Calculus

Problem 9601

An object drops from $159 \mathrm{ft}$ ; height $s=159-16t^2$ . Find velocity, time to hit ground, and impact velocity.

See Solution

Problem 9602

Minimize the cost of a rectangular container with a square base and volume $2816 \mathrm{ft}^{3}$ , given material costs.

See Solution

Problem 9603

Estimate the solution to $e^{4 x}+x-2=0$ using Newton's method with initial guess $x_{0}=0$ for one iteration.

See Solution

Problem 9604

Find the approximate value of $x_{0}$ using the tangent line at $(6,7)$ for the function $f$ with given properties.

See Solution

Problem 9605

Find the next approximation $x_{1}$ using Newton's method for $f(x)=0$ with initial guess $x_{0}=4$ and tangent line $y=3x-6$ .

See Solution

Problem 9606

Show that $f(x)=2 x^{3}-3$ has a zero in $[1,2]$ using the Intermediate Value Theorem and apply 3 bisection steps for an estimate.

See Solution

Problem 9607

Design a rectangular container with a square base and volume $50000 \mathrm{ft}^{3}$ . Minimize costs: top at \$6/ft², sides at \$10/ft², bottom at \$2/ft². Find dimensions.

See Solution

Problem 9608

Find if $\lim_{x \rightarrow 0} \frac{\sin^{-1}(2x)}{\tan^{-1}(5x)}$ exists, and if so, determine its value.

See Solution

Problem 9609

Zeigen Sie, dass $F(x)$ eine Stammfunktion von $f$ ist. Berechnen Sie die Fläche zwischen $-2$ und $0$ von $f$ und der $x$ -Achse.

See Solution

Problem 9610

Find the limit as $\theta$ approaches 0: $\lim _{\theta \rightarrow 0} \frac{\tan \theta}{\theta^{2} \cot 3 \theta}$ .

See Solution

Problem 9611

Find the maximum sales per day for the function $s(d)=d^{3}+10$ on the interval $0<d \leq 3$ .

See Solution

Problem 9612

Gegeben ist $f(x)=\frac{1}{2} x^{2}$ . Bestimme: a) mittlere Änderungsrate auf $[0; 2]$ , b) lokale Änderungsrate bei $x_{0}=-1$ , c) lokale Änderungsrate bei $x_{0}=2$ .

See Solution

Problem 9613

Find the absolute extrema of $f(x)=2 x^{3}-15 x^{2}-84 x$ on $[-6,9]$ . Provide $(x, f(x))$ as an ordered pair.

See Solution

Problem 9614

Find values of $x$ where the rate of change of $f(x)=-3(x-5)(x-2)$ is decreasing.

See Solution

Problem 9615

Is the function $f(x)=\left\{\begin{array}{ll} x, & -1 \leq x<0 \\ \tan x, & 0 \leq x \leq \pi / 4 \end{array}\right.$ continuous and differentiable at $x=0$ ?

See Solution

Problem 9616

Find the average cost function for $C(x)=2 x^{2}+54 x+98$ and determine its minimum value per unit.

See Solution

Problem 9617

Find the highest point on the intersection of the cone $x^{2}+y^{2}-z^{2}=0$ and the plane $x+2z=4$ using Lagrange multipliers.

See Solution

Problem 9618

Find $y' (2)$ for $y=\log _{6}\left(x^{4}-4 x^{3}+1\right)^{5}+4^{x^{2}}-1200 x$ . Round to 3 decimal places.

See Solution

Problem 9619

Bestimme den Wert von a, sodass die Aussage $\int_{0}^{2} e^{2 x} d x=\frac{e-1}{2}$ wahr ist.

See Solution

Problem 9620

Find the tangent line equation for the function $f(x)=x^{2}+8$ at the point where $x=-5$ .

See Solution

Problem 9621

Determine conclusions from the Intermediate Value Theorem for $f(x)$ on $(-7,0)$ , given $f(-7)=5$ and $f(0)=2$ .

See Solution

Problem 9622

Determine conclusions from the Intermediate Value Theorem for $f(x)$ on $(-4,2)$ given $f(-4)=-5$ and $f(2)=-3$ .

See Solution

Problem 9623

Untersuchen Sie, ob $F(x)$ und $F^{}(x)$ zu $f(x)$ gehören. Geben Sie $g g f . f(x)$ an.
a) $F(x)=2 e^{x}(e^{x}+1)$ ; $F^{}(x)=e^{2 x}+2 e^{x}+4$ b) $F(x)=(x+1)^{3}$ ; $F^{\prime \prime}(x)=x^{3}+3 x(x+1)$ c) $F(x)=\sin x$ ; $F^{}(x)=\cos (x-\pi)$ d) $F(x)=x^{2}+\frac{1}{x^{2}}$ ; $F^{}(x)=\frac{(x^{2}-1)^{2}}{x^{2}}$

See Solution

Problem 9624

Determine conclusions using the Intermediate Value Theorem for $f(x)$ on $(-6,0)$ , given $f(-6)=3$ and $f(0)=-3$ .

