Calculus

Problem 401

Find the Cartesian equation x=f(y)x=f(y) from the parametric equations x(t)=2t2x(t)=2t^2 and y(t)=9+4ty(t)=-9+4t.

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Problem 402

Find the derivative of the inverse of f(x)=4(x1/3)f(x)=4(x^{-1/3}) for x>0x>0, expressing the result with xx as the independent variable.

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Problem 403

Find the value of the function P(t)=531,000e0.016tP(t)=531,000 e^{-0.016 t} when t=12t=12.

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Problem 404

Express the limit as a definite integral on [1,9][1, 9]: limni=1nxi(xi)2+8Δx\lim_{n \to \infty} \sum_{i=1}^{n} \frac{x_i^*}{(x_i^*)^2 + 8} \Delta x

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Problem 405

Find the maxima, minima, and intervals where y=xlnxy=x \ln x is increasing or decreasing, then sketch the graph.

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Problem 406

Use Trapezoid Rule with 4 subintervals to approximate 0πxcos(x)dx\int_{0}^{\pi} x \cos (x) dx.

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Problem 407

Find the vertex of the parabolic equation h=4.9t2+23.4th=-4.9 t^{2}+23.4 t describing the height of a golf ball in meters, rounding each coordinate to the nearest tenth. The vertex is (,)(\square, \square).

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Problem 408

Finde die Ableitungsfunktion f(x)f'(x) von f(x)=3x24xf(x)=3x^2-4x und bestimme f(5)f'(5), f(2)f'(-2), f(0)f'(0), f(0.5)f'(0.5). Finde den Punkt bb mit f(b)=8f'(b)=8 und den Punkt, an dem der Graph von ff die Steigung 5 hat.

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Problem 409

Find all cc in the interval (2,7)(2,7) such that f(c)=f(7)f(2)72f'(c) = \frac{f(7) - f(2)}{7 - 2} where f(x)=914xf(x) = 9 - \frac{14}{x}.

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Problem 410

Find two points, other than the origin, on the graph of the function t(x)=94x5t(x) = -\frac{9}{4}x^5 that fit within the [10,10][-10,10] by [10,10][-10,10] grid.

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Problem 411

Find the domain and range of the quadratic function y=3x2y=-3x^2.

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Problem 412

Find the simplified limit of (p(q+h)p(q))/h(p(q+h) - p(q))/h as hh approaches 0, where p(q)=q2+2q5p(q) = q^2 + 2q - 5.

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Problem 413

Find the range of xx where xexx e^{-x} is positive.

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Problem 414

Find the value of e2.72e^{2.72} and round to four decimal places.

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Problem 415

Find equivalent expressions for log(107)\log(10^7). Options: A. 7log107 \cdot \log 10, B. 7107 \cdot 10, C. 1, D. 7.

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Problem 416

Find the value of xx that satisfies lnx=3.1\ln x = 3.1, rounded to two decimal places.

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Problem 417

Find the sum of the infinite series n=17(38)n\sum_{n=1}^{\infty}-7\left(\frac{3}{8}\right)^{n}.

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Problem 418

Find the logarithm of 11024\frac{1}{1024} in base 4.

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Problem 419

Find the endpoint and range of y=x8+3y=\sqrt{x-8}+3 function.

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Problem 420

Find the area bounded by the graphs of y=2xy=\frac{2}{x} and y=0y=0 over the interval 1x4e1 \leq x \leq 4e.

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Problem 421

Sketch the inverse of exponential functions f(x)=4x,8x,(13)x,(15)xf(x) = 4^x, 8^x, \left(\frac{1}{3}\right)^x, \left(\frac{1}{5}\right)^x. Write the inverse functions in exponential and logarithmic form.

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Problem 422

Find the sin8(x)dx\int \sin^8(x)\,dx or ddxsin8(x)\frac{d}{dx}\sin^8(x).

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Problem 423

Find fx(x,y)f_x(x,y) and fy(x,y)f_y(x,y) for f(x,y)=6x5xy65y3f(x,y)=6x^5-xy^6-5y^3.

