Calculus

Problem 20701

Find how many people were ill at the start of the epidemic using $N(0)=\frac{20,000}{1+20 e^{-2.5\cdot 0}}$ .

See Solution

Problem 20702

Find the slope of $y=2 \cos x \sin 2 x$ at $x=\frac{\pi}{2}$ and the tangent line to $y=x^{2} \sin (2 x)$ at $x=-\pi$ .

See Solution

Problem 20703

Given $y=x^{3}+p x^{2}$ , show the origin is stationary, find other points in terms of $p$ , and determine their nature. For $y=x^{3}+p x^{2}+p x$ , find $p$ values with no stationary points.

See Solution

Problem 20704

Find the second derivatives of $z=(x^{3}+2xy)(y-x^{2})$ at $(x=2, y=1)$ : (a) $\frac{\partial^{2} z}{\partial x^{2}}$ ; (b) $\frac{\partial^{2} z}{\partial y \partial x}$ .

See Solution

Problem 20705

Find the second derivatives of the function $z=(x^{3}+2xy)(y-x^{2})$ at $x=2$ , $y=1$ . Round to two decimals. (a) $\frac{\partial^{2} z}{\partial x^{2}}=$ (b) $\frac{\partial^{2} z}{\partial y \partial x}=$

See Solution

Problem 20706

Find the integral of the function $\frac{1}{3} x$ .

See Solution

Problem 20707

Evaluate the limits using L'Hôpital's Rule:
1) $\lim _{x \rightarrow 0} \frac{\sin (14 x)}{\sin (8 x)}=$ 2) $\lim _{x \rightarrow 0} \frac{\sin (a x)}{\sin (b x)}=\square$ (with $b \neq 0$ )

See Solution

Problem 20708

Evaluate the limit using L'Hôpital's Rule: $\lim _{x \rightarrow \infty}\left(1+x^{5}\right)^{1 / x}=\square$ . Enter INF, -INF, or DNE.

See Solution

Problem 20709

Calculate the area between $f(x)=9 x+8$ and $g(x)=x^{2}+6 x+2$ from $x=0$ to $x=2$ .

See Solution

Problem 20710

Berechnen Sie die Integrale:
1) $\int_{0}^{2}(2+x)^{3} d x$ 2) $\int_{2}^{3}\left(1+\frac{1}{x^{2}}\right) d x$ 3) $\int_{2}^{4} \frac{-1}{x^{3}} d x$ 4) $\int_{0}^{9} \frac{2}{5} \sqrt{x} d x$
Verwenden Sie den Hauptsatz.

See Solution

Problem 20711

Bestimmen Sie die Ableitung von $f(x)=3 x \cdot e^{-\frac{x^{2}}{2}}$ .

See Solution

Problem 20712

Find $\frac{d x}{d t}$ , $\frac{d y}{d t}$ , and $\frac{d y}{d x}$ for $x=4 t^{3}+3 t$ , $y=6 t-5 t^{2}$ .

See Solution

Problem 20713

Find $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ for $x=t^{2}+7$ , $y=t^{2}+7t$ . When is the curve concave upward?

See Solution

Problem 20714

Evaluate the limits using L'Hôpital's Rule. Enter INF for $\infty$ , -INF for $-\infty$ , or DNE if it doesn't exist. a) $\lim _{x \rightarrow \pi / 2} 5 \tan x \cos x=\square$ b) $\lim _{x \rightarrow \pi / 2} 4 \tan x \sin (2 x)=\square$

See Solution

Problem 20715

Vereinfachen Sie die Funktion: $f^{\prime}(x)=3 \cdot e^{-\frac{x^{2}}{2}} + 3x \cdot e^{-\frac{x^{2}}{2}} \cdot (-x)$

See Solution

Problem 20716

Find the area between the curve $y=9x^2$ and the $x$ -axis on the interval $[0, b]$ using a definite integral.

See Solution

Problem 20717

Find the area between the curve defined by $x=\sin ^{2}(t)$ and $y=8 \cos (t)$ and the $y$ -axis.

See Solution

Problem 20718

Berechne die Integrale: b) $\int_{2}^{3}\left(1+\frac{1}{x^{2}}\right) d x$ , c) $\int_{2}^{4} \frac{-1}{x^{3}} d x$ .

See Solution

Problem 20719

Calculate the area between $y=\sin(x)$ and $y=\cos(x)$ from $x=\frac{\pi}{3}$ to $x=\frac{3\pi}{4}$ .

See Solution

Problem 20720

Find the Riemann sum formula for $f(x)=5 x^{2}$ on $[0,3]$ : $S_{n}=\square$ (Use $n$ as the variable.)

See Solution

Problem 20721

Determine which statement about the piecewise function $f(x)$ is false: (A) $f$ is continuous at $x=1$ . (B) $f$ is continuous at $x=2$ . (C) $f$ is continuous at $x=3$ .

See Solution

Problem 20722

Identify the first error in confirming the continuity of $f(x)=\frac{x^{2}-5 x+4}{x^{2}-6 x+8}$ at $x=4$ . Steps provided.

See Solution

Problem 20723

Express the limit $\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n}\left(c_{k}^{5}-3 c_{k}\right) \Delta x_{k}$ as a definite integral.

