Calculus

Problem 501

Find the value of 14650e0.157(2)14650 e^{-0.157 \cdot(-2)}.

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Problem 502

Differentiate the function f(x)=x74e8xf(x) = x^7 - 4e^{8x} to find f(x)f'(x).

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Problem 503

Estimate the numerical limit of limx0ln(5x+10)ln(10)5\lim_{x \to 0} \frac{\ln(5x+10) - \ln(10)}{5} or state that it does not exist. Give the answer to at least three decimal places.

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Problem 504

Find the derivative yy' of y=cot1(t)y=\cot^{-1}(\sqrt{t}).

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Problem 505

Find the derivative of y=lntanxy=\ln \sqrt{\tan x} evaluated at x=π4x=\frac{\pi}{4}.

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Problem 506

Find the rate of change of volume, in cubic centimeters per minute, when a spherical balloon's radius increases at 3 cm/min and is 10 cm.
V=(4/3)πr3V = (4/3) \pi r^3, where rr is the radius.

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Problem 507

Find the indefinite integral of 5dx5 dx.

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Problem 508

Solve for xx in the equation 4=ln(x+5)34=\frac{\ln (x+5)}{3}.

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Problem 509

Find the vertical asymptote of the logarithmic function g(x)=log4(x+7)g(x) = \log_4(x+7).

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Problem 510

Find the limit of g(x)g(x) as xx approaches 1, given that 2xg(x)x4x2+22x \leq g(x) \leq x^4 - x^2 + 2 for all x0x \geq 0.

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Problem 511

Find the end behavior of the quadratic function D(x)=49x2D(x) = 49 - x^2.

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Problem 512

Rewrite the exponential function x(t)=(1.45)t2x(t)=(1.45)^{\frac{t}{2}} in the form y=a(1+r)ty=a(1+r)^{t} or y=a(1r)ty=a(1-r)^{t} to determine growth or decay. Round aa and rr to nearest hundredth.

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Problem 513

Sketch the graph of g(t)=(0.5)tg(t) = (0.5)^t to determine the sign of g(2)g'(2). Then, use a small interval to estimate g(2)g'(2) rounded to three decimal places.

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Problem 514

Find the antiderivative of the rational function 2x5+10x2+7x4\frac{2 x^{5}+10 x^{2}+7}{x^{4}}.

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Problem 515

Find the value of e0.5e^{-0.5}.

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Problem 516

Find the displacement of a mass suspended on a spring, with velocity v(t)=7sin(t)+7cos(t)v(t) = 7 \sin (t) + 7 \cos (t), between t=3t = 3 and t=8t = 8 seconds. Give the answer to at least three decimal places.

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Problem 517

Find the interval on which the function f(x)=4+log2(42x)f(x) = 4 + \log_2(4 - 2x) is decreasing.

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Problem 518

Find the value of tt where the logistic function y=M1+cekMty=\frac{M}{1+c e^{-k M t}} equals M2\frac{M}{2}.

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Problem 519

Find the value of xx when y=a+blnxy = a + b \ln x, given a=12.3301088919a = 12.3301088919 and b=8.22097130319b = -8.22097130319, and y=16y = -16.

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Problem 520

Complete the exponential decay table: Original Amount: 19,00019,000, Decay Rate: 13%13\% per year, Years: 1818, Final Amount after 1818 years of decay.

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Problem 521

Find the demand function D(p)=2p26p+300D(p) = -2p^2 - 6p + 300 and its rate of change w.r.t. price pp. Interpret the rate of change when p=$11p = \$11.

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Problem 522

Find the time (in seconds) when Ana, who dives from a high springboard, hits the water given the height model h(x)=5(x+1)(x3)h(x) = -5(x+1)(x-3).

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Problem 523

Evaluate limx1x1x1\lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1} using the provided table of xx and f(x)f(x) values.

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Problem 524

Solve for time tt in the equation r=r0+12at2r = r_0 + \frac{1}{2}at^2, where r0r_0 and aa are known constants.

