Calculus

Problem 17201

Find an equation of a curve that intersects every curve of the family $y=\frac{1}{x}+k$ at right angles. Options: (A) $y=-x$ (B) $y=-x^{2}$ (C) $y=-\frac{1}{3} x^{3}$ (D) $y=\frac{1}{3} x^{3}$ (E) $y=\ln x$

See Solution

Problem 17202

A 10 ft long trough has triangular ends (3 ft wide, 1 ft high). Water fills at $12 \mathrm{ft}^{3} / \mathrm{min}$ . Find the rise rate at 6 in deep.

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Problem 17203

Calculate the limit: $\lim _{n \rightarrow \infty}\left(1+\frac{\sqrt{n}}{2^{n+1}}\right)^{2 n}$ .

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Problem 17204

Calculate the limit: $\lim _{n \rightarrow \infty}\left(1+\frac{\pi}{2^{n+1}}\right)^{2 n}$ .

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Problem 17205

Evaluate the left and right Riemann sums for $n=10$ over $[0,5]$ using the values of $f(x)$ provided.

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Problem 17206

Estimate the alternating harmonic series using the first 10 terms for $S_{10}$ , then show how to rearrange $2 \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ .

See Solution

Problem 17207

A balloon 150 ft away rises at 8 ft/s. How fast is the distance to the balloon increasing when it's 50 ft high?

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Problem 17208

Evaluate these limits: (a) $\lim _{x \rightarrow \infty} \frac{x^{3}+3}{2 x^{3}+4 x+1}$ (b) $\lim _{x \rightarrow 0} \frac{\sin^{-1} x}{x}$ (c) $\lim _{x \rightarrow 0} \frac{\sin^{-1} x}{\sin x}$ (d) $\lim _{x \rightarrow 1} \frac{1+\cos n x}{\tan^{2} n x}$ (e) $\lim _{x \rightarrow 0} \frac{\sec x-\cos x}{\sin x}$ (f) $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}$ (g) $\lim _{x \rightarrow \infty} \frac{x+1}{x+2}$ (h) $\lim _{x \rightarrow \infty} |\sqrt{x+1}-\sqrt{x}|$ Find discontinuities of: (a) $\frac{x^{3}+4 x+6}{x^{2}-6 x+8}$ (b) $\sec x$ (c) $\frac{\sin x}{\sqrt{x}}$

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Problem 17209

Estimate the area under $f(x)=\sqrt{16-x^{2}}$ from $-4$ to $4$ using a Riemann sum with $n=5$ . Compute the error.

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Problem 17210

Use the left endpoints of 4 subintervals to find the displacement for $v=\frac{1}{2t+4}$ over $0 \leq t \leq 8$ . Round to two decimal places.

See Solution

Problem 17211

Find points $c \in (a, b)$ for the mean value theorem for the given functions and intervals.

See Solution

Problem 17212

Estimate the area under $f(x)=\sqrt{9-x^{2}}$ from $-3$ to $3$ using a Riemann sum with $n=4$ midpoints. Error?

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Problem 17213

An object moves at $16 \mathrm{~m/s}$ for $0 \leq t \leq 2 \mathrm{~s}$ and $26 \mathrm{~m/s}$ for $2<t \leq 5 \mathrm{~s}$ . Find displacement for $0 \leq t \leq 5$ .

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Problem 17214

1. Find max/min points of $y=x^{3}-9 x^{2}+24 x$ .
2. Evaluate $\int \sin 3 x \sin x \, dx$ .
3. Evaluate $\int x^{4} \cos 2 x \, dx$ .
4. Evaluate $\int \frac{2 x^{3}-x^{3}-x}{2 x-3} \, dx$ .
5. Show $y=e^{2 x}$ satisfies $y^{\prime \prime}-2 y^{\prime}=0$ .

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Problem 17215

Approximate the displacement of an object with velocity $v = \frac{1}{3t+1}$ (m/s) from $t=0$ to $t=8$ using 4 subintervals. Displacement is $\square$ m.

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Problem 17216

Approximate the displacement of an object with velocity $v=\frac{1}{2t+4}$ from $t=0$ to $t=8$ using 4 subintervals.

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Problem 17217

Find the limit as $x$ approaches 25 for the expression $\frac{\sqrt{x}-5}{25-x}$ .

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Problem 17218

Find the limit as $h$ approaches 0 for $\frac{(h-2)^{2}-4}{h}$ .

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Problem 17219

Given $f(7)=3$ , $f^{\prime}(7)=10$ , $f^{\prime \prime}(7)=5$ , find $h^{\prime}(7)$ for $h(x)=\sqrt{1+f^{\prime}(x)^{2}}$ . Round to three decimal places.

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Problem 17220

Find $\frac{dy}{dx}$ at $(5,2)$ for $f(x)^{2}+y^{2}=6y-7f(x)$ , given $f(5)=1$ , $f'(5)=2$ . Round to three decimals.

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Problem 17221

Given $f(6)=5$ , $f^{\prime}(6)=6$ , and $f^{\prime \prime}(6)=5$ , find $h^{\prime}(6)$ for $h(x)=\sqrt{1+f^{\prime}(x)^{2}}$ . Round to three decimal places.

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Problem 17222

Find $K$ in the formula $\frac{d \omega}{d \theta}=K \frac{\cos (2 \theta) \sin (2 \theta)}{\sin (6 \omega)}$ from the equation $9 \cos (6 \omega)=8+9[\cos (2 \theta)]^{2}$ . Round $K$ to three decimal places.