See Solution

Problem 20724

Find the Riemann sum formula for $f(x)=5 x^{2}$ on $[0,3]$ using right endpoints, then take the limit as $n \to \infty$ .

See Solution

Problem 20725

Evaluate the limits using L'Hôpital's Rule. Enter INF for $\infty$ , -INF for $-\infty$ , or DNE if it doesn't exist. a) $\lim _{x \rightarrow \pi / 2} 5 \tan x \cos x=5$ b) $\lim _{x \rightarrow \pi / 2} 4 \tan x \sin (2 x)=\square$

See Solution

Problem 20726

Given the piecewise function $f(x)$ , which statement is false about its continuity at the specified points? (A) $x=1$ , (B) $x=2$ , (C) $x=3$ , (D) $x=4$ .

See Solution

Problem 20727

Given the function $f(x)=\frac{x^{2}+2 x-12}{8 \cos \left(\frac{\pi}{2} x\right)+2 x^{2}}$ , why is $f$ not continuous at $x=2$ ?

See Solution

Problem 20728

Identify the first error in confirming the continuity of $f(x)=\frac{x^{2}-5 x+4}{x^{2}-6 x+8}$ at $x=4$ . Steps given.

See Solution

Problem 20729

Calculate the area between $y=\sin (x)$ and $y=\cos (x)$ from $x=\frac{\pi}{3}$ to $x=\frac{3 \pi}{4}$ .

See Solution

Problem 20730

Write a Riemann sum formula for $f(x)=5 x^{2}$ on $[0,4]$ : $s_{n}=\sum_{i=1}^{n} f\left(\frac{4i}{n}\right) \cdot \frac{4}{n}$

See Solution

Problem 20731

A pebble creates ripples with an area increasing at $21 \mathrm{~cm}^{2} / \mathrm{s}$ . Find the radius change rate at $6 \mathrm{~cm}$ .

See Solution

Problem 20732

Find the limit using L'Hôpital's Rule: $\lim _{x \rightarrow 0^{+}}(10 x)^{x}=\square$ . Enter INF, -INF, or DNE.

See Solution

Problem 20733

An airplane is $10,000 \mathrm{ft}$ high, moving at $700 \mathrm{ft/s}$ . Find the rate the angle to an observer decreases when $40,000 \mathrm{ft}$ away.

See Solution

Problem 20734

Find the Riemann sum formula for $f(x)=5 x^{2}$ on $[0,4]$ using right endpoints. Then, find the limit as $n \to \infty$ .

See Solution

Problem 20735

Evaluate the integral $\int(5 x-4)^{4} d x$ using the substitution $u=5 x-4$ .

See Solution

Problem 20736

Bestimme die Stammfunktionen für die folgenden Funktionen mit gegebenen Bedingungen: a) $f(x)=x^{2}$ durch $P(1 \mid 1)$ . b) $f(x)=1-x^{2}$ bei $y=4$ . c) $f(x)=2+x$ mit Nullstelle bei $x=1$ .

See Solution

Problem 20737

Find the area between the curves $f(x)=260+13x-13x^2$ and $g(x)=13x^2-65x$ .

See Solution

Problem 20738

Estimez $\int_{2}^{5} f(x) \, dx$ avec trois méthodes : (a) points médians, (b) trapèzes, (c) Simpson. Donnez les résultats.

See Solution

Problem 20739

Find the max and min points of $y=\sin(x)$ on the interval $[0, 2\pi]$ .

See Solution

Problem 20740

Use substitution to solve the integral: $\int 4.4 e^{(-3 x+4)} d x$ .

See Solution

Problem 20741

Find the integral: $\int \frac{x}{(x^{2}+5)^{1.3}} \, dx$

See Solution

Problem 20742

Find the number $c$ that satisfies the Mean Value Theorem for $f(x)=\sqrt{4x-3}$ on $1 \leq x \leq 3$ . What is $c$ ? (A) 1.5 (B) 1.75 (C) 2 (D) 2.25

See Solution

Problem 20743

Determine if the sequence $a_{n}=\frac{4 \sqrt{n}}{\sqrt{n}+3}$ converges or diverges. Find the limit if it converges.

See Solution

Problem 20744

Find the number $c$ that satisfies the Mean Value Theorem for $g(x)=x^{3}+12x^{2}+36x$ on $[-8, -2]$ . What is $c$ ?
Choose 1 answer: (A) -7 (B) -6 (C) -3 (D) -1

See Solution

Problem 20745

Find the sum of the series given the partial sums $s_{n}=8-3(0.8)^{n}$ .

See Solution

Problem 20746

Find $h^{\prime}(1)$ for $h(x)=\ln \left[(f(x))^{2}+1\right]$ given $f(1)=-1$ and $f^{\prime}(1)=1$ .

See Solution

Problem 20747

Given $f(x)=\frac{x+7}{x}$ : a) Show why it has an inverse using the derivative. b) Find $f^{-1}$ . c) Determine domains and ranges of $f$ and $f^{-1}$ .

See Solution

Problem 20748

Find the vertical asymptotes of $f(x)=\frac{-5}{\cos \left(\frac{3 \pi}{2} x\right)}$ from the options given.

See Solution

Problem 20749

A 2.5 kg object is dropped from 10.0 m. Find its speed when it hits the ground.