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Problem 525

Differentiate the function f(θ)=secθ5+secθf(\theta) = \frac{\sec \theta}{5 + \sec \theta}.

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Problem 526

Find the derivative of 5x24\sqrt{5 x^{2}-4} with respect to xx.

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Problem 527

Evaluate the definite integral of 4ye2y\frac{4y}{e^{2y}} from 0 to 1.

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Problem 528

Solve the following exponential equations: a. e0.2t=9e^{-0.2 t}=9, b. ekt=15e^{k t}=\frac{1}{5}, c. e(ln0.4)t=0.7e^{(\ln 0.4) t}=0.7.

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Problem 529

Prove there is at least one negative solution to ex=xe^{x} = -x using the Intermediate Value Theorem. Approximate this solution using the bisection method with error at most 0.01.

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Problem 530

Find the derivative of f(x)=5x4(x32)f(x) = 5x^4(x^3 - 2) and identify the correct application of the product rule.

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Problem 531

Find the equation of a curve with dydx=42yx13\frac{dy}{dx} = 42yx^{13} and yy-intercept at 2.

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Problem 532

Show that for positive real numbers uu and vv, and a positive real number a1a \neq 1, loga(uv)=logaulogav\log_{a}\left(\frac{u}{v}\right) = \log_{a}u - \log_{a}v.

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Problem 533

Find the logarithm of 1.6.

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Problem 534

Find the derivative of f(x)=(x3+x2)tanxf(x) = (x^{3} + x^{2})^{\tan x} using logarithmic differentiation.

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Problem 535

(a) Find the largest δ\delta such that x1<δ|x-1|<\delta implies 4x4<0.5|4x-4|<0.5. (b) Find the largest δ\delta such that x1<δ|x-1|<\delta implies 4x4<0.05|4x-4|<0.05.

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Problem 536

Quantity with initial value of 8900 grows exponentially at 65%65\% per hour. Find the value after 411 minutes, rounded to nearest hundredth.

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Problem 537

Find the equation of the secant line for the natural logarithm function f(x)=ln(x)f(x) = \ln(x) on the interval [1,7][1, 7].

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Problem 538

Find the function y(x)y(x) that satisfies the first-order differential equation y=ysinxy' = y \sin x, where y=πecosxy = \pi e^{-\cos x} is a solution.

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Problem 539

Find the antiderivative of R(x)=100ex+5(1+ex+5)2R(x) = \frac{100 e^{x+5}}{(1+e^{x+5})^2}, where xx is the natural log of a histamine dose in mM. Evaluate the antiderivative at x=8.9x=-8.9 and round to 3 decimal places.

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Problem 540

Evaluate the indefinite integral 2arccos(7x)149x2dx=\int \frac{2 \arccos (7 x)}{\sqrt{1-49 x^{2}}} d x = C$

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Problem 541

Find the maximum value of the quadratic function y=3x2+12x9y = -3x^2 + 12x - 9. The vertex is at xvertex=b/2ax_{\text{vertex}} = -b/2a, and the maximum value is y=3y = 3.

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Problem 542

Find the value of cosh(3ln3)sinh(3ln3)\cosh(3\ln 3) - \sinh(3\ln 3).

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Problem 543

Exercise 1: Limits 3) Write the reference limits of the function lnx\ln x 4) Using these limits, calculate the limits of the following functions at the bounds of their domain: a) f(x)=x22+lnxf(x)=x^{2}-2+\ln x b) g(x)=x2+(2lnx)2g(x)=x^{2}+(2-\ln x)^{2} c) h(x)=ln(x+2)1+x2h(x)=\frac{\ln (x+2)}{1+x^{2}} d) l(x)=x(lnx)2l(x)=x(\ln x)^{2}
Exercise 2: Derivative and sign study of the derivative Calculate the derivative function of the following functions and study the sign of the derivative: 4) f(x)=1+(lnx)2f(x)=-1+(\ln x)^{2} 5) g(x)=lnxx+1g(x)=\ln \frac{x}{x+1} 6) h(x)=lnx2x+4h(x)=\ln \left|\frac{x-2}{x+4}\right|
Exercise 3: Complex numbers 3) Solve the equation z22iz2=0z^{2}-2 i z-2=0 in the set C\mathbb{C} of complex numbers 4) Let z1z_{1} and z2z_{2} be the solutions of this equation such that Re(z1)>Re(z2)\operatorname{Re}\left(z_{1}\right)>\operatorname{Re}\left(z_{2}\right).