See Solution

Problem 17223

Find marginal utilities for $u=x^{0.2} y^{0.8}$ and maximize utility with $4x + 2y = 180$ .

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Problem 17224

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{4^{x}-1}{2^{3+x}-8}$ .

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Problem 17225

Relation entre prix $p$ et quantité $q$ : $10 p^{2}+p q+\frac{q^{2}}{2}=C$ . Trouver $\frac{d p}{d q}$ pour $C=1000$ .

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Problem 17226

Evaluate the integral $\int_{3}^{+\infty} \frac{d x}{x \sqrt{1+x^{2}}}$ .

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Problem 17227

Find $h^{\prime}(5)$ for $h(x)=\sqrt{1+f^{\prime}(x)^{2}}$ given $f(5)=3$ , $f^{\prime}(5)=4$ , $f^{\prime \prime}(5)=8$ .

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Problem 17228

Find the derivative $\theta^{\prime}(n)$ of $\theta(n)=\tan^{-1}(0.42/n)$ and compute $\theta^{\prime}(0.42)$ . Round to three decimal places.

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Problem 17229

Find $\frac{d y}{d x}$ at $(4,2)$ for $f(x)^{2}+y^{2}=6 y-7 f(x)$ , given $f(4)=1$ , $f^{\prime}(4)=2.4$ . Round to three decimals.

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Problem 17230

Find the slope of the tangent line at $(1,1)$ for the curve $9 x^{2}+4 x y-9 y^{2}=4$ . Round to three decimal places.

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Problem 17231

Differentiate $3 \cos (2 \omega)=7+8[\cos (4 \theta)]^{2}$ to find $\frac{d \omega}{d \theta}$ and determine $K$ as a decimal.

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Problem 17232

Find $\frac{d y}{d x}$ for $y=9+\log_{10}(6/x)$ and evaluate at $x=0.4$ . Round to three decimal places.

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Problem 17233

Find the derivative $\theta^{\prime}(n)$ of $\theta(n)=\tan^{-1}(0.38/n)$ and compute $\theta^{\prime}(0.38)$ . Round to three decimal places.

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Problem 17234

Find $\frac{d y}{d x}$ at $(7,2)$ for $f(x)^{2}+y^{2}=6y-7f(x)$ , given $f(7)=1$ , $f'(7)=2.7$ .

See Solution

Problem 17235

Find $h^{\prime}(3)$ for $h(x)=\sqrt{1+f^{\prime}(x)^{2}}$ given $f(3)=3$ , $f^{\prime}(3)=2$ , $f^{\prime \prime}(3)=6$ .

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Problem 17236

Find $\frac{d y}{d x}$ for $y=9+\log_{10}(4/x)$ and evaluate at $x=0.3$ . Round to three decimal places.

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Problem 17237

Find the value of $K$ in the formula $\frac{d \omega}{d \theta}=K \frac{\cos (2 \theta) \sin (2 \theta)}{\sin (6 \omega)}$ from the equation $9 \cos (6 \omega)=8+9[\cos (2 \theta)]^{2}$ . Round $K$ to three decimal places.

See Solution

Problem 17238

Find $h^{\prime}(5)$ for $h(x)=\sqrt{1+f^{\prime}(x)^{2}}$ given $f(5)=8$ , $f^{\prime}(5)=2$ , $f^{\prime \prime}(5)=8$ . Round to three decimal places.

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Problem 17239

Find the derivative $\theta'(n)$ of $\theta(n) = \tan^{-1}(0.42/n)$ and compute $\theta'(0.42)$ .

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Problem 17240

Calculate the average rate of change of $f(x)=\sqrt{x}$ between $x=4$ and $x=36$ .

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Problem 17241

Find $h^{\prime}(7)$ if $h(x)=\sqrt{1+f^{\prime}(x)^{2}}$ with $f(7)=9$ , $f^{\prime}(7)=4$ , $f^{\prime \prime}(7)=6$ .

See Solution

Problem 17242

Find $\frac{d y}{d x}$ for $y=2+\log_{10}(2/x)$ and evaluate at $x=0.4$ . Round to three decimal places.

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Problem 17243

Find $\frac{d y}{d x}$ at $(10,2)$ given $f(10)=1$ , $f^{\prime}(10)=1.7$ , and $f(x)^{2}+y^{2}=6 y-7 f(x)$ . Round to three decimals.

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Problem 17244

Find the initial velocity and velocity after 5 seconds for $v(t)=52(1-e^{-0.19 t})$ in $\mathrm{m/s}$ .

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Problem 17245

Find the derivative $\theta^{\prime}(n)$ of $\theta(n)=\tan^{-1}(0.2/n)$ and compute $\theta^{\prime}(0.2)$ . Round to three decimals.

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Problem 17246

Given $f$ with $f(6)=5$ , $f^{\prime}(6)=6$ , and $f^{\prime \prime}(6)=5$ , find $h^{\prime}(6)$ for $h(x)=\sqrt{1+f^{\prime}(x)^{2}}$ . Round to three decimal places.

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Problem 17247

Find $\frac{d y}{d x}$ at $(4,2)$ given $f(4)=1$ , $f^{\prime}(4)=2.1$ , and $f(x)^{2}+y^{2}=6 y-7 f(x)$ . Round to three decimal places.

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Problem 17248

Find $h'(7)$ for $h(x)=\sqrt{1+f'(x)^{2}}$ given $f(7)=9$ , $f'(7)=4$ , $f''(7)=6$ . Round to three decimal places.

