Calculus

Problem 27801

Differentiate the integral $\int_{x^{2}}^{\sin x}(\sqrt[3]{\ln |t+4|} d t)$ with respect to $x$ .

See Solution

Problem 27802

Finde Werte für $a, b, c, d$ in $f(t)=a \cdot(t-b) \cdot e^{-c t}+d$ für die Aussagen (1)-(5) zur Übernachtungszahlen.

See Solution

Problem 27803

Sophie hat Fieber, beschrieben durch $f(t)=1,5 \cdot t \cdot e^{-0,5 t+1}+37$ . Bestimme max. Temperatur, Abnahmezeit, langfristige Temp., und mehr.

See Solution

Problem 27804

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=\sqrt{2}$ , $x=-\frac{\pi}{4}$ , $x=\frac{\pi}{4}$ around $y=-\sqrt{2}$ .

See Solution

Problem 27805

Find the limit: $\lim _{x \rightarrow 2^{-}} \frac{3}{x-2}$ .

See Solution

Problem 27806

Find the distance a particle moves from $t=0$ to $t=3$ with velocity $v(t)=e^{t}$ . Show your calculator input.

See Solution

Problem 27807

Evaluate the derivative: $\frac{d}{d x} \int_{3}^{x^{2}-3 x}(t-1)^{2} d t$

See Solution

Problem 27808

Find the function $F(x)=\int \frac{3}{x+5} dx$ given $F(-4)=-7$ .

See Solution

Problem 27809

Estimate the velocity after 5 seconds using the trapezoid rule with acceleration $a(t)$ values at $t = 0, 1, 2, 3, 4, 5$ .

See Solution

Problem 27810

Find $f(5)$ if $f$ is the antiderivative of $\frac{x^{3}}{x^{2}+3}$ and $f(1)=7$ . Show your calculator input.

See Solution

Problem 27811

Object has acceleration $a(t)=2t-8$ . Given $v(0)=15$ and $x(3)=12$ . Find $v(t)$ , when at rest, $x(t)$ , and displacement from $[3,9]$ .

See Solution

Problem 27812

Find the inhaled air volume from $t=0$ to $t=8$ using $f(t)=\frac{1}{2} \sin \left(\frac{2 \pi}{5} t\right)$ . Round to three decimals.

See Solution

Problem 27813

Find the average rate of change of the function $f(x)=-x^{2}-3x+5$ over the interval $-3 \leq x \leq 4$ .

See Solution

Problem 27814

Deposit \ $1000 at an 8.5% interest rate, compounded continuously. Find the balance after 5 years using$ F=\ $1000e^{0.085 \cdot 5}$ . Round to the nearest cent.

See Solution

Problem 27815

Given continuous functions $f$ and $g$ , with $\int_{1}^{2} f(x) d x=-4$ , $\int_{1}^{5} f(x) d x=6$ , and $\int_{1}^{5} g(x) d x=8$ , find:
a. $\int_{2}^{2} g(x) d x$ b. $\int_{5}^{1} g(x) d x$ c. $\int_{1}^{2} 3 f(x) d x$ d. $\int_{2}^{5} f(x) d x$ e. $\int_{1}^{5}[f(x)-g(x)] d x$ f. $\int_{1}^{5}[4 f(x)-g(x)] d x$

See Solution

Problem 27816

Find $\int_{3}^{4} f(t) dt$ and $\int_{4}^{3} f(t) dt$ given $\int_{0}^{3} f(t) dt=3$ and $\int_{0}^{4} f(t) dt=7$ . Also, identify the false statement among: (A) $\int_{3}^{7} h(x) k(x) dx=15$ , (B) $\int_{3}^{7}[h(x)+k(x)] dx=8$ , (C) $\int_{3}^{7} 2 h(x) dx=10$ , (D) $\int_{3}^{7}[h(x)-k(x)] dx=3$ , (E) $\int_{7}^{3}[k(x)-h(x)] dx=2$ , (F) $\int_{3}^{3}[h(x)+k(x)] dx=0$ .

See Solution

Problem 27817

For a continuous function $g$ where $\int_{-2}^{1} g(x) \, dx = 2$ and $\int_{1}^{3} g(x) \, dx = -6$ , find $\int_{3}^{-2} g(x) \, dx$ .
For a graph of $f(x)$ with areas of 3, calculate: (a) $\int_{-4}^{2}[4 f(x)-5] \, dx$ , (b) $\int_{-4}^{2}|f(x)| \, dx$ , (c) $\int_{2}^{-4} f(x) \, dx$ , $\int_{-2}^{2} f(|x|) \, dx$ , $\left|\int_{-4}^{2} f(x) \, dx\right|$ , $\int_{-2}^{4} f(-x) \, dx$ .

See Solution

Problem 27818

A bakery sells 640 bagels daily at RM 8.00. For every RM 0.20 price drop, sales increase by 40. Find the price for max revenue using differentiation.

See Solution

Problem 27819

Find the velocity $v(t)$ and acceleration $a(t)$ from the position function $s(t) = t^{3} - 11t^{2} + 24t$ .

See Solution

Problem 27820

A roller coaster moves at $42.5 \frac{m}{s}$ at $1.5 m$ high. Find: height at $8.3 \frac{m}{s}$ , speed at lowest point, energy loss at $14 \frac{m}{s}$ , and work to stop at $7.5 m$ .

