Calculus

Problem 24401

Find and simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=x^{3}-4x+1$ .

See Solution

Problem 24402

Evaluate the integral from -1 to 0 of $(6x - 1)$ . Verify with a graphing utility.

See Solution

Problem 24403

Find points on the curve $y=x^{3}+2x+2$ where the tangent is parallel to $3x-y=1$ . Are there multiple points?

See Solution

Problem 24404

Find the derivative of $w=z^{-6}-\frac{1}{z}$ .

See Solution

Problem 24405

Find the interval where the function $f(x)=300 x-x^{3}$ is increasing. Choose from the given options.

See Solution

Problem 24406

Calculate the integral $\int_{-10}^{8} \sqrt{80-2 x-x^{2}} \, dx = \square$ (use $\pi$ if necessary).

See Solution

Problem 24407

Evaluate the integral using geometry: $\int_{-10}^{8} \sqrt{80-2 x-x^{2}} d x=\square$ (exact answer with $\pi$ ).

See Solution

Problem 24408

Evaluate the integral from 1 to 2 of $(x^{2}+x+2)(2 x^{3}+3 x^{2}+12 x)$ .

See Solution

Problem 24409

Determine the curve's equation passing through $(-1,3)$ with slope $3-x^{2}$ .

See Solution

Problem 24410

Untersuchen Sie, wo die Funktion $f$ die Steigung $m$ hat: a) $f(x)=\frac{2}{x^{2}}, m=0,5$ b) $f(x)=\frac{4}{x}, m=-\frac{1}{9}$ c) $f(x)=\frac{4}{x^{3}}, m=-0,75$

See Solution

Problem 24411

Find the derivative $y^{\prime}$ of the function $y=(4 x-5)(5 x+1)$ .

See Solution

Problem 24412

Compute the second partial derivative $\frac{\partial^{2}}{\partial x^{2}}\left(x^{3}+y^{4}+e^{x^{2}+y^{2}}\right)$ .

See Solution

Problem 24413

Evaluate the expression $\frac{\frac{\pi}{3} \int_{0}^{3} y^{2} d y}{\frac{\pi}{3} \int_{0}^{3} y d y}$ .

See Solution

Problem 24414

Calculate the growth rate $\frac{\mathrm{dL}}{\mathrm{dt}}$ at $t=5, 10, 15$ weeks for $L=-37.57+3.73t-6.35 \times 10^{-4}t^{3}$ .

See Solution

Problem 24415

Compute the partial derivative $\frac{\partial}{\partial x} e^{2 x^{2}+9 x y+y^{2}}$ .

See Solution

Problem 24416

Find the derivative $\frac{\partial}{\partial x} x^{2} \ln \left(x y+3 x+y^{2}\right)$ .

See Solution

Problem 24417

Calculate fetal growth rate $\frac{\mathrm{dH}}{\mathrm{dt}}$ using $H=-29.69+1.669 t^{2}-0.7218 t^{2} \log t$ . Compare at $t=8$ and $t=36$ weeks.

See Solution

Problem 24418

Find the derivative of $(x+4) y^{5} e^{x y}$ with respect to $y$ .

See Solution

Problem 24419

Find the tangent line equation for $y=f(x)=x^{2}+1$ at the point $(-4, 17)$ .

See Solution

Problem 24420

Find $b$ in terms of $c$ and $d$ such that $\int_{c}(x+b) dx=0$ for $0<c<d$ . $b=\square$

See Solution

Problem 24421

Find the curve equation through $(1,-2)$ with slope given by $x\left(x^{2}+1\right)^{2}$ .

See Solution

Problem 24422

Find the derivative of $y=5 x^{4}+8 x^{3}+1$ . Which option is correct? A) $4 x^{3}+3 x^{2}$ B) $20 x^{3}+24 x^{2}$ C) $4 x^{3}+3 x^{2}-7$ D) $20 x^{3}+24 x^{2}-7$

See Solution

Problem 24423

Find the slope of the tangent line for $s=4 t^{4}-2 t^{3}$ at $t=-1$ .

See Solution

Problem 24424

Evaluate the integral $\int_{1}^{7} f(x) dx$ where $f(x)=2x$ for $1 \leq x \leq 5$ and $f(x)=12-2x$ for $5<x \leq 7$ .

See Solution

Problem 24425

Determine the definiteness of the Hessian for $Q(K, L)=K^{1/2}L^{1/2}$ when $K, L > 0$ . Options: a. Positive semidefinite b. Negative semidefinite c. Negative definite d. None.

See Solution

Problem 24426

A person throws a ball upward, passing a window at $14 \frac{\mathrm{m}}{\mathrm{s}}$ and $18 \mathrm{~m}$ high. Find total air time.

See Solution

Problem 24427

Bestimme die Ableitungsfunktion $f^{\prime}$ für die Funktionen a) $f(x)=\frac{1}{2} x^{2}$ , b) $f(x)=2 x-1$ , c) $f(x)=x-x^{2}$ mit $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .

