2x2+10)f(x)=\log _{5}(x^{3}-2 x^{2}+10)f(x)=log5​(x3−2x2+10).

Problem 8621

Find f′(x)f^{\prime}(x)f′(x) for f(x)=log⁡4(x3−4x2+9)f(x)=\log _{4}(x^{3}-4 x^{2}+9)f(x)=log4​(x3−4x2+9).

Problem 8622

Find the derivative of 1x2\frac{1}{x^{2}}x21​.

Problem 8623

Differentiate f(x)=exx15f(x)=e^{\sqrt{x}} x^{15}f(x)=ex​x15 without simplifying. Find f′(x)=f'(x)=f′(x)=

Problem 8624

Differentiate the function y=75x3+2x2−3y=7^{5 x^{3}+2 x^{2}-3}y=75x3+2x2−3.

Problem 8625

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=ln⁡(x15e−3x)y=\ln \left(x^{15} e^{-3 x}\right)y=ln(x15e−3x).

Problem 8626

Given f(x)=ln⁡((−3x3+1)6(3x2−5)7)f(x)=\ln \left(\frac{\left(-3 x^{3}+1\right)^{6}}{\left(3 x^{2}-5\right)^{7}}\right)f(x)=ln((3x2−5)7(−3x3+1)6​), find an equivalent form and f′(x)f^{\prime}(x)f′(x).

Problem 8627

Find the limit: lim⁡x→01−1−x2x\lim_{x \rightarrow 0} \frac{1-\sqrt{1-x^{2}}}{x}limx→0​x1−1−x2​​

Problem 8628

Compute the integral: ∫tan⁡7(x)sec⁡4(x) dx=□+C\int \tan^{7}(x) \sec^{4}(x) \, dx = \square + C∫tan7(x)sec4(x)dx=□+C

Problem 8629

Evaluate the integral using uuu-substitution: ∫cos⁡5(6x)sin⁡(6x) dx=+C\int \cos^{5}(6x) \sin(6x) \, dx = +C∫cos5(6x)sin(6x)dx=+C

Problem 8630

Find dxdt\frac{d x}{d t}dtdx​ at x=−1x=-1x=−1 given y=−5x2−3y=-5 x^{2}-3y=−5x2−3 and dydt=4\frac{d y}{d t}=4dtdy​=4.

Problem 8631

Find dxdt\frac{d x}{d t}dtdx​ at x=1x=1x=1 given y=2x2+5y=2 x^{2}+5y=2x2+5 and dydt=4\frac{d y}{d t}=4dtdy​=4.

Problem 8632

Evaluate the integral using a double-angle formula: ∫cos⁡2(11x) dx=□+C\int \cos^{2}(11x) \, dx = \square + C∫cos2(11x)dx=□+C

Problem 8633

Problem 8634

Explain why lim⁡x→0sin⁡∣x∣x\lim _{x \rightarrow 0} \frac{\sin |x|}{x}limx→0​xsin∣x∣​ does not exist, given that lim⁡x→0sin⁡(x)x=1\lim _{x \rightarrow 0} \frac{\sin (x)}{x}=1limx→0​xsin(x)​=1.

Problem 8635

Evaluate the integral from -2π to -π: ∫−2π−πsin⁡(2x)cos⁡(x) dx=\int_{-2 \pi}^{-\pi} \sin(2x) \cos(x) \, dx =∫−2π−π​sin(2x)cos(x)dx=

Problem 8636

Problem 8637

Check if the intermediate value theorem shows a zero for g(x)=2x3−13x2+18x+5g(x)=2 x^{3}-13 x^{2}+18 x+5g(x)=2x3−13x2+18x+5 in intervals: a. [1,2][1,2][1,2], b. [2,3][2,3][2,3], c. [3,4][3,4][3,4], d. [4,5][4,5][4,5].

Problem 8638

Find the limit as xxx approaches infinity: lim⁡x→∞11+9x28+9x\lim _{x \rightarrow \infty} \frac{\sqrt{11+9 x^{2}}}{8+9 x}limx→∞​8+9x11+9x2​​.

Problem 8639

Find the limits: (a) lim⁡x→∞11+2x26+7x\lim _{x \rightarrow \infty} \frac{\sqrt{11+2 x^{2}}}{6+7 x}limx→∞​6+7x11+2x2​​ and (b) lim⁡x→−∞11+2x26+7x\lim _{x \rightarrow-\infty} \frac{\sqrt{11+2 x^{2}}}{6+7 x}limx→−∞​6+7x11+2x2​​.