See Solution

Problem 27821

Find the area between the curves $y=\sqrt{x}$ and $y=x^{2}$ . What is the area? Options: none, $\frac{1}{4}$ , $\frac{1}{2}$ , $\frac{1}{3}$ , $\frac{1}{6}$ .

See Solution

Problem 27822

Find the volume of the solid of rotation (without evaluating) for these cases:
(a) Disk method for $y=4-x^2$ about $x$ -axis. (b) Washer method for $y=x^2 +1$ and $y=2$ about $x$ -axis. (c) Shell method for $x=y^2$ and $x=9$ about $x$ -axis. (d) Shell method for $x=6-y$ , $x=y^2$ , and $y=2$ about $y=2$ .

See Solution

Problem 27823

Find the volume of revolution for the region in the first quadrant between $x=y^{2}$ and $x=6-y$ .
(a) About the $y$ -axis using the washer method. (b) About $x=-1$ using the washer method. (c) About the $x$ -axis using the shell method.

See Solution

Problem 27824

Find the volume of the solid of revolution for the region between $y=x^{3}+3$ , $y=4$ , and the $y$ -axis using different methods.

See Solution

Problem 27825

Find the value of $\alpha$ if the differential of $f(x)=\alpha \sqrt{x}$ from 1 to 4 is 12. Choices: a. 0 b. None c. 4 d. $2x$ e. 8.

See Solution

Problem 27826

Determine where the graph of $y=x^{3}+x^{2}-27 x$ is concave down: a. $x \geq \frac{-1}{3}$ , b. $x \leq 0$ , c. $x \geq 0$ , d. $-6 \geq x \geq 2$ , e. None.

See Solution

Problem 27827

Find the constant $\alpha$ if the differential of $f(x)=\alpha \sqrt{x}$ from 1 to 4 equals 12.

See Solution

Problem 27828

You and a friend bike to a restaurant; you ride east at $16 \mathrm{mph}$ and your friend north at $12 \mathrm{mph}$ . After $4 \mathrm{mi}$ , find the rate of distance change between you.

See Solution

Problem 27829

Find the volume of the solid formed by revolving the area between $y=x^{3}+3$ , $y=4$ , and the $y$ -axis:
1. (a) Around the $y$ -axis: $V=\int_{3}^{4} \pi\left(R^{2}(y)\right) d y$
(b) Around the $x$ -axis: $V=\int_{0}^{1} \pi\left[16-(x^{3}+3)^{2}\right] d x$
(c) Around the line $y=1$ : Use the shell method.

See Solution

Problem 27830

Minimize Marginal Cost: Given $C(x) = 0.0001 x^{3} - 0.06 x^{2} + 4x + 100$ , is marginal cost at $x=100$ increasing, decreasing, or constant? Find minimum marginal cost.

See Solution

Problem 27831

Identify which function is continuous: A) $f(x)=\frac{4}{x^{3}}$ , B) $g(x)=\frac{1}{5-x}$ , C) $h(x)=\frac{x^{2}}{11}$ .

See Solution

Problem 27832

Find the singular solution of the equation $\frac{d y}{d x}=(y-3)^{2}$ .

See Solution

Problem 27833

Find the derivative of $y = x^{2} \sqrt{2x - 3}$ .

See Solution

Problem 27834

Find the maximum height $m$ from which a camera can be dropped without exceeding a speed of $16.0 \mathrm{~m/s}$ .

See Solution

Problem 27835

Find the area between the curves $y=3 \sqrt{x}$ and $y=3 x^{2}$ .

See Solution

Problem 27836

Given the function $f(x)=\frac{x^{4}+1}{x^{2}}$ , determine which statement is true about its properties.

See Solution

Problem 27837

Find the critical points of the function $f(x) = x^{3} + 3x^{2} - x - 3$ by calculating its derivative and solving $f'(x) = 0$ .

See Solution

Problem 27838

A particle's position is given by $s(t)=2 t^{3}-11 t^{2}+12 t-13$ . Find velocity, acceleration, rest times, and direction changes.

See Solution

Problem 27839

Find the average rate of change of the function $f(x)=0.00186 x^{3}-0.0921 x^{2}+2.67 x-1.73$ from 2005 to 2015.

See Solution

Problem 27840

Analyze the function $f(x)=2 x^{3}+3 x^{2}-36 x$ for increasing/decreasing intervals, local extrema, and concavity.

See Solution

Problem 27841

Analyze the function $f(x)=x^{3}-12 x+2$ for increase/decrease, max/min values, concavity, and sketch it. Also do the same for $f(x)=36 x+3 x^{2}-2$ .

See Solution

Problem 27842

Find intervals of increase/decrease, local max/min, concavity, inflection points, and sketch the graph for $f(x)=36 x+3 x^{2}-2 x^{3}$ .

See Solution

Problem 27843

Analyze $f(x)=x^{3}-12 x+2$ and $f(x)=36 x+3 x^{2}-2 x^{3}$ for increase/decrease, max/min, concavity, and sketch the graphs.

See Solution

Problem 27844

Find the dimension of $d c / d w$ and compute $d c / d w$ for $c=\sum_{i=1}^{N} w_{i}(x_{i}-y_{i})^{2}$ .