See Solution

Problem 20750

A 10-foot ladder leans against a wall. If the bottom slides away at 4 ft/sec, how fast is the top sliding down when it's 8 ft from the wall?

See Solution

Problem 20751

Find the indefinite integral: $\int \frac{e^{4 x}-e^{2 x}+1}{e^{x}} d x$ (Use $C$ for the constant of integration).

See Solution

Problem 20752

Calculate the maximum height of a projectile launched upward at 128 ft/s, ignoring air resistance.

See Solution

Problem 20753

Find the tangent line equation for $h(x)=-x e^{9-x}$ at the point $(9,-9)$ .

See Solution

Problem 20754

Solve the equation: $\int(4 x+6)dx = 8 x + 4$ .

See Solution

Problem 20755

Bestimme die Ableitung der Funktion $f(x)=\frac{x}{2 \cdot \sqrt{x}}$ .

See Solution

Problem 20756

Can the Mean Value Theorem apply to $f(x)=x^{2/3}$ on $[1, 27]$ ? Select all that apply.

See Solution

Problem 20757

Given the function $f(x)=x^{2}-2x+9$ , find critical numbers, intervals of increase/decrease, and relative extrema.

See Solution

Problem 20758

Find the car's velocity when $s(t)=0$ and $s(t)=14$ for $s(t)=t^{2}-3t-4$ .

See Solution

Problem 20759

Find the inflection point of $f(x)=x^{3}-12 x^{2}$ and describe concavity in interval notation.

See Solution

Problem 20760

A hummingbird's position is $s(t)=3 t^{3}-7 t$ . Find velocity at $t=1$ sec, acceleration at $t=1$ sec, and when velocity=0.

See Solution

Problem 20761

Given functions for a particle's position, find velocity and acceleration, then identify intervals for speeding up or slowing down.
1. $s(t)=2 t^{3}-3 t^{2}-12 t+8$
2. $s(t)=2 t^{3}-15 t^{2}+36 t-10$

See Solution

Problem 20762

Bacteria grow exponentially at 3.8/hour. How many hours until the population doubles?

See Solution

Problem 20763

Find the specific solution for the equation $f^{\prime \prime}(x)=3 \cos (x)$ with $f^{\prime}(0)=4$ , $f(0)=-5$ .

See Solution

Problem 20764

Finde die Tangentengleichung von $f$ , die parallel zu $g$ ist. Wähle aus: a) bis f) mit verschiedenen $f$ und $g$ .

See Solution

Problem 20765

Bestimme den Wert der Reihe $\sum_{k=0}^{\infty} \frac{1}{\left((-1)^{k}+4\right)^{k}}.$

See Solution

Problem 20766

Determine if the series $\sum_{k=1}^{\infty}(-1)^{k+1}\left(\frac{7 k^{3}+k}{3 k^{3}+k+1}\right)^{k}$ converges or diverges using tests.

See Solution

Problem 20767

Calculate the sum $\sum_{k=1}^{n} 5\left(\frac{4 k}{n}\right)^{2} \frac{4}{n}$ .

See Solution

Problem 20768

Bacteria growth is modeled by $n(t)=925 e^{0.4 t}$ . Find (a) relative growth rate, (b) initial population, (c) population at $t=5$ .

See Solution

Problem 20769

Use the Root Test to check if the series $\sum_{k=1}^{\infty}\left(-\frac{7 k+6}{6 k}\right)^{k}$ converges or diverges.

See Solution

Problem 20770

Use the Root Test to check if the series $\sum_{k=1}^{\infty}\left(1+\frac{17}{k}\right)^{k^{2}}$ converges or diverges.

See Solution

Problem 20771

Untersuche die Funktionen $f_{k}(x)=x^{4}-k x^{2}$ auf Extrem- und Wendepunkte für $k=-2$ und $k=2$ . Bestimme die Tiefpunkte für $k>0$ und zeige, dass $\frac{x_{e}}{x_{w}}$ unabhängig von $k$ ist.

See Solution

Problem 20772

Find the intervals where the function $f(x)=\frac{x^{2}}{x-3}$ is decreasing.

See Solution

Problem 20773

Use the Ratio Test to check if the series $\sum_{k=1}^{\infty} \frac{2^{k}}{k^{149}}$ converges or diverges.

See Solution

Problem 20774

Bestimme die Ableitung von $f(x)=7^{2 x}-3^{x}$ .

See Solution

Problem 20775

Use the Ratio Test to check if the series $\sum_{k=1}^{\infty} \frac{8(k !)^{2}}{(2 k) !}$ converges or diverges.

See Solution

Problem 20776

Determine if the series $\sum_{k=1}^{\infty}\left(\frac{4 k^{3}}{5 k^{3}+1}\right)^{k}$ converges absolutely, conditionally, or diverges.

See Solution

Problem 20777

Find values of $p$ for which the series $\sum_{k=2}^{\infty} \frac{(\ln k)^{2}}{k^{p}}$ converges.

See Solution

Problem 20778

Use the Ratio Test to check if the series $\sum_{k=1}^{\infty} k e^{-6 k}$ converges or diverges. Choose A, B, or C.