See Solution

Problem 24428

Find the partial derivative $\frac{\partial}{\partial y} \left( x^{3} \ln(2xy + 3x + 3y^{3}) \right)$ .

See Solution

Problem 24429

Find the nature of the stationary point of the function with Hessian matrix $\mathbf{H} f(x, y)=\begin{bmatrix} x^{2}+2 & -1 \\ -1 & 1 \end{bmatrix}$ . Select one: a. global minimum b. local minimum c. saddle point d. global maximum.

See Solution

Problem 24430

Compute the derivative $\frac{\partial}{\partial y} x^{3} \ln (2 x y + 3 x + 3 y^{3})$ .

See Solution

Problem 24431

Compute the second partial derivative $\frac{\partial^{2}}{\partial x^{2}}\left(x^{3}+y^{4}+e^{x^{2}+y^{2}}\right)$ .

See Solution

Problem 24432

Which expression does not need the chain rule to find $\frac{d y}{d x}$ ? (A) $y=\sin^{-1}(3x^2-4)$ (B) $3x^6-4y^2=2xy^5+7$ (C) $y=3x^2-\sqrt{x}+\frac{2}{x}$ (D) $\cos(x+y)+2^y-x=0$

See Solution

Problem 24433

Find the tangent and normal lines to the curve $\frac{x^{2}}{25}+\frac{y^{2}}{36}=1$ at the point $\left(1, \frac{12}{5} \sqrt{6}\right)$ .

See Solution

Problem 24434

Calculate the area under the line $y=6x-1$ from $x=1$ to $x=3$ .

See Solution

Problem 24435

Gravel forms an inverted cone with height to radius ratio of $3:2$ . If delivered at 20 ft³/min, find height increase rate at radius 10 ft.

See Solution

Problem 24436

Solve the equation $y^{\prime}=x y-x^{2}$ for: (a) when $y^{\prime}=0$ , (b) when $y^{\prime}>0$ , (c) when $y^{\prime}<0$ .

See Solution

Problem 24437

Find the velocity and acceleration at $t=4$ s for $s(t)=t^{2}-3t$ . Calculate $v(4)=\square$ .

See Solution

Problem 24438

Gravel forms an inverted cone with height to radius ratio of $3:2$ . If gravel is added at 20 ft³/min, find radius increase after 15 sec.

See Solution

Problem 24439

Find values of $a$ and $b$ for the piecewise function $f(x)$ to be differentiable at $x=1$ .

See Solution

Problem 24440

Find the velocity and acceleration at $t=2$ s for $s(t)=t^{2}-5t$ . Calculate $v(2)=\square$ .

See Solution

Problem 24441

Calculate the area in the first quadrant under the curve $y=8x-x^{4}$ .

See Solution

Problem 24442

Find the velocity and acceleration at $t=2$ s for $s(t)=t^{2}-5t$ and $s(t)=\sqrt{t^{2}+5}$ .

See Solution

Problem 24443

A scientist models the density $D(x) = \frac{x+7}{4x^2 - 6x + 5}$ .
(a) Find $x$ that maximizes density and the maximum density, rounding to the nearest hundredth.
(b) For large $x$ , density appears to be undefined.

See Solution

Problem 24444

Find the volume of the solid formed by revolving the region bounded by $y=x$ , $y=4x$ , and $y=8$ about the $y$ -axis. Set up the integral for the volume.

See Solution

Problem 24445

Find $x$ that maximizes $D(x)=\frac{x+7}{4 x^{2}-6 x+5}$ and the maximum density. Round to the nearest hundredth.

See Solution

Problem 24446

Find the growth rate of the fungus given by $L(t)=3.3 t+0.6 \cos \left(\frac{2 \pi t}{24}\right)$ .
(a) Calculate $\frac{\mathrm{dL}}{\mathrm{dt}}$ . (b) Determine the largest and smallest growth rates.

See Solution

Problem 24447

Find the area under the curve $y=x \sqrt[3]{9-x^{2}}$ from $x=0$ to $x=3$ using direct integration.

See Solution

Problem 24448

Calculate the integral from -4 to -3 of $x^{-2}$ : $\int_{-4}^{-3} x^{-2} d x=\square$

See Solution

Problem 24449

Evaluate the integral $\int_{-4}^{-3} x^{-2} \, dx = \square$ using the Fundamental Theorem of Calculus.

See Solution

Problem 24450

Find the time $t$ when the maximum number of people infected, given $P(t)=4 e^{t / 20}-e^{t / 10}$ .

See Solution

Problem 24451

Calculate the integral $\int\left(-3 \sec x \tan x+5 \sec ^{2} x\right) d x$ .

See Solution

Problem 24452

Differentiate the function $f(x)=e^{x}+x^{3}-5 \ln x$ .