See Solution

Problem 27845

Find the limit: $\lim _{x \rightarrow-4} \frac{x^{2}-16}{x-1}$ .

See Solution

Problem 27846

At noon, ship A is 180 km west of ship B. A sails east at 30 km/h, B sails north at 25 km/h. Find the distance change rate at 4 pm.

See Solution

Problem 27847

Find the derivative of the function $y = \sqrt[3]{x(x+1)(x+2)}$ .

See Solution

Problem 27848

A cup of coffee cools from $95^{\circ} \mathrm{C}$ to $77^{\circ} \mathrm{C}$ at $1^{\circ} \mathrm{C}$ per minute. Find time to reach $77^{\circ} \mathrm{C}$ . Round to two decimal places.

See Solution

Problem 27849

Find the derivatives of these functions: (1) $y=x \ln x$ , (2) $y=\sin(x^{2})$ .

See Solution

Problem 27850

Prove that for $x>0$ , the inequalities $x-\frac{x^{2}}{2}<\ln (1+x)<x$ hold true.

See Solution

Problem 27851

Determine the global max and min of $f(x)=x^{5}+(1-x)^{5}$ for $x$ in the interval $[0,1]$ .

See Solution

Problem 27852

Find the limit: $\lim _{n \rightarrow \infty} \sqrt[n]{3}$ .

See Solution

Problem 27853

Find the limit: $\lim _{n \rightarrow \infty}(\sqrt{n+1}-\sqrt{n})$ .

See Solution

Problem 27854

Find the derivative of $y = \sqrt[3]{x(x+1)(x+2)}$ .

See Solution

Problem 27855

Find the limit: $\lim _{n \rightarrow \infty} \frac{2^{n}}{n !}$ .

See Solution

Problem 27856

Find the limit: $\lim _{x \rightarrow-3} \frac{x^{2}+x-6}{x^{2}+4 x+3}$ .

See Solution

Problem 27857

Jeriel invested \ $72,000 at$ 7.25\% $continuous interest. Amelia invested \$72,000 at$ 6.75\%$ daily. How much longer for Amelia's money to double?

See Solution

Problem 27858

Find the tangent line approximation of $f(x)=2 \cos x + 1$ at $x=\frac{\pi}{2}$ to estimate $f(1.5)$ .

See Solution

Problem 27859

Find the limit as $t$ approaches 0 for the expression $\frac{1}{t}-\frac{1}{t^{2}+t}$ .

See Solution

Problem 27860

Find the maximum acceleration for the velocity $v(t)=\frac{2}{3} t^{3}-4 t^{2}+8 t-2$ on $0 \leq t \leq 3$ .

See Solution

Problem 27861

Find the time $t$ when the particle at $x(t)=e^{-t} \cos t$ is farthest right for $0 \leq t \leq 2 \pi$ .

See Solution

Problem 27862

Find critical numbers, increasing/decreasing intervals, local extrema, and concavity for $f(x)=x^{3}-6x^{2}+9x+1$ .

See Solution

Problem 27863

Calculate the integral: $\int \frac{9 x+2}{x^{2}+x-6} d x$

See Solution

Problem 27864

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{4}+1}}{x^{2}-3 x+5}$ .

See Solution

Problem 27865

Find the volume of the solid of revolution for the region between $y=x^{3}+3$ , $y=4$ , and the $y$ -axis, using the disk method around the $y$ -axis.

See Solution

Problem 27866

Ein Dozent will den besten Weg von $P_{1}=(10,20)$ nach $P_{2}=(10,0)$ finden, indem er Fahrrad und S-Bahn kombiniert. Berechnen Sie den Punkt auf der Geraden $g(x)=10-x$ , der die kürzeste Reisezeit ergibt.

See Solution

Problem 27867

Ein Dozent plant seine Wege zur Hochschule. Bestimme den Punkt auf der Geraden $g(x)=10-x$ , der die Reisezeit minimiert.

See Solution

Problem 27868

Untersuchen Sie die Funktion $f(x)=\frac{x^{2}}{x^{2}-4}$ vollständig und skizzieren Sie den Graphen im passenden Intervall.

See Solution

Problem 27869

Find the volume of the solid formed by revolving the region in the first quadrant between $y=x^{3}+3$ , $y=4$ , and the $y$ -axis around $x=-1$ using the shell method.

See Solution

Problem 27870

Find the best linear approximation, $g$ , to the function $f(x)=4 \sqrt{x}$ on the interval $[0,1]$ .

See Solution

Problem 27871

Find the limit: $\lim _{x \rightarrow-2} \frac{x+2}{\sqrt{x+3}-1}$ .

See Solution

Problem 27872

Find the limit as $x$ approaches 4 from the left for $\frac{2x^2 - x + 5}{(x-4)(x-4)}$ .

See Solution

Problem 27873

Calculate the following integrals given that each area under $f(x)$ is 3: a. $\int_{-4}^{2}[4 f(x)-5] d x$ , b. $\int_{-4}^{2}|f(x)| d x$ , c. $\int_{2}^{-4} f(x) d x$ , d. $\int_{-2}^{2} f(|x|) d x$ , e. $\left|\int_{-4}^{2} f(x) d x\right|$ .