See Solution

Problem 20779

Determine if the series $\sum_{k=1}^{\infty} \frac{k !}{(2 k+2) !}$ converges absolutely, conditionally, or diverges.

See Solution

Problem 20780

Ein Körper bewegt sich mit $s(t)=4 t^{2}$ . Bestimmen Sie die Änderungsrate von $s$ für $t_{0}=1$ und $t_{1}=5$ . Was bedeutet diese Rate?

See Solution

Problem 20781

Lamar's blood pressure is modeled by $p(t)=87+16 \cos (113 \pi t)$ . Find heartbeats/min, min BP, and cycle time.

See Solution

Problem 20782

Determine the values of $x$ for which the series $\sum_{k=1}^{\infty} \frac{x^{k}}{k !}$ converges using the Ratio or Root Test.

See Solution

Problem 20783

Find the Gini coefficient for countries A and B using their income functions: $f_A(x)=0.7x+0.27x^2+0.03x^3$ and $f_B(x)=0.6x+0.05x^2+0.35x^3$ . Round to three decimal places.

See Solution

Problem 20784

Gegeben sind die Funktionen $f(x)=0,5 x^{2}$ und $g(x)=3 x^{3}+1$ . Bestimme den Differenzenquotienten für die Intervalle: a) $I=[0 ; 2]$ , b) $I=[-1 ; 3]$ , c) $I=[-1 ; 1]$ , d) $I=[-2 ;-1]$ .

See Solution

Problem 20785

Calculate the consumers' surplus for the demand function $D(x)=\frac{750}{x+10}$ at $x_{E}=16$ units, rounding to the nearest cent.

See Solution

Problem 20786

Calculate the consumers' surplus for the demand function $D(x)=\frac{750}{6+x}$ at $x_{E}=1$ units, rounded to the nearest cent.

See Solution

Problem 20787

Find the distance between the tortoise and hare from when they match speeds to one hour later using $H(t)=-8 t^{2}+35 t$ and $T(t)=3 t$ .

See Solution

Problem 20788

Bestimmen Sie die Ableitungen der Funktionen a) $f(x)=4^{x}$ , b) $f(x)=3 \cdot 2^{x}-5$ , c) $f(x)=7^{2x}-3^{x}$ , d) $f(x)=2^{-x}+2^{x}$ .

See Solution

Problem 20789

Find the consumers' surplus for the demand function $D(x)=\frac{-2}{9} x^{2}+324$ at $x_{E}=27$ units.

See Solution

Problem 20790

Find the consumers' surplus for the demand function $D(x)=\frac{-1}{3} x^{2}+300$ at $x_{K}=15$ units.

See Solution

Problem 20791

Determine if the integral $\int_{6}^{\infty} \frac{2}{(x+4)^{3 / 2}} d x$ is convergent or divergent and evaluate if convergent.

See Solution

Problem 20792

Untersuchen Sie $f(x)=e^{-0,5x^2}$ auf Achsenschnittpunkte, Extrema, Wendepunkte und Symmetrie. Skizzieren Sie $f$ für $-3 \leq x \leq 3$ . Bestimmen Sie Tangente und Normale im Punkt $P(1|f(1))$ und berechnen Sie den Flächeninhalt und Umfang des Dreiecks.

See Solution

Problem 20793

Evaluate the integral: $\int 14 \ln \sqrt[3]{x} \, dx$ (Include constant $C$ )

See Solution

Problem 20794

Evaluate the integral: $\int 17 \ln \sqrt[3]{x} \, dx$

See Solution

Problem 20795

Untersuchen Sie die Funktion $f(x)=e^{-0.5x^2}$ auf Schnittpunkte, Extrema, Wendepunkte und Symmetrie. Skizzieren Sie $f$ für $-3 \leq x \leq 3$ . Bestimmen Sie die Tangente $t$ und Normale $n$ in $P(1|f(1))$ und berechnen Sie den Flächeninhalt und Umfang des Dreiecks, das von $t$ , $n$ und der $y$ -Achse gebildet wird.

See Solution

Problem 20796

Determine if the function $f(x)$ is continuous and differentiable at $x=1$ where $f(x)=\begin{cases} 7x^2-12x & \text{if } x \geq 1 \\ x^2-6 & \text{if } x < 1 \end{cases}$ .

See Solution

Problem 20797

A space station with a radius of $2400 \mathrm{~m}$ needs to create $1 \mathrm{~g}$ of centripetal acceleration. Find the tangential velocity and rotation time.

See Solution

Problem 20798

Evaluate the integral: $\int \frac{x-9}{(x+6)(x-3)} d x$

See Solution

Problem 20799

Zeichnen Sie die Tangente an $f(x)=-x^{3}+3 x^{2}$ im Punkt $\mathrm{P}(1 \mid 2)$ und bestimmen Sie die Steigung.

See Solution

Problem 20800

Find the tangent line to $f(x) = \sqrt{3x - 1}$ parallel to $g(x) = 1.5x - 2$ .

See Solution

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Calculus

Problem 20701

Find how many people were ill at the start of the epidemic using N(0)=20,0001+20e−2.5⋅0N(0)=\frac{20,000}{1+20 e^{-2.5\cdot 0}}N(0)=1+20e−2.5⋅020,000​.

