Calculus

Problem 15701

Find the first derivative of the function $E(t)=2 t\left(e^{\frac{-t}{10}}\right)$ .

See Solution

Problem 15702

Find the limit using L'Hôpital's Rule: $\lim _{x \rightarrow 0} \frac{\sin (x)}{x}=$

See Solution

Problem 15703

Approximate $(26.5)^{\frac{1}{3}}$ using linearization at $a=27$ . Answer to three decimal places and state if it's an overestimate or underestimate.

See Solution

Problem 15704

How many hours should Ryan study for maximum effectiveness given the function $E(t)=2 t\left(e^{\frac{-t}{10}}\right)$ ?

See Solution

Problem 15705

Find the $y$ -coordinates for points on $y=x^{2}$ , calculate secant slopes, and estimate the tangent slope at $P(3,9)$ .

See Solution

Problem 15706

Find the linearization $L(x)$ of $f(x)=\sqrt[4]{13+x}$ at $a=2$ and use it to approximate $\sqrt[4]{15.5}$ .

See Solution

Problem 15707

Find the profit from drilling a well 300 feet deep, given $P^{\prime}(x)=\sqrt[4]{x}$ . Set up the integral:
$P(300)=\int_{0}^{300}(\sqrt[4]{x}) d x$
Total profit is \$ \square. (Round to two decimal places.)

See Solution

Problem 15708

Calculate $f(8)$ and $f(9)$ for the function $f(x)=-x^{3}+8 x^{2}+5 x+10$ . Is there a real zero between 8 and 9? YES or NO?

See Solution

Problem 15709

Azote $(\mathrm{N})$ et hydrogène $(\mathrm{H})$ forment l'ammoniac : $Q(t)=100-\frac{1000}{10+t}$ . Trouvez :
a) $Q(0)$ et $Q(20)$ . b) Variation de $Q$ sur $[10 s, 20 s]$ . c) Taux de variation moyen sur $[10 s, 20 s]$ et $[20 s, 30 s]$ . d) Évaluez $\lim _{h \rightarrow 0^{+}} \frac{Q(h+0)-Q(0)}{h}$ . e) Trouvez la fonction de variation de $Q$ .

See Solution

Problem 15710

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos (x)}{x^{2}}$ .

See Solution

Problem 15711

Untersuchen Sie die Funktionen auf lokale Extrempunkte und skizzieren Sie die Graphen: a) $f(x)=x-2+e^{-x}$ , b) $f(x)=x^{2} \cdot e^{x+1}$ .

See Solution

Problem 15712

Find the average acceleration of a particle with velocity $v(t)=(2.3 \mathrm{~m/s})+(4.1 \mathrm{~m/s}^2)t-(6.2 \mathrm{~m/s}^3)t^2$ from $t=1.0 \mathrm{~s}$ to $t=2.0 \mathrm{~s}$ .

See Solution

Problem 15713

Find Newton's method formula for $f(x)=4-\tan(4x)$ with $x_{0}=1$ and compute $x_{1}$ and $x_{2}$ . Which formula is correct? A. $x_{n}=x_{n+1}+\frac{4-\tan(4x_{n+1})}{4\sec^2(4x_{n+1})}$ B. $x_{n+1}=x_{n}-\frac{4-\tan(4x_{n})}{4\sec^2(4x_{n})}$ C. $x_{n+1}=x_{n}+\frac{4\sec^2(4x_{n})}{4-\tan(4x_{n})}$ D. $x_{n+1}=x_{n}+\frac{4-\tan(4x_{n})}{4\sec^2(4x_{n})}$

See Solution

Problem 15714

Fällt der Fahrer in die Barriere zwischen A(2,1) und B(2,2), wenn das Motorrad der Funktion $f(x)=(1-x) \cdot e^{x}$ folgt?

See Solution

Problem 15715

Bestimme die Tangentengleichung $t$ von $f(x)=0,05 x^{3}+0,25 x^{2}-0,2 x+1$ bei $P(-5 \mid 2$ und die Fläche $A$ zwischen $f$ und $t$ .

See Solution

Problem 15716

Find Newton's method formula for $f(x)=4-\tan(4x)$ with $x_{0}=1$ and compute $x_{1}$ and $x_{2}$ .

See Solution

Problem 15717

Find the increase in national debt from 1996 to 2005 using the model $D^{\prime}(t)=849+824.7 t-163.8 t^{2}+13.08 t^{3}$ .

See Solution

Problem 15718

Find the increase in national debt (in billions) from 1996 to 2001 using the model $D^{\prime}(t)=858.03+819.52 t-191.73 t^{2}+11.52 t^{3}$ .

See Solution

Problem 15719

Find the indefinite integral $\int \frac{x^{3}}{\left(x^{4}-4\right)^{4}} d x$ using the substitution $u=\square$ and $d u=\square d x$ .

See Solution

Problem 15720

Evaluate the integral $\int x^{2}(x^{3}-4)^{40} dx$ using the substitution $u=$ .