See Solution

Problem 27874

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

See Solution

Problem 27875

Given that $\int_{-2}^{1} g(x) \, dx = 2$ and $\int_{1}^{3} g(x) \, dx = -6$ , find $\int_{3}^{-2} g(x) \, dx$ .
For the areas, if each region under $f(x)$ has area 3, find: a. $\int_{-4}^{2}[4 f(x) - 5] \, dx$ b. $\int_{-4}^{2}|f(x)| \, dx$ c. $\int_{2}^{-4} f(x) \, dx$ d. $\int_{-2}^{2} f(|x|) \, dx$ e. $\left|\int_{-4}^{2} f(x) \, dx\right|$ f. $\int_{-2}^{4} f(-x) \, dx$

See Solution

Problem 27876

Investigate the convergence of these series: 1. $\sum_{n=1}^{\infty}\left(\frac{2+(-1)^{n}}{\pi}\right)^{n}$ ; 2. $\sum_{n=1}^{\infty}\left(\frac{3 n-2}{3 n-1}\right)^{2}-3$ .

See Solution

Problem 27877

1. Calculate $\int_{0}^{1} 3x \, dx$ .
2. Find $\int_{-2}^{3}(x-5) \, dx$ .
3. Given $g(x)=\int_{\pi}^{x} \frac{1}{1+t^{4}} \, dt$ , determine $g^{\prime}(2)$ .
4. Evaluate $\int_{-2}^{-1}\left(x-\frac{1}{x^{2}}\right) \, dx$ .
5. Solve $\int_{1}^{9}\left(\frac{x-2}{\sqrt{x}}\right) \, dx$ .

See Solution

Problem 27878

An 8-foot ladder leans against a wall. If the top slides down at 2 ft/s, how fast does the bottom move when it's 4 ft from the wall?

See Solution

Problem 27879

Find the value of the series: $\sum_{n=1}^{\infty}\left(\frac{3n-2}{3n-1}\right)^{n^2-3}$ .

See Solution

Problem 27880

Let $f_{m}(x)=\frac{m x^{2}-2 x-7}{x+2}$ .
Part A:
1. Show $C_{m}$ passes through a fixed point $F$ .
2. Find $m$ for two extremums.
Part B ( $m=1$ ):
1. Find $a, b, c$ for $f_{1}(x)=a x+b+\frac{c}{x+2}$ .
2. Show $I(-2;-6)$ is a symmetry center of $C_{1}$ .
3. a) Calculate limits at domain bounds. b) Does $C_{1}$ have a vertical asymptote? c) Show $y=x-4$ is an asymptote to $C_{1}$ . d) Compare $C_{1}$ and $(\Delta)$ .
4. a) Calculate $f_{1}^{\prime}(x)$ and its sign. b) Create the variation table for $f_{1}$ .
5. Find tangent $(T)$ to $C_{1}$ at $F$ .
6. Plot $(\Delta)$ , $(T)$ , and $C_{1}$ .
7. Find tangents to $C_{1}$ parallel to $y=\frac{3}{4} x+1$ .
8. a) Find $h$ for line $(D)$ cutting $C_{1}$ at $M$ and $N$ . b) Find midpoint $J$ of $[M N]$ and its locus.

See Solution

Problem 27881

Find the relative extrema of $y=\frac{x^{4}}{4}-\frac{x^{3}}{3}$ using its derivative and critical points.

See Solution

Problem 27882

Find the relative extrema of the function $y=\frac{x^{4}}{4}-\frac{x^{3}}{3}$ .

See Solution

Problem 27883

Evaluate the series $\sum_{n=1}^{\infty} a n^{2} x^{n}$ using the Cauchy product with $a_{i}=x^{i}$ and $b_{j}=a j x^{j}$ .

See Solution

Problem 27884

Evaluate the series $\sum_{n=1}^{\infty} a n^{2} x^{n}$ using the Cauchy product for $a \neq 0$ and $|x|<1$ .

See Solution

Problem 27885

A rock is thrown up at $25.0 \mathrm{~m/s}$ from $17.0 \mathrm{~m}$ , landing in an $80.0 \mathrm{~m}$ lake. Find time $s$ .

See Solution

Problem 27886

Investigate the absolute convergence and convergence of these series:
1. $-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}-\frac{1}{7}+\ldots$
2. $1-\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{6}-\frac{1}{8}+\frac{1}{5}-\frac{1}{10}-\frac{1}{12}+\ldots$

See Solution

Problem 27887

Find the local extrema (max/min) of the function $f(x)=x^{4}-8x^{3}+1$ .

See Solution

Problem 27888

Find where the function $f(x)=|2x+6|$ is not differentiable.

See Solution

Problem 27889

Find $(f \circ g)^{\prime}(2)$ given $f(x)=x^{4}$ , $g(2)=1$ , and $g^{\prime}(2)=2$ .

See Solution

Problem 27890

Identify the horizontal asymptote of $f(x)=\frac{7}{x^{2}-7 x+9}$ and if it crosses it.

See Solution

Problem 27891

A cart of mass $m$ kg moves at 6 m/s in a circle with radius 3 m. Find the centripetal acceleration.

See Solution

Problem 27892

A fireworks shell is launched at $256 \mathrm{ft/sec}$ . Find $s(t)$ and the time it's visible above $768 \mathrm{ft}$ .