Problem 20702

Find the slope of y=2cos⁡xsin⁡2xy=2 \cos x \sin 2 xy=2cosxsin2x at x=π2x=\frac{\pi}{2}x=2π​ and the tangent line to y=x2sin⁡(2x)y=x^{2} \sin (2 x)y=x2sin(2x) at x=−πx=-\pix=−π.

Problem 20703

Given y=x3+px2y=x^{3}+p x^{2}y=x3+px2, show the origin is stationary, find other points in terms of ppp, and determine their nature. For y=x3+px2+pxy=x^{3}+p x^{2}+p xy=x3+px2+px, find ppp values with no stationary points.

Problem 20704

Find the second derivatives of z=(x3+2xy)(y−x2)z=(x^{3}+2xy)(y-x^{2})z=(x3+2xy)(y−x2) at (x=2,y=1)(x=2, y=1)(x=2,y=1): (a) ∂2z∂x2\frac{\partial^{2} z}{\partial x^{2}}∂x2∂2z​; (b) ∂2z∂y∂x\frac{\partial^{2} z}{\partial y \partial x}∂y∂x∂2z​.

Problem 20705

Find the second derivatives of the function z=(x3+2xy)(y−x2)z=(x^{3}+2xy)(y-x^{2})z=(x3+2xy)(y−x2) at x=2x=2x=2, y=1y=1y=1. Round to two decimals. (a) ∂2z∂x2=\frac{\partial^{2} z}{\partial x^{2}}=∂x2∂2z​= (b) ∂2z∂y∂x=\frac{\partial^{2} z}{\partial y \partial x}=∂y∂x∂2z​=

Problem 20706

Find the integral of the function 13x\frac{1}{3} x31​x.

Problem 20707

Problem 20708

Evaluate the limit using L'Hôpital's Rule: lim⁡x→∞(1+x5)1/x=□\lim _{x \rightarrow \infty}\left(1+x^{5}\right)^{1 / x}=\squarex→∞lim​(1+x5)1/x=□. Enter INF, -INF, or DNE.

Problem 20709

Calculate the area between f(x)=9x+8f(x)=9 x+8f(x)=9x+8 and g(x)=x2+6x+2g(x)=x^{2}+6 x+2g(x)=x2+6x+2 from x=0x=0x=0 to x=2x=2x=2.

Problem 20710

Problem 20711

Bestimmen Sie die Ableitung von f(x)=3x⋅e−x22f(x)=3 x \cdot e^{-\frac{x^{2}}{2}}f(x)=3x⋅e−2x2​.

Problem 20712

Find dxdt\frac{d x}{d t}dtdx​, dydt\frac{d y}{d t}dtdy​, and dydx\frac{d y}{d x}dxdy​ for x=4t3+3tx=4 t^{3}+3 tx=4t3+3t, y=6t−5t2y=6 t-5 t^{2}y=6t−5t2.

Problem 20713

Find dydx\frac{d y}{d x}dxdy​ and d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ for x=t2+7x=t^{2}+7x=t2+7, y=t2+7ty=t^{2}+7ty=t2+7t. When is the curve concave upward?

Problem 20714

Problem 20715

Vereinfachen Sie die Funktion: f′(x)=3⋅e−x22+3x⋅e−x22⋅(−x)f^{\prime}(x)=3 \cdot e^{-\frac{x^{2}}{2}} + 3x \cdot e^{-\frac{x^{2}}{2}} \cdot (-x)f′(x)=3⋅e−2x2​+3x⋅e−2x2​⋅(−x)

Problem 20716

Find the area between the curve y=9x2y=9x^2y=9x2 and the xxx-axis on the interval [0,b][0, b][0,b] using a definite integral.

Problem 20717

Find the area between the curve defined by x=sin⁡2(t)x=\sin ^{2}(t)x=sin2(t) and y=8cos⁡(t)y=8 \cos (t)y=8cos(t) and the yyy-axis.

Problem 20718

Berechne die Integrale: b) ∫23(1+1x2)dx\int_{2}^{3}\left(1+\frac{1}{x^{2}}\right) d x∫23​(1+x21​)dx, c) ∫24−1x3dx\int_{2}^{4} \frac{-1}{x^{3}} d x∫24​x3−1​dx.

Problem 20719

Calculate the area between y=sin⁡(x)y=\sin(x)y=sin(x) and y=cos⁡(x)y=\cos(x)y=cos(x) from x=π3x=\frac{\pi}{3}x=3π​ to x=3π4x=\frac{3\pi}{4}x=43π​.

Problem 20720

Find the Riemann sum formula for f(x)=5x2f(x)=5 x^{2}f(x)=5x2 on [0,3][0,3][0,3]: Sn=□S_{n}=\squareSn​=□ (Use nnn as the variable.)

Problem 20721

Determine which statement about the piecewise function f(x)f(x)f(x) is false: (A) fff is continuous at x=1x=1x=1. (B) fff is continuous at x=2x=2x=2. (C) fff is continuous at x=3x=3x=3.

Problem 20722

Identify the first error in confirming the continuity of f(x)=x2−5x+4x2−6x+8f(x)=\frac{x^{2}-5 x+4}{x^{2}-6 x+8}f(x)=x2−6x+8x2−5x+4​ at x=4x=4x=4. Steps provided.