See Solution

Problem 9625

Find the first derivative of the vehicle's acceleration function: $a(t) = (-t^{2}+6t) \cdot e^{-t}$ .

See Solution

Problem 9626

Determine where the function $f(x)=(x-7)e^{-9x}$ is increasing, decreasing, and find its local extrema.

See Solution

Problem 9627

Estimate the distance (in feet) a braking car travels using a midpoint sum with six equal intervals from the given velocity graph.

See Solution

Problem 9628

Berechnen Sie die Ableitungen der Funktionen: $f_{t}(x)=\frac{-1}{t^{2} x^{2}}$ , $f_{a}(x)=(3 a^{2} x^{3}-x^{2})(-2 x)$ und $f_{t}(x)=t^{-1}+2 x-\sqrt{t x^{\pi}}$ . Bestimmen Sie Nullstellen und Tangente von $f_{a}(x)=x^{3}-ax$ .

See Solution

Problem 9629

Find lower and upper estimates for $\int_{10}^{30} f(x) dx$ using the table values with five equal subintervals.

See Solution

Problem 9630

Solve the initial value problem: $y^{(4)} - 5y''' + 4y'' = 3x$ , with $y(0)=0$ , $y'(0)=0$ , $y''(0)=0$ , $y'''(0)=0$ .

See Solution

Problem 9631

Approximate the integral $\int_{1}^{3} \frac{x}{x^{2}+3} d x$ using the Midpoint Rule with $n=5$ . Round to four decimal places.

See Solution

Problem 9632

Show that the polynomial $f(x)=x^{3}+x^{2}-2x+5$ has a real zero between -3 and -1 using the Intermediate Value Theorem.

See Solution

Problem 9633

Solve the IVP: $\frac{d y}{d x}=\frac{x \sin x}{y}$ with initial condition $y(0)=-1$ .

See Solution

Problem 9634

Find the derivative $p^{\prime}(t)$ of $p(t)=137 t^{2}+1,080 t+14,911$ . Then, calculate consumption for 2026 and its rate of change.

See Solution

Problem 9635

Find the volume using cylindrical shells for the region bounded by $y=x^{4}$ , $y=0$ , $x=1$ rotated about $x=2$ .

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Problem 9636

Find the limit of the sequence $a_{n}=\frac{1-n}{1-2 \sqrt{n}}$ or prove it diverges.

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Problem 9637

Find the limit of these sequences if they converge; otherwise, prove divergence: a) $1/n$ , c) $a_n=\frac{1-n}{1-2\sqrt{n}}$ .

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Problem 9638

Find the second derivative $f^{\prime \prime}(x)$ for the function $f(x)=(x^{2}+1)^{5}$ .

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Problem 9639

Invest \$25000 at 4\% compounded continuously. (a) Find value after 5 years. (b) When will it reach \$48000?

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Problem 9640

Find the $x$ and $y$ coordinates of all inflection points for the function $f(x)=x^{3}+39x^{2}$ .

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Problem 9641

Differentiate the function: $f(t)=t e^{t} \cot (t)$ and find $f^{\prime}(t)$ .

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Problem 9642

Find the limit of $\frac{1-n}{1-2 \sqrt{n}}$ as $n$ approaches infinity.

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Problem 9643

Find the $x$ and $y$ coordinates of all inflection points for the function $f(x)=x^{3}+27 x^{2}$ .

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Problem 9644

Find the derivative of $y=2 e^{x} \sin (x)$ using the product and chain rules.

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Problem 9645

Invest \$25000 at 4\% interest compounded continuously. (a) Find value after 9 years. (b) When will it reach \$48000?

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Problem 9646

Find the average rate of change of the function $f(x)=3 x^{2}+4$ on the interval $[2, 2+h]$ .

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Problem 9647

Find the derivative of $j(y) = 31y^6 - 10\frac{1}{y^9}$ .

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Problem 9648

Find the derivative of $F(x)=\frac{x^{3}}{\cos (x)+x}$ .

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Problem 9649

Invest \$10000 at 5\% continuous compounding. (a) Find value after 9 years. (b) When will it reach \$24000?

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Problem 9650

Find the derivative of $k(x)=x^{11}-0.2 x^{10}+\frac{\pi}{3} x^{2}+\pi^{2}$ .

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Problem 9651

Find the limit of the sequence $a_{n}=\frac{1+\sqrt[99]{n}}{1+\sqrt[100]{n}}$ or prove it diverges.