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Problem 17249

Find the initial population and the population after 8 years for $P(t)=\frac{790}{1+6 e^{-0.4 t}}$ . Round to whole numbers.

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Problem 17250

Find $\frac{d y}{d x}$ at $(10,2)$ for $f(x)^{2}+y^{2}=6 y-7 f(x)$ , given $f(10)=1$ and $f^{\prime}(10)=1.7$ .

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Problem 17251

Find the constant $K$ in the formula $\frac{d \omega}{d \theta}=K \cos (10 \theta) \sin (10 \theta) / \sin (4 \omega)$ from the equation $4 \cos (4 \omega)=6+7[\cos (10 \theta)]^{2}$ .

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Problem 17252

A company models price $p$ and demand $x$ as $p=1155-0.11 x^{2}$ and cost as $C(x)=800+384 x$ . Find profit $P(x)$ and analyze its maxima, minima, and concavity.

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Problem 17253

Find $\frac{dy}{dx}$ at $(10,2)$ for $f(x)^2 + y^2 = 6y - 7f(x)$ , given $f(10)=1$ and $f'(10)=1.7$ .

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Problem 17254

Invest \$16,563 at 6.1% interest, compounded continuously. Find the function, balances after 1, 2, 5, 10 years, and doubling time.

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Problem 17255

Find the constant $K$ in the formula $\frac{d \omega}{d \theta}=K \cos (2 \theta) \sin (2 \theta) / \sin (6 \omega)$ from the equation $9 \cos (6 \omega)=8+9[\cos (2 \theta)]^{2}$ . Round $K$ to three decimal places.

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Problem 17256

A red sports car's speed is $f(t)=\frac{44000}{(t+20)^{2}}$ for $0 \leq t \leq 6$ .
a. Explain why $\sum_{k=1}^{n} \frac{6}{n} f\left(\frac{6 k}{n}\right)$ estimates the distance in 6 seconds.
b. Compare $\sum_{k=0}^{n-1} \frac{6}{n} f\left(\frac{6 k}{n}\right)$ to the previous sum.
c. What does $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{6}{n} f\left(\frac{6 k}{n}\right)$ signify? What was its value from a previous exercise?

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Problem 17257

The demand for a fitness video game is given by $p=110-37.73\sqrt{x}$ , and cost is $C=-0.04x^2+10x+150$ .
(a) Write profit $P(x)$ as a function of $x$ .
(b) Find the production level for maximum profit and the maximum profit value.
(c) Determine the production level where profit decreases least rapidly.

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Problem 17258

Find the derivative $\theta^{\prime}(n)$ of $\theta(n)=\tan^{-1}(0.07/n)$ and calculate $\theta^{\prime}(0.07)$ . Round to three decimal places.

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Problem 17259

Define integrals for: a) distance with speed $f(t)$ from $t=0$ to $t=6$ ; b) time from $t=3$ to $t=5$ ; c) $\lim_{n \to \infty} \sum_{k=1}^{n} \sin\left(\frac{k \pi}{n}\right)\left(\frac{\pi}{n}\right)$ and $\lim_{n \to \infty} \sum_{k=1}^{n}\left(\frac{2k}{n}\right)^{2}\left(\frac{2}{n}\right)$ .

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Problem 17260

Use the Fundamental Theorem of Calculus to find the area under $y=9-x^{2}$ from $x=0$ to $x=3$ .

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Problem 17261

Evaluate the integral: $\int \ln \left(x^{9}\right) d x$ .

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Problem 17262

Evaluate the integral: $\int 3 x e^{5 x} dx =$ 圆园 $+ C$

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Problem 17263

Evaluate the integral: $\int \frac{1}{x \ln x} d x$ . What is the result?

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Problem 17264

Find the integral of $x(x^{2}+3)^{\frac{5}{4}} \, dx$ and include the constant $C$ .

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Problem 17265

Find the limits of $g(t)$ as $t$ approaches 0, 3, and 4 from both sides, where $g(t) = \begin{cases} t-4, & t<0 \\ t^{2}, & 0 \leq t \leq 4 \\ 4t, & t>4 \end{cases}$ .

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Problem 17266

Find the integral for $\lim_{n \to \infty} \sum_{n=1}^{n} \sin\left(\frac{k \pi}{n}\right)\left(\frac{\pi}{n}\right)$ and $\lim_{n \to \infty} \sum_{n=1}^{n}\left(\frac{2k}{n}\right)^{2}\left(\frac{2}{n}\right)$ . Apply the Fundamental Theorem to $y=9-x^{2}$ .

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Problem 17267

Calculate the integral $\int_{1}^{2}\left(8 x^{3}+\frac{1}{2 x^{2}}\right) d x$ .

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Problem 17268

Find the integral equivalents for: i) $\lim_{n \to \infty} \sum_{n=1}^{n} \sin\left(\frac{k \pi}{n}\right)\left(\frac{\pi}{n}\right)$ ii) $\lim_{n \to \infty} \sum_{n=1}^{n}\left(\frac{2k}{n}\right)^{2}\left(\frac{2}{n}\right)$ . Also, use the Fundamental Theorem of Calculus for the area under $y=9-x^{2}$ .

See Solution

Problem 17269

Calculate the integral $\int_{0}^{2} x^{2} dx$ using $n=4$ .

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Problem 17270

Calculate the integral from 1 to 4 of the function $\sqrt{t} \ln t$ .