See Solution

Problem 15721

Berechnen Sie das Integral $A=\int_{a}^{b}\left|x^{3}-x^{2}-4 x+3+x^{2}-3\right| d x$ für gegebene Werte von $a$ und $b$ .

See Solution

Problem 15722

Find the indefinite integral: $\int x^{4} \sqrt{7+x^{5}} \, dx + C$

See Solution

Problem 15723

Find the dimensions of a lidless rectangular flower box with length $4w$ and surface area $5200 \mathrm{~cm}^{2}$ to maximize volume.

See Solution

Problem 15724

Evaluate the integral $\int \frac{d x}{(2 x+2)^{2}}$ using the substitution $u=$ .

See Solution

Problem 15725

Determine if $f(x)=6x^{2}+48x+98$ has a max, min, or neither. Find the value and location if it does.

See Solution

Problem 15726

A particle $P$ moves on the $x$ -axis with displacement $s=5 t^{2}-t^{3}$ . Find: a) Change in $s$ from $t=2$ to $t=4$ . b) Change in $s$ during the third second.

See Solution

Problem 15727

Joan has \$4687 and needs \$5579. If she invests at 4.1% annual interest, how long to reach \$5579? Round to 2 decimal places.

See Solution

Problem 15728

Find $C(r) = 0.16 \pi r^{2} + \frac{24}{r}$ . Calculate $C'(r)=0$ to find $r$ , then use it to find $h$ .

See Solution

Problem 15729

Find the can dimensions that minimize cost for a volume of 300 cm³, with costs of 0.04 and 0.08 cents per cm².
Use: $r=2.879, \quad h=\square$
Minimum cost: $C=$ cents. Round to the nearest hundredth.

See Solution

Problem 15730

Find the linearization $L(x)$ of the function $f(x) = 8^{x}$ at the point $a=0$ .

See Solution

Problem 15731

Given the population model $P(t)=500 e^{\beta t}$ , find $\beta$ if $P(5)=1000$ . Then use linear approximation at $t=5$ to estimate $P(10)$ and $P(0)$ .

See Solution

Problem 15732

Write Newton's method formula and compute $x_{1}$ and $x_{2}$ for $f(x)=x^{2}-12$ with $x_{0}=3$ . Choose the correct formula.

See Solution

Problem 15733

Find the general anti-derivative of $h(x) = e^{x} + x^{5}$ , using $C$ as the arbitrary constant.

See Solution

Problem 15734

Approximate $(26.5)^{\frac{1}{3}}$ using the linearization of the cube root function at $a=27$ .

See Solution

Problem 15735

Find the surface area of the curve $3 x=\left(y^{2}+2\right)^{\frac{3}{2}}$ from $y=0$ to $y=1$ revolved around the $x$ -axis.

See Solution

Problem 15736

Find the general antiderivative of the function $f(x)=\frac{5}{x}$ for all $x \neq 0$ .

See Solution

Problem 15737

Approximate $(26.5)^{\frac{1}{3}}$ using linearization at $a=27$ to three decimal places. Is it an overestimate or underestimate?

See Solution

Problem 15738

Find the antiderivative of $f(x)=\sqrt{x}$ , $f(x)=x^{n}$ for $n \neq -1$ , and $f(x)=\frac{1}{x}$ for $x>0$ .

See Solution

Problem 15739

Find the function $f(x)$ such that $f^{\prime}(x) = x^{5}$ and $f(0) = 3$ . What is $f(x)$ ?

See Solution

Problem 15740

Find the function $f(x)$ such that $f'(x)=x^2$ and $f(0)=9$ .

See Solution

Problem 15741

Find the $x$ -values for the global maximum and minimum of $h(x)$ on $3 \leq x \leq 12$ given $h'(6)$ is undefined.

See Solution

Problem 15742

Find the 4th-degree Taylor polynomial $L_{4}(x)$ for $f(x)=\mathrm{e}^{6 x}+\ln (2 x)$ .

See Solution

Problem 15743

Find the tangent line equation for $f(x)=x$ at $x=4$ . First, determine the slope using the derivative.

See Solution

Problem 15744

Determine the intervals of concavity for $f(x)=-4 x^{3}+24 x^{2}+165 x-6$ and identify any inflection points.

See Solution

Problem 15745

Find the tangent line equation for $f(x)=x^{2}$ at $x=4$ . First, calculate the derivative for the slope.

See Solution

Problem 15746

Find the function $f(x)$ such that $f'(x) = x^5$ and $f(0) = 9$ .

See Solution

Problem 15747

Given the population model $P(t)=500 e^{\beta t}$ , find $\beta$ if $P(5)=2000$ . Then predict $P(12)$ and $P(0)$ using linear approximation at $a=5$ .

See Solution

Problem 15748

Find the general antiderivative of $f(x)=\frac{7}{x}$ for $x \neq 0$ . Use $C$ as the constant. $F(x)=$

See Solution

Problem 15749

Find the limits: a) $\lim _{x \rightarrow 0} \frac{4 \tan x}{x}$ using $\tan x = \frac{\sin x}{\cos x}$ ; b) $\lim _{x \rightarrow \frac{5 \pi}{3}} \sqrt{\cos x}$ .