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Problem 424

Evaluate f(x)=lnx9f(x)=\frac{\ln x}{9} when x=2.197225x=2.197225. Round the answer to three decimal places.

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Problem 425

Given f(x)=x+9+1f(x) = -\sqrt{x+9} + 1, find the domain, range, x-intercept, y-intercept, and min/max values in [9,7][-9, 7].

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Problem 426

Solve the equation 6ln(x4)=306 \ln (x-4)=30 for xx.

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Problem 427

Find critical values, intervals of increase/decrease, local max/min, concavity, and inflection points of f(x)=3x2ln(x)f(x) = 3x^2 \ln(x) for x>0x > 0.

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Problem 428

Find the linear and quadratic approximating polynomials for f(x)=1xf(x)=\frac{1}{x} at x=1x=1 and use them to approximate 11.03\frac{1}{1.03}.

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Problem 429

Solve for the value of xx where the natural logarithm of xx is equal to 3.

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Problem 430

Find the partial derivative of 200x14y34200 x^{\frac{1}{4}} y^{\frac{3}{4}} with respect to xx.

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Problem 431

Find the value(s) of xx where the graph of g(x)g(x) has a relative maximum, given that g(x)=3x(x+2)2g'(x) = -3x(x+2)^2.

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Problem 432

Solve the equation x34x=2xx^{3}-4 x=2^{x} and find the fourth solution, rounded to the nearest tenth.

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Problem 433

Interpret the vertex of the equation h=16t2+48th = -16t^2 + 48t, where hh is the height in feet and tt is the time in seconds after Jevonte kicks a football.

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Problem 434

Find the derivative of arctanx2+1\arctan \sqrt{x^{2}+1}.

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Problem 435

Find the value of xx that satisfies the equation ln(x5)=0\ln(x-5) = 0.

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Problem 436

Find the first and second derivatives of the linear function y=7x+1y = 7x + 1, where dydx=\frac{dy}{dx} = \square and d2ydx2=\frac{d^2y}{dx^2} = \square.

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Problem 437

Suppose f(0)=3f(0)=-3 and f(x)9f'(x) \leq 9 for all xx. What is the largest possible value for f(3)f(3)?
The largest possible value for f(3)f(3) is 00.

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Problem 438

Find the partial fraction decomposition of 74x8x210x+3\frac{74 x}{8 x^{2}-10 x+3} in the form f(x)2x1+g(x)4x3\frac{f(x)}{2 x-1}+\frac{g(x)}{4 x-3}, where f(x)f(x) and g(x)g(x) are to be determined.

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Problem 439

Find the derivative of y=f(x)=x24y = f(x) = x^2 - 4 at x=1x = 1, evaluate f(1)f(1), and graph y=f(x)y = f(x) and the tangent line at (1,f(1))(1, f(1)).

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Problem 440

Use Maclaurin series to estimate ee from exe^x, rounding to 4 decimal places. e2.7183e \approx \boxed{2.7183}

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Problem 441

Differentiate and simplify these exponential functions: f(x)=e7xf(x)=e^{7x}, f(x)=3e2xf(x)=3\cdot e^{2x}, f(x)=e2xf(x)=e^{2x}, f(x)=0.2e5x1f(x)=0.2\cdot e^{5x-1}, f(x)=0.5e4xf(x)=0.5\cdot e^{4x}, f(x)=e2xf(x)=e^{-2x}, f(x)=4e12xf(x)=4\cdot e^{\frac{1}{2}x}, f(x)=e0.2xf(x)=e^{0.2x}, f(x)=13e3xf(x)=\frac{1}{3}\cdot e^{-3x}, f(x)=110e5xf(x)=-\frac{1}{10}\cdot e^{5x}, f(x)=3e4x+2f(x)=3\cdot e^{-4x+2}, f(x)=5e3x2f(x)=5\cdot e^{-3x-2}. Also differentiate and simplify: f(x)=2x+e2xf(x)=2x+e^{2x}, f(x)=e2x5e15xf(x)=e^{2x}-5\cdot e^{\frac{1}{5}x}, f(x)=x2+e3x1f(x)=x^{2}+e^{3x-1}, f(x)=20e0.1x+x+5f(x)=20\cdot e^{0.1x}+x+5, f(x)=13x32e0.25xf(x)=\frac{1}{3}x^{3}-2\cdot e^{-0.25x}, f(x)=2x2ex+3+x3f(x)=2x^{2}-e^{-x+3}+x^{3}.