See Solution

Problem 24453

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{-\pi / 2}^{\pi / 2}(\cos x-4) d x=\square$

See Solution

Problem 24454

Calculate the integral $\int\left(-8 \sec x \tan x-3 \sec ^{2} x\right) d x = \square$

See Solution

Problem 24455

Ein Hubschrauber steigt. Gegeben sind $h(3) = 120 \mathrm{~m}$ und $h'(3) = 20 \frac{\mathrm{m}}{\mathrm{s}}$ .
a) Finde $h(7)$ und $h'(3)$ . b) Schätze $h(3,5)$ mit linearer Näherung.

See Solution

Problem 24456

Find the area, rounded to 2 decimal places, between the polar curves $r=5 \cos (\theta)$ and $r=3-\cos (\theta)$ .

See Solution

Problem 24457

Integrate: a) $\int(e^{x}-x^{2}-5) \, dx=$

See Solution

Problem 24458

Calculate the area between the curve $f(x)=e^{x}-10$ and the $x$ -axis from $0$ to $\ln 10$ .

See Solution

Problem 24459

Calculate the indefinite integral $\int(5 \sec x \tan x + 7 \sec^2 x) \, dx$ .

See Solution

Problem 24460

Find the derivative of the integral $\frac{d}{d x} \int_{x}^{5} \sqrt{t^{3}+2} d t$ and solve for the result.

See Solution

Problem 24461

Finde die Punkte, wo die Tangente von $f$ den Steigungswinkel $21,8^{\circ}$ hat für die Funktionen: a) $f(x)=5 x^{2}$ b) $f(x)=-\frac{40}{x}$ c) $f(x)=\frac{5}{6} x^{3}$ d) $f(x)=0,15 x^{2}-0,2 x$

See Solution

Problem 24462

Integrate: $\int_{-1}^{1}(t^{2}-t^{4}) dt=$

See Solution

Problem 24463

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{1}^{5} \frac{9}{w^{2}} d w=\square$

See Solution

Problem 24464

Calculate the integral: $\int_{-1}^{1}\left(t^{2}-t^{4}\right) dt$ .

See Solution

Problem 24465

Find the tangent line equation for $f(x)=\sqrt{2x+1}$ at the point where $x=0$ .

See Solution

Problem 24466

Simplify the expression: $\frac{d}{d x} \int_{x}^{5} \sqrt{t^{3}+2} d t = \square$

See Solution

Problem 24467

Finde die Punkte, wo die Tangente der Funktion $f$ parallel zur Linie $y=2x-3$ ist. a) $f(x)=4x^{3}-x$ b) $f(x)=\frac{1}{3}x^{3}+\frac{1}{2}x^{2}-10x$

See Solution

Problem 24468

Calculate the area between the curves $y=6 x^{2}-6$ and $y=-x^{2}+1$ from $x=-1$ to $x=1$ .

See Solution

Problem 24469

Finde die Punkte des Graphen von $f$ , wo die Tangente einen Steigungswinkel von $21,8^{\circ}$ hat: a) $f(x)=5 x^{2}$ b) $f(x)=-\frac{40}{x}$

See Solution

Problem 24470

Find $\frac{d^{2} y}{d x^{2}}$ at the point $(-1,3)$ given $-3 y=-3+2 x y$ .

See Solution

Problem 24471

Calculate the area under the curve $y=x^{2}-4x+5$ between $x=-1$ and $x=3$ .

See Solution

Problem 24472

Find $\frac{d^{2} y}{d x^{2}}$ at the point $(-1,3)$ for the equation $-3 y=-3+2 x y$ .

See Solution

Problem 24473

Simplify the expression: $\frac{d}{d x} \int_{-x}^{x} \sqrt{4+t^{8}} d t$ . Hint: Use $\int_{-x}^{x} \sqrt{1+t^{2}} d t$ .

See Solution

Problem 24474

Find critical points and points of inflection for the function $g(x)=x^{3}+x^{2}-5x+3$ .

See Solution

Problem 24475

Find the limit using L'Hopital's Rule: $\lim _{x \rightarrow-1} \frac{2 x^{2}-x-3}{x+1}$ .

See Solution

Problem 24476

Find $f(x)$ such that $\int_{0}^{x} f(t) dt = 4 \cos x + 3x - 4$ . Verify by substitution.

See Solution

Problem 24477

Find the area of region $R$ under the curve $f(x)=x^{2}(x-11)$ from $x=-1$ to $x=12$ . Round to the nearest hundredth.

See Solution

Problem 24478

Find the velocity of the object at time $t_0=4$ for the position $x(t)=(t^2-t)(t^2+t)$ . Choices: a) 240 b) 248 c) -248 d) 2 e) 2032

See Solution

Problem 24479

Find the first derivative of $y=\sin x \cos x$ . Simplify your answer and show all work.