See Solution

Problem 15750

Find the linearization $L(x)$ of $f(x)=\sqrt[9]{2+x}$ at $a=5$ and use it to approximate $\sqrt[9]{7.2}$ .

See Solution

Problem 15751

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

See Solution

Problem 15752

Compute the integral $\int_{\gamma} x^{2} d \ell$ along the path $y=\ln x$ for $0<x \leq \sqrt{3}$ .

See Solution

Problem 15753

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{\left(x_{0}+\Delta x\right)^{2} e^{-\left(x_{0}+\Delta x\right)} - x_{0}^{2} e^{-x_{0}}}{\Delta x}$ .

See Solution

Problem 15754

Find the limit: $\lim _{x \rightarrow 0} \frac{\cos 2 x-\cos ^{2} x}{\cos x-1}$ .

See Solution

Problem 15755

Find the moments of inertia $I_{x}=\int_{\gamma} y^{2} d \ell$ and $I_{y}=\int_{\gamma} x^{2} d \ell$ for a semicircular wire.

See Solution

Problem 15756

Find the limit using L'Hôpital's Rule: $\lim _{x \rightarrow 0} \frac{\sin (x)}{x}=$

See Solution

Problem 15757

Find the point of diminishing returns $(x, y)$ for $R(x) = 11,000 - x^3 + 39x^2 + 700x$ , where $0 \leq x \leq 20$ .

See Solution

Problem 15758

Find the point of diminishing returns $(x, y)$ for the revenue function $R(x)=11,000-x^{3}+36 x^{2}+800 x$ where $0 \leq x \leq 20$ . The point is $\square$ .

See Solution

Problem 15759

Approximate the displacement of an object with $v=1(4t+1)$ m/s from $t=0$ to $t=8$ using 4 subintervals. Result: $\square$ m.

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Problem 15760

Find the largest volume of a box made from a 3 ft square cardboard by cutting out squares from the corners.

See Solution

Problem 15761

Find the point of diminishing returns $(x, y)$ for $R(x) = 10,000 - x^3 + 42x^2 + 600x$ where $0 \leq x \leq 20$ .

See Solution

Problem 15762

Use Newton's method to find a root of $f(x)=3 x^{2}-\sqrt{x}-4$ starting from $x_{0}=2$ . Show your steps.

See Solution

Problem 15763

Find $\frac{d Q}{d t}$ at $Q=70 \mathrm{~g}$ for the function $Q(t) = 100 - \frac{1000}{10 + t}$ .

See Solution

Problem 15764

A farmer wants to fence 24 million sq ft. Let $y$ be the side perpendicular to the dividing fence and $x$ the parallel side. Write $F(x)$ for total fencing length. Find $F'(x)$ and critical numbers. What are the dimensions to minimize fencing cost? Smaller value: $\mathrm{ft}$ , larger value: $\mathrm{ft}$ .

See Solution

Problem 15765

A farmer fences 24 million sq ft in a rectangle. Let $x$ and $y$ be side lengths. Find $F(x)$ , $F'(x)$ , and critical numbers.
$F(x)=$
$F^{\prime}(x)=$
$x=$
Smaller side: 4000 ft, larger side: 6000 ft.

See Solution

Problem 15766

Use Newton's method to find a root of $f(x)=3 x^{2}-\sqrt{x}-4$ starting from $x_{0}=2$ . Show your work in a table.

See Solution

Problem 15767

A farmer wants to fence 24 million sq ft in a rectangle. Write $F(x)$ for fencing length: $F(x)=2\left(\frac{24,000,000}{x}\right)+3 x$ . Find $F^{\prime}(x)$ : $F^{\prime}(x)=-\frac{48,000,000}{x^{2}}+3$ . Find critical numbers: $x=$ for min cost. Lengths: 4000 ft, 6000 ft.

See Solution

Problem 15768

Find the cost per ton, $y$ , for an oil tanker of $x$ thousand tons using $\bar{C}(x)=\frac{220,000}{x+480}$ .
a. Calculate $\overline{\mathrm{C}}(25)$ , $\overline{\mathrm{C}}(50)$ , $\overline{\mathrm{C}}(100)$ , $\overline{\mathrm{C}}(200)$ , $\overline{\mathrm{C}}(300)$ , $\overline{\mathrm{C}}(400)$ . b. Identify the horizontal and vertical asymptotes.