Problem 8640

Problem 8641

Evaluate the limit as xxx approaches infinity: lim⁡x→∞3+11x11−3x\lim _{x \rightarrow \infty} \frac{3+11 x}{11-3 x}limx→∞​11−3x3+11x​.

Problem 8642

Calculate the average rate of change of $f(x)=\sin \left(\frac{x}{2}\right)$ from 0 to $\frac{\pi}{2}$ .

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

Bestimme die Änderungsrate der Blutgeschwindigkeit $v=C \cdot (R^{2}-r^{2})$ bezüglich $r$ und erkläre die negative Änderungsrate. Wie beeinflusst die Erweiterung der Blutgefäße die Geschwindigkeit?

Ein Wasserbecken wird durch $x=4$ , $y=4$ und $f(x)=-10 x \cdot e^{-x-1}$ begrenzt. Bestimmen Sie die Länge der Beckenränder und die Flächeninhalte.

Ein Ölfleck hat anfangs eine Fläche von $30 \mathrm{~cm}^{2}$ und wächst um $6 \mathrm{~cm}^{2}$ pro Minute.
a) Finde $A(t)$ für die Fläche in Abhängigkeit von $t$ in Minuten. b) Bestimme $r(t)$ für den Radius und berechne $r^{\prime}(12)$ .

Find the derivative of $y=\ln \left(\mathrm{e}^{-9 x}+9 x \mathrm{e}^{-9 x}\right)$ .

Find $f^{\prime}(1)$ for $f(x)=\frac{\ln (x)}{x^{4}}$ . Options: 1, 0, $\frac{1}{\ln (4)}$ , 5.

What is the rate of temperature change if water goes from $18^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ in 6 minutes?

Differentiate $y=3^{5x^3 + 5x + 2}$ . Find $\frac{dy}{dx}$ .

Evaluate the derivative of $\sin \left(\cot \frac{\pi}{6}\right)$ with respect to $x$ .

Find the derivative of $y=-3 e^{x}$ . What is $\frac{d y}{d x}=?$

Differentiate the function $y=4^{4 x^{3}+4}$ .

Differentiate $k(x)=\mathrm{e}^{\left(-1-4 x^{4}\right)^{4}}$ and find $k^{\prime}(x)$ . No simplification needed.

Differentiate $g(x)=\frac{8 x^{6}}{e^{x^{4}}}$ . Find $g^{\prime}(x)$ .

Find the derivative of $f(x)=\ln(x^{21})$ with respect to $x$ .

Find the derivative using the limit definition for the function $f(x) = \frac{1}{x^{2}}$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\log _{5}(x^{3}-2 x^{2}+10)$ .

Find $f^{\prime}(x)$ for $f(x)=\log _{4}(x^{3}-4 x^{2}+9)$ .

Find the derivative of $\frac{1}{x^{2}}$ .

Differentiate $f(x)=e^{\sqrt{x}} x^{15}$ without simplifying. Find $f'(x)=$

Differentiate the function $y=7^{5 x^{3}+2 x^{2}-3}$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=\ln \left(x^{15} e^{-3 x}\right)$ .

Given $f(x)=\ln \left(\frac{\left(-3 x^{3}+1\right)^{6}}{\left(3 x^{2}-5\right)^{7}}\right)$ , find an equivalent form and $f^{\prime}(x)$ .

Find the limit: $\lim_{x \rightarrow 0} \frac{1-\sqrt{1-x^{2}}}{x}$

Compute the integral: $\int \tan^{7}(x) \sec^{4}(x) \, dx = \square + C$

Evaluate the integral using $u$ -substitution: $\int \cos^{5}(6x) \sin(6x) \, dx = +C$

Find $\frac{d x}{d t}$ at $x=-1$ given $y=-5 x^{2}-3$ and $\frac{d y}{d t}=4$ .

Find $\frac{d x}{d t}$ at $x=1$ given $y=2 x^{2}+5$ and $\frac{d y}{d t}=4$ .

Evaluate the integral using a double-angle formula: $\int \cos^{2}(11x) \, dx = \square + C$

Explain why $\lim _{x \rightarrow 0} \frac{\sin |x|}{x}$ does not exist, given that $\lim _{x \rightarrow 0} \frac{\sin (x)}{x}=1$ .