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Problem 24480

Consider the function $g(x)=\int_{0}^{x} 8 \sin^{2} t \, dt$ .
a. Graph $8 \sin^{2} t$ .
b. Find $g^{\prime}(x)$ .
c. Graph $g$ .

See Solution

Problem 24481

A manufacturer needs a square base box with $1 \mathrm{~kL}$ capacity.
a) Find height $y$ in terms of base $x$ . b) Show total steel cost is $C(x)=24 x^{2}+\frac{96}{x}$ . c) Find $C^{\prime}(x)$ . d) Determine dimensions that minimize cost and the minimum cost.

See Solution

Problem 24482

Calculate the work done by a constant force on a block displaced 5.0 meters. Options: A) 4.0 J, B) 0 J, C) 20 J, D) 0.80 J.

See Solution

Problem 24483

A 10-foot ladder leans against a wall. If the base moves away at 9 ft/s, how fast is the triangle's area changing when it's $\gamma$ ft from the wall?

See Solution

Problem 24484

Find the rate of change of area $A=\frac{1}{2} r^{2} \theta$ with respect to $r$ at $r=3$ when $\theta$ is constant.

See Solution

Problem 24485

Calculate the area under the curve $y=\sqrt{x}$ from $x=4$ to $x=5$ .

See Solution

Problem 24486

Find the new limits for the integral $\int_{5}^{8} f(x) d x$ after the change $u=x^{2}-1$ . Lower limit is 24, upper limit is $\square$ .

See Solution

Problem 24487

Find the new limits of integration for $u=x^{2}-1$ in $\int_{5}^{8} f(x) d x$ . New lower limit is $\square$ .

See Solution

Problem 24488

Given a population growth rate of $4\%$ , with $N(5) = 100$ , estimate $N(5.2)$ using linear approximation. Find $N(5.2) = \square$ .

See Solution

Problem 24489

A rectangle in a circle of radius 5 in. has a length decreasing at 2 in/sec. Find the area change rate when length is 5 in.

See Solution

Problem 24490

Find the area $A = 2 \int_{0}^{5} \sqrt{5-y} \, dy$ . Evaluate $A$ . Answer: $A \simeq 14.9 \mathrm{~m}^{2}$ .

See Solution

Problem 24491

Evaluate the integral using substitution $u=x^{2}+7$ :
$\int 2 x\left(x^{2}+7\right)^{4} d x=\int\left(u^{4}\right) d u$
Find the result. $\int 2 x\left(x^{2}+7\right)^{4} d x=\square$

See Solution

Problem 24492

A snowball's radius decreases from 36 cm to 9 cm in 30 min. Find the volume change rate when the radius is 5 cm.

See Solution

Problem 24493

Evaluate the integral using the substitution $u=x^{2}+7$ : $\int 2 x\left(x^{2}+7\right)^{4} d x=\int(\square) d u$

See Solution

Problem 24494

A cube's volume decreases at 6 m³/min. Find the edge's change rate when volume is 8 m³. Options: a) $\frac{3}{2}$ b) $-\frac{1}{2}$ c) $6$ d) $-12$ e) $0-2$

See Solution

Problem 24495

Find the rate of change of the $x$ -coordinate when the particle at $(-1,4)$ has a $y$ -change of 3 units/sec on $y^{2}=4(x+5)$ .

See Solution

Problem 24496

Find the limit using l'Hôpital's Rule: $\lim _{x \rightarrow 9} \frac{x-9}{\sqrt{x}-3}=\square($ Simplify your answer.)

See Solution

Problem 24497

Find the time intervals where the object moves right, given $x(t)=\frac{1}{2} t^{4}-\frac{8}{3} t^{3}+3 t^{2}$ . a) (0,1) ∪ (3, ∞) b) (1,3) c) (0, 2/3) ∪ (2, ∞) d) (0, 1/2) ∪ (3/2, ∞) e) (0,1) ∪ (9/2, ∞)

See Solution

Problem 24498

Find the limit as $x$ approaches 5 for the expression $\frac{(x-5)}{25-x^{2}}$ .

See Solution

Problem 24499

Evaluate the integral using $u=4 x^{2}+5 x$ : $\int(8 x+5) \sqrt{4 x^{2}+5 x} d x$ . What is the result?

See Solution

Problem 24500

Evaluate the integral using $u=4 x^{2}+5 x$ : $\int(8 x+5) \sqrt{4 x^{2}+5 x} d x=\int(\square) d u$

See Solution

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Calculus

Problem 24401

Find and simplify the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for the function f(x)=x3−4x+1f(x)=x^{3}-4x+1f(x)=x3−4x+1.

Problem 24402

Evaluate the integral from -1 to 0 of (6x−1)(6x - 1)(6x−1). Verify with a graphing utility.

Problem 24403

Find points on the curve y=x3+2x+2y=x^{3}+2x+2y=x3+2x+2 where the tangent is parallel to 3x−y=13x-y=13x−y=1. Are there multiple points?

