Calculus

Problem 14701

The position $r$ (in cm) of an object at time $t$ (in s) is $r=17-24 t^{2}+4 t^{3}$ .
a. Initial location? $r=17$ cm. b. When returns to initial location? $t=6$ s. c. Velocity and acceleration at return? velocity= $\square$ , acceleration= $\square$ . d. When not moving? $t=0,6$ s. e. When zero acceleration? $t=0,1$ s. f. When moving backwards and slowing down? $t \in(0,1)$ s.

See Solution

Problem 14702

Find the object's location $r=17-24 t^{2}+4 t^{3}$ in cm at time $t$ in s. Answer questions a-f with 3 significant figures.

See Solution

Problem 14703

What is the maximum height of a stomp rocket launched at 30.38 m/s? Use the formula for maximum height: $H = \frac{v^2}{2g}$ .

See Solution

Problem 14704

Find $\frac{d y}{d x}$ for the equation $x^{2}+y^{2}=x y+19$ at $x=5$ , $y=2$ .

See Solution

Problem 14705

Find how fast the area of a circle is decreasing when the radius is 5 inches, given the radius decreases at 4 in/sec.

See Solution

Problem 14706

A 30-foot ladder leans against a wall. If the base moves away at $3 \mathrm{ft/sec}$ , how fast does the top descend when 10 ft away?

See Solution

Problem 14707

Find $\frac{d y}{d x}$ for the equation $x^{2}+y^{2}=x y+39$ at $x=7, y=2$ .

See Solution

Problem 14708

A circle's radius is decreasing at 4 in/sec. Find the rate of area decrease when the radius is 5 inches.

See Solution

Problem 14709

Determine the horizontal asymptote of the function $f(x)=9 \cdot 6^{x+7}+9$ .

See Solution

Problem 14710

Calculate the average value of $f(x)=\frac{1}{\sqrt{4 x-3}}$ over the interval $[3, 21]$ .

See Solution

Problem 14711

Evaluate the integral: $\int \sec 9 \theta \tan 9 \theta d \theta$

See Solution

Problem 14712

Find the center of mass coordinates $(\bar{x}, \bar{y})$ for the region between the axes and lines $x=2$ , $y=x+2$ with density $\rho=3x$ .

See Solution

Problem 14713

Find the center of mass coordinates $(\bar{x}, \bar{y})$ for the region between the axes and lines $x=2$ , $y=x+2$ , with density $\rho=3x$ .

See Solution

Problem 14714

Find the tangent line equation to the curve at $t=-1$ : $x=t^{9}+1$ , $y=t^{10}+t$ .

See Solution

Problem 14715

For the function $f(x)=x^{4}-4 x^{3}+4$ , find: (a) domain, (b) $f'(x)$ , (c) increase/decrease regions, (d) local extrema, (e) $f''(x)$ , (f) concavity, (g) inflection points.

See Solution

Problem 14716

Find the derivative $\frac{dy}{dx}$ for the equation $xy - 5x = 8$ .

See Solution

Problem 14717

Find the difference quotient, $\frac{f(x+h)-f(x)}{h}$ , for the function $f(x)=-7 x^{2}-2 x+2$ .

See Solution

Problem 14718

Find the average rate of change of $y=\cos x$ from $x=\frac{3 \pi}{4}$ to $x=\frac{7 \pi}{4}$ .

See Solution

Problem 14719

Find the key point and horizontal asymptote of $f(x)=e^{x+2}-4$ . Options: A, B, C, D, or E.

See Solution

Problem 14720

Find all critical points of the function $f(x)=2 x^{2}-1000 \sqrt{x}$ .

See Solution

Problem 14721

Calculate the integral $\int 5 x e^{3-9 x} d x$ and specify if you used substitution or integration by parts.

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Problem 14722

Show that if $f, g: \mathbb{R} \rightarrow \mathbb{R}$ are differentiable and $g$ is one-to-one, then $\frac{d}{d x} f(g(x))=g^{\prime}(x) f^{\prime}(g(x))$ .

See Solution

Problem 14723

Evaluate the integral: $\int \frac{\cos x}{\sin ^{8} x} \, dx$

See Solution

Problem 14724

Find the average rate of change of $g(x)=x^{2}$ from $x=3$ to $x=9$ . Options: (1) 8, (2) 12, (3) 15, (4) 18.