Problem 20723

Express the limit lim⁡∥P∥→0∑k=1n(ck5−3ck)Δxk\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n}\left(c_{k}^{5}-3 c_{k}\right) \Delta x_{k}lim∥P∥→0​∑k=1n​(ck5​−3ck​)Δxk​ as a definite integral.

Problem 20724

Find the Riemann sum formula for f(x)=5x2f(x)=5 x^{2}f(x)=5x2 on [0,3][0,3][0,3] using right endpoints, then take the limit as n→∞n \to \inftyn→∞.

Problem 20725

Problem 20726

Given the piecewise function f(x)f(x)f(x), which statement is false about its continuity at the specified points? (A) x=1x=1x=1, (B) x=2x=2x=2, (C) x=3x=3x=3, (D) x=4x=4x=4.

Problem 20727

Given the function f(x)=x2+2x−128cos⁡(π2x)+2x2f(x)=\frac{x^{2}+2 x-12}{8 \cos \left(\frac{\pi}{2} x\right)+2 x^{2}}f(x)=8cos(2π​x)+2x2x2+2x−12​, why is fff not continuous at x=2x=2x=2?

Problem 20728

Identify the first error in confirming the continuity of f(x)=x2−5x+4x2−6x+8f(x)=\frac{x^{2}-5 x+4}{x^{2}-6 x+8}f(x)=x2−6x+8x2−5x+4​ at x=4x=4x=4. Steps given.

Problem 20729

Calculate the area between y=sin⁡(x)y=\sin (x)y=sin(x) and y=cos⁡(x)y=\cos (x)y=cos(x) from x=π3x=\frac{\pi}{3}x=3π​ to x=3π4x=\frac{3 \pi}{4}x=43π​.

Problem 20730

Write a Riemann sum formula for f(x)=5x2f(x)=5 x^{2}f(x)=5x2 on [0,4][0,4][0,4]: sn=∑i=1nf(4in)⋅4ns_{n}=\sum_{i=1}^{n} f\left(\frac{4i}{n}\right) \cdot \frac{4}{n}sn​=i=1∑n​f(n4i​)⋅n4​

Problem 20731

A pebble creates ripples with an area increasing at 21 cm2/s21 \mathrm{~cm}^{2} / \mathrm{s}21 cm2/s. Find the radius change rate at 6 cm6 \mathrm{~cm}6 cm.

Problem 20732

Find the limit using L'Hôpital's Rule: lim⁡x→0+(10x)x=□\lim _{x \rightarrow 0^{+}}(10 x)^{x}=\squarex→0+lim​(10x)x=□. Enter INF, -INF, or DNE.

Problem 20733

An airplane is 10,000ft10,000 \mathrm{ft}10,000ft high, moving at 700ft/s700 \mathrm{ft/s}700ft/s. Find the rate the angle to an observer decreases when 40,000ft40,000 \mathrm{ft}40,000ft away.

Problem 20734

Find the Riemann sum formula for f(x)=5x2f(x)=5 x^{2}f(x)=5x2 on [0,4][0,4][0,4] using right endpoints. Then, find the limit as n→∞n \to \inftyn→∞.

Problem 20735

Evaluate the integral ∫(5x−4)4dx\int(5 x-4)^{4} d x∫(5x−4)4dx using the substitution u=5x−4u=5 x-4u=5x−4.

Problem 20736

Bestimme die Stammfunktionen für die folgenden Funktionen mit gegebenen Bedingungen: a) f(x)=x2f(x)=x^{2}f(x)=x2 durch P(1∣1)P(1 \mid 1)P(1∣1). b) f(x)=1−x2f(x)=1-x^{2}f(x)=1−x2 bei y=4y=4y=4. c) f(x)=2+xf(x)=2+xf(x)=2+x mit Nullstelle bei x=1x=1x=1.

Problem 20737

Find the area between the curves f(x)=260+13x−13x2f(x)=260+13x-13x^2f(x)=260+13x−13x2 and g(x)=13x2−65xg(x)=13x^2-65xg(x)=13x2−65x.

Problem 20738

Estimez ∫25f(x) dx\int_{2}^{5} f(x) \, dx∫25​f(x)dx avec trois méthodes : (a) points médians, (b) trapèzes, (c) Simpson. Donnez les résultats.

Problem 20739

Find the max and min points of y=sin⁡(x)y=\sin(x)y=sin(x) on the interval [0,2π][0, 2\pi][0,2π].

Problem 20740

Use substitution to solve the integral: ∫4.4e(−3x+4)dx\int 4.4 e^{(-3 x+4)} d x∫4.4e(−3x+4)dx.

Problem 20741

Find the integral: ∫x(x2+5)1.3 dx\int \frac{x}{(x^{2}+5)^{1.3}} \, dx∫(x2+5)1.3x​dx

Problem 20742

Find how many people were ill at the start of the epidemic using $N(0)=\frac{20,000}{1+20 e^{-2.5\cdot 0}}$ .

Find the slope of $y=2 \cos x \sin 2 x$ at $x=\frac{\pi}{2}$ and the tangent line to $y=x^{2} \sin (2 x)$ at $x=-\pi$ .