Problem 24404

Find the derivative of w=z−6−1zw=z^{-6}-\frac{1}{z}w=z−6−z1​.

Problem 24405

Find the interval where the function f(x)=300x−x3f(x)=300 x-x^{3}f(x)=300x−x3 is increasing. Choose from the given options.

Problem 24406

Calculate the integral ∫−10880−2x−x2 dx=□\int_{-10}^{8} \sqrt{80-2 x-x^{2}} \, dx = \square∫−108​80−2x−x2​dx=□ (use π\piπ if necessary).

Problem 24407

Evaluate the integral using geometry: ∫−10880−2x−x2dx=□\int_{-10}^{8} \sqrt{80-2 x-x^{2}} d x=\square∫−108​80−2x−x2​dx=□ (exact answer with π\piπ).

Problem 24408

Evaluate the integral from 1 to 2 of (x2+x+2)(2x3+3x2+12x)(x^{2}+x+2)(2 x^{3}+3 x^{2}+12 x)(x2+x+2)(2x3+3x2+12x).

Problem 24409

Determine the curve's equation passing through (−1,3)(-1,3)(−1,3) with slope 3−x23-x^{2}3−x2.

Problem 24410

Untersuchen Sie, wo die Funktion fff die Steigung mmm hat: a) f(x)=2x2,m=0,5f(x)=\frac{2}{x^{2}}, m=0,5f(x)=x22​,m=0,5 b) f(x)=4x,m=−19f(x)=\frac{4}{x}, m=-\frac{1}{9}f(x)=x4​,m=−91​ c) f(x)=4x3,m=−0,75f(x)=\frac{4}{x^{3}}, m=-0,75f(x)=x34​,m=−0,75

Problem 24411

Find the derivative y′y^{\prime}y′ of the function y=(4x−5)(5x+1)y=(4 x-5)(5 x+1)y=(4x−5)(5x+1).

Problem 24412

Compute the second partial derivative ∂2∂x2(x3+y4+ex2+y2)\frac{\partial^{2}}{\partial x^{2}}\left(x^{3}+y^{4}+e^{x^{2}+y^{2}}\right)∂x2∂2​(x3+y4+ex2+y2).

Problem 24413

Evaluate the expression π3∫03y2dyπ3∫03ydy\frac{\frac{\pi}{3} \int_{0}^{3} y^{2} d y}{\frac{\pi}{3} \int_{0}^{3} y d y}3π​∫03​ydy3π​∫03​y2dy​.

Problem 24414

Calculate the growth rate dLdt\frac{\mathrm{dL}}{\mathrm{dt}}dtdL​ at t=5,10,15t=5, 10, 15t=5,10,15 weeks for L=−37.57+3.73t−6.35×10−4t3L=-37.57+3.73t-6.35 \times 10^{-4}t^{3}L=−37.57+3.73t−6.35×10−4t3.

Problem 24415

Compute the partial derivative ∂∂xe2x2+9xy+y2\frac{\partial}{\partial x} e^{2 x^{2}+9 x y+y^{2}}∂x∂​e2x2+9xy+y2.

Problem 24416

Find the derivative ∂∂xx2ln⁡(xy+3x+y2)\frac{\partial}{\partial x} x^{2} \ln \left(x y+3 x+y^{2}\right)∂x∂​x2ln(xy+3x+y2).

Problem 24417

Calculate fetal growth rate dHdt\frac{\mathrm{dH}}{\mathrm{dt}}dtdH​ using H=−29.69+1.669t2−0.7218t2log⁡tH=-29.69+1.669 t^{2}-0.7218 t^{2} \log tH=−29.69+1.669t2−0.7218t2logt. Compare at t=8t=8t=8 and t=36t=36t=36 weeks.

Problem 24418

Find the derivative of (x+4)y5exy(x+4) y^{5} e^{x y}(x+4)y5exy with respect to yyy.

Problem 24419

Find the tangent line equation for y=f(x)=x2+1y=f(x)=x^{2}+1y=f(x)=x2+1 at the point (−4,17)(-4, 17)(−4,17).

Problem 24420

Find bbb in terms of ccc and ddd such that ∫c(x+b)dx=0\int_{c}(x+b) dx=0∫c​(x+b)dx=0 for 0<c<d0<c<d0<c<d. b=□b=\squareb=□

Problem 24421

Find the curve equation through (1,−2)(1,-2)(1,−2) with slope given by x(x2+1)2x\left(x^{2}+1\right)^{2}x(x2+1)2.

Problem 24422

Find the derivative of y=5x4+8x3+1y=5 x^{4}+8 x^{3}+1y=5x4+8x3+1. Which option is correct? A) 4x3+3x24 x^{3}+3 x^{2}4x3+3x2 B) 20x3+24x220 x^{3}+24 x^{2}20x3+24x2 C) 4x3+3x2−74 x^{3}+3 x^{2}-74x3+3x2−7 D) 20x3+24x2−720 x^{3}+24 x^{2}-720x3+24x2−7

Problem 24423

Find the slope of the tangent line for s=4t4−2t3s=4 t^{4}-2 t^{3}s=4t4−2t3 at t=−1t=-1t=−1.