See Solution

Problem 14725

When is the bear population $p(t)=250+30 \cos t$ decreasing? Choose from: a. $0<t<\frac{\pi}{2}$ , b. $\frac{3 \pi}{2}<t<2 \pi$ , c. $\pi<t<2 \pi$ , d. $0<t<\pi$ .

See Solution

Problem 14726

A red sports car's speed is $f(t)=\frac{44000}{(t+20)^{2}}$ for $0 \leq t \leq 6$ . Analyze distance estimates and improvements.

See Solution

Problem 14727

Find $\frac{d h}{d p}$ at $p=200$ for the equation $20 h^{5}+6,000,000=p^{3}$ . Interpret your answer.

See Solution

Problem 14728

Find the indefinite integral of (20x + 17) ln(7x) dx.

See Solution

Problem 14729

Find the indefinite integral of (16x + 3) ln(3x) dx.

See Solution

Problem 14730

Given the derivative $f'(x) = 6 - 5x^2$ , determine where $f$ is increasing/decreasing, local extrema, and concavity.

See Solution

Problem 14731

A function $f$ has $f(5)=28$ , $f^{\prime}(5)=1.4$ , and $f^{\prime \prime}(x)<0$ for $x \geq 5$ . Determine if: (a) $f(5.4)=28.56$ is possible? (b) $f(5.4)=28.28$ is possible? (c) $f(5.4)=29.12$ is possible?

See Solution

Problem 14732

Calculate the total cost for 520 cards using the cost function $C^{\prime}(x)=-0.05 x+75$ for $x \leq 1000$ .

See Solution

Problem 14733

Find the critical points of the function $f(x)=x^{3}-6x^{2}+2x+16$ .

See Solution

Problem 14734

Determine where the function $g(t)=-4 t^{2}+5 t+5$ is increasing or decreasing and find its local extrema.

See Solution

Problem 14735

Find the second derivatives of $f(x) = x^3-6x^2+2x+16$ at $c=\frac{6+\sqrt{30}}{3}$ and $c=\frac{6-\sqrt{30}}{3}$ . Determine local extrema.

See Solution

Problem 14736

Find the four second partial derivatives of $z=\sqrt{64-x^{2}-y^{2}}$ .

See Solution

Problem 14737

Find the average rate of change of $f(x)=3 x^{2}+3$ over the interval $[3,5]$ .

See Solution

Problem 14738

Find the quantity that minimizes average cost for $C(q)=q^{3}-11 q^{2}+52 q$ and $R(q)=-3 q^{2}+2600 q$ .

See Solution

Problem 14739

At noon, ship A is 170 km west of ship B. A sails east at 35 km/h, B north at 30 km/h. Find distance change rate at 4 p.m.

See Solution

Problem 14740

Find the rate of area increase (in $\mathrm{cm}^{2} / \mathrm{s}$ ) for a square with side increasing at $4 \mathrm{~cm/s}$ when area is $16 \mathrm{~cm}^{2}$ .

See Solution

Problem 14741

Find the second partial derivatives of the function $v=\frac{xy}{x-y}$ .

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Problem 14742

Is the statement "The slope of tangent lines to $h(x)=x^{2n+1}$ is always non-negative" true, sometimes true, or never true? Explain and provide 2 examples.

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Problem 14743

Evaluate the integral: $\int \frac{\tan^{-1}(9x)}{1+81x^{2}} \, dx$

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Problem 14744

Show that for the curve $x^{2} y - y - x = 4$ , $\frac{d y}{d x} = \frac{1 - 2 x y}{x^{2} - 1}$ .

See Solution

Problem 14745

Find the derivatives: a. $y=10 \cdot 15^{x}-x^{8}$ . b. $f(x)=\frac{x^{10}}{e^{12}}+8^{x}$ . c. $g(w)=\frac{w^{5} 4^{w}}{\pi^{5}}$ .

See Solution

Problem 14746

Find when the slope of the tangent line of a function is zero. $(\mathrm{Com}=3)$

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Problem 14747

Find the derivative $h^{\prime}(x)$ of the function $h(x)=\ln(\sec(x^{4}))$ .

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Problem 14748

Differentiate a function using the product and chain rule instead of the quotient rule. $(\mathrm{Com}=3)$

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Problem 14749

Differentiate a function using the product rule instead of the chain rule. Provide an example.