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Problem 17271

Calculate the integral $\int_{0}^{8} \sqrt[3]{x} \, dx$ with $n=8$ .

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Problem 17272

Create a table for $f(x)=\frac{x-4}{\sqrt{x}-2}$ at $x=4.1,4.01,4.001,3.9,3.99,3.999$ to estimate $\lim_{x \to 4} f(x)$ and factor to find the limit.

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Problem 17273

Calculate the integral $\int_{0}^{2} x^{3} \, dx$ using $n=4$ .

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Problem 17274

Untersuche die Vergessenskurve $s(t)=\frac{15 t+95}{150 t+95} \cdot 100$ für $t=0, 1, 24$ und finde den Zeitpunkt für 50\% Vergessen. Bestimme den langfristigen Prozentsatz und skizziere den Graphen.

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Problem 17275

Evaluate the integral: $\int \cos^{6} x \sin x \, dx$

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Problem 17276

Approximate the integral of $f(x)=3x-x^2$ from $x=0$ to $x=3$ using $n=3$ .

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Problem 17277

Untersuche die Vergessenskurve $s(t)=\frac{15 t+95}{150 t+95} \cdot 100$ . Berechne $s(0)$ , $s(1)$ , $s(24)$ und wann $s(t)=50\%$ . Finde den Grenzwert für $t \to \infty$ und skizziere den Graphen.

See Solution

Problem 17278

Johanna trinkt Kaffee. Bestimme die Temperatur $T(0)$ , $T(5)$ , Wartezeit für $T=50^{\circ}C$ und Endtemperatur bei langer Zeit.

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Problem 17279

Bestimme die Steigung der Funktion $f(x)=e^{-0,5 x^{2}}$ bei $x_{0}=1$ .

See Solution

Problem 17280

Bestimme die Tangentengleichung von $f(x)=\sqrt{2x+2}$ bei $x_0=1$ .

See Solution

Problem 17281

Find the values of $f(-1)$ , $f^{\prime}(-1)$ , and $f^{\prime \prime}(-1)$ given inflection points at $x=-3.8, -1, -0.3, 3.2$ .

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Problem 17282

Evaluate the integral: $\int_{-\infty}^{\infty} \frac{9 x^{2}}{(x^{2}+2 x+2)(x^{2}+1)^{2}} dx$

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Problem 17283

Bestimmen Sie die Ableitung von $f(x)=\left(1-3 x^{4}\right)^{2}$ mit der Kettenregel.

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Problem 17284

Berechne die mittlere Steigung von $f(x)=x^{2}$ auf $[2, a]$ mit $a>2$ und bestimme $a$ , wenn die Steigung 6 ist.

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Problem 17285

Gegeben ist $f(x)=x^{3}+3 x^{2}-4$ . Finde den Wendepunkt, die Wendetangente, die Länge $\overline{\mathrm{PQ}}$ und den Flächeninhalt des Dreiecks.

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Problem 17286

Calculate the integral $A_{1}=\int_{-1}^{0} x^{3} dx$ .

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Problem 17287

Finde die Extrempunkte und deren Art für die Funktion $m(x)=x^{4}-4 x^{2}+3$ .

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Problem 17288

Finde die Extrempunkte und deren Art für die Funktion $f(x)=x^{3}-3x$ . Berechne $f'(x)$ und $f''(x)$ .

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Problem 17289

Find the limit: $\lim _{n \rightarrow \infty} n^{2}\left(\sqrt[3]{n^{3}+1}-n\right)$ .

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Problem 17290

Bestimme die Ableitung für: a) $f(x)=1$ , b) $g(x)=2x-10$ , c) $h(x)=4x^{2}$ , d) $k(x)=3(x-1)^{2}$ .

See Solution

Problem 17291

Calculate the integral from 0 to 1 of $\frac{1}{1+x^{2}}$ with respect to $x$ .

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Problem 17292

Bestimmen Sie für $f(x)=\sin(x)$ im Intervall $[-\pi; \pi]$ : a) Steigung bei $\pi$ , b) Stellen mit Steigung 1, c) Steigungswinkel bei 0, d) Stellen mit $20^{\circ}$ , e) x-Werte für monoton steigendes Verhalten.

See Solution

Problem 17293

Finde die Extrempunkte und deren Art für die Funktion $h(x)=x^{3}-3 x^{2}+1$ .

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Problem 17294

Bestimme die erste Ableitung der Funktion $f(x)=\frac{1}{4} x^{4}-\frac{1}{4} x^{3}-\frac{35}{16} x^{2}$ .

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Problem 17295

Evaluate the integral $\int_{0}^{1} \frac{1}{1+x^{2}} d x$ using Simpson's rule.

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Problem 17296

Gegeben ist die Funktion $f_{a}(x)=-a x^{3}+4 a x$ .
a) Zeigen Sie die Punktsymmetrie zum Ursprung. b) Beweisen Sie, dass die Punkte $P(-2,0)$ und $Q(2,0)$ auf den Graphen liegen. c) Nachweisen, dass es einen Hoch- und einen Tiefpunkt gibt. d) Bestimmen Sie die Wendetangente und den Wert von $a$ für die Steigung $m=8$ .

See Solution

Problem 17297

Approximate the integral $\int_{0}^{1} \frac{1}{1+x^{2}} dx$ using Simpson's Rule with an even number of intervals (n).

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Problem 17298

Approximate $\int_{0}^{2} \frac{1}{16+x^{2}} dx$ using the Trapezoidal rule with 6 subintervals.