Problem 24424

Evaluate the integral ∫17f(x)dx\int_{1}^{7} f(x) dx∫17​f(x)dx where f(x)=2xf(x)=2xf(x)=2x for 1≤x≤51 \leq x \leq 51≤x≤5 and f(x)=12−2xf(x)=12-2xf(x)=12−2x for 5<x≤75<x \leq 75<x≤7.

Problem 24425

Determine the definiteness of the Hessian for Q(K,L)=K1/2L1/2Q(K, L)=K^{1/2}L^{1/2}Q(K,L)=K1/2L1/2 when K,L>0K, L > 0K,L>0. Options: a. Positive semidefinite b. Negative semidefinite c. Negative definite d. None.

Problem 24426

A person throws a ball upward, passing a window at 14ms14 \frac{\mathrm{m}}{\mathrm{s}}14sm​ and 18 m18 \mathrm{~m}18 m high. Find total air time.

Problem 24427

Problem 24428

Find the partial derivative ∂∂y(x3ln⁡(2xy+3x+3y3))\frac{\partial}{\partial y} \left( x^{3} \ln(2xy + 3x + 3y^{3}) \right)∂y∂​(x3ln(2xy+3x+3y3)).

Problem 24429

Find the nature of the stationary point of the function with Hessian matrix Hf(x,y)=[x2+2−1−11]\mathbf{H} f(x, y)=\begin{bmatrix} x^{2}+2 & -1 \\ -1 & 1 \end{bmatrix}Hf(x,y)=[x2+2−1​−11​]. Select one: a. global minimum b. local minimum c. saddle point d. global maximum.

Problem 24430

Compute the derivative ∂∂yx3ln⁡(2xy+3x+3y3)\frac{\partial}{\partial y} x^{3} \ln (2 x y + 3 x + 3 y^{3})∂y∂​x3ln(2xy+3x+3y3).

Problem 24431

Compute the second partial derivative ∂2∂x2(x3+y4+ex2+y2)\frac{\partial^{2}}{\partial x^{2}}\left(x^{3}+y^{4}+e^{x^{2}+y^{2}}\right)∂x2∂2​(x3+y4+ex2+y2).

Problem 24432

Problem 24433

Find the tangent and normal lines to the curve x225+y236=1\frac{x^{2}}{25}+\frac{y^{2}}{36}=125x2​+36y2​=1 at the point (1,1256)\left(1, \frac{12}{5} \sqrt{6}\right)(1,512​6​).

Problem 24434

Calculate the area under the line y=6x−1y=6x-1y=6x−1 from x=1x=1x=1 to x=3x=3x=3.

Problem 24435

Gravel forms an inverted cone with height to radius ratio of 3:23:23:2. If delivered at 20 ft³/min, find height increase rate at radius 10 ft.

Problem 24436

Solve the equation y′=xy−x2y^{\prime}=x y-x^{2}y′=xy−x2 for: (a) when y′=0y^{\prime}=0y′=0, (b) when y′>0y^{\prime}>0y′>0, (c) when y′<0y^{\prime}<0y′<0.

Problem 24437

Find the velocity and acceleration at t=4t=4t=4 s for s(t)=t2−3ts(t)=t^{2}-3ts(t)=t2−3t. Calculate v(4)=□v(4)=\squarev(4)=□.

Problem 24438

Gravel forms an inverted cone with height to radius ratio of 3:23:23:2. If gravel is added at 20 ft³/min, find radius increase after 15 sec.

Problem 24439

Find values of aaa and bbb for the piecewise function f(x)f(x)f(x) to be differentiable at x=1x=1x=1.

Problem 24440

Find the velocity and acceleration at t=2t=2t=2 s for s(t)=t2−5ts(t)=t^{2}-5ts(t)=t2−5t. Calculate v(2)=□v(2)=\squarev(2)=□.

Problem 24441

Find and simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=x^{3}-4x+1$ .

Evaluate the integral from -1 to 0 of $(6x - 1)$ . Verify with a graphing utility.

Find points on the curve $y=x^{3}+2x+2$ where the tangent is parallel to $3x-y=1$ . Are there multiple points?

Find the derivative of $w=z^{-6}-\frac{1}{z}$ .

Find the interval where the function $f(x)=300 x-x^{3}$ is increasing. Choose from the given options.

Calculate the integral $\int_{-10}^{8} \sqrt{80-2 x-x^{2}} \, dx = \square$ (use $\pi$ if necessary).

Evaluate the integral using geometry: $\int_{-10}^{8} \sqrt{80-2 x-x^{2}} d x=\square$ (exact answer with $\pi$ ).