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Problem 14750

Use the product rule to differentiate $y=\frac{(3+x)}{(2 x-9)}$ instead of the quotient rule.

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Problem 14751

Explain the meaning of a derivative of a function: $(\mathrm{Com}=3)$ .

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Problem 14752

Find the slope and concavity of the curve where $x=\cos \theta$ and $y=3 \sin \theta$ at $\theta=0$ .

See Solution

Problem 14753

Explain how to apply the product rule to $y=\frac{(3+x)}{(2 x-9)}$ instead of the quotient rule.

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Problem 14754

In a class of 30, model student understanding with $P(t)=\frac{150}{5+25 e^{-0.45 t}}$ .
A) Initial students understanding? B) Days for 10 students to understand? Choose: - f) $\mathrm{t}=(\ln 5 / 2) / 0.45$ - g) $\mathrm{t}=(\ln 10) / 0.45$ - h) $\mathrm{t}=0.45 /(\ln 5 / 2)$ - j) $\mathrm{t}=0.45 /(\ln 10)$

See Solution

Problem 14755

A man 2 m tall walks towards a wall 12 m away at 2.5 m/s. How fast is his shadow on the wall decreasing when 4 m away?

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Problem 14756

Find the derivative of the function $f(x)=\tan^{-1}(\sqrt{8x^2-1})$ . What is $f^{\prime}(x)$ ?

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Problem 14757

A rocket's height is given by $h(t)=-4.9 t^2+9.8 t$ . Find when it hits the ground, speeds, and average speed in 1 second.

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Problem 14758

Find the foraging time $F$ that minimizes energy expenditure $E = 0.25F + \frac{1.7}{F^2}$ . Round to three decimal places.

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Problem 14759

Find the normal equation to the tangent of $y=\frac{3 x}{(x^{2}-1)}$ at $x=3$ .

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Problem 14760

Compute the derivative of the function $f(x)=e^{x}-e^{-x}$ .

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Problem 14761

Evaluate the limits: a) $\lim _{x \rightarrow -4} \frac{x^{2}+12x+32}{x+4}$ b) $\lim _{x \rightarrow 4} \frac{4-\sqrt{12+x}}{x-4}$

See Solution

Problem 14762

Find the derivative of $g(x) = \frac{x}{1 - e^{x}}$ .

See Solution

Problem 14763

Find the average rate of change of $g(x)=\frac{2}{x+3}$ from $x=1$ to $x=3$ .

See Solution

Problem 14764

Gravel is dumped at $30 \mathrm{ft}^{3}/\mathrm{min}$ , forming a cone. Find height increase rate when height is $5 \mathrm{ft}$ .

See Solution

Problem 14765

Find the derivative of $p(x)=\frac{x^{2}}{4^{x}}$ .

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Problem 14766

A car moves north at $30 \mathrm{~m/s}$ and a drone flies east at $7 \mathrm{~m/s}$ , 20 m high. Find distance after 4 seconds.

See Solution

Problem 14767

Find the derivative of $h(x) = \cos^2(x)e^x$ .

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Problem 14768

A car moves north at $30 \mathrm{~m/s}$ and a drone flies east at $7 \mathrm{~m/s}$ at $20 \mathrm{~m}$ . Find the rate of distance change 4 seconds later.

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Problem 14769

A coin is dropped from 443.2 m. Find its velocity on impact and the time it takes to hit the ground.

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Problem 14770

Calculate the coin's velocity and time to hit the ground when dropped from 354m, with gravity at 9.8 m/s $^2$ .

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Problem 14771

Sam is building two adjacent rectangular pens against a wall with a total area of 300 sq ft. Find the minimum fence length using $x$ for width and $y$ for length. Use calculus to verify.

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Problem 14772

A disk's radius increases at $0.03 \mathrm{~mm/sec}$ . Find the area change rate when the radius is $200 \mathrm{~mm}$ .
A cylinder's height decreases at $0.2 \mathrm{~cm/sec}$ . Find the radius change rate when radius is $3 \mathrm{~cm}$ and height is $4 \mathrm{~cm}$ .

See Solution

Problem 14773

Find the grams of carbon-14 left after 7136 years using the model $A=16 e^{-0.000121 t}$ . Answer: $\square$ grams.