See Solution

Problem 17299

Finde die Tangenten- und Normalengleichung an $f(x)=x^3-3x^2$ bei $x=0,5$ .

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Problem 17300

Ein Objekt fällt aus $140 \mathrm{~m}$ Höhe. Bestimme die Zeit, bis es $20 \mathrm{~m}$ und $70 \mathrm{~m}$ erreicht, und berechne die Durchschnittsgeschwindigkeit.

See Solution

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Calculus

Problem 17201

Find an equation of a curve that intersects every curve of the family y=1x+ky=\frac{1}{x}+ky=x1​+k at right angles. Options: (A) y=−xy=-xy=−x (B) y=−x2y=-x^{2}y=−x2 (C) y=−13x3y=-\frac{1}{3} x^{3}y=−31​x3 (D) y=13x3y=\frac{1}{3} x^{3}y=31​x3 (E) y=ln⁡xy=\ln xy=lnx

Problem 17202

A 10 ft long trough has triangular ends (3 ft wide, 1 ft high). Water fills at 12ft3/min12 \mathrm{ft}^{3} / \mathrm{min}12ft3/min. Find the rise rate at 6 in deep.

Problem 17203

Calculate the limit: lim⁡n→∞(1+n2n+1)2n\lim _{n \rightarrow \infty}\left(1+\frac{\sqrt{n}}{2^{n+1}}\right)^{2 n}limn→∞​(1+2n+1n​​)2n.

Problem 17204

Calculate the limit: lim⁡n→∞(1+π2n+1)2n\lim _{n \rightarrow \infty}\left(1+\frac{\pi}{2^{n+1}}\right)^{2 n}limn→∞​(1+2n+1π​)2n.

Problem 17205

Evaluate the left and right Riemann sums for n=10n=10n=10 over [0,5][0,5][0,5] using the values of f(x)f(x)f(x) provided.

Problem 17206

Estimate the alternating harmonic series using the first 10 terms for S10S_{10}S10​, then show how to rearrange 2∑n=1∞(−1)n+1n2 \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}2∑n=1∞​n(−1)n+1​.

Problem 17207

A balloon 150 ft away rises at 8 ft/s. How fast is the distance to the balloon increasing when it's 50 ft high?

Problem 17208

Problem 17209

Estimate the area under f(x)=16−x2f(x)=\sqrt{16-x^{2}}f(x)=16−x2​ from −4-4−4 to 444 using a Riemann sum with n=5n=5n=5. Compute the error.

Problem 17210

Use the left endpoints of 4 subintervals to find the displacement for v=12t+4v=\frac{1}{2t+4}v=2t+41​ over 0≤t≤80 \leq t \leq 80≤t≤8. Round to two decimal places.

Problem 17211

Find points c∈(a,b)c \in (a, b)c∈(a,b) for the mean value theorem for the given functions and intervals.

Problem 17212

Estimate the area under f(x)=9−x2f(x)=\sqrt{9-x^{2}}f(x)=9−x2​ from −3-3−3 to 333 using a Riemann sum with n=4n=4n=4 midpoints. Error?

Problem 17213

An object moves at 16 m/s16 \mathrm{~m/s}16 m/s for 0≤t≤2 s0 \leq t \leq 2 \mathrm{~s}0≤t≤2 s and 26 m/s26 \mathrm{~m/s}26 m/s for 2<t≤5 s2<t \leq 5 \mathrm{~s}2<t≤5 s. Find displacement for 0≤t≤50 \leq t \leq 50≤t≤5.

Problem 17214

Problem 17215

Approximate the displacement of an object with velocity v=13t+1v = \frac{1}{3t+1}v=3t+11​ (m/s) from t=0t=0t=0 to t=8t=8t=8 using 4 subintervals. Displacement is □\square□ m.

Problem 17216

Approximate the displacement of an object with velocity v=12t+4v=\frac{1}{2t+4}v=2t+41​ from t=0t=0t=0 to t=8t=8t=8 using 4 subintervals.

Problem 17217

Find the limit as xxx approaches 25 for the expression x−525−x\frac{\sqrt{x}-5}{25-x}25−xx​−5​.

Problem 17218

Find the limit as hhh approaches 0 for (h−2)2−4h\frac{(h-2)^{2}-4}{h}h(h−2)2−4​.

Problem 17219

Given f(7)=3f(7)=3f(7)=3, f′(7)=10f^{\prime}(7)=10f′(7)=10, f′′(7)=5f^{\prime \prime}(7)=5f′′(7)=5, find h′(7)h^{\prime}(7)h′(7) for h(x)=1+f′(x)2h(x)=\sqrt{1+f^{\prime}(x)^{2}}h(x)=1+f′(x)2​. Round to three decimal places.

Problem 17220

Find dydx\frac{dy}{dx}dxdy​ at (5,2)(5,2)(5,2) for f(x)2+y2=6y−7f(x)f(x)^{2}+y^{2}=6y-7f(x)f(x)2+y2=6y−7f(x), given f(5)=1f(5)=1f(5)=1, f′(5)=2f'(5)=2f′(5)=2. Round to three decimals.

Problem 17221

Given f(6)=5f(6)=5f(6)=5, f′(6)=6f^{\prime}(6)=6f′(6)=6, and f′′(6)=5f^{\prime \prime}(6)=5f′′(6)=5, find h′(6)h^{\prime}(6)h′(6) for h(x)=1+f′(x)2h(x)=\sqrt{1+f^{\prime}(x)^{2}}h(x)=1+f′(x)2​. Round to three decimal places.