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Problem 544

Find the largest possible value of f(15)f(15) if f(x)f(x) is continuous and differentiable on [6,15][6,15], f(6)=2f(6)=-2, and f(x)10f'(x) \leq 10 for all x[6,15]x \in [6,15].

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Problem 545

Find the minimum degree of a polynomial with exactly two inflection points.

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Problem 546

Find the value of f(81)f(81) for a continuous function ff that satisfies 0x4f(t)dt+0xf(t4)dt=x\int_{0}^{x^{4}} f(t) dt + \int_{0}^{x} f(t^{4}) dt = x for all xx.

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Problem 547

Find the limit of the expression (4x5x)/(3x4x)(4^x - 5^x) / (3^x - 4^x) as xx approaches infinity.

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Problem 548

Evaluate the definite integral 107x1+x4dx\int_{-1}^{0} \frac{7 x}{1+x^{4}} d x and select the correct answer.

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Problem 549

Find the linear approximation of cos8\cos 8 about x0=12x_0 = \frac{1}{2} for the function f(x)=cos(2x)f(x) = \cos(2x).

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Problem 550

Find the value of 2sinh(2lnx)2 \sinh (2 \ln x) when x=ex=e.

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Problem 551

Find the linear approximation of f(x)=x1/4f(x) = x^{1/4} at x0=2x_0 = 2. Select the correct answer.

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Problem 552

Particle travels along x-axis with velocity v(t)=t0.3+4cos(2t)v(t) = t^{0.3} + 4 \cos(2t). Given x=5x = -5 at t=3.5t = 3.5, find x(4)x(4), v(4)v(4), a(4)a(4) and analyze motion. Round to nearest thousandth.

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Problem 553

Evaluate the exponential function f(x)=52xf(x) = 5^{2x} when x=1x = -1.

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Problem 554

Find the derivative of f(x)=x11f(x) = x^{11}.

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Problem 555

Find the derivative f(x)f'(x) of f(x)=x2ln(124x2)f(x) = x^2 \ln(12 - 4x^2) and its domain.

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Problem 556

Find f(x)f(x) when f(x)=f'(x) = x22x+1 x^2 - 2x + 1 anddeterminethebehaviorof and determine the behavior of f(x)$.

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Problem 557

Find the derivative of f(x)=g(x)x2f(x) = \frac{g(x)}{x^{2}} when g(2)=2g(2) = -2 and g(2)=2g'(2) = 2.

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Problem 558

Evaluate the indefinite integral 9x2lnxdx\int 9 x^{2} \ln x d x.

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Problem 559

Find the second derivative of the secant function with respect to xx.

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Problem 560

Rewrite the exponential equation P=493(0.38)tP=493(0.38)^{t} in the form P=P0ektP=P_{0} e^{k t}. Find the values of P0P_{0} and kk.

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Problem 561

Evaluate the indefinite integral 2x2+5x52+1xdx\int \frac{2 x^{2}+5 x^{\frac{5}{2}}+1}{x} d x.

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Problem 562

Solve the exponential equation 10e4x5=1310e^{-\frac{4x}{5}} = 13 for xx.

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Problem 563

Describe how f(2)=4f'(2) = 4 relates to the graph of f(x)=x32x2+5f(x) = x^3 - 2x^2 + 5, where f(x)f'(x) is the derivative.

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Problem 564

Use logarithmic differentiation to find the derivative dydx\frac{dy}{dx} of y=(2+x)3/xy = (2 + x)^{3/x}.