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Problem 14774

Approximate the area under $y=x^{3}$ from $x=3$ to $x=6$ using a Right Endpoint method with 6 subdivisions.

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Problem 14775

Find the derivative of $y$ where $y=\cos^{-1}(2x^7)$ .

See Solution

Problem 14776

Find the derivative of $y$ with respect to $x$ for $y=\ln \left(\frac{(x^{2}+5)^{5}}{\sqrt{9-x}}\right)$ .

See Solution

Problem 14777

Find the rate of area increase of a heated circular plate when its radius is $41 \mathrm{~cm}$ and increases at $0.02 \mathrm{~cm/min}$ .

See Solution

Problem 14778

Find the normal line equation to the curve $f(x)=\frac{2x}{x+1}$ at $x=-3$ .

See Solution

Problem 14779

Find $dy$ for the equation $y = x \sqrt{14 - x^4}$ .

See Solution

Problem 14780

Find $\int_{0}^{4} f(x) d x$ given $\int_{0}^{1} f(x) d x=-3$ , $\int_{0}^{2} f(x) d x=6$ , $\int_{1}^{4} f(x) d x=9$ .

See Solution

Problem 14781

Find the point(s) on the curve $x=6 t^{2}+8, y=t^{3}-6$ where the tangent line slope is $\frac{1}{2}$ .

See Solution

Problem 14782

Find the tangent line equation for $f(x)=(4x^{2}-3x+3)^{2}$ at the point $(1,16)$ .

See Solution

Problem 14783

Evaluate the integrals $\int_{0}^{2} g(t) dt$ and $\int_{1}^{5} g(t) dt$ for $g(x)=-|x-3|+2$ .

See Solution

Problem 14784

Estimate the distance traveled by a vehicle in 84 seconds using $L_{6}$ , $R_{6}$ , and $M_{3}$ from speedometer data.

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Problem 14785

Find the radius and height of a cone with maximum volume formed from a circular sheet of radius 29.

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Problem 14786

A plane at 1 mile altitude flies at $530 \mathrm{mi/h}$ . Find the distance rate from the plane to a radar station 2 miles away.

See Solution

Problem 14787

Find the tangent line equation for $f(x)=\frac{5 x}{(x^{2}-2)^{3}}$ at the point $(0,0)$ .

See Solution

Problem 14788

Evaluate the limit: $\lim _{x \rightarrow 2} \frac{e^{x^{2}}-e^{4}}{x-2}=$

See Solution

Problem 14789

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{e^{2 x}-6-13 x}{(13 x)^{2}}=$

See Solution

Problem 14790

Find the derivative of $y$ with respect to $x$ for $y=\ln \left(\frac{(x^{2}+2)^{5}}{\sqrt{8-x}}\right)$ .

See Solution

Problem 14791

Approximate $\int_{1}^{10} f(x) dx$ using a right Riemann sum with intervals $[1,3],[3,5],[5,8],[8,10]$ . Choices: (A) 96, (B) 116, (C) 132, (D) 159.

See Solution

Problem 14792

Find which option represents the Riemann sum: $\frac{1}{20}\sum_{k=1}^{20}\left(\frac{k}{20}\right)^{2}$ . Options are integrals of $x^{2}$ or $\frac{1}{x^{2}}$ .

See Solution

Problem 14793

Find $\frac{d x}{d t}$ at $x=-1$ given $y=-4 x^{2}-5$ and $\frac{d y}{d t}=1$ .

See Solution

Problem 14794

A light is 19 feet from a wall, rotating every 5 seconds. Find the speed of its projection on the wall at 10 degrees.

See Solution

Problem 14795

Compare the average rates of change for $f(x)=0.1 x^{2}$ and $g(x)=0.3 x^{2}$ over the interval $1 \leq x \leq 4$ .

See Solution

Problem 14796

Calculate the integral: $\int_{0}^{8} \frac{x e^{x}}{(1+x)^{2}} d x$

See Solution

Problem 14797

Compare the average rates of change for $f(x) = -2x^2$ and $g(x) = -4x^2$ over the interval $[-4, -2]$ .

See Solution

Problem 14798

Find the volume $V$ using cylindrical shells for the region bounded by $y=e^{-x}$ , $y=0$ , $x=-3$ , $x=0$ rotated about $x=1$ .