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Problem 9652

Find the area represented by the integral $\int_{-3}^{8}(10-5 x) d x$ .

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Problem 9653

Find the area represented by the integral $\int_{-8}^{0}\left(5+\sqrt{64-x^{2}}\right) d x$ .

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Problem 9654

Determine the asymptotes for the function $f(x)=\frac{x^{2}-1}{\sqrt{x+1}}$ .

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Problem 9655

Find $\int_{1}^{5} f(x) d x$ given $\int_{1}^{7} f(x) d x=8.2$ and $\int_{5}^{7} f(x) d x=3.7$ .

See Solution

Problem 9656

Find the third derivative of $f(x)=\cos ^{2} x$ . Options: A. $-\cos ^{2} x$ , B. $2 \sin 2 x$ , C. $8 \sin x \cos x$ , D. $\sin 2 x$ , E. None.

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Problem 9657

Find $f^{\prime \prime}(\sqrt{e})$ if $f(x)=(\ln x)^{2}$ .

See Solution

Problem 9658

Die Firma Meier verkauft Schokolade. Gegeben ist die Funktion $f(t)=-0,0001 t^{3}+0,15 t^{2}+15 t$ für $0 \leq t \leq 1500$ .
a. Verkaufszahlen nach 700 Tagen? b. An welchem Tag die meisten Schokoladen verkauft werden? c. Wann ist die Zunahme der täglichen Verkaufszahlen am größten?

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Problem 9659

Find the limit as $\theta$ approaches $\frac{\pi}{2}$ from the left of $(\tan \theta)^{\cos \theta}$ .

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Problem 9660

Berechne die Ableitung von $f(x)=\frac{1}{8} x^{2}(x+3)(x-2)$ .

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Problem 9661

Find the derivative $\frac{d}{d x}(f(\ln x))$ where $f(x)=x^{2}+2 x$ .

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Problem 9662

Find the derivative of the function $f(x)=\ln \sqrt{x^{2}+1}$ . What is $f^{\prime}(x)$ ?

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Problem 9663

Evaluate the limit: $\lim _{x \rightarrow \frac{\pi}{2}^{-}} \frac{\tan x}{\left(\frac{11}{2 x-\pi}\right)}$ using l'Hôpital's Rule.

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Problem 9664

Evaluate the limit: $\lim _{x \rightarrow \frac{\pi}{2}^{-}} \frac{\tan x}{\left(\frac{10}{2 x-\pi}\right)}$

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Problem 9665

The drug concentration $C$ is given by $C=\frac{35 \cdot t}{24+t^{2}}$ . Find $C$ after 2.5 hours, time to reach $0.5\%$ , and end behavior as $t \to \infty$ .

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Problem 9666

Find $f^{\prime}\left(\frac{\pi}{3}\right)$ if $f(x)=\sec ^{2} x$ . Choices: 4, $4 \sqrt{3}$ , $8 \sqrt{3}$ , $\sqrt{3}$ , 12.

See Solution

Problem 9667

Evaluate the limit: $\lim _{x \rightarrow-1} \frac{2 x^{3}-3 x^{2}-12 x-7}{3 x^{4}+3 x^{3}-4 x^{2}-5 x-1}$ . How to proceed? A. Direct substitution B. l'Hôpital's Rule once C. l'Hôpital's Rule multiple times D. Multiply by a unit fraction.

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Problem 9668

Find the difference quotient for $f(x)=\frac{4}{x}$ : 1) without simplification, 2) with simplification.

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Problem 9669

Find the rate of change of oil volume $V(t)=10(10-t)^{2}$ at $t=5$ minutes. Choices: -100, -50, -150, -200 litres/min.

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Problem 9670

Find the derivative of the function $f(x)=4 x^{3}+2 x^{2}$ and evaluate it at $x=3$ . What is $f^{\prime}(3)$ ?

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Problem 9671

Find the volume of a solid with cross-section discs of radius $e^{-x}$ for $1 \leq x < \infty$ . Round to four decimal places.

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Problem 9672

Find the volume of the solid defined by $0 \leq r \leq R$ , $0 \leq \theta \leq 2 \pi$ , $0 \leq z \leq \theta$ .

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Problem 9673

Evaluate the limit: $\lim _{v \rightarrow 7} \frac{v-6-\sqrt{v^{2}-48}}{v-7}$ using l'Hôpital's Rule.

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Problem 9674

Find $x$ where the slope of the tangent line for $y=x^{3}+6x^{2}$ equals 36.

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Problem 9675

Find the max and min of $f(x)=\frac{x-7}{x+5}$ on the interval $(-5,1]$ .

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Problem 9676

Find the Linearization of $y=(16+x)^{-1/2}$ at $x=9$ .