Given $y=x^{3}+p x^{2}$ , show the origin is stationary, find other points in terms of $p$ , and determine their nature. For $y=x^{3}+p x^{2}+p x$ , find $p$ values with no stationary points.

Find the second derivatives of $z=(x^{3}+2xy)(y-x^{2})$ at $(x=2, y=1)$ : (a) $\frac{\partial^{2} z}{\partial x^{2}}$ ; (b) $\frac{\partial^{2} z}{\partial y \partial x}$ .

Find the second derivatives of the function $z=(x^{3}+2xy)(y-x^{2})$ at $x=2$ , $y=1$ . Round to two decimals. (a) $\frac{\partial^{2} z}{\partial x^{2}}=$ (b) $\frac{\partial^{2} z}{\partial y \partial x}=$

Find the integral of the function $\frac{1}{3} x$ .

Evaluate the limit using L'Hôpital's Rule: $\lim _{x \rightarrow \infty}\left(1+x^{5}\right)^{1 / x}=\square$ . Enter INF, -INF, or DNE.

Calculate the area between $f(x)=9 x+8$ and $g(x)=x^{2}+6 x+2$ from $x=0$ to $x=2$ .

Bestimmen Sie die Ableitung von $f(x)=3 x \cdot e^{-\frac{x^{2}}{2}}$ .

Find $\frac{d x}{d t}$ , $\frac{d y}{d t}$ , and $\frac{d y}{d x}$ for $x=4 t^{3}+3 t$ , $y=6 t-5 t^{2}$ .

Find $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ for $x=t^{2}+7$ , $y=t^{2}+7t$ . When is the curve concave upward?

Vereinfachen Sie die Funktion: $f^{\prime}(x)=3 \cdot e^{-\frac{x^{2}}{2}} + 3x \cdot e^{-\frac{x^{2}}{2}} \cdot (-x)$

Find the area between the curve $y=9x^2$ and the $x$ -axis on the interval $[0, b]$ using a definite integral.

Find the area between the curve defined by $x=\sin ^{2}(t)$ and $y=8 \cos (t)$ and the $y$ -axis.

Berechne die Integrale: b) $\int_{2}^{3}\left(1+\frac{1}{x^{2}}\right) d x$ , c) $\int_{2}^{4} \frac{-1}{x^{3}} d x$ .

Calculate the area between $y=\sin(x)$ and $y=\cos(x)$ from $x=\frac{\pi}{3}$ to $x=\frac{3\pi}{4}$ .

Find the Riemann sum formula for $f(x)=5 x^{2}$ on $[0,3]$ : $S_{n}=\square$ (Use $n$ as the variable.)

Determine which statement about the piecewise function $f(x)$ is false: (A) $f$ is continuous at $x=1$ . (B) $f$ is continuous at $x=2$ . (C) $f$ is continuous at $x=3$ .

Identify the first error in confirming the continuity of $f(x)=\frac{x^{2}-5 x+4}{x^{2}-6 x+8}$ at $x=4$ . Steps provided.

Express the limit $\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n}\left(c_{k}^{5}-3 c_{k}\right) \Delta x_{k}$ as a definite integral.

Find the Riemann sum formula for $f(x)=5 x^{2}$ on $[0,3]$ using right endpoints, then take the limit as $n \to \infty$ .

Given the piecewise function $f(x)$ , which statement is false about its continuity at the specified points? (A) $x=1$ , (B) $x=2$ , (C) $x=3$ , (D) $x=4$ .

Given the function $f(x)=\frac{x^{2}+2 x-12}{8 \cos \left(\frac{\pi}{2} x\right)+2 x^{2}}$ , why is $f$ not continuous at $x=2$ ?

Identify the first error in confirming the continuity of $f(x)=\frac{x^{2}-5 x+4}{x^{2}-6 x+8}$ at $x=4$ . Steps given.

Calculate the area between $y=\sin (x)$ and $y=\cos (x)$ from $x=\frac{\pi}{3}$ to $x=\frac{3 \pi}{4}$ .

Write a Riemann sum formula for $f(x)=5 x^{2}$ on $[0,4]$ : $s_{n}=\sum_{i=1}^{n} f\left(\frac{4i}{n}\right) \cdot \frac{4}{n}$

A pebble creates ripples with an area increasing at $21 \mathrm{~cm}^{2} / \mathrm{s}$ . Find the radius change rate at $6 \mathrm{~cm}$ .

Find the limit using L'Hôpital's Rule: $\lim _{x \rightarrow 0^{+}}(10 x)^{x}=\square$ . Enter INF, -INF, or DNE.

An airplane is $10,000 \mathrm{ft}$ high, moving at $700 \mathrm{ft/s}$ . Find the rate the angle to an observer decreases when $40,000 \mathrm{ft}$ away.

Find the Riemann sum formula for $f(x)=5 x^{2}$ on $[0,4]$ using right endpoints. Then, find the limit as $n \to \infty$ .

Evaluate the integral $\int(5 x-4)^{4} d x$ using the substitution $u=5 x-4$ .

Bestimme die Stammfunktionen für die folgenden Funktionen mit gegebenen Bedingungen: a) $f(x)=x^{2}$ durch $P(1 \mid 1)$ . b) $f(x)=1-x^{2}$ bei $y=4$ . c) $f(x)=2+x$ mit Nullstelle bei $x=1$ .