See Solution

Problem 27893

Estimate the area under $h$ from $x=-6$ to $x=14$ using left and right Riemann sums with 5 subdivisions. Order the areas:
Left Riemann sum, Right Riemann sum, Actual area.

See Solution

Problem 27894

Berechne die durchschnittliche Änderungsrate von $f(x)=\sin(x)+3x$ für $x \in [1 ; 5]$ .

See Solution

Problem 27895

Berechne die durchschnittliche Änderungsrate von $f(x)=\sin (x)+3 x$ für $x \in[1 ; 5]$ .

See Solution

Problem 27896

Find the integral of $\frac{\tan^{-1} x}{x^{2}}$ with respect to $x$ .

See Solution

Problem 27897

Differentiate the function $\left(\frac{1}{e^{\sqrt{x}}}\right)$ .

See Solution

Problem 27898

Finde die Stelle, an der der Graph von $f(x)=\frac{1}{4} x^{4}+x+5$ eine waagerechte Tangente hat.

See Solution

Problem 27899

Approximate $f(x)=\ln x$ for $x=9\pi$ and round to four decimal places. Choose the correct answer: a, b, c, or d.

See Solution

Problem 27900

Find the limits: (a) $\lim _{x \rightarrow 1}\left(\frac{x^{2}+x-2}{x^{2}+2 x-3}\right)$ , (b) $\lim _{x \rightarrow 2}\left(\frac{\sqrt{x^{2}+3}-\sqrt{7}}{x-2}\right)$ , (c) $\lim _{x \rightarrow-5} \frac{\frac{1}{5}+\frac{1}{x}}{5+x}$ . Justify without graphs or tables.

See Solution

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Calculus

Problem 27801

Differentiate the integral ∫x2sin⁡x(ln⁡∣t+4∣3dt)\int_{x^{2}}^{\sin x}(\sqrt[3]{\ln |t+4|} d t)∫x2sinx​(3ln∣t+4∣​dt) with respect to xxx.

Problem 27802

Finde Werte für a,b,c,da, b, c, da,b,c,d in f(t)=a⋅(t−b)⋅e−ct+df(t)=a \cdot(t-b) \cdot e^{-c t}+df(t)=a⋅(t−b)⋅e−ct+d für die Aussagen (1)-(5) zur Übernachtungszahlen.

Problem 27803

Sophie hat Fieber, beschrieben durch f(t)=1,5⋅t⋅e−0,5t+1+37f(t)=1,5 \cdot t \cdot e^{-0,5 t+1}+37f(t)=1,5⋅t⋅e−0,5t+1+37. Bestimme max. Temperatur, Abnahmezeit, langfristige Temp., und mehr.

Problem 27804

Find the volume of the solid formed by revolving the area between y=sec⁡xy=\sec xy=secx, y=2y=\sqrt{2}y=2​, x=−π4x=-\frac{\pi}{4}x=−4π​, x=π4x=\frac{\pi}{4}x=4π​ around y=−2y=-\sqrt{2}y=−2​.

Problem 27805

Find the limit: lim⁡x→2−3x−2\lim _{x \rightarrow 2^{-}} \frac{3}{x-2}limx→2−​x−23​.

Problem 27806

Find the distance a particle moves from t=0t=0t=0 to t=3t=3t=3 with velocity v(t)=etv(t)=e^{t}v(t)=et. Show your calculator input.

Problem 27807

Evaluate the derivative: ddx∫3x2−3x(t−1)2dt\frac{d}{d x} \int_{3}^{x^{2}-3 x}(t-1)^{2} d tdxd​∫3x2−3x​(t−1)2dt

Problem 27808

Find the function F(x)=∫3x+5dxF(x)=\int \frac{3}{x+5} dxF(x)=∫x+53​dx given F(−4)=−7F(-4)=-7F(−4)=−7.

Problem 27809

Estimate the velocity after 5 seconds using the trapezoid rule with acceleration a(t)a(t)a(t) values at t=0,1,2,3,4,5t = 0, 1, 2, 3, 4, 5t=0,1,2,3,4,5.

Problem 27810

Find f(5)f(5)f(5) if fff is the antiderivative of x3x2+3\frac{x^{3}}{x^{2}+3}x2+3x3​ and f(1)=7f(1)=7f(1)=7. Show your calculator input.

Problem 27811

Object has acceleration a(t)=2t−8a(t)=2t-8a(t)=2t−8. Given v(0)=15v(0)=15v(0)=15 and x(3)=12x(3)=12x(3)=12. Find v(t)v(t)v(t), when at rest, x(t)x(t)x(t), and displacement from [3,9][3,9][3,9].

Problem 27812

Find the inhaled air volume from t=0t=0t=0 to t=8t=8t=8 using f(t)=12sin⁡(2π5t)f(t)=\frac{1}{2} \sin \left(\frac{2 \pi}{5} t\right)f(t)=21​sin(52π​t). Round to three decimals.

Problem 27813

Find the average rate of change of the function f(x)=−x2−3x+5f(x)=-x^{2}-3x+5f(x)=−x2−3x+5 over the interval −3≤x≤4-3 \leq x \leq 4−3≤x≤4.

Problem 27814

Deposit \1000atan8.51000 at an 8.5% interest rate, compounded continuously. Find the balance after 5 years using 1000atan8.5F=\1000e0.085⋅51000e^{0.085 \cdot 5}1000e0.085⋅5. Round to the nearest cent.