Problem 17222

Problem 17223

Find marginal utilities for u=x0.2y0.8u=x^{0.2} y^{0.8}u=x0.2y0.8 and maximize utility with 4x+2y=1804x + 2y = 1804x+2y=180.

Problem 17224

Evaluate the limit: lim⁡x→04x−123+x−8\lim _{x \rightarrow 0} \frac{4^{x}-1}{2^{3+x}-8}limx→0​23+x−84x−1​.

Problem 17225

Relation entre prix ppp et quantité qqq: 10p2+pq+q22=C10 p^{2}+p q+\frac{q^{2}}{2}=C10p2+pq+2q2​=C. Trouver dpdq\frac{d p}{d q}dqdp​ pour C=1000C=1000C=1000.

Problem 17226

Evaluate the integral ∫3+∞dxx1+x2\int_{3}^{+\infty} \frac{d x}{x \sqrt{1+x^{2}}}∫3+∞​x1+x2​dx​.

Problem 17227

Find h′(5)h^{\prime}(5)h′(5) for h(x)=1+f′(x)2h(x)=\sqrt{1+f^{\prime}(x)^{2}}h(x)=1+f′(x)2​ given f(5)=3f(5)=3f(5)=3, f′(5)=4f^{\prime}(5)=4f′(5)=4, f′′(5)=8f^{\prime \prime}(5)=8f′′(5)=8.

Problem 17228

Find the derivative θ′(n)\theta^{\prime}(n)θ′(n) of θ(n)=tan⁡−1(0.42/n)\theta(n)=\tan^{-1}(0.42/n)θ(n)=tan−1(0.42/n) and compute θ′(0.42)\theta^{\prime}(0.42)θ′(0.42). Round to three decimal places.

Problem 17229

Find dydx\frac{d y}{d x}dxdy​ at (4,2)(4,2)(4,2) for f(x)2+y2=6y−7f(x)f(x)^{2}+y^{2}=6 y-7 f(x)f(x)2+y2=6y−7f(x), given f(4)=1f(4)=1f(4)=1, f′(4)=2.4f^{\prime}(4)=2.4f′(4)=2.4. Round to three decimals.

Problem 17230

Find the slope of the tangent line at (1,1)(1,1)(1,1) for the curve 9x2+4xy−9y2=49 x^{2}+4 x y-9 y^{2}=49x2+4xy−9y2=4. Round to three decimal places.

Problem 17231

Differentiate 3cos⁡(2ω)=7+8[cos⁡(4θ)]23 \cos (2 \omega)=7+8[\cos (4 \theta)]^{2}3cos(2ω)=7+8[cos(4θ)]2 to find dωdθ\frac{d \omega}{d \theta}dθdω​ and determine KKK as a decimal.

Problem 17232

Find dydx\frac{d y}{d x}dxdy​ for y=9+log⁡10(6/x)y=9+\log_{10}(6/x)y=9+log10​(6/x) and evaluate at x=0.4x=0.4x=0.4. Round to three decimal places.

Problem 17233

Find the derivative θ′(n)\theta^{\prime}(n)θ′(n) of θ(n)=tan⁡−1(0.38/n)\theta(n)=\tan^{-1}(0.38/n)θ(n)=tan−1(0.38/n) and compute θ′(0.38)\theta^{\prime}(0.38)θ′(0.38). Round to three decimal places.

Problem 17234

Find dydx\frac{d y}{d x}dxdy​ at (7,2)(7,2)(7,2) for f(x)2+y2=6y−7f(x)f(x)^{2}+y^{2}=6y-7f(x)f(x)2+y2=6y−7f(x), given f(7)=1f(7)=1f(7)=1, f′(7)=2.7f'(7)=2.7f′(7)=2.7.

Problem 17235

Find h′(3)h^{\prime}(3)h′(3) for h(x)=1+f′(x)2h(x)=\sqrt{1+f^{\prime}(x)^{2}}h(x)=1+f′(x)2​ given f(3)=3f(3)=3f(3)=3, f′(3)=2f^{\prime}(3)=2f′(3)=2, f′′(3)=6f^{\prime \prime}(3)=6f′′(3)=6.

Problem 17236

Find dydx\frac{d y}{d x}dxdy​ for y=9+log⁡10(4/x)y=9+\log_{10}(4/x)y=9+log10​(4/x) and evaluate at x=0.3x=0.3x=0.3. Round to three decimal places.

Problem 17237

Problem 17238

Find h′(5)h^{\prime}(5)h′(5) for h(x)=1+f′(x)2h(x)=\sqrt{1+f^{\prime}(x)^{2}}h(x)=1+f′(x)2​ given f(5)=8f(5)=8f(5)=8, f′(5)=2f^{\prime}(5)=2f′(5)=2, f′′(5)=8f^{\prime \prime}(5)=8f′′(5)=8. Round to three decimal places.

Problem 17239

Find the derivative θ′(n)\theta'(n)θ′(n) of θ(n)=tan⁡−1(0.42/n)\theta(n) = \tan^{-1}(0.42/n)θ(n)=tan−1(0.42/n) and compute θ′(0.42)\theta'(0.42)θ′(0.42).

Problem 17240

Calculate the average rate of change of f(x)=xf(x)=\sqrt{x}f(x)=x​ between x=4x=4x=4 and x=36x=36x=36.