Evaluate the integral from 1 to 2 of $(x^{2}+x+2)(2 x^{3}+3 x^{2}+12 x)$ .

Determine the curve's equation passing through $(-1,3)$ with slope $3-x^{2}$ .

Untersuchen Sie, wo die Funktion $f$ die Steigung $m$ hat: a) $f(x)=\frac{2}{x^{2}}, m=0,5$ b) $f(x)=\frac{4}{x}, m=-\frac{1}{9}$ c) $f(x)=\frac{4}{x^{3}}, m=-0,75$

Find the derivative $y^{\prime}$ of the function $y=(4 x-5)(5 x+1)$ .

Compute the second partial derivative $\frac{\partial^{2}}{\partial x^{2}}\left(x^{3}+y^{4}+e^{x^{2}+y^{2}}\right)$ .

Evaluate the expression $\frac{\frac{\pi}{3} \int_{0}^{3} y^{2} d y}{\frac{\pi}{3} \int_{0}^{3} y d y}$ .

Calculate the growth rate $\frac{\mathrm{dL}}{\mathrm{dt}}$ at $t=5, 10, 15$ weeks for $L=-37.57+3.73t-6.35 \times 10^{-4}t^{3}$ .

Compute the partial derivative $\frac{\partial}{\partial x} e^{2 x^{2}+9 x y+y^{2}}$ .

Find the derivative $\frac{\partial}{\partial x} x^{2} \ln \left(x y+3 x+y^{2}\right)$ .

Calculate fetal growth rate $\frac{\mathrm{dH}}{\mathrm{dt}}$ using $H=-29.69+1.669 t^{2}-0.7218 t^{2} \log t$ . Compare at $t=8$ and $t=36$ weeks.

Find the derivative of $(x+4) y^{5} e^{x y}$ with respect to $y$ .

Find the tangent line equation for $y=f(x)=x^{2}+1$ at the point $(-4, 17)$ .

Find $b$ in terms of $c$ and $d$ such that $\int_{c}(x+b) dx=0$ for $0<c<d$ . $b=\square$

Find the curve equation through $(1,-2)$ with slope given by $x\left(x^{2}+1\right)^{2}$ .

Find the derivative of $y=5 x^{4}+8 x^{3}+1$ . Which option is correct? A) $4 x^{3}+3 x^{2}$ B) $20 x^{3}+24 x^{2}$ C) $4 x^{3}+3 x^{2}-7$ D) $20 x^{3}+24 x^{2}-7$

Find the slope of the tangent line for $s=4 t^{4}-2 t^{3}$ at $t=-1$ .

Evaluate the integral $\int_{1}^{7} f(x) dx$ where $f(x)=2x$ for $1 \leq x \leq 5$ and $f(x)=12-2x$ for $5<x \leq 7$ .

Determine the definiteness of the Hessian for $Q(K, L)=K^{1/2}L^{1/2}$ when $K, L > 0$ . Options: a. Positive semidefinite b. Negative semidefinite c. Negative definite d. None.

A person throws a ball upward, passing a window at $14 \frac{\mathrm{m}}{\mathrm{s}}$ and $18 \mathrm{~m}$ high. Find total air time.

Find the partial derivative $\frac{\partial}{\partial y} \left( x^{3} \ln(2xy + 3x + 3y^{3}) \right)$ .

Find the nature of the stationary point of the function with Hessian matrix $\mathbf{H} f(x, y)=\begin{bmatrix} x^{2}+2 & -1 \\ -1 & 1 \end{bmatrix}$ . Select one: a. global minimum b. local minimum c. saddle point d. global maximum.

Compute the derivative $\frac{\partial}{\partial y} x^{3} \ln (2 x y + 3 x + 3 y^{3})$ .

Compute the second partial derivative $\frac{\partial^{2}}{\partial x^{2}}\left(x^{3}+y^{4}+e^{x^{2}+y^{2}}\right)$ .

Find the tangent and normal lines to the curve $\frac{x^{2}}{25}+\frac{y^{2}}{36}=1$ at the point $\left(1, \frac{12}{5} \sqrt{6}\right)$ .

Calculate the area under the line $y=6x-1$ from $x=1$ to $x=3$ .

Gravel forms an inverted cone with height to radius ratio of $3:2$ . If delivered at 20 ft³/min, find height increase rate at radius 10 ft.

Solve the equation $y^{\prime}=x y-x^{2}$ for: (a) when $y^{\prime}=0$ , (b) when $y^{\prime}>0$ , (c) when $y^{\prime}<0$ .

Find the velocity and acceleration at $t=4$ s for $s(t)=t^{2}-3t$ . Calculate $v(4)=\square$ .

Gravel forms an inverted cone with height to radius ratio of $3:2$ . If gravel is added at 20 ft³/min, find radius increase after 15 sec.

Find values of $a$ and $b$ for the piecewise function $f(x)$ to be differentiable at $x=1$ .