See Solution

Problem 15769

The function $f(x)$ increases on $(-\infty, 0)$ and decreases on $(0,+\infty)$ . What is true about $f(x)$ ? A) max at $x=0$ B) min at $x=0$ C) domain $(0,+\infty)$ D) range $(0,+\infty)$

See Solution

Problem 15770

Which limit equals $\int_{3}^{5} x^{4} d x$ ? (A) $\lim_{n \to \infty} \sum_{k=1}^{n} \left(3+\frac{k}{n}\right)^{4} \frac{1}{n}$ (B) $\lim_{n \to \infty} \sum_{k=1}^{n} \left(3+\frac{k}{n}\right)^{4} \frac{2}{n}$ (C) $\lim_{n \to \infty} \sum_{k=1}^{n} \left(3+\frac{2k}{n}\right)^{4} \frac{1}{n}$ (D) $\lim_{n \to \infty} \sum_{k=1}^{n} \left(3+\frac{2k}{n}\right)^{4} \frac{2}{n}$

See Solution

Problem 15771

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(2 x^{6}+8)^{5}$ .

See Solution

Problem 15772

Calculate the energy change in kJ for ammonia synthesis when volume changes from 7.8 L to 4.9 L at 45.0 atm.

See Solution

Problem 15773

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(3-9 x)^{7}$ .

See Solution

Problem 15774

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{2 x-1}{9 x+7}$ .

See Solution

Problem 15775

Find the derivative $f^{\prime}(x)$ for the function $f(x)=-2 x^{3}$ . What is $f^{\prime}(x)=\square$ ?

See Solution

Problem 15776

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

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Problem 15777

What could the instantaneous velocity be between points B and C if the velocities at A, B, and C are $5 \mathrm{~m/s}$ , $15 \mathrm{~m/s}$ , and $10 \mathrm{~m/s}$ ? Options: less than $5 \mathrm{~m/s}$ , less than $10 \mathrm{~m/s}$ , greater than $10 \mathrm{~m/s}$ , greater than $15 \mathrm{~m/s}$ .

See Solution

Problem 15778

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(6-8 x)^{15}$ .

See Solution

Problem 15779

Find the derivative using the quotient rule for $y=\frac{2 x^{2}+1}{x^{2}+5}$ . What is $y^{\prime}=\square$ ?

See Solution

Problem 15780

Find the derivative using the product rule for $y=(4 x^{2}+5)(3 x-2)$ . What is $y^{\prime}=$ ?

See Solution

Problem 15781

Find the antiderivative $F$ of $f(t)=\csc^2 t$ such that $F\left(\frac{\pi}{4}\right)=5$ .

See Solution

Problem 15782

Solve the initial value problem: $f'(x) = 6x - 5$ , with $f(0) = 9$ .

See Solution

Problem 15783

Find the derivative of the function $f(x)=\tan ^{2}\left(3 x^{2}\right)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 15784

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

See Solution

Problem 15785

Differentiate the function $y=x^{10} e^{x}$ . What is $\frac{d y}{d x}=\square$ ?

See Solution

Problem 15786

Find the derivative $f^{\prime}(x)$ of $f(x)=x^{2}+3x-6$ and calculate $f^{\prime}(1)$ , $f^{\prime}(2)$ , $f^{\prime}(3)$ .

See Solution

Problem 15787

Differentiate $y=\ln(5x)$ and show your work.

See Solution

Problem 15788

Differentiate the function $f(x)=-4-5 x+8 e^{x}$ . Find $f^{\prime}(x)=\square$ .

See Solution

Problem 15789

Find the derivative of the function $g(x)=2 x \sqrt{x^{2}+7}$ . What is $g^{\prime}(x)$ ?

See Solution

Problem 15790

Find the second derivative of the function $f(x)=\sin(x^{2})$ , i.e., $f^{\prime \prime}(x)=?$

See Solution

Problem 15791

Find the derivative of $g(x)=\frac{1}{\sqrt{\tan(3x^2)}}$ . What is $g'(x)$ ?

See Solution

Problem 15792

Find the position function for an object with acceleration $a(t)=-40$ , initial velocity $v(0)=22$ , and initial position $s(0)=0$ .

See Solution

Problem 15793

A softball is popped up with an initial velocity of $34 \mathrm{~m/s}$ . Find velocity, position, max height time, and ground strike time.

See Solution

Problem 15794

Find the derivative $r^{\prime}(-1)$ for the function $r(t)=\frac{3}{(t^{2}-2 t)^{2}}$ .

See Solution

Problem 15795

Find the object's velocity, position, highest point time/height, and ground strike time after being released from $400 \mathrm{~m}$ at $11 \mathrm{~m/s}$ .

See Solution

Problem 15796

Find the $y$ -coordinates for points $Q_1(3.5,y)$ to $Q_4(3.001,y)$ on $y=x^2$ , then calculate slopes to $P(3,9)$ . Repeat for $Q_5(2.5,y)$ to $Q_8(2.999,y)$ and estimate tangent slope at $P$ .

See Solution

Problem 15797

Find the $y$ -coordinates for points on $y=x^{2}$ and calculate slopes of secants to estimate tangent slope at $P(3,9)$ .

See Solution

Problem 15798

Find $h^{\prime}(2)$ for $h(x)=f(g(x))$ using values from the given table for $f(x)$ and $g(x)$ .