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Problem 565

Find h(2)h'(2) where h(x)=f(g(x))h(x)=f(g(x)), ff and gg are differentiable, and f(2)=3f(2)=-3, f(2)=4f'(2)=4, f(5)=4f(5)=-4, f(5)=5f'(5)=5, g(2)=5g(2)=5, g(2)=5g'(2)=-5, g(4)=4g(-4)=4, g(4)=4g'(-4)=-4.

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Problem 566

Calculate R=ln(h1/h2)R=\ln(h_1/h_2) given h1=24.1±0.1 cmh_1=24.1\pm0.1\mathrm{~cm} and h2=12.8±0.1 cmh_2=12.8\pm0.1\mathrm{~cm}. Select the correct value of RR.

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Problem 567

Sami measured physical quantities X±ΔX,Y±ΔY,Z±ΔZX \pm \Delta X, Y \pm \Delta Y, Z \pm \Delta Z. To calculate R=4(Z7Y4)/(X7)R=4 (Z^{7} Y^{4})/(X^{7}), ΔR/R\Delta R/R is given by option a: 14ΔZ/Z+16ΔY/Y14ΔX/X14 \Delta Z/Z + 16 \Delta Y/Y - 14 \Delta X/X.

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Problem 568

Solve the exponential equation ey=x+2e^{y}=x+2 for yy in terms of xx.

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Problem 569

Find the rate of change of monthly revenue RR with respect to the number of completed responses xx when x=1000x=1000, given R(x)=12000+155x2+30xR(x)=-12000+15\sqrt{5x^2+30x} and the number of responses is increasing by 5 per month.

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Problem 570

Find the minimum marginal cost of the cost function C(x)=2x35x2+7x+6C(x) = 2x^3 - 5x^2 + 7x + 6. The minimum marginal cost is $\$ \square.

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Problem 571

Find the equation of the line passing through (2,2) and parallel to the tangent line of y=3x32xy=-3 x^{3}-2 x at (1,5)(-1,5), where y=12x2y=\frac{1}{2}-x^{2} and the tangents are perpendicular.

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Problem 572

Solve the exponential equation ex5+4=7e^{\frac{x}{5}}+4=7 for the value of xx.

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Problem 573

Solve the equation 2e3x2+4=162 e^{3x - 2} + 4 = 16

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Problem 574

Find the derivative of y=ekx2y=e^{-k x^{2}} with respect to xx, where kk is a constant.

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Problem 575

Find the derivative of the inverse function f1(x)f^{-1}(x) of f(x)=3+2x+exf(x)=3+2x+e^x evaluated at x=4x=4.

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Problem 576

Find yy' given sin(x)=ey\sin(x) = e^y, where options are tan(x)-\tan(x), tan(x)\tan(x), cot(x)\cot(x), cot(x)-\cot(x).

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Problem 577

Find the value of g(f(x))g(f(x)) when f(x)=exf(x)=e^{x} and g(x)=ln(x)g(x)=\ln(x).

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Problem 578

Find the 99th derivative of y=sin(x)y = \sin(x).

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Problem 579

Find the derivative of x2+42x3\frac{\sqrt{x^{2}+4}}{2x-3} with respect to xx.

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Problem 580

Find the net change and average rate of change between points (1,1)(-1,1) and (5,1)(5,1) on the given function's graph.

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Problem 581

Find the derivative of f(x)f(x) at x=π/4x = \pi/4 if ef(x)=cos(x)e^{f(x)} = \cos(x).

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Problem 582

Calculate the total profit of a 250 ft deep well using the marginal profit function P(x)=4x4P'(x) = 4 \sqrt[4]{x}. Set up the integral and solve for the total profit.
P(250)=02504x4dx P(250) = \int_{0}^{250} 4 \sqrt[4]{x} \, dx
The total profit is $2,000.00\$2,000.00.

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Problem 583

Differentiate y=x5exy = x^{5} e^{x} to find yy'.

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Problem 584

Find the derivative of F(x)=(g(3x))F(x) = (g(3x)) at x=2x = 2, given f(3)=6f(3) = 6, f(3)=8f'(3) = 8, g(6)=8g(6) = -8, and g(6)=3g'(6) = 3.