See Solution

Problem 14799

Find the limit: $\lim _{x \rightarrow \infty}\left(\frac{x}{x+4}\right)^{x}=$ (Exact answer or DNE if it doesn't exist.)

See Solution

Problem 14800

Approximate $\sqrt{25.1}$ using linear approximation with $f(x)=\sqrt{x}$ at $x=25$ . Find $L(x)$ and give 9 significant figures.

See Solution

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Calculus

Problem 14701

Problem 14702

Find the object's location r=17−24t2+4t3r=17-24 t^{2}+4 t^{3}r=17−24t2+4t3 in cm at time ttt in s. Answer questions a-f with 3 significant figures.

Problem 14703

What is the maximum height of a stomp rocket launched at 30.38 m/s? Use the formula for maximum height: H=v22gH = \frac{v^2}{2g}H=2gv2​.

Problem 14704

Find dydx\frac{d y}{d x}dxdy​ for the equation x2+y2=xy+19x^{2}+y^{2}=x y+19x2+y2=xy+19 at x=5x=5x=5, y=2y=2y=2.

Problem 14705

Find how fast the area of a circle is decreasing when the radius is 5 inches, given the radius decreases at 4 in/sec.

Problem 14706

A 30-foot ladder leans against a wall. If the base moves away at 3ft/sec3 \mathrm{ft/sec}3ft/sec, how fast does the top descend when 10 ft away?

Problem 14707

Find dydx\frac{d y}{d x}dxdy​ for the equation x2+y2=xy+39x^{2}+y^{2}=x y+39x2+y2=xy+39 at x=7,y=2x=7, y=2x=7,y=2.

Problem 14708

A circle's radius is decreasing at 4 in/sec. Find the rate of area decrease when the radius is 5 inches.

Problem 14709

Determine the horizontal asymptote of the function f(x)=9⋅6x+7+9f(x)=9 \cdot 6^{x+7}+9f(x)=9⋅6x+7+9.

Problem 14710

Calculate the average value of f(x)=14x−3f(x)=\frac{1}{\sqrt{4 x-3}}f(x)=4x−3​1​ over the interval [3,21][3, 21][3,21].

Problem 14711

Evaluate the integral: ∫sec⁡9θtan⁡9θdθ\int \sec 9 \theta \tan 9 \theta d \theta∫sec9θtan9θdθ

Problem 14712

Find the center of mass coordinates (xˉ,yˉ)(\bar{x}, \bar{y})(xˉ,yˉ​) for the region between the axes and lines x=2x=2x=2, y=x+2y=x+2y=x+2 with density ρ=3x\rho=3xρ=3x.

Problem 14713

Find the center of mass coordinates (xˉ,yˉ)(\bar{x}, \bar{y})(xˉ,yˉ​) for the region between the axes and lines x=2x=2x=2, y=x+2y=x+2y=x+2, with density ρ=3x\rho=3xρ=3x.

Problem 14714

Find the tangent line equation to the curve at t=−1t=-1t=−1: x=t9+1x=t^{9}+1x=t9+1, y=t10+ty=t^{10}+ty=t10+t.

Problem 14715

For the function f(x)=x4−4x3+4f(x)=x^{4}-4 x^{3}+4f(x)=x4−4x3+4, find: (a) domain, (b) f′(x)f'(x)f′(x), (c) increase/decrease regions, (d) local extrema, (e) f′′(x)f''(x)f′′(x), (f) concavity, (g) inflection points.

Problem 14716

Find the derivative dydx\frac{dy}{dx}dxdy​ for the equation xy−5x=8xy - 5x = 8xy−5x=8.

Problem 14717

Find 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)=−7x2−2x+2f(x)=-7 x^{2}-2 x+2f(x)=−7x2−2x+2.

Problem 14718

Find the average rate of change of y=cos⁡xy=\cos xy=cosx from x=3π4x=\frac{3 \pi}{4}x=43π​ to x=7π4x=\frac{7 \pi}{4}x=47π​.

Problem 14719

Find the key point and horizontal asymptote of f(x)=ex+2−4f(x)=e^{x+2}-4f(x)=ex+2−4. Options: A, B, C, D, or E.

Problem 14720

Find all critical points of the function f(x)=2x2−1000xf(x)=2 x^{2}-1000 \sqrt{x}f(x)=2x2−1000x​.

Problem 14721

Calculate the integral ∫5xe3−9xdx\int 5 x e^{3-9 x} d x∫5xe3−9xdx and specify if you used substitution or integration by parts.