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Problem 442

The function f(t)=930(1.009)t/10f(t)=930(1.009)^{t/10} represents a quantity over tt years. The constant 1.009 indicates the quantity changes by %\square \% per year.

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Problem 443

Find the elasticity of demand E(p)E(p) using the price-demand equation x=30002p2x=3000-2p^2.

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Problem 444

Solve for the value of xx given the natural logarithm equation lnx=6\ln x = 6.

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Problem 445

Find the derivative of g(x)=x1sec(t3)dtg(x) = \int_{x}^{1} \sec(t^{3}) dt, where g(x)=sec(1)tan(1)+sec(x3)tan(x3)3x2sec(x3)g'(x) = -\sec(1)\tan(1) + \sec(x^{3})\tan(x^{3}) - 3x^{2}\sec(x^{3}).

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Problem 446

Find the derivative of log5x\log_5\sqrt{x}.

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Problem 447

Find d2ydx2\frac{d^{2} y}{d x^{2}} at (4,3)(4,3) given x2+y2=25x^{2}+y^{2}=25. Options: (A) 2527-\frac{25}{27} (B) 727-\frac{7}{27} (C) 727\frac{7}{27} (D) 34\frac{3}{4} (E) 2527\frac{25}{27}

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Problem 448

A passenger jet is descending. Find the slope of the descent graph in feet per minute given y=20y=20 at x=10x=10 and y=1y=1 at x=2x=2.

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Problem 449

Find points with given slope: For functions f(x)f(x), find where the slope is mm. a) f(x)=14x46x,m=2f(x)=\frac{1}{4} x^{4}-6 x, m=2 b) f(x)=16x3+x2,m=2.5f(x)=-\frac{1}{6} x^{3}+x^{2}, m=-2.5 c) f(x)=2xx,m=3f(x)=\frac{2}{x}-x, m=-3 d) f(x)=3x,m=3f(x)=3 \sqrt{x}, m=3

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Problem 450

Solve for xx in the equation ln(x+4)ln(x1)=1\ln (x+4) - \ln (x-1) = 1.

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Problem 451

Solve for xx in the equation df=(x+c)210df = \frac{(x+c)^{2}}{10}.

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Problem 452

Find the value of f(3)f(3) if f(x)=(18)(8x)f(x)=\left(\frac{1}{8}\right)\left(8^{x}\right). A. 164\frac{1}{64} B. 512 C. 1512\frac{1}{512} D. 64

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Problem 453

Finde die Umkehrfunktion von y=16x4y=16x^4 für x>0x>0.

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Problem 454

Approximate the area under the function f(x)=x44x3+8x+18f(x) = x^4 - 4x^3 + 8x + 18 between x=2x = -2 and x=4x = 4 using the Mid-Ordinate rule with a step size of 1.

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Problem 455

Decompose h(x)=5(x3)9h(x)=\frac{-5}{(-x-3)^{9}} into h(x)=f(g(x))h(x)=f(g(x)) where ff and gg are component functions.

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Problem 456

1a) Solve 2x1=82^{x-1}=8. 1b) Solve 32x=2433^{2x}=243. 2) Show f(x+3)=8f(x)f(x+3)=8f(x) for f(x)=2xf(x)=2^x. 3) Find f(4)f(4) and domain of g(x)=log3(x24x+3)g(x)=\log_3(x^2-4x+3). 4) Find inverse of f(x)=e3x1f(x)=e^{3x-1}. 5) Find g(x)g(x) s.t. (fg)(x)=(gf)(x)=x(f\circ g)(x)=(g\circ f)(x)=x for f(x)=exf(x)=e^{\sqrt{x}}. 6) Convert angles a) 315315^\circ b) 40-40^\circ c) 330330^\circ to radians. 7) Sketch f(x)=x35x2+6xx24f(x)=\frac{x^3-5x^2+6x}{x^2-4}.