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Problem 9677

Solve the equation $x^{2} y + 4 y = 8$ . Find $\frac{d y}{d y}$ in terms of $x$ and $y$ .

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Problem 9678

Evaluate the limit:
$\lim _{v \rightarrow 6} \frac{v-4-\sqrt{v^{2}-32}}{v-6}$
Choose the method: A. l'Hôpital's Rule once, B. direct substitution, X. l'Hôpital's Rule multiple times.
Find the limit value: $\lim _{v \rightarrow 6} \frac{v-4-\sqrt{v^{2}-32}}{v-6}=\square$

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Problem 9679

Find the volume of a solid with disc cross-sections of radius $e^{-x}$ for $1 \leq x < \infty$ . Enter as X.XXXX.

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Problem 9680

Explain Newton's method: how does the iteration formula approximate a function's root using tangent lines?

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Problem 9681

Find the volume of the region under $y=\ln \sqrt{x}$ from $x=1$ to $x=e$ when revolved around the $x$ -axis.

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Problem 9682

Find Newton's method formula and calculate $x_{1}$ for $f(x)=x^{2}-12$ with $x_{0}=3$ .

See Solution

Problem 9683

When to stop Newton's method? Choose the correct option about digit agreement for accuracy.

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Problem 9684

Find the integral to compute the volume of the region $D$ revolved around $x=5$ : $y=1-x^{4}$ , first quadrant.

See Solution

Problem 9685

Find Newton's method formula for $f(x)=x^{2}-12$ using $x_{0}=3$ . Choose the correct formula from options A-D.

See Solution

Problem 9686

Find Newton's method formula for $f(x)=x^{2}-2x-8$ with $x_{0}=3$ . Choose the correct option for $x_{n+1}$ .

See Solution

Problem 9687

Find the $y$ -intercept of the tangent line $L$ to $f(x)=\csc^{-1} x$ at $x=2$ .

See Solution

Problem 9688

Find critical numbers of $f(x)=-5 \sin (x) \cos (x)$ on $(-\pi, \pi)$ and intervals where $f$ is increasing/decreasing.

See Solution

Problem 9689

Find critical numbers of $f(x)=-5 \sin (x) \cos (x)$ on $(-\pi, \pi)$ . Identify intervals where $f$ is increasing and decreasing.

See Solution

Problem 9690

Find the critical numbers of $f(x)=-5 \sin (x) \cos (x)$ on $(-\pi, \pi)$ and determine where it's increasing and decreasing.

See Solution

Problem 9691

Find $\frac{d y}{d x}$ for the curve defined by $x y + y^{2} = 4$ where $y \geq 0$ .

See Solution

Problem 9692

Find the correct Newton's method formula and compute $x_{1}$ and $x_{2}$ for $f(x)=5-\tan(6x)$ , $x_{0}=1$ .

See Solution

Problem 9693

Approximate the integral $\int_{1}^{3} \frac{x}{x^{2}+3} d x$ using the Midpoint Rule with $n=5$ . Round to four decimal places.

See Solution

Problem 9694

Which formula represents Newton's method for the function? A. $x_{n}=x_{n+1}+\frac{5-\tan \left(6 x_{n+1}\right)}{6 \sec ^{2}\left(6 x_{n+1}\right)}$ B. $x_{n+1}=x_{n}+\frac{5-\tan \left(6 x_{n}\right)}{6 \sec ^{2}\left(6 x_{n}\right)}$ C. $x_{n+1}=x_{n}-\frac{5-\tan \left(6 x_{n}\right)}{6 \sec ^{2}\left(6 x_{n}\right)}$ D. $x_{n+1}=x_{n}+\frac{6 \sec ^{2}\left(6 x_{n}\right)}{5-\tan \left(6 x_{n}\right)}$

See Solution

Problem 9695

Calculate the integral $\int_{0}^{4} f(x) \, dx$ for the piecewise function defined by the given line segments.

See Solution

Problem 9696

Rearrange the steps for logarithmic differentiation of $y=(3 x)^{\sqrt{x}}$ . Identify the correct order of calculations and explanations.

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Problem 9697

Identify cases where logarithmic differentiation is needed for $\frac{d y}{d x}$ from these functions:
1. $y=(7 x^{2}+4 x^{-1})^{12}$
2. $y=\ln(x^{3}+4 x^{2})$
3. $y=(7 x)^{\tan x}$
4. $y=(3 x^{4}+x)^{2 x}$
5. $y=2^{\sin x}$
6. $y=(\sin x)^{x}$

See Solution

Problem 9698

Find the derivative of $f(t)=4 \sqrt{6 t^{2}+9}$ . What is $u=g(t)$ where $u=6 t^{2}+9$ ?

See Solution

Problem 9699

Evaluate the integral from -9 to 9: $\int_{-9}^{9} e \, dx$ .