Problem 17241

Find h′(7)h^{\prime}(7)h′(7) if h(x)=1+f′(x)2h(x)=\sqrt{1+f^{\prime}(x)^{2}}h(x)=1+f′(x)2​ with f(7)=9f(7)=9f(7)=9, f′(7)=4f^{\prime}(7)=4f′(7)=4, f′′(7)=6f^{\prime \prime}(7)=6f′′(7)=6.

Problem 17242

Find an equation of a curve that intersects every curve of the family $y=\frac{1}{x}+k$ at right angles. Options: (A) $y=-x$ (B) $y=-x^{2}$ (C) $y=-\frac{1}{3} x^{3}$ (D) $y=\frac{1}{3} x^{3}$ (E) $y=\ln x$

A 10 ft long trough has triangular ends (3 ft wide, 1 ft high). Water fills at $12 \mathrm{ft}^{3} / \mathrm{min}$ . Find the rise rate at 6 in deep.

Calculate the limit: $\lim _{n \rightarrow \infty}\left(1+\frac{\sqrt{n}}{2^{n+1}}\right)^{2 n}$ .

Calculate the limit: $\lim _{n \rightarrow \infty}\left(1+\frac{\pi}{2^{n+1}}\right)^{2 n}$ .

Evaluate the left and right Riemann sums for $n=10$ over $[0,5]$ using the values of $f(x)$ provided.

Estimate the alternating harmonic series using the first 10 terms for $S_{10}$ , then show how to rearrange $2 \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ .

Estimate the area under $f(x)=\sqrt{16-x^{2}}$ from $-4$ to $4$ using a Riemann sum with $n=5$ . Compute the error.

Use the left endpoints of 4 subintervals to find the displacement for $v=\frac{1}{2t+4}$ over $0 \leq t \leq 8$ . Round to two decimal places.

Find points $c \in (a, b)$ for the mean value theorem for the given functions and intervals.

Estimate the area under $f(x)=\sqrt{9-x^{2}}$ from $-3$ to $3$ using a Riemann sum with $n=4$ midpoints. Error?

An object moves at $16 \mathrm{~m/s}$ for $0 \leq t \leq 2 \mathrm{~s}$ and $26 \mathrm{~m/s}$ for $2<t \leq 5 \mathrm{~s}$ . Find displacement for $0 \leq t \leq 5$ .

Approximate the displacement of an object with velocity $v = \frac{1}{3t+1}$ (m/s) from $t=0$ to $t=8$ using 4 subintervals. Displacement is $\square$ m.

Approximate the displacement of an object with velocity $v=\frac{1}{2t+4}$ from $t=0$ to $t=8$ using 4 subintervals.

Find the limit as $x$ approaches 25 for the expression $\frac{\sqrt{x}-5}{25-x}$ .

Find the limit as $h$ approaches 0 for $\frac{(h-2)^{2}-4}{h}$ .

Given $f(7)=3$ , $f^{\prime}(7)=10$ , $f^{\prime \prime}(7)=5$ , find $h^{\prime}(7)$ for $h(x)=\sqrt{1+f^{\prime}(x)^{2}}$ . Round to three decimal places.

Find $\frac{dy}{dx}$ at $(5,2)$ for $f(x)^{2}+y^{2}=6y-7f(x)$ , given $f(5)=1$ , $f'(5)=2$ . Round to three decimals.

Given $f(6)=5$ , $f^{\prime}(6)=6$ , and $f^{\prime \prime}(6)=5$ , find $h^{\prime}(6)$ for $h(x)=\sqrt{1+f^{\prime}(x)^{2}}$ . Round to three decimal places.

Find marginal utilities for $u=x^{0.2} y^{0.8}$ and maximize utility with $4x + 2y = 180$ .

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{4^{x}-1}{2^{3+x}-8}$ .

Relation entre prix $p$ et quantité $q$ : $10 p^{2}+p q+\frac{q^{2}}{2}=C$ . Trouver $\frac{d p}{d q}$ pour $C=1000$ .

Evaluate the integral $\int_{3}^{+\infty} \frac{d x}{x \sqrt{1+x^{2}}}$ .

Find $h^{\prime}(5)$ for $h(x)=\sqrt{1+f^{\prime}(x)^{2}}$ given $f(5)=3$ , $f^{\prime}(5)=4$ , $f^{\prime \prime}(5)=8$ .

Find the derivative $\theta^{\prime}(n)$ of $\theta(n)=\tan^{-1}(0.42/n)$ and compute $\theta^{\prime}(0.42)$ . Round to three decimal places.

Find $\frac{d y}{d x}$ at $(4,2)$ for $f(x)^{2}+y^{2}=6 y-7 f(x)$ , given $f(4)=1$ , $f^{\prime}(4)=2.4$ . Round to three decimals.

Find the slope of the tangent line at $(1,1)$ for the curve $9 x^{2}+4 x y-9 y^{2}=4$ . Round to three decimal places.

Differentiate $3 \cos (2 \omega)=7+8[\cos (4 \theta)]^{2}$ to find $\frac{d \omega}{d \theta}$ and determine $K$ as a decimal.

Find $\frac{d y}{d x}$ for $y=9+\log_{10}(6/x)$ and evaluate at $x=0.4$ . Round to three decimal places.

Find the derivative $\theta^{\prime}(n)$ of $\theta(n)=\tan^{-1}(0.38/n)$ and compute $\theta^{\prime}(0.38)$ . Round to three decimal places.