Problem 27815

Problem 27816

Problem 27817

Problem 27818

A bakery sells 640 bagels daily at RM 8.00. For every RM 0.20 price drop, sales increase by 40. Find the price for max revenue using differentiation.

Problem 27819

Find the velocity v(t) v(t) v(t) and acceleration a(t) a(t) a(t) from the position function s(t)=t3−11t2+24t s(t) = t^{3} - 11t^{2} + 24t s(t)=t3−11t2+24t.

Problem 27820

A roller coaster moves at 42.5ms42.5 \frac{m}{s}42.5sm​ at 1.5m1.5 m1.5m high. Find: height at 8.3ms8.3 \frac{m}{s}8.3sm​, speed at lowest point, energy loss at 14ms14 \frac{m}{s}14sm​, and work to stop at 7.5m7.5 m7.5m.

Problem 27821

Find the area between the curves y=xy=\sqrt{x}y=x​ and y=x2y=x^{2}y=x2. What is the area? Options: none, 14\frac{1}{4}41​, 12\frac{1}{2}21​, 13\frac{1}{3}31​, 16\frac{1}{6}61​.

Problem 27822

Problem 27823

Find the volume of revolution for the region in the first quadrant between x=y2x=y^{2}x=y2 and x=6−yx=6-yx=6−y. (a) About the yyy-axis using the washer method. (b) About x=−1x=-1x=−1 using the washer method. (c) About the xxx-axis using the shell method.

Problem 27824

Find the volume of the solid of revolution for the region between y=x3+3y=x^{3}+3y=x3+3, y=4y=4y=4, and the yyy-axis using different methods.

Problem 27825

Find the value of α\alphaα if the differential of f(x)=αxf(x)=\alpha \sqrt{x}f(x)=αx​ from 1 to 4 is 12. Choices: a. 0 b. None c. 4 d. 2x2x2x e. 8.

Problem 27826

Determine where the graph of y=x3+x2−27xy=x^{3}+x^{2}-27 xy=x3+x2−27x is concave down: a. x≥−13x \geq \frac{-1}{3}x≥3−1​, b. x≤0x \leq 0x≤0, c. x≥0x \geq 0x≥0, d. −6≥x≥2-6 \geq x \geq 2−6≥x≥2, e. None.

Problem 27827

Find the constant α\alphaα if the differential of f(x)=αxf(x)=\alpha \sqrt{x}f(x)=αx​ from 1 to 4 equals 12.

Problem 27828

You and a friend bike to a restaurant; you ride east at 16mph16 \mathrm{mph}16mph and your friend north at 12mph12 \mathrm{mph}12mph. After 4mi4 \mathrm{mi}4mi, find the rate of distance change between you.

Problem 27829

Problem 27830

Minimize Marginal Cost: Given C(x)=0.0001x3−0.06x2+4x+100C(x) = 0.0001 x^{3} - 0.06 x^{2} + 4x + 100C(x)=0.0001x3−0.06x2+4x+100, is marginal cost at x=100x=100x=100 increasing, decreasing, or constant? Find minimum marginal cost.

Problem 27831

Identify which function is continuous: A) f(x)=4x3f(x)=\frac{4}{x^{3}}f(x)=x34​, B) g(x)=15−xg(x)=\frac{1}{5-x}g(x)=5−x1​, C) h(x)=x211h(x)=\frac{x^{2}}{11}h(x)=11x2​.

Problem 27832

Find the singular solution of the equation dydx=(y−3)2\frac{d y}{d x}=(y-3)^{2}dxdy​=(y−3)2.

Problem 27833

Find the derivative of y=x22x−3y = x^{2} \sqrt{2x - 3}y=x22x−3​.

Problem 27834

Find the maximum height mmm from which a camera can be dropped without exceeding a speed of 16.0 m/s16.0 \mathrm{~m/s}16.0 m/s.

Problem 27835

Find the area between the curves y=3xy=3 \sqrt{x}y=3x​ and y=3x2y=3 x^{2}y=3x2.

Problem 27836

Given the function f(x)=x4+1x2f(x)=\frac{x^{4}+1}{x^{2}}f(x)=x2x4+1​, determine which statement is true about its properties.

Problem 27837

Find the critical points of the function f(x)=x3+3x2−x−3f(x) = x^{3} + 3x^{2} - x - 3f(x)=x3+3x2−x−3 by calculating its derivative and solving f′(x)=0f'(x) = 0f′(x)=0.

Problem 27838

A particle's position is given by s(t)=2t3−11t2+12t−13s(t)=2 t^{3}-11 t^{2}+12 t-13s(t)=2t3−11t2+12t−13. Find velocity, acceleration, rest times, and direction changes.

Problem 27839

Find the average rate of change of the function f(x)=0.00186x3−0.0921x2+2.67x−1.73f(x)=0.00186 x^{3}-0.0921 x^{2}+2.67 x-1.73f(x)=0.00186x3−0.0921x2+2.67x−1.73 from 2005 to 2015.

Problem 27840

Analyze the function f(x)=2x3+3x2−36xf(x)=2 x^{3}+3 x^{2}-36 xf(x)=2x3+3x2−36x for increasing/decreasing intervals, local extrema, and concavity.