Find the area between the curves $f(x)=260+13x-13x^2$ and $g(x)=13x^2-65x$ .

Estimez $\int_{2}^{5} f(x) \, dx$ avec trois méthodes : (a) points médians, (b) trapèzes, (c) Simpson. Donnez les résultats.

Find the max and min points of $y=\sin(x)$ on the interval $[0, 2\pi]$ .

Use substitution to solve the integral: $\int 4.4 e^{(-3 x+4)} d x$ .

Find the integral: $\int \frac{x}{(x^{2}+5)^{1.3}} \, dx$

Find the number $c$ that satisfies the Mean Value Theorem for $f(x)=\sqrt{4x-3}$ on $1 \leq x \leq 3$ . What is $c$ ? (A) 1.5 (B) 1.75 (C) 2 (D) 2.25

Determine if the sequence $a_{n}=\frac{4 \sqrt{n}}{\sqrt{n}+3}$ converges or diverges. Find the limit if it converges.

Find the number $c$ that satisfies the Mean Value Theorem for $g(x)=x^{3}+12x^{2}+36x$ on $[-8, -2]$ . What is $c$ ?
Choose 1 answer: (A) -7 (B) -6 (C) -3 (D) -1

Find the sum of the series given the partial sums $s_{n}=8-3(0.8)^{n}$ .

Find $h^{\prime}(1)$ for $h(x)=\ln \left[(f(x))^{2}+1\right]$ given $f(1)=-1$ and $f^{\prime}(1)=1$ .

Given $f(x)=\frac{x+7}{x}$ : a) Show why it has an inverse using the derivative. b) Find $f^{-1}$ . c) Determine domains and ranges of $f$ and $f^{-1}$ .

Find the vertical asymptotes of $f(x)=\frac{-5}{\cos \left(\frac{3 \pi}{2} x\right)}$ from the options given.

Find the indefinite integral: $\int \frac{e^{4 x}-e^{2 x}+1}{e^{x}} d x$ (Use $C$ for the constant of integration).

Find the tangent line equation for $h(x)=-x e^{9-x}$ at the point $(9,-9)$ .

Solve the equation: $\int(4 x+6)dx = 8 x + 4$ .

Bestimme die Ableitung der Funktion $f(x)=\frac{x}{2 \cdot \sqrt{x}}$ .

Can the Mean Value Theorem apply to $f(x)=x^{2/3}$ on $[1, 27]$ ? Select all that apply.

Given the function $f(x)=x^{2}-2x+9$ , find critical numbers, intervals of increase/decrease, and relative extrema.

Find the car's velocity when $s(t)=0$ and $s(t)=14$ for $s(t)=t^{2}-3t-4$ .

Find the inflection point of $f(x)=x^{3}-12 x^{2}$ and describe concavity in interval notation.

A hummingbird's position is $s(t)=3 t^{3}-7 t$ . Find velocity at $t=1$ sec, acceleration at $t=1$ sec, and when velocity=0.

Given functions for a particle's position, find velocity and acceleration, then identify intervals for speeding up or slowing down.
1. $s(t)=2 t^{3}-3 t^{2}-12 t+8$
2. $s(t)=2 t^{3}-15 t^{2}+36 t-10$

Find the specific solution for the equation $f^{\prime \prime}(x)=3 \cos (x)$ with $f^{\prime}(0)=4$ , $f(0)=-5$ .

Finde die Tangentengleichung von $f$ , die parallel zu $g$ ist. Wähle aus: a) bis f) mit verschiedenen $f$ und $g$ .

Bestimme den Wert der Reihe $\sum_{k=0}^{\infty} \frac{1}{\left((-1)^{k}+4\right)^{k}}.$

Determine if the series $\sum_{k=1}^{\infty}(-1)^{k+1}\left(\frac{7 k^{3}+k}{3 k^{3}+k+1}\right)^{k}$ converges or diverges using tests.

Calculate the sum $\sum_{k=1}^{n} 5\left(\frac{4 k}{n}\right)^{2} \frac{4}{n}$ .

Bacteria growth is modeled by $n(t)=925 e^{0.4 t}$ . Find (a) relative growth rate, (b) initial population, (c) population at $t=5$ .

Use the Root Test to check if the series $\sum_{k=1}^{\infty}\left(-\frac{7 k+6}{6 k}\right)^{k}$ converges or diverges.

Use the Root Test to check if the series $\sum_{k=1}^{\infty}\left(1+\frac{17}{k}\right)^{k^{2}}$ converges or diverges.

Untersuche die Funktionen $f_{k}(x)=x^{4}-k x^{2}$ auf Extrem- und Wendepunkte für $k=-2$ und $k=2$ . Bestimme die Tiefpunkte für $k>0$ und zeige, dass $\frac{x_{e}}{x_{w}}$ unabhängig von $k$ ist.

Find the intervals where the function $f(x)=\frac{x^{2}}{x-3}$ is decreasing.

Use the Ratio Test to check if the series $\sum_{k=1}^{\infty} \frac{2^{k}}{k^{149}}$ converges or diverges.

Bestimme die Ableitung von $f(x)=7^{2 x}-3^{x}$ .