Find $\frac{d y}{d x}$ at $(7,2)$ for $f(x)^{2}+y^{2}=6y-7f(x)$ , given $f(7)=1$ , $f'(7)=2.7$ .

Find $h^{\prime}(3)$ for $h(x)=\sqrt{1+f^{\prime}(x)^{2}}$ given $f(3)=3$ , $f^{\prime}(3)=2$ , $f^{\prime \prime}(3)=6$ .

Find $\frac{d y}{d x}$ for $y=9+\log_{10}(4/x)$ and evaluate at $x=0.3$ . Round to three decimal places.

Find $h^{\prime}(5)$ for $h(x)=\sqrt{1+f^{\prime}(x)^{2}}$ given $f(5)=8$ , $f^{\prime}(5)=2$ , $f^{\prime \prime}(5)=8$ . Round to three decimal places.

Find the derivative $\theta'(n)$ of $\theta(n) = \tan^{-1}(0.42/n)$ and compute $\theta'(0.42)$ .

Calculate the average rate of change of $f(x)=\sqrt{x}$ between $x=4$ and $x=36$ .

Find $h^{\prime}(7)$ if $h(x)=\sqrt{1+f^{\prime}(x)^{2}}$ with $f(7)=9$ , $f^{\prime}(7)=4$ , $f^{\prime \prime}(7)=6$ .

Find $\frac{d y}{d x}$ for $y=2+\log_{10}(2/x)$ and evaluate at $x=0.4$ . Round to three decimal places.

Find $\frac{d y}{d x}$ at $(10,2)$ given $f(10)=1$ , $f^{\prime}(10)=1.7$ , and $f(x)^{2}+y^{2}=6 y-7 f(x)$ . Round to three decimals.

Find the initial velocity and velocity after 5 seconds for $v(t)=52(1-e^{-0.19 t})$ in $\mathrm{m/s}$ .

Find the derivative $\theta^{\prime}(n)$ of $\theta(n)=\tan^{-1}(0.2/n)$ and compute $\theta^{\prime}(0.2)$ . Round to three decimals.

Given $f$ with $f(6)=5$ , $f^{\prime}(6)=6$ , and $f^{\prime \prime}(6)=5$ , find $h^{\prime}(6)$ for $h(x)=\sqrt{1+f^{\prime}(x)^{2}}$ . Round to three decimal places.

Find $\frac{d y}{d x}$ at $(4,2)$ given $f(4)=1$ , $f^{\prime}(4)=2.1$ , and $f(x)^{2}+y^{2}=6 y-7 f(x)$ . Round to three decimal places.

Find $h'(7)$ for $h(x)=\sqrt{1+f'(x)^{2}}$ given $f(7)=9$ , $f'(7)=4$ , $f''(7)=6$ . Round to three decimal places.

Find the initial population and the population after 8 years for $P(t)=\frac{790}{1+6 e^{-0.4 t}}$ . Round to whole numbers.

Find $\frac{d y}{d x}$ at $(10,2)$ for $f(x)^{2}+y^{2}=6 y-7 f(x)$ , given $f(10)=1$ and $f^{\prime}(10)=1.7$ .

A company models price $p$ and demand $x$ as $p=1155-0.11 x^{2}$ and cost as $C(x)=800+384 x$ . Find profit $P(x)$ and analyze its maxima, minima, and concavity.

Find $\frac{dy}{dx}$ at $(10,2)$ for $f(x)^2 + y^2 = 6y - 7f(x)$ , given $f(10)=1$ and $f'(10)=1.7$ .

Find the derivative $\theta^{\prime}(n)$ of $\theta(n)=\tan^{-1}(0.07/n)$ and calculate $\theta^{\prime}(0.07)$ . Round to three decimal places.

Use the Fundamental Theorem of Calculus to find the area under $y=9-x^{2}$ from $x=0$ to $x=3$ .

Evaluate the integral: $\int \ln \left(x^{9}\right) d x$ .

Evaluate the integral: $\int 3 x e^{5 x} dx =$ 圆园 $+ C$

Evaluate the integral: $\int \frac{1}{x \ln x} d x$ . What is the result?

Find the integral of $x(x^{2}+3)^{\frac{5}{4}} \, dx$ and include the constant $C$ .

Find the limits of $g(t)$ as $t$ approaches 0, 3, and 4 from both sides, where $g(t) = \begin{cases} t-4, & t<0 \\ t^{2}, & 0 \leq t \leq 4 \\ 4t, & t>4 \end{cases}$ .

Calculate the integral $\int_{1}^{2}\left(8 x^{3}+\frac{1}{2 x^{2}}\right) d x$ .

Calculate the integral $\int_{0}^{2} x^{2} dx$ using $n=4$ .

Calculate the integral from 1 to 4 of the function $\sqrt{t} \ln t$ .

Calculate the integral $\int_{0}^{8} \sqrt[3]{x} \, dx$ with $n=8$ .

Create a table for $f(x)=\frac{x-4}{\sqrt{x}-2}$ at $x=4.1,4.01,4.001,3.9,3.99,3.999$ to estimate $\lim_{x \to 4} f(x)$ and factor to find the limit.

Calculate the integral $\int_{0}^{2} x^{3} \, dx$ using $n=4$ .

Untersuche die Vergessenskurve $s(t)=\frac{15 t+95}{150 t+95} \cdot 100$ für $t=0, 1, 24$ und finde den Zeitpunkt für 50\% Vergessen. Bestimme den langfristigen Prozentsatz und skizziere den Graphen.