Problem 27841

Analyze the function f(x)=x3−12x+2f(x)=x^{3}-12 x+2f(x)=x3−12x+2 for increase/decrease, max/min values, concavity, and sketch it. Also do the same for f(x)=36x+3x2−2f(x)=36 x+3 x^{2}-2f(x)=36x+3x2−2.

Problem 27842

Find intervals of increase/decrease, local max/min, concavity, inflection points, and sketch the graph for f(x)=36x+3x2−2x3f(x)=36 x+3 x^{2}-2 x^{3}f(x)=36x+3x2−2x3.

Differentiate the integral $\int_{x^{2}}^{\sin x}(\sqrt[3]{\ln |t+4|} d t)$ with respect to $x$ .

Finde Werte für $a, b, c, d$ in $f(t)=a \cdot(t-b) \cdot e^{-c t}+d$ für die Aussagen (1)-(5) zur Übernachtungszahlen.

Sophie hat Fieber, beschrieben durch $f(t)=1,5 \cdot t \cdot e^{-0,5 t+1}+37$ . Bestimme max. Temperatur, Abnahmezeit, langfristige Temp., und mehr.

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=\sqrt{2}$ , $x=-\frac{\pi}{4}$ , $x=\frac{\pi}{4}$ around $y=-\sqrt{2}$ .

Find the limit: $\lim _{x \rightarrow 2^{-}} \frac{3}{x-2}$ .

Find the distance a particle moves from $t=0$ to $t=3$ with velocity $v(t)=e^{t}$ . Show your calculator input.

Evaluate the derivative: $\frac{d}{d x} \int_{3}^{x^{2}-3 x}(t-1)^{2} d t$

Find the function $F(x)=\int \frac{3}{x+5} dx$ given $F(-4)=-7$ .

Estimate the velocity after 5 seconds using the trapezoid rule with acceleration $a(t)$ values at $t = 0, 1, 2, 3, 4, 5$ .

Find $f(5)$ if $f$ is the antiderivative of $\frac{x^{3}}{x^{2}+3}$ and $f(1)=7$ . Show your calculator input.

Object has acceleration $a(t)=2t-8$ . Given $v(0)=15$ and $x(3)=12$ . Find $v(t)$ , when at rest, $x(t)$ , and displacement from $[3,9]$ .

Find the inhaled air volume from $t=0$ to $t=8$ using $f(t)=\frac{1}{2} \sin \left(\frac{2 \pi}{5} t\right)$ . Round to three decimals.

Find the average rate of change of the function $f(x)=-x^{2}-3x+5$ over the interval $-3 \leq x \leq 4$ .

Deposit \ $1000 at an 8.5% interest rate, compounded continuously. Find the balance after 5 years using$ F=\ $1000e^{0.085 \cdot 5}$ . Round to the nearest cent.

Find the velocity $v(t)$ and acceleration $a(t)$ from the position function $s(t) = t^{3} - 11t^{2} + 24t$ .

A roller coaster moves at $42.5 \frac{m}{s}$ at $1.5 m$ high. Find: height at $8.3 \frac{m}{s}$ , speed at lowest point, energy loss at $14 \frac{m}{s}$ , and work to stop at $7.5 m$ .

Find the area between the curves $y=\sqrt{x}$ and $y=x^{2}$ . What is the area? Options: none, $\frac{1}{4}$ , $\frac{1}{2}$ , $\frac{1}{3}$ , $\frac{1}{6}$ .

Find the volume of revolution for the region in the first quadrant between $x=y^{2}$ and $x=6-y$ .
(a) About the $y$ -axis using the washer method. (b) About $x=-1$ using the washer method. (c) About the $x$ -axis using the shell method.

Find the volume of the solid of revolution for the region between $y=x^{3}+3$ , $y=4$ , and the $y$ -axis using different methods.

Find the value of $\alpha$ if the differential of $f(x)=\alpha \sqrt{x}$ from 1 to 4 is 12. Choices: a. 0 b. None c. 4 d. $2x$ e. 8.

Determine where the graph of $y=x^{3}+x^{2}-27 x$ is concave down: a. $x \geq \frac{-1}{3}$ , b. $x \leq 0$ , c. $x \geq 0$ , d. $-6 \geq x \geq 2$ , e. None.

Find the constant $\alpha$ if the differential of $f(x)=\alpha \sqrt{x}$ from 1 to 4 equals 12.

You and a friend bike to a restaurant; you ride east at $16 \mathrm{mph}$ and your friend north at $12 \mathrm{mph}$ . After $4 \mathrm{mi}$ , find the rate of distance change between you.

Minimize Marginal Cost: Given $C(x) = 0.0001 x^{3} - 0.06 x^{2} + 4x + 100$ , is marginal cost at $x=100$ increasing, decreasing, or constant? Find minimum marginal cost.

Identify which function is continuous: A) $f(x)=\frac{4}{x^{3}}$ , B) $g(x)=\frac{1}{5-x}$ , C) $h(x)=\frac{x^{2}}{11}$ .

Find the singular solution of the equation $\frac{d y}{d x}=(y-3)^{2}$ .

Find the derivative of $y = x^{2} \sqrt{2x - 3}$ .

Find the maximum height $m$ from which a camera can be dropped without exceeding a speed of $16.0 \mathrm{~m/s}$ .