See Solution

Problem 15799

Analyze the motion of a payload dropped from $400 \mathrm{~m}$ at $11 \mathrm{~m/s}$ :
a. Find $v(t)$ . b. Find $s(t)$ . c. Determine the time and height at the highest point. d. Find when it hits the ground.

See Solution

Problem 15800

Find $p^{\prime}(5)$ for $p(x)=f(x) \cdot g(f(x))$ using given values of $f$ and $g$ .

See Solution

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Calculus

Problem 15701

Find the first derivative of the function E(t)=2t(e−t10)E(t)=2 t\left(e^{\frac{-t}{10}}\right)E(t)=2t(e10−t​).

Problem 15702

Find the limit using L'Hôpital's Rule: lim⁡x→0sin⁡(x)x=\lim _{x \rightarrow 0} \frac{\sin (x)}{x}=limx→0​xsin(x)​=

Problem 15703

Approximate (26.5)13(26.5)^{\frac{1}{3}}(26.5)31​ using linearization at a=27a=27a=27. Answer to three decimal places and state if it's an overestimate or underestimate.

Problem 15704

How many hours should Ryan study for maximum effectiveness given the function E(t)=2t(e−t10)E(t)=2 t\left(e^{\frac{-t}{10}}\right)E(t)=2t(e10−t​)?

Problem 15705

Find the yyy-coordinates for points on y=x2y=x^{2}y=x2, calculate secant slopes, and estimate the tangent slope at P(3,9)P(3,9)P(3,9).

Problem 15706

Find the linearization L(x)L(x)L(x) of f(x)=13+x4f(x)=\sqrt[4]{13+x}f(x)=413+x​ at a=2a=2a=2 and use it to approximate 15.54\sqrt[4]{15.5}415.5​.

Problem 15707

Find the profit from drilling a well 300 feet deep, given P′(x)=x4P^{\prime}(x)=\sqrt[4]{x}P′(x)=4x​. Set up the integral: P(300)=∫0300(x4)dx P(300)=\int_{0}^{300}(\sqrt[4]{x}) d x P(300)=∫0300​(4x​)dx Total profit is \$ \square. (Round to two decimal places.)

Problem 15708

Calculate f(8)f(8)f(8) and f(9)f(9)f(9) for the function f(x)=−x3+8x2+5x+10f(x)=-x^{3}+8 x^{2}+5 x+10f(x)=−x3+8x2+5x+10. Is there a real zero between 8 and 9? YES or NO?

Problem 15709

Problem 15710

Find the limit: lim⁡x→01−cos⁡(x)x2\lim _{x \rightarrow 0} \frac{1-\cos (x)}{x^{2}}limx→0​x21−cos(x)​.

Problem 15711

Untersuchen Sie die Funktionen auf lokale Extrempunkte und skizzieren Sie die Graphen: a) f(x)=x−2+e−xf(x)=x-2+e^{-x}f(x)=x−2+e−x, b) f(x)=x2⋅ex+1f(x)=x^{2} \cdot e^{x+1}f(x)=x2⋅ex+1.

Problem 15712

Find the average acceleration of a particle with velocity v(t)=(2.3 m/s)+(4.1 m/s2)t−(6.2 m/s3)t2v(t)=(2.3 \mathrm{~m/s})+(4.1 \mathrm{~m/s}^2)t-(6.2 \mathrm{~m/s}^3)t^2v(t)=(2.3 m/s)+(4.1 m/s2)t−(6.2 m/s3)t2 from t=1.0 st=1.0 \mathrm{~s}t=1.0 s to t=2.0 st=2.0 \mathrm{~s}t=2.0 s.

Problem 15713

Problem 15714

Fällt der Fahrer in die Barriere zwischen A(2,1) und B(2,2), wenn das Motorrad der Funktion f(x)=(1−x)⋅exf(x)=(1-x) \cdot e^{x}f(x)=(1−x)⋅ex folgt?

Problem 15715

Bestimme die Tangentengleichung ttt von f(x)=0,05x3+0,25x2−0,2x+1f(x)=0,05 x^{3}+0,25 x^{2}-0,2 x+1f(x)=0,05x3+0,25x2−0,2x+1 bei P(−5∣2P(-5 \mid 2P(−5∣2 und die Fläche AAA zwischen fff und ttt.

Problem 15716

Find Newton's method formula for f(x)=4−tan⁡(4x)f(x)=4-\tan(4x)f(x)=4−tan(4x) with x0=1x_{0}=1x0​=1 and compute x1x_{1}x1​ and x2x_{2}x2​.

Problem 15717

Find the increase in national debt from 1996 to 2005 using the model D′(t)=849+824.7t−163.8t2+13.08t3D^{\prime}(t)=849+824.7 t-163.8 t^{2}+13.08 t^{3}D′(t)=849+824.7t−163.8t2+13.08t3.

Problem 15718

Find the increase in national debt (in billions) from 1996 to 2001 using the model D′(t)=858.03+819.52t−191.73t2+11.52t3D^{\prime}(t)=858.03+819.52 t-191.73 t^{2}+11.52 t^{3}D′(t)=858.03+819.52t−191.73t2+11.52t3.