Find the velocity and acceleration at $t=2$ s for $s(t)=t^{2}-5t$ . Calculate $v(2)=\square$ .

Calculate the area in the first quadrant under the curve $y=8x-x^{4}$ .

Find the velocity and acceleration at $t=2$ s for $s(t)=t^{2}-5t$ and $s(t)=\sqrt{t^{2}+5}$ .

A scientist models the density $D(x) = \frac{x+7}{4x^2 - 6x + 5}$ .
(a) Find $x$ that maximizes density and the maximum density, rounding to the nearest hundredth.
(b) For large $x$ , density appears to be undefined.

Find the volume of the solid formed by revolving the region bounded by $y=x$ , $y=4x$ , and $y=8$ about the $y$ -axis. Set up the integral for the volume.

Find $x$ that maximizes $D(x)=\frac{x+7}{4 x^{2}-6 x+5}$ and the maximum density. Round to the nearest hundredth.

Find the growth rate of the fungus given by $L(t)=3.3 t+0.6 \cos \left(\frac{2 \pi t}{24}\right)$ .
(a) Calculate $\frac{\mathrm{dL}}{\mathrm{dt}}$ . (b) Determine the largest and smallest growth rates.

Find the area under the curve $y=x \sqrt[3]{9-x^{2}}$ from $x=0$ to $x=3$ using direct integration.

Calculate the integral from -4 to -3 of $x^{-2}$ : $\int_{-4}^{-3} x^{-2} d x=\square$

Evaluate the integral $\int_{-4}^{-3} x^{-2} \, dx = \square$ using the Fundamental Theorem of Calculus.

Find the time $t$ when the maximum number of people infected, given $P(t)=4 e^{t / 20}-e^{t / 10}$ .

Calculate the integral $\int\left(-3 \sec x \tan x+5 \sec ^{2} x\right) d x$ .

Differentiate the function $f(x)=e^{x}+x^{3}-5 \ln x$ .

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{-\pi / 2}^{\pi / 2}(\cos x-4) d x=\square$

Calculate the integral $\int\left(-8 \sec x \tan x-3 \sec ^{2} x\right) d x = \square$

Ein Hubschrauber steigt. Gegeben sind $h(3) = 120 \mathrm{~m}$ und $h'(3) = 20 \frac{\mathrm{m}}{\mathrm{s}}$ .
a) Finde $h(7)$ und $h'(3)$ . b) Schätze $h(3,5)$ mit linearer Näherung.

Find the area, rounded to 2 decimal places, between the polar curves $r=5 \cos (\theta)$ and $r=3-\cos (\theta)$ .

Integrate: a) $\int(e^{x}-x^{2}-5) \, dx=$

Calculate the area between the curve $f(x)=e^{x}-10$ and the $x$ -axis from $0$ to $\ln 10$ .

Calculate the indefinite integral $\int(5 \sec x \tan x + 7 \sec^2 x) \, dx$ .

Find the derivative of the integral $\frac{d}{d x} \int_{x}^{5} \sqrt{t^{3}+2} d t$ and solve for the result.

Integrate: $\int_{-1}^{1}(t^{2}-t^{4}) dt=$

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{1}^{5} \frac{9}{w^{2}} d w=\square$

Calculate the integral: $\int_{-1}^{1}\left(t^{2}-t^{4}\right) dt$ .

Find the tangent line equation for $f(x)=\sqrt{2x+1}$ at the point where $x=0$ .

Simplify the expression: $\frac{d}{d x} \int_{x}^{5} \sqrt{t^{3}+2} d t = \square$

Finde die Punkte, wo die Tangente der Funktion $f$ parallel zur Linie $y=2x-3$ ist. a) $f(x)=4x^{3}-x$ b) $f(x)=\frac{1}{3}x^{3}+\frac{1}{2}x^{2}-10x$

Calculate the area between the curves $y=6 x^{2}-6$ and $y=-x^{2}+1$ from $x=-1$ to $x=1$ .

Finde die Punkte des Graphen von $f$ , wo die Tangente einen Steigungswinkel von $21,8^{\circ}$ hat: a) $f(x)=5 x^{2}$ b) $f(x)=-\frac{40}{x}$

Find $\frac{d^{2} y}{d x^{2}}$ at the point $(-1,3)$ given $-3 y=-3+2 x y$ .

Calculate the area under the curve $y=x^{2}-4x+5$ between $x=-1$ and $x=3$ .

Find $\frac{d^{2} y}{d x^{2}}$ at the point $(-1,3)$ for the equation $-3 y=-3+2 x y$ .

Simplify the expression: $\frac{d}{d x} \int_{-x}^{x} \sqrt{4+t^{8}} d t$ . Hint: Use $\int_{-x}^{x} \sqrt{1+t^{2}} d t$ .

Find critical points and points of inflection for the function $g(x)=x^{3}+x^{2}-5x+3$ .