Problem 14722

Show that if f,g:R→Rf, g: \mathbb{R} \rightarrow \mathbb{R}f,g:R→R are differentiable and ggg is one-to-one, then ddxf(g(x))=g′(x)f′(g(x))\frac{d}{d x} f(g(x))=g^{\prime}(x) f^{\prime}(g(x))dxd​f(g(x))=g′(x)f′(g(x)).

Problem 14723

Evaluate the integral: ∫cos⁡xsin⁡8x dx\int \frac{\cos x}{\sin ^{8} x} \, dx∫sin8xcosx​dx

Problem 14724

Find the average rate of change of g(x)=x2g(x)=x^{2}g(x)=x2 from x=3x=3x=3 to x=9x=9x=9. Options: (1) 8, (2) 12, (3) 15, (4) 18.

Problem 14725

When is the bear population p(t)=250+30cos⁡tp(t)=250+30 \cos tp(t)=250+30cost decreasing? Choose from: a. 0<t<π20<t<\frac{\pi}{2}0<t<2π​, b. 3π2<t<2π\frac{3 \pi}{2}<t<2 \pi23π​<t<2π, c. π<t<2π\pi<t<2 \piπ<t<2π, d. 0<t<π0<t<\pi0<t<π.

Problem 14726

A red sports car's speed is f(t)=44000(t+20)2f(t)=\frac{44000}{(t+20)^{2}}f(t)=(t+20)244000​ for 0≤t≤60 \leq t \leq 60≤t≤6. Analyze distance estimates and improvements.

Problem 14727

Find dhdp\frac{d h}{d p}dpdh​ at p=200p=200p=200 for the equation 20h5+6,000,000=p320 h^{5}+6,000,000=p^{3}20h5+6,000,000=p3. Interpret your answer.

Problem 14728

Find the indefinite integral of (20x + 17) ln(7x) dx.

Problem 14729

Find the indefinite integral of (16x + 3) ln(3x) dx.

Problem 14730

Given the derivative f′(x)=6−5x2f'(x) = 6 - 5x^2f′(x)=6−5x2, determine where fff is increasing/decreasing, local extrema, and concavity.

Problem 14731

Problem 14732

Calculate the total cost for 520 cards using the cost function C′(x)=−0.05x+75C^{\prime}(x)=-0.05 x+75C′(x)=−0.05x+75 for x≤1000x \leq 1000x≤1000.

Problem 14733

Find the critical points of the function f(x)=x3−6x2+2x+16f(x)=x^{3}-6x^{2}+2x+16f(x)=x3−6x2+2x+16.

Problem 14734

Determine where the function g(t)=−4t2+5t+5g(t)=-4 t^{2}+5 t+5g(t)=−4t2+5t+5 is increasing or decreasing and find its local extrema.

Problem 14735

Find the second derivatives of f(x)=x3−6x2+2x+16f(x) = x^3-6x^2+2x+16f(x)=x3−6x2+2x+16 at c=6+303c=\frac{6+\sqrt{30}}{3}c=36+30​​ and c=6−303c=\frac{6-\sqrt{30}}{3}c=36−30​​. Determine local extrema.

Problem 14736

Find the four second partial derivatives of z=64−x2−y2z=\sqrt{64-x^{2}-y^{2}}z=64−x2−y2​.

Problem 14737

Find the average rate of change of f(x)=3x2+3f(x)=3 x^{2}+3f(x)=3x2+3 over the interval [3,5][3,5][3,5].

Problem 14738

Find the quantity that minimizes average cost for C(q)=q3−11q2+52qC(q)=q^{3}-11 q^{2}+52 qC(q)=q3−11q2+52q and R(q)=−3q2+2600qR(q)=-3 q^{2}+2600 qR(q)=−3q2+2600q.

Problem 14739

At noon, ship A is 170 km west of ship B. A sails east at 35 km/h, B north at 30 km/h. Find distance change rate at 4 p.m.

Problem 14740

Find the rate of area increase (in cm2/s\mathrm{cm}^{2} / \mathrm{s}cm2/s) for a square with side increasing at 4 cm/s4 \mathrm{~cm/s}4 cm/s when area is 16 cm216 \mathrm{~cm}^{2}16 cm2.