Problem 15719

Find the indefinite integral ∫x3(x4−4)4dx\int \frac{x^{3}}{\left(x^{4}-4\right)^{4}} d x∫(x4−4)4x3​dx using the substitution u=□u=\squareu=□ and du=□dxd u=\square d xdu=□dx.

Problem 15720

Evaluate the integral ∫x2(x3−4)40dx\int x^{2}(x^{3}-4)^{40} dx∫x2(x3−4)40dx using the substitution u=u=u=.

Problem 15721

Berechnen Sie das Integral A=∫ab∣x3−x2−4x+3+x2−3∣dxA=\int_{a}^{b}\left|x^{3}-x^{2}-4 x+3+x^{2}-3\right| d xA=∫ab​∣∣​x3−x2−4x+3+x2−3∣∣​dx für gegebene Werte von aaa und bbb.

Problem 15722

Find the indefinite integral: ∫x47+x5 dx+C\int x^{4} \sqrt{7+x^{5}} \, dx + C∫x47+x5​dx+C

Problem 15723

Find the dimensions of a lidless rectangular flower box with length 4w4w4w and surface area 5200 cm25200 \mathrm{~cm}^{2}5200 cm2 to maximize volume.

Problem 15724

Evaluate the integral ∫dx(2x+2)2\int \frac{d x}{(2 x+2)^{2}}∫(2x+2)2dx​ using the substitution u=u=u=.

Problem 15725

Determine if f(x)=6x2+48x+98f(x)=6x^{2}+48x+98f(x)=6x2+48x+98 has a max, min, or neither. Find the value and location if it does.

Problem 15726

A particle PPP moves on the xxx-axis with displacement s=5t2−t3s=5 t^{2}-t^{3}s=5t2−t3. Find: a) Change in sss from t=2t=2t=2 to t=4t=4t=4. b) Change in sss during the third second.

Problem 15727

Joan has \$4687 and needs \$5579. If she invests at 4.1% annual interest, how long to reach \$5579? Round to 2 decimal places.

Problem 15728

Find C(r)=0.16πr2+24rC(r) = 0.16 \pi r^{2} + \frac{24}{r}C(r)=0.16πr2+r24​. Calculate C′(r)=0C'(r)=0C′(r)=0 to find rrr, then use it to find hhh.

Problem 15729

Find the can dimensions that minimize cost for a volume of 300 cm³, with costs of 0.04 and 0.08 cents per cm². Use: r=2.879,h=□ r=2.879, \quad h=\square r=2.879,h=□Minimum cost: C=C=C= cents. Round to the nearest hundredth.

Problem 15730

Find the linearization L(x)L(x)L(x) of the function f(x)=8xf(x) = 8^{x}f(x)=8x at the point a=0a=0a=0.

Problem 15731

Given the population model P(t)=500eβtP(t)=500 e^{\beta t}P(t)=500eβt, find β\betaβ if P(5)=1000P(5)=1000P(5)=1000. Then use linear approximation at t=5t=5t=5 to estimate P(10)P(10)P(10) and P(0)P(0)P(0).

Problem 15732

Write Newton's method formula and compute x1x_{1}x1​ and x2x_{2}x2​ for f(x)=x2−12f(x)=x^{2}-12f(x)=x2−12 with x0=3x_{0}=3x0​=3. Choose the correct formula.

Problem 15733

Find the general anti-derivative of h(x)=ex+x5h(x) = e^{x} + x^{5}h(x)=ex+x5, using CCC as the arbitrary constant.

Problem 15734

Approximate (26.5)13(26.5)^{\frac{1}{3}}(26.5)31​ using the linearization of the cube root function at a=27a=27a=27.

Problem 15735

Find the surface area of the curve 3x=(y2+2)323 x=\left(y^{2}+2\right)^{\frac{3}{2}}3x=(y2+2)23​ from y=0y=0y=0 to y=1y=1y=1 revolved around the xxx-axis.

Problem 15736

Find the general antiderivative of the function f(x)=5xf(x)=\frac{5}{x}f(x)=x5​ for all x≠0x \neq 0x=0.

Problem 15737

Approximate (26.5)13(26.5)^{\frac{1}{3}}(26.5)31​ using linearization at a=27a=27a=27 to three decimal places. Is it an overestimate or underestimate?

Problem 15738

Find the antiderivative of f(x)=xf(x)=\sqrt{x}f(x)=x​, f(x)=xnf(x)=x^{n}f(x)=xn for n≠−1n \neq -1n=−1, and f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ for x>0x>0x>0.

Problem 15739

Find the function f(x)f(x)f(x) such that f′(x)=x5f^{\prime}(x) = x^{5}f′(x)=x5 and f(0)=3f(0) = 3f(0)=3. What is f(x)f(x)f(x)?

Problem 15740

Find the function f(x)f(x)f(x) such that f′(x)=x2f'(x)=x^2f′(x)=x2 and f(0)=9f(0)=9f(0)=9.