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Problem 9700

Find the derivative of $g(x) = \int_{1}^{x} \ln(6 + t^2) \, dt$ using the fundamental theorem of calculus.

See Solution

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Calculus

Problem 9601

An object drops from 159ft159 \mathrm{ft}159ft; height s=159−16t2s=159-16t^2s=159−16t2. Find velocity, time to hit ground, and impact velocity.

Problem 9602

Minimize the cost of a rectangular container with a square base and volume 2816ft32816 \mathrm{ft}^{3}2816ft3, given material costs.

Problem 9603

Estimate the solution to e4x+x−2=0e^{4 x}+x-2=0e4x+x−2=0 using Newton's method with initial guess x0=0x_{0}=0x0​=0 for one iteration.

Problem 9604

Find the approximate value of x0x_{0}x0​ using the tangent line at (6,7)(6,7)(6,7) for the function fff with given properties.

Problem 9605

Find the next approximation x1x_{1}x1​ using Newton's method for f(x)=0f(x)=0f(x)=0 with initial guess x0=4x_{0}=4x0​=4 and tangent line y=3x−6y=3x-6y=3x−6.

Problem 9606

Show that f(x)=2x3−3f(x)=2 x^{3}-3f(x)=2x3−3 has a zero in [1,2][1,2][1,2] using the Intermediate Value Theorem and apply 3 bisection steps for an estimate.

Problem 9607

Design a rectangular container with a square base and volume 50000ft350000 \mathrm{ft}^{3}50000ft3. Minimize costs: top at \$6/ft², sides at \$10/ft², bottom at \$2/ft². Find dimensions.

Problem 9608

Find if lim⁡x→0sin⁡−1(2x)tan⁡−1(5x)\lim_{x \rightarrow 0} \frac{\sin^{-1}(2x)}{\tan^{-1}(5x)}limx→0​tan−1(5x)sin−1(2x)​ exists, and if so, determine its value.

Problem 9609

Zeigen Sie, dass F(x)F(x)F(x) eine Stammfunktion von fff ist. Berechnen Sie die Fläche zwischen −2-2−2 und 000 von fff und der xxx-Achse.

Problem 9610

Find the limit as θ\thetaθ approaches 0: lim⁡θ→0tan⁡θθ2cot⁡3θ\lim _{\theta \rightarrow 0} \frac{\tan \theta}{\theta^{2} \cot 3 \theta}limθ→0​θ2cot3θtanθ​.

Problem 9611

Find the maximum sales per day for the function s(d)=d3+10s(d)=d^{3}+10s(d)=d3+10 on the interval 0<d≤30<d \leq 30<d≤3.

Problem 9612

Gegeben ist f(x)=12x2f(x)=\frac{1}{2} x^{2}f(x)=21​x2. Bestimme: a) mittlere Änderungsrate auf [0;2][0; 2][0;2], b) lokale Änderungsrate bei x0=−1x_{0}=-1x0​=−1, c) lokale Änderungsrate bei x0=2x_{0}=2x0​=2.

Problem 9613

Find the absolute extrema of f(x)=2x3−15x2−84xf(x)=2 x^{3}-15 x^{2}-84 xf(x)=2x3−15x2−84x on [−6,9][-6,9][−6,9]. Provide (x,f(x))(x, f(x))(x,f(x)) as an ordered pair.

Problem 9614

Find values of xxx where the rate of change of f(x)=−3(x−5)(x−2)f(x)=-3(x-5)(x-2)f(x)=−3(x−5)(x−2) is decreasing.

Problem 9615

Is the function f(x)={x,−1≤x<0tan⁡x,0≤x≤π/4f(x)=\left\{\begin{array}{ll} x, & -1 \leq x<0 \\ \tan x, & 0 \leq x \leq \pi / 4 \end{array}\right.f(x)={x,tanx,​−1≤x<00≤x≤π/4​ continuous and differentiable at x=0x=0x=0?

Problem 9616

Find the average cost function for C(x)=2x2+54x+98C(x)=2 x^{2}+54 x+98C(x)=2x2+54x+98 and determine its minimum value per unit.

Problem 9617

Find the highest point on the intersection of the cone x2+y2−z2=0x^{2}+y^{2}-z^{2}=0x2+y2−z2=0 and the plane x+2z=4x+2z=4x+2z=4 using Lagrange multipliers.

Problem 9618

Find y′(2)y' (2)y′(2) for y=log⁡6(x4−4x3+1)5+4x2−1200xy=\log _{6}\left(x^{4}-4 x^{3}+1\right)^{5}+4^{x^{2}}-1200 xy=log6​(x4−4x3+1)5+4x2−1200x. Round to 3 decimal places.