Problem 14741

Find the object's location $r=17-24 t^{2}+4 t^{3}$ in cm at time $t$ in s. Answer questions a-f with 3 significant figures.

What is the maximum height of a stomp rocket launched at 30.38 m/s? Use the formula for maximum height: $H = \frac{v^2}{2g}$ .

Find $\frac{d y}{d x}$ for the equation $x^{2}+y^{2}=x y+19$ at $x=5$ , $y=2$ .

A 30-foot ladder leans against a wall. If the base moves away at $3 \mathrm{ft/sec}$ , how fast does the top descend when 10 ft away?

Find $\frac{d y}{d x}$ for the equation $x^{2}+y^{2}=x y+39$ at $x=7, y=2$ .

Determine the horizontal asymptote of the function $f(x)=9 \cdot 6^{x+7}+9$ .

Calculate the average value of $f(x)=\frac{1}{\sqrt{4 x-3}}$ over the interval $[3, 21]$ .

Evaluate the integral: $\int \sec 9 \theta \tan 9 \theta d \theta$

Find the center of mass coordinates $(\bar{x}, \bar{y})$ for the region between the axes and lines $x=2$ , $y=x+2$ with density $\rho=3x$ .

Find the center of mass coordinates $(\bar{x}, \bar{y})$ for the region between the axes and lines $x=2$ , $y=x+2$ , with density $\rho=3x$ .

Find the tangent line equation to the curve at $t=-1$ : $x=t^{9}+1$ , $y=t^{10}+t$ .

For the function $f(x)=x^{4}-4 x^{3}+4$ , find: (a) domain, (b) $f'(x)$ , (c) increase/decrease regions, (d) local extrema, (e) $f''(x)$ , (f) concavity, (g) inflection points.

Find the derivative $\frac{dy}{dx}$ for the equation $xy - 5x = 8$ .

Find the difference quotient, $\frac{f(x+h)-f(x)}{h}$ , for the function $f(x)=-7 x^{2}-2 x+2$ .

Find the average rate of change of $y=\cos x$ from $x=\frac{3 \pi}{4}$ to $x=\frac{7 \pi}{4}$ .

Find the key point and horizontal asymptote of $f(x)=e^{x+2}-4$ . Options: A, B, C, D, or E.

Find all critical points of the function $f(x)=2 x^{2}-1000 \sqrt{x}$ .

Calculate the integral $\int 5 x e^{3-9 x} d x$ and specify if you used substitution or integration by parts.

Show that if $f, g: \mathbb{R} \rightarrow \mathbb{R}$ are differentiable and $g$ is one-to-one, then $\frac{d}{d x} f(g(x))=g^{\prime}(x) f^{\prime}(g(x))$ .

Evaluate the integral: $\int \frac{\cos x}{\sin ^{8} x} \, dx$

Find the average rate of change of $g(x)=x^{2}$ from $x=3$ to $x=9$ . Options: (1) 8, (2) 12, (3) 15, (4) 18.

When is the bear population $p(t)=250+30 \cos t$ decreasing? Choose from: a. $0<t<\frac{\pi}{2}$ , b. $\frac{3 \pi}{2}<t<2 \pi$ , c. $\pi<t<2 \pi$ , d. $0<t<\pi$ .

A red sports car's speed is $f(t)=\frac{44000}{(t+20)^{2}}$ for $0 \leq t \leq 6$ . Analyze distance estimates and improvements.

Find $\frac{d h}{d p}$ at $p=200$ for the equation $20 h^{5}+6,000,000=p^{3}$ . Interpret your answer.

Given the derivative $f'(x) = 6 - 5x^2$ , determine where $f$ is increasing/decreasing, local extrema, and concavity.

Calculate the total cost for 520 cards using the cost function $C^{\prime}(x)=-0.05 x+75$ for $x \leq 1000$ .

Find the critical points of the function $f(x)=x^{3}-6x^{2}+2x+16$ .

Determine where the function $g(t)=-4 t^{2}+5 t+5$ is increasing or decreasing and find its local extrema.

Find the second derivatives of $f(x) = x^3-6x^2+2x+16$ at $c=\frac{6+\sqrt{30}}{3}$ and $c=\frac{6-\sqrt{30}}{3}$ . Determine local extrema.

Find the four second partial derivatives of $z=\sqrt{64-x^{2}-y^{2}}$ .