Problem 15741

Find the first derivative of the function $E(t)=2 t\left(e^{\frac{-t}{10}}\right)$ .

Find the limit using L'Hôpital's Rule: $\lim _{x \rightarrow 0} \frac{\sin (x)}{x}=$

Approximate $(26.5)^{\frac{1}{3}}$ using linearization at $a=27$ . Answer to three decimal places and state if it's an overestimate or underestimate.

How many hours should Ryan study for maximum effectiveness given the function $E(t)=2 t\left(e^{\frac{-t}{10}}\right)$ ?

Find the $y$ -coordinates for points on $y=x^{2}$ , calculate secant slopes, and estimate the tangent slope at $P(3,9)$ .

Find the linearization $L(x)$ of $f(x)=\sqrt[4]{13+x}$ at $a=2$ and use it to approximate $\sqrt[4]{15.5}$ .

Find the profit from drilling a well 300 feet deep, given $P^{\prime}(x)=\sqrt[4]{x}$ . Set up the integral:
$P(300)=\int_{0}^{300}(\sqrt[4]{x}) d x$
Total profit is \$ \square. (Round to two decimal places.)

Calculate $f(8)$ and $f(9)$ for the function $f(x)=-x^{3}+8 x^{2}+5 x+10$ . Is there a real zero between 8 and 9? YES or NO?

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos (x)}{x^{2}}$ .

Untersuchen Sie die Funktionen auf lokale Extrempunkte und skizzieren Sie die Graphen: a) $f(x)=x-2+e^{-x}$ , b) $f(x)=x^{2} \cdot e^{x+1}$ .

Find the average acceleration of a particle with velocity $v(t)=(2.3 \mathrm{~m/s})+(4.1 \mathrm{~m/s}^2)t-(6.2 \mathrm{~m/s}^3)t^2$ from $t=1.0 \mathrm{~s}$ to $t=2.0 \mathrm{~s}$ .

Fällt der Fahrer in die Barriere zwischen A(2,1) und B(2,2), wenn das Motorrad der Funktion $f(x)=(1-x) \cdot e^{x}$ folgt?

Bestimme die Tangentengleichung $t$ von $f(x)=0,05 x^{3}+0,25 x^{2}-0,2 x+1$ bei $P(-5 \mid 2$ und die Fläche $A$ zwischen $f$ und $t$ .

Find Newton's method formula for $f(x)=4-\tan(4x)$ with $x_{0}=1$ and compute $x_{1}$ and $x_{2}$ .

Find the increase in national debt from 1996 to 2005 using the model $D^{\prime}(t)=849+824.7 t-163.8 t^{2}+13.08 t^{3}$ .

Find the increase in national debt (in billions) from 1996 to 2001 using the model $D^{\prime}(t)=858.03+819.52 t-191.73 t^{2}+11.52 t^{3}$ .

Find the indefinite integral $\int \frac{x^{3}}{\left(x^{4}-4\right)^{4}} d x$ using the substitution $u=\square$ and $d u=\square d x$ .

Evaluate the integral $\int x^{2}(x^{3}-4)^{40} dx$ using the substitution $u=$ .

Berechnen Sie das Integral $A=\int_{a}^{b}\left|x^{3}-x^{2}-4 x+3+x^{2}-3\right| d x$ für gegebene Werte von $a$ und $b$ .

Find the indefinite integral: $\int x^{4} \sqrt{7+x^{5}} \, dx + C$

Find the dimensions of a lidless rectangular flower box with length $4w$ and surface area $5200 \mathrm{~cm}^{2}$ to maximize volume.

Evaluate the integral $\int \frac{d x}{(2 x+2)^{2}}$ using the substitution $u=$ .

Determine if $f(x)=6x^{2}+48x+98$ has a max, min, or neither. Find the value and location if it does.

A particle $P$ moves on the $x$ -axis with displacement $s=5 t^{2}-t^{3}$ . Find: a) Change in $s$ from $t=2$ to $t=4$ . b) Change in $s$ during the third second.

Find $C(r) = 0.16 \pi r^{2} + \frac{24}{r}$ . Calculate $C'(r)=0$ to find $r$ , then use it to find $h$ .

Find the can dimensions that minimize cost for a volume of 300 cm³, with costs of 0.04 and 0.08 cents per cm².
Use: $r=2.879, \quad h=\square$
Minimum cost: $C=$ cents. Round to the nearest hundredth.

Find the linearization $L(x)$ of the function $f(x) = 8^{x}$ at the point $a=0$ .

Given the population model $P(t)=500 e^{\beta t}$ , find $\beta$ if $P(5)=1000$ . Then use linear approximation at $t=5$ to estimate $P(10)$ and $P(0)$ .

Write Newton's method formula and compute $x_{1}$ and $x_{2}$ for $f(x)=x^{2}-12$ with $x_{0}=3$ . Choose the correct formula.

Find the general anti-derivative of $h(x) = e^{x} + x^{5}$ , using $C$ as the arbitrary constant.