Problem 9619

Bestimme den Wert von a, sodass die Aussage ∫02e2xdx=e−12\int_{0}^{2} e^{2 x} d x=\frac{e-1}{2}∫02​e2xdx=2e−1​ wahr ist.

Problem 9620

Find the tangent line equation for the function f(x)=x2+8f(x)=x^{2}+8f(x)=x2+8 at the point where x=−5x=-5x=−5.

Problem 9621

Determine conclusions from the Intermediate Value Theorem for f(x)f(x)f(x) on (−7,0)(-7,0)(−7,0), given f(−7)=5f(-7)=5f(−7)=5 and f(0)=2f(0)=2f(0)=2.

Problem 9622

Determine conclusions from the Intermediate Value Theorem for f(x)f(x)f(x) on (−4,2)(-4,2)(−4,2) given f(−4)=−5f(-4)=-5f(−4)=−5 and f(2)=−3f(2)=-3f(2)=−3.

Problem 9623

Problem 9624

Determine conclusions using the Intermediate Value Theorem for f(x)f(x)f(x) on (−6,0)(-6,0)(−6,0), given f(−6)=3f(-6)=3f(−6)=3 and f(0)=−3f(0)=-3f(0)=−3.

Problem 9625

Find the first derivative of the vehicle's acceleration function: a(t)=(−t2+6t)⋅e−ta(t) = (-t^{2}+6t) \cdot e^{-t}a(t)=(−t2+6t)⋅e−t.

Problem 9626

Determine where the function f(x)=(x−7)e−9xf(x)=(x-7)e^{-9x}f(x)=(x−7)e−9x is increasing, decreasing, and find its local extrema.

Problem 9627

Estimate the distance (in feet) a braking car travels using a midpoint sum with six equal intervals from the given velocity graph.

Problem 9628

Problem 9629

Find lower and upper estimates for ∫1030f(x)dx\int_{10}^{30} f(x) dx∫1030​f(x)dx using the table values with five equal subintervals.

Problem 9630

Solve the initial value problem: y(4)−5y′′′+4y′′=3xy^{(4)} - 5y''' + 4y'' = 3xy(4)−5y′′′+4y′′=3x, with y(0)=0y(0)=0y(0)=0, y′(0)=0y'(0)=0y′(0)=0, y′′(0)=0y''(0)=0y′′(0)=0, y′′′(0)=0y'''(0)=0y′′′(0)=0.

Problem 9631

Approximate the integral ∫13xx2+3dx\int_{1}^{3} \frac{x}{x^{2}+3} d x∫13​x2+3x​dx using the Midpoint Rule with n=5n=5n=5. Round to four decimal places.

Problem 9632

Show that the polynomial f(x)=x3+x2−2x+5f(x)=x^{3}+x^{2}-2x+5f(x)=x3+x2−2x+5 has a real zero between -3 and -1 using the Intermediate Value Theorem.

Problem 9633

Solve the IVP: dydx=xsin⁡xy\frac{d y}{d x}=\frac{x \sin x}{y}dxdy​=yxsinx​ with initial condition y(0)=−1y(0)=-1y(0)=−1.

Problem 9634

Find the derivative p′(t)p^{\prime}(t)p′(t) of p(t)=137t2+1,080t+14,911p(t)=137 t^{2}+1,080 t+14,911p(t)=137t2+1,080t+14,911. Then, calculate consumption for 2026 and its rate of change.

Problem 9635

Find the volume using cylindrical shells for the region bounded by y=x4y=x^{4}y=x4, y=0y=0y=0, x=1x=1x=1 rotated about x=2x=2x=2.

Problem 9636

Find the limit of the sequence an=1−n1−2na_{n}=\frac{1-n}{1-2 \sqrt{n}}an​=1−2n​1−n​ or prove it diverges.

Problem 9637

Find the limit of these sequences if they converge; otherwise, prove divergence: a) 1/n1/n1/n, c) an=1−n1−2na_n=\frac{1-n}{1-2\sqrt{n}}an​=1−2n​1−n​.

Problem 9638

Find the second derivative f′′(x)f^{\prime \prime}(x)f′′(x) for the function f(x)=(x2+1)5f(x)=(x^{2}+1)^{5}f(x)=(x2+1)5.

Problem 9639

Invest \$25000 at 4\% compounded continuously. (a) Find value after 5 years. (b) When will it reach \$48000?

Problem 9640

Find the xxx and yyy coordinates of all inflection points for the function f(x)=x3+39x2f(x)=x^{3}+39x^{2}f(x)=x3+39x2.

Problem 9641

An object drops from $159 \mathrm{ft}$ ; height $s=159-16t^2$ . Find velocity, time to hit ground, and impact velocity.

Minimize the cost of a rectangular container with a square base and volume $2816 \mathrm{ft}^{3}$ , given material costs.