Find the average rate of change of $f(x)=3 x^{2}+3$ over the interval $[3,5]$ .

Find the quantity that minimizes average cost for $C(q)=q^{3}-11 q^{2}+52 q$ and $R(q)=-3 q^{2}+2600 q$ .

Find the rate of area increase (in $\mathrm{cm}^{2} / \mathrm{s}$ ) for a square with side increasing at $4 \mathrm{~cm/s}$ when area is $16 \mathrm{~cm}^{2}$ .

Find the second partial derivatives of the function $v=\frac{xy}{x-y}$ .

Is the statement "The slope of tangent lines to $h(x)=x^{2n+1}$ is always non-negative" true, sometimes true, or never true? Explain and provide 2 examples.

Evaluate the integral: $\int \frac{\tan^{-1}(9x)}{1+81x^{2}} \, dx$

Show that for the curve $x^{2} y - y - x = 4$ , $\frac{d y}{d x} = \frac{1 - 2 x y}{x^{2} - 1}$ .

Find the derivatives: a. $y=10 \cdot 15^{x}-x^{8}$ . b. $f(x)=\frac{x^{10}}{e^{12}}+8^{x}$ . c. $g(w)=\frac{w^{5} 4^{w}}{\pi^{5}}$ .

Find when the slope of the tangent line of a function is zero. $(\mathrm{Com}=3)$

Find the derivative $h^{\prime}(x)$ of the function $h(x)=\ln(\sec(x^{4}))$ .

Differentiate a function using the product and chain rule instead of the quotient rule. $(\mathrm{Com}=3)$

Use the product rule to differentiate $y=\frac{(3+x)}{(2 x-9)}$ instead of the quotient rule.

Explain the meaning of a derivative of a function: $(\mathrm{Com}=3)$ .

Find the slope and concavity of the curve where $x=\cos \theta$ and $y=3 \sin \theta$ at $\theta=0$ .

Explain how to apply the product rule to $y=\frac{(3+x)}{(2 x-9)}$ instead of the quotient rule.

Find the derivative of the function $f(x)=\tan^{-1}(\sqrt{8x^2-1})$ . What is $f^{\prime}(x)$ ?

A rocket's height is given by $h(t)=-4.9 t^2+9.8 t$ . Find when it hits the ground, speeds, and average speed in 1 second.

Find the foraging time $F$ that minimizes energy expenditure $E = 0.25F + \frac{1.7}{F^2}$ . Round to three decimal places.

Find the normal equation to the tangent of $y=\frac{3 x}{(x^{2}-1)}$ at $x=3$ .

Compute the derivative of the function $f(x)=e^{x}-e^{-x}$ .

Evaluate the limits: a) $\lim _{x \rightarrow -4} \frac{x^{2}+12x+32}{x+4}$ b) $\lim _{x \rightarrow 4} \frac{4-\sqrt{12+x}}{x-4}$

Find the derivative of $g(x) = \frac{x}{1 - e^{x}}$ .

Find the average rate of change of $g(x)=\frac{2}{x+3}$ from $x=1$ to $x=3$ .

Gravel is dumped at $30 \mathrm{ft}^{3}/\mathrm{min}$ , forming a cone. Find height increase rate when height is $5 \mathrm{ft}$ .

Find the derivative of $p(x)=\frac{x^{2}}{4^{x}}$ .

A car moves north at $30 \mathrm{~m/s}$ and a drone flies east at $7 \mathrm{~m/s}$ , 20 m high. Find distance after 4 seconds.

Find the derivative of $h(x) = \cos^2(x)e^x$ .

A car moves north at $30 \mathrm{~m/s}$ and a drone flies east at $7 \mathrm{~m/s}$ at $20 \mathrm{~m}$ . Find the rate of distance change 4 seconds later.

Calculate the coin's velocity and time to hit the ground when dropped from 354m, with gravity at 9.8 m/s $^2$ .

Sam is building two adjacent rectangular pens against a wall with a total area of 300 sq ft. Find the minimum fence length using $x$ for width and $y$ for length. Use calculus to verify.

Find the grams of carbon-14 left after 7136 years using the model $A=16 e^{-0.000121 t}$ . Answer: $\square$ grams.

Approximate the area under $y=x^{3}$ from $x=3$ to $x=6$ using a Right Endpoint method with 6 subdivisions.