Approximate $(26.5)^{\frac{1}{3}}$ using the linearization of the cube root function at $a=27$ .

Find the surface area of the curve $3 x=\left(y^{2}+2\right)^{\frac{3}{2}}$ from $y=0$ to $y=1$ revolved around the $x$ -axis.

Find the general antiderivative of the function $f(x)=\frac{5}{x}$ for all $x \neq 0$ .

Approximate $(26.5)^{\frac{1}{3}}$ using linearization at $a=27$ to three decimal places. Is it an overestimate or underestimate?

Find the antiderivative of $f(x)=\sqrt{x}$ , $f(x)=x^{n}$ for $n \neq -1$ , and $f(x)=\frac{1}{x}$ for $x>0$ .

Find the function $f(x)$ such that $f^{\prime}(x) = x^{5}$ and $f(0) = 3$ . What is $f(x)$ ?

Find the function $f(x)$ such that $f'(x)=x^2$ and $f(0)=9$ .

Find the $x$ -values for the global maximum and minimum of $h(x)$ on $3 \leq x \leq 12$ given $h'(6)$ is undefined.

Find the 4th-degree Taylor polynomial $L_{4}(x)$ for $f(x)=\mathrm{e}^{6 x}+\ln (2 x)$ .

Find the tangent line equation for $f(x)=x$ at $x=4$ . First, determine the slope using the derivative.

Determine the intervals of concavity for $f(x)=-4 x^{3}+24 x^{2}+165 x-6$ and identify any inflection points.

Find the tangent line equation for $f(x)=x^{2}$ at $x=4$ . First, calculate the derivative for the slope.

Find the function $f(x)$ such that $f'(x) = x^5$ and $f(0) = 9$ .

Given the population model $P(t)=500 e^{\beta t}$ , find $\beta$ if $P(5)=2000$ . Then predict $P(12)$ and $P(0)$ using linear approximation at $a=5$ .

Find the general antiderivative of $f(x)=\frac{7}{x}$ for $x \neq 0$ . Use $C$ as the constant. $F(x)=$

Find the limits: a) $\lim _{x \rightarrow 0} \frac{4 \tan x}{x}$ using $\tan x = \frac{\sin x}{\cos x}$ ; b) $\lim _{x \rightarrow \frac{5 \pi}{3}} \sqrt{\cos x}$ .

Find the linearization $L(x)$ of $f(x)=\sqrt[9]{2+x}$ at $a=5$ and use it to approximate $\sqrt[9]{7.2}$ .

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

Compute the integral $\int_{\gamma} x^{2} d \ell$ along the path $y=\ln x$ for $0<x \leq \sqrt{3}$ .

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{\left(x_{0}+\Delta x\right)^{2} e^{-\left(x_{0}+\Delta x\right)} - x_{0}^{2} e^{-x_{0}}}{\Delta x}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\cos 2 x-\cos ^{2} x}{\cos x-1}$ .

Find the moments of inertia $I_{x}=\int_{\gamma} y^{2} d \ell$ and $I_{y}=\int_{\gamma} x^{2} d \ell$ for a semicircular wire.

Find the limit using L'Hôpital's Rule: $\lim _{x \rightarrow 0} \frac{\sin (x)}{x}=$

Find the point of diminishing returns $(x, y)$ for $R(x) = 11,000 - x^3 + 39x^2 + 700x$ , where $0 \leq x \leq 20$ .

Find the point of diminishing returns $(x, y)$ for the revenue function $R(x)=11,000-x^{3}+36 x^{2}+800 x$ where $0 \leq x \leq 20$ . The point is $\square$ .

Approximate the displacement of an object with $v=1(4t+1)$ m/s from $t=0$ to $t=8$ using 4 subintervals. Result: $\square$ m.

Find the point of diminishing returns $(x, y)$ for $R(x) = 10,000 - x^3 + 42x^2 + 600x$ where $0 \leq x \leq 20$ .

Use Newton's method to find a root of $f(x)=3 x^{2}-\sqrt{x}-4$ starting from $x_{0}=2$ . Show your steps.

Find $\frac{d Q}{d t}$ at $Q=70 \mathrm{~g}$ for the function $Q(t) = 100 - \frac{1000}{10 + t}$ .

A farmer fences 24 million sq ft in a rectangle. Let $x$ and $y$ be side lengths. Find $F(x)$ , $F'(x)$ , and critical numbers.
$F(x)=$
$F^{\prime}(x)=$
$x=$
Smaller side: 4000 ft, larger side: 6000 ft.

Use Newton's method to find a root of $f(x)=3 x^{2}-\sqrt{x}-4$ starting from $x_{0}=2$ . Show your work in a table.

The function $f(x)$ increases on $(-\infty, 0)$ and decreases on $(0,+\infty)$ . What is true about $f(x)$ ? A) max at $x=0$ B) min at $x=0$ C) domain $(0,+\infty)$ D) range $(0,+\infty)$

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(2 x^{6}+8)^{5}$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(3-9 x)^{7}$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{2 x-1}{9 x+7}$ .