Estimate the solution to $e^{4 x}+x-2=0$ using Newton's method with initial guess $x_{0}=0$ for one iteration.

Find the approximate value of $x_{0}$ using the tangent line at $(6,7)$ for the function $f$ with given properties.

Find the next approximation $x_{1}$ using Newton's method for $f(x)=0$ with initial guess $x_{0}=4$ and tangent line $y=3x-6$ .

Show that $f(x)=2 x^{3}-3$ has a zero in $[1,2]$ using the Intermediate Value Theorem and apply 3 bisection steps for an estimate.

Design a rectangular container with a square base and volume $50000 \mathrm{ft}^{3}$ . Minimize costs: top at \$6/ft², sides at \$10/ft², bottom at \$2/ft². Find dimensions.

Find if $\lim_{x \rightarrow 0} \frac{\sin^{-1}(2x)}{\tan^{-1}(5x)}$ exists, and if so, determine its value.

Zeigen Sie, dass $F(x)$ eine Stammfunktion von $f$ ist. Berechnen Sie die Fläche zwischen $-2$ und $0$ von $f$ und der $x$ -Achse.

Find the limit as $\theta$ approaches 0: $\lim _{\theta \rightarrow 0} \frac{\tan \theta}{\theta^{2} \cot 3 \theta}$ .

Find the maximum sales per day for the function $s(d)=d^{3}+10$ on the interval $0<d \leq 3$ .

Gegeben ist $f(x)=\frac{1}{2} x^{2}$ . Bestimme: a) mittlere Änderungsrate auf $[0; 2]$ , b) lokale Änderungsrate bei $x_{0}=-1$ , c) lokale Änderungsrate bei $x_{0}=2$ .

Find the absolute extrema of $f(x)=2 x^{3}-15 x^{2}-84 x$ on $[-6,9]$ . Provide $(x, f(x))$ as an ordered pair.

Find values of $x$ where the rate of change of $f(x)=-3(x-5)(x-2)$ is decreasing.

Is the function $f(x)=\left\{\begin{array}{ll} x, & -1 \leq x<0 \\ \tan x, & 0 \leq x \leq \pi / 4 \end{array}\right.$ continuous and differentiable at $x=0$ ?

Find the average cost function for $C(x)=2 x^{2}+54 x+98$ and determine its minimum value per unit.

Find the highest point on the intersection of the cone $x^{2}+y^{2}-z^{2}=0$ and the plane $x+2z=4$ using Lagrange multipliers.

Find $y' (2)$ for $y=\log _{6}\left(x^{4}-4 x^{3}+1\right)^{5}+4^{x^{2}}-1200 x$ . Round to 3 decimal places.

Bestimme den Wert von a, sodass die Aussage $\int_{0}^{2} e^{2 x} d x=\frac{e-1}{2}$ wahr ist.

Find the tangent line equation for the function $f(x)=x^{2}+8$ at the point where $x=-5$ .

Determine conclusions from the Intermediate Value Theorem for $f(x)$ on $(-7,0)$ , given $f(-7)=5$ and $f(0)=2$ .

Determine conclusions from the Intermediate Value Theorem for $f(x)$ on $(-4,2)$ given $f(-4)=-5$ and $f(2)=-3$ .

Determine conclusions using the Intermediate Value Theorem for $f(x)$ on $(-6,0)$ , given $f(-6)=3$ and $f(0)=-3$ .

Find the first derivative of the vehicle's acceleration function: $a(t) = (-t^{2}+6t) \cdot e^{-t}$ .

Determine where the function $f(x)=(x-7)e^{-9x}$ is increasing, decreasing, and find its local extrema.

Find lower and upper estimates for $\int_{10}^{30} f(x) dx$ using the table values with five equal subintervals.

Solve the initial value problem: $y^{(4)} - 5y''' + 4y'' = 3x$ , with $y(0)=0$ , $y'(0)=0$ , $y''(0)=0$ , $y'''(0)=0$ .

Approximate the integral $\int_{1}^{3} \frac{x}{x^{2}+3} d x$ using the Midpoint Rule with $n=5$ . Round to four decimal places.

Show that the polynomial $f(x)=x^{3}+x^{2}-2x+5$ has a real zero between -3 and -1 using the Intermediate Value Theorem.

Solve the IVP: $\frac{d y}{d x}=\frac{x \sin x}{y}$ with initial condition $y(0)=-1$ .

Find the derivative $p^{\prime}(t)$ of $p(t)=137 t^{2}+1,080 t+14,911$ . Then, calculate consumption for 2026 and its rate of change.

Find the volume using cylindrical shells for the region bounded by $y=x^{4}$ , $y=0$ , $x=1$ rotated about $x=2$ .