Evaluate the integral from -2π to -π: $\int_{-2 \pi}^{-\pi} \sin(2x) \cos(x) \, dx =$

Check if the intermediate value theorem shows a zero for $g(x)=2 x^{3}-13 x^{2}+18 x+5$ in intervals: a. $[1,2]$ , b. $[2,3]$ , c. $[3,4]$ , d. $[4,5]$ .

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sqrt{11+9 x^{2}}}{8+9 x}$ .

Find the limits: (a) $\lim _{x \rightarrow \infty} \frac{\sqrt{11+2 x^{2}}}{6+7 x}$ and (b) $\lim _{x \rightarrow-\infty} \frac{\sqrt{11+2 x^{2}}}{6+7 x}$ .

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{3+11 x}{11-3 x}$ .

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{(4-x)(7+8 x)}{(3-3 x)(2+7 x)}$ .

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sqrt{7+9 x^{2}}}{(2+11 x)}$ .

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{3 x^{3}-3 x^{2}-6 x}{11-4 x-9 x^{3}}$ .

Find the limits: (a) $\lim _{x \rightarrow \infty} \frac{2+11 x}{6-3 x}$ and (b) $\lim _{x \rightarrow-\infty} \frac{2+11 x}{6-3 x}$ .

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sqrt{9+4 x^{2}}}{4+10 x}$ .

Find the horizontal asymptote of $r(x)=\frac{-4 x^{2}+5 x-1}{x^{2}+2}$ and where it crosses, if applicable.

Find the displacement at time $t$ and total distance traveled for $v(t)=4-2t$ from $t=0$ to $t=2$ .

Evaluate the limit as $x$ approaches 2 for $\frac{2 x^{3}-16}{x-2}$ .

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation $x^{2}+x y+y^{2}=3$ .

Find the second derivative $y^{\prime \prime}$ of the function $y=\frac{\csc x}{2}$ .

Find the limit as $x$ approaches 2 for the piecewise function $f(x)$ defined as: $(x+1)^{2}$ for $x \leq 2$ and $x+7$ for $x > 2$ .

Find the derivative $h^{\prime}(x)$ for $h(x)=\ln \frac{x(x-1)^{3}}{\sqrt{x-2}}$ . No simplification needed.

Find the derivative $\frac{d y}{d x}$ using implicit differentiation for the equation $2 x^{2}-y^{2}=9$ .

Find the derivative $y^{\prime}$ for the function $y=-4 e^{\sec x}$ .

Find the differential $d G$ for $G(u, v, w) = \frac{3 u w^{2}}{v}$ and simplify $d G / G$ .

Find the tangent line equation for $f(x)=x(1-x)^{4}$ at $x=2$ .

Find the average rate of change of $f(x)=x^{2}$ from $x=6$ to $x=7$ and illustrate it graphically.

Calculate the average rate of change of $f(x)=\sqrt{2 x}$ from $x_{1}=2$ to $x_{2}=8$ . Options: A. 2 B. 7 C. $\frac{1}{3}$ D. $-\frac{3}{10}$

A sensitive gravimeter at the top of Sears Tower reads $2.18 \times 10^{-3} \mathrm{~m/s}^{2}$ less gravity. Find the height.

Given $f(4)=6$ , $f'(4)=7$ , $f(6)=10$ , and $f'(6)=-5$ , find $\left(f^{-1}\right)'(10)$ .

Find the derivative of the inverse function at -12 for $f(x)=\frac{1}{3} x^{3}+\frac{5}{3} x+2$ .

Find the derivative of the inverse function at -12 for $f(x)=\frac{1}{3} x^{3}+\frac{5}{3} x+2$ .

Find $f^{\prime \prime}\left(\frac{\pi}{3}\right)$ for the function $f(x)=\sec (x)$ .

Find the first and second derivatives of the function $H(\theta) = \theta \cos(\theta)$ . What are $H^{\prime}(\theta)$ and $H^{\prime \prime}(\theta)$ ?

Find $\left(f^{-1}\right)^{\prime}(-12)$ for the function $f(x)=\frac{1}{3} x^{3}+\frac{5}{3} x+2$ .

Find $d f$ for $f(x, y, z)=x^{2} y^{1/3} z^{2/3}$ and use it to approximate $(3.02)^{2}(\sqrt[3]{(1.04)(8.05)^{2}})$ .