Find the area between the curves $y=3 \sqrt{x}$ and $y=3 x^{2}$ .

Given the function $f(x)=\frac{x^{4}+1}{x^{2}}$ , determine which statement is true about its properties.

Find the critical points of the function $f(x) = x^{3} + 3x^{2} - x - 3$ by calculating its derivative and solving $f'(x) = 0$ .

A particle's position is given by $s(t)=2 t^{3}-11 t^{2}+12 t-13$ . Find velocity, acceleration, rest times, and direction changes.

Find the average rate of change of the function $f(x)=0.00186 x^{3}-0.0921 x^{2}+2.67 x-1.73$ from 2005 to 2015.

Analyze the function $f(x)=2 x^{3}+3 x^{2}-36 x$ for increasing/decreasing intervals, local extrema, and concavity.

Analyze the function $f(x)=x^{3}-12 x+2$ for increase/decrease, max/min values, concavity, and sketch it. Also do the same for $f(x)=36 x+3 x^{2}-2$ .

Find intervals of increase/decrease, local max/min, concavity, inflection points, and sketch the graph for $f(x)=36 x+3 x^{2}-2 x^{3}$ .

Analyze $f(x)=x^{3}-12 x+2$ and $f(x)=36 x+3 x^{2}-2 x^{3}$ for increase/decrease, max/min, concavity, and sketch the graphs.

Find the dimension of $d c / d w$ and compute $d c / d w$ for $c=\sum_{i=1}^{N} w_{i}(x_{i}-y_{i})^{2}$ .

Find the limit: $\lim _{x \rightarrow-4} \frac{x^{2}-16}{x-1}$ .

Find the derivative of the function $y = \sqrt[3]{x(x+1)(x+2)}$ .

A cup of coffee cools from $95^{\circ} \mathrm{C}$ to $77^{\circ} \mathrm{C}$ at $1^{\circ} \mathrm{C}$ per minute. Find time to reach $77^{\circ} \mathrm{C}$ . Round to two decimal places.

Find the derivatives of these functions: (1) $y=x \ln x$ , (2) $y=\sin(x^{2})$ .

Prove that for $x>0$ , the inequalities $x-\frac{x^{2}}{2}<\ln (1+x)<x$ hold true.

Determine the global max and min of $f(x)=x^{5}+(1-x)^{5}$ for $x$ in the interval $[0,1]$ .

Find the limit: $\lim _{n \rightarrow \infty} \sqrt[n]{3}$ .

Find the limit: $\lim _{n \rightarrow \infty}(\sqrt{n+1}-\sqrt{n})$ .

Find the derivative of $y = \sqrt[3]{x(x+1)(x+2)}$ .

Find the limit: $\lim _{n \rightarrow \infty} \frac{2^{n}}{n !}$ .

Find the limit: $\lim _{x \rightarrow-3} \frac{x^{2}+x-6}{x^{2}+4 x+3}$ .

Jeriel invested \ $72,000 at$ 7.25\% $continuous interest. Amelia invested \$72,000 at$ 6.75\%$ daily. How much longer for Amelia's money to double?

Find the tangent line approximation of $f(x)=2 \cos x + 1$ at $x=\frac{\pi}{2}$ to estimate $f(1.5)$ .

Find the limit as $t$ approaches 0 for the expression $\frac{1}{t}-\frac{1}{t^{2}+t}$ .

Find the maximum acceleration for the velocity $v(t)=\frac{2}{3} t^{3}-4 t^{2}+8 t-2$ on $0 \leq t \leq 3$ .

Find the time $t$ when the particle at $x(t)=e^{-t} \cos t$ is farthest right for $0 \leq t \leq 2 \pi$ .

Find critical numbers, increasing/decreasing intervals, local extrema, and concavity for $f(x)=x^{3}-6x^{2}+9x+1$ .

Calculate the integral: $\int \frac{9 x+2}{x^{2}+x-6} d x$

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{4}+1}}{x^{2}-3 x+5}$ .

Find the volume of the solid of revolution for the region between $y=x^{3}+3$ , $y=4$ , and the $y$ -axis, using the disk method around the $y$ -axis.

Ein Dozent will den besten Weg von $P_{1}=(10,20)$ nach $P_{2}=(10,0)$ finden, indem er Fahrrad und S-Bahn kombiniert. Berechnen Sie den Punkt auf der Geraden $g(x)=10-x$ , der die kürzeste Reisezeit ergibt.

Ein Dozent plant seine Wege zur Hochschule. Bestimme den Punkt auf der Geraden $g(x)=10-x$ , der die Reisezeit minimiert.

Untersuchen Sie die Funktion $f(x)=\frac{x^{2}}{x^{2}-4}$ vollständig und skizzieren Sie den Graphen im passenden Intervall.

Find the volume of the solid formed by revolving the region in the first quadrant between $y=x^{3}+3$ , $y=4$ , and the $y$ -axis around $x=-1$ using the shell method.

Find the best linear approximation, $g$ , to the function $f(x)=4 \sqrt{x}$ on the interval $[0,1]$ .

Find the limit: $\lim _{x \rightarrow-2} \frac{x+2}{\sqrt{x+3}-1}$ .

Find the limit as $x$ approaches 4 from the left for $\frac{2x^2 - x + 5}{(x-4)(x-4)}$ .

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