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Problem 585

Find the derivative of a function expressed as a composite of other functions using the Chain Rule. a. Given y=(u/2)+6y=(u/2)+6 and u=2x12u=2x-12, find dydu\frac{dy}{du}. dydu=12\frac{dy}{du}=\frac{1}{2}

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Problem 586

Find the values of aa, bb, and cc such that the integral 01(x+3)ex2+6x+5dx\int_{0}^{1}(x+3) e^{x^{2}+6 x+5} d x can be expressed as abceudu\int_{a}^{b} c e^{u} d u.

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Problem 587

Find the derivative of the function g(u)=u+12u1g(u) = \frac{u+1}{2u-1} using the definition of derivative. State the domain of g(u)g(u) and its derivative in interval notation.

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Problem 588

Find the derivative of 3x(2x+1)33x(2x+1)^3.

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Problem 589

Find the derivative of y with respect to x evaluated at x=1 for the following functions: (21) y=2x1x+3y=\frac{2 x-1}{x+3}, (22) y=5x2x2+3y=\frac{5 x-2}{x^{2}+3}, (23) y=(3x+2x)(x5+1)y=\left(\frac{3 x+2}{x}\right)\left(x^{-5}+1\right), (24) y=(2x7x2)(x1x+1)y=\left(2 x^{7}-x^{2}\right)\left(\frac{x-1}{x+1}\right).

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Problem 590

The point is moving along x3y2=72x^{3}y^{2}=72. When at (2,3)(2,3), the xx-coordinate changes at -5 units/min. Find the rate of change of the yy-coordinate.
dydt=\frac{dy}{dt}= (Type an integer or a fraction. Simplify your answer.)

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Problem 591

What is the value of xx that satisfies 7lnx=2.47 \cdot \ln x=2.4? (Round to two decimal places.) A. x=18.48x=18.48 B. x=99.48x=99.48 C. x=1.98x=1.98 D. x=1.41x=1.41

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Problem 592

Calculate g(4)g(4) and g(4)g'(4) where g(x)g(x) is the inverse of f(x)=x2+6xf(x)=\sqrt{x^2+6x} for x0x \geq 0.

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Problem 593

Find the indefinite integral of 6x+109x\frac{6}{x}+\frac{10}{9\sqrt{x}} on the positive real numbers. Translate the password using sin(2x)\sin(2x) and a\sqrt{a}.

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Problem 594

Convert exponential equation y=2.2(3.74)xy=2.2(3.74)^x to y=aebxy=a e^{b x}. Round aa to 2.20 and bb to 1.32.

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Problem 595

An arrow is shot upward at 80 ft/s from a 25 ft platform. The path is h=16t2+80t+25h=-16t^{2}+80t+25, where hh is height and tt is time. Find the maximum height.

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Problem 596

Find the values of AA and BB that make the piecewise function f(x)f(x) differentiable at x=0x=0, where f(x)=x2+1f(x) = x^2 + 1 for x0x \geq 0, and f(x)=Asinx+Bcosxf(x) = A \sin x + B \cos x for x<0x < 0.

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Problem 597

Evaluate the integral 98+9xdx\int \frac{9}{8+9 x} d x for x89x \neq -\frac{8}{9}. Type the exact answer.

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Problem 598

Determine the inflection points and extrema of the function f(x)=18x3+34x2f(x) = \frac{1}{8} x^{3} + \frac{3}{4} x^{2}. Sketch the graph.

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Problem 599

Find the derivative of f(x)=xf(x)=\sqrt{x} using the definition of the derivative.

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Problem 600

Find the unit tangent vector and length of the curve r(t)=6t3i+2t3j3t3k\mathbf{r}(t) = 6t^3\mathbf{i} + 2t^3\mathbf{j} - 3t^3\mathbf{k} for 1t21 \leq t \leq 2. The unit tangent vector is (67)i+(27)j+(37)k(\frac{6}{7})\mathbf{i} + (\frac{2}{7})\mathbf{j} + (-\frac{3}{7})\mathbf{k}, and the length is 979\sqrt{7} units.

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