Find the limit of the sequence $a_{n}=\frac{1-n}{1-2 \sqrt{n}}$ or prove it diverges.

Find the limit of these sequences if they converge; otherwise, prove divergence: a) $1/n$ , c) $a_n=\frac{1-n}{1-2\sqrt{n}}$ .

Find the second derivative $f^{\prime \prime}(x)$ for the function $f(x)=(x^{2}+1)^{5}$ .

Find the $x$ and $y$ coordinates of all inflection points for the function $f(x)=x^{3}+39x^{2}$ .

Differentiate the function: $f(t)=t e^{t} \cot (t)$ and find $f^{\prime}(t)$ .

Find the limit of $\frac{1-n}{1-2 \sqrt{n}}$ as $n$ approaches infinity.

Find the $x$ and $y$ coordinates of all inflection points for the function $f(x)=x^{3}+27 x^{2}$ .

Find the derivative of $y=2 e^{x} \sin (x)$ using the product and chain rules.

Find the average rate of change of the function $f(x)=3 x^{2}+4$ on the interval $[2, 2+h]$ .

Find the derivative of $j(y) = 31y^6 - 10\frac{1}{y^9}$ .

Find the derivative of $F(x)=\frac{x^{3}}{\cos (x)+x}$ .

Find the derivative of $k(x)=x^{11}-0.2 x^{10}+\frac{\pi}{3} x^{2}+\pi^{2}$ .

Find the limit of the sequence $a_{n}=\frac{1+\sqrt[99]{n}}{1+\sqrt[100]{n}}$ or prove it diverges.

Find the area represented by the integral $\int_{-3}^{8}(10-5 x) d x$ .

Find the area represented by the integral $\int_{-8}^{0}\left(5+\sqrt{64-x^{2}}\right) d x$ .

Determine the asymptotes for the function $f(x)=\frac{x^{2}-1}{\sqrt{x+1}}$ .

Find $\int_{1}^{5} f(x) d x$ given $\int_{1}^{7} f(x) d x=8.2$ and $\int_{5}^{7} f(x) d x=3.7$ .

Find the third derivative of $f(x)=\cos ^{2} x$ . Options: A. $-\cos ^{2} x$ , B. $2 \sin 2 x$ , C. $8 \sin x \cos x$ , D. $\sin 2 x$ , E. None.

Find $f^{\prime \prime}(\sqrt{e})$ if $f(x)=(\ln x)^{2}$ .

Find the limit as $\theta$ approaches $\frac{\pi}{2}$ from the left of $(\tan \theta)^{\cos \theta}$ .

Berechne die Ableitung von $f(x)=\frac{1}{8} x^{2}(x+3)(x-2)$ .

Find the derivative $\frac{d}{d x}(f(\ln x))$ where $f(x)=x^{2}+2 x$ .

Find the derivative of the function $f(x)=\ln \sqrt{x^{2}+1}$ . What is $f^{\prime}(x)$ ?

Evaluate the limit: $\lim _{x \rightarrow \frac{\pi}{2}^{-}} \frac{\tan x}{\left(\frac{11}{2 x-\pi}\right)}$ using l'Hôpital's Rule.

Evaluate the limit: $\lim _{x \rightarrow \frac{\pi}{2}^{-}} \frac{\tan x}{\left(\frac{10}{2 x-\pi}\right)}$

The drug concentration $C$ is given by $C=\frac{35 \cdot t}{24+t^{2}}$ . Find $C$ after 2.5 hours, time to reach $0.5\%$ , and end behavior as $t \to \infty$ .

Find $f^{\prime}\left(\frac{\pi}{3}\right)$ if $f(x)=\sec ^{2} x$ . Choices: 4, $4 \sqrt{3}$ , $8 \sqrt{3}$ , $\sqrt{3}$ , 12.

Find the difference quotient for $f(x)=\frac{4}{x}$ : 1) without simplification, 2) with simplification.

Find the rate of change of oil volume $V(t)=10(10-t)^{2}$ at $t=5$ minutes. Choices: -100, -50, -150, -200 litres/min.

Find the derivative of the function $f(x)=4 x^{3}+2 x^{2}$ and evaluate it at $x=3$ . What is $f^{\prime}(3)$ ?

Find the volume of a solid with cross-section discs of radius $e^{-x}$ for $1 \leq x < \infty$ . Round to four decimal places.

Find the volume of the solid defined by $0 \leq r \leq R$ , $0 \leq \theta \leq 2 \pi$ , $0 \leq z \leq \theta$ .

Evaluate the limit: $\lim _{v \rightarrow 7} \frac{v-6-\sqrt{v^{2}-48}}{v-7}$ using l'Hôpital's Rule.

Find $x$ where the slope of the tangent line for $y=x^{3}+6x^{2}$ equals 36.