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Problem 457

Find the maximum value of Z|Z| if Z4Z=2\left|Z-\frac{4}{Z}\right|=2.

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Problem 458

Find the increasing and decreasing intervals of f(x)=x+5x4f(x) = \frac{x+5}{x-4}.

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Problem 459

Calculate the area of the surface generated by rotating the curve y=2x+1y=\sqrt{2x+1} on [0,3][0,3] around the xx-axis.

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Problem 460

Find the range of y=2x+13x6y=\frac{2 x+1}{3 x-6}, excluding which real number(s)?

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Problem 461

(10 points) Find the domain, asymptotes, critical points, intervals of increasing/decreasing, and concavity for f(x)=x2+16xf(x)=x^{2}+\frac{16}{x}. Sketch the curve.

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Problem 462

Find the second derivative of f(x)=xsinxf(x) = x \sin x. The second derivative is f(x)=2cosx+xsinxf''(x) = -2 \cos x + x \sin x.

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Problem 463

Find the minimum value of the parabolic function y=x2+4y = x^2 + 4. Express the solution as a simplified fraction or integer.

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Problem 464

Find the marginal revenue function. R(x)=8x0.07x2R(x)=8x-0.07x^2, then R(x)=R'(x)=

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Problem 465

Find the derivative of y=ecosh(2x)y=e^{\cosh (2 x)}.

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Problem 466

Gravel is dumped at 30ft3/min30 \mathrm{ft}^{3} / \mathrm{min} onto a conical pile with equal base diameter and height. Find the rate of height increase when the pile is 10ft10 \mathrm{ft} high.

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Problem 467

Hangi seçenek f(x)=xxf(x) = \frac{|x|}{x} için doğrudur? A) x=1x=1 noktasında kaldırılabilir süreksizliği vardır B) x=1x=1 noktasında sıçrama süreksizliği vardır C) x=0x=0 noktasında silinmez süreksizliği vardır D) x=0x=0 noktasında kaldırılabilir süreksizliği vardır E) x=0x=0 noktasında sıçrama süreksizliği vardır

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Problem 468

Find the range of the function h(x)=5+log5xh(x) = 5 + \log_5 x. Simplify your answer in interval notation.

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Problem 469

Evaluate the sum of f(3)f(-3) and f(2)f(-2) for the piecewise function f(x)={32x,x3x2+7,x<3f(x) = \begin{cases} -3-2x, & x \geq -3 \\ -x^2+7, & x < -3 \end{cases}.

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Problem 470

Find the value of tt when 6e9t=966e^{9t} = 96.

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Problem 471

Evaluate the indefinite integral of sin(lnx)\sin(\ln x) with respect to xx.

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Problem 472

Find the solutions to the equation ln(x2)=1\ln(x-2)=1, where x>2x>2.

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Problem 473

Find the current ii at time t=3t=3 seconds given q=74t+740e10tq=\frac{7}{4} t+\frac{7}{40} e^{-10 t} where i=dqdti=\frac{d q}{d t}.

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Problem 474

Analyze the end behavior of the function h(x)=5x311x2h(x)=-\frac{5 x^{3}}{11 x-2} using limit notation.

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Problem 475

Find the antiderivative F(x)F(x) of f(x)=2x+7f(x) = 2x + 7 given F(1)=8F(1) = 8, then calculate the exact value of F(2)F(2).

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Problem 476

Find the derivative of f(x)=xx16f(x) = \frac{x}{x-16} and simplify. Identify the correct application of the quotient rule.

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Problem 477

Find the value of xx given the equation 13lnx=y-\frac{1}{3} \ln x = y.

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Problem 478

Find the domain and range of the inverse function f1(x)f^{-1}(x) where f(x)=lnx+2f(x) = \sqrt{\ln x} + 2.

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Problem 479

Approximate 122312^{\frac{2}{3}} using a calculator and select the correct answer.

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Problem 480

Evaluate limits and find horizontal asymptotes of f(x)=5x39x4+7x2f(x) = \frac{5 x^{3}-9}{x^{4}+7 x^{2}}. Determine limxf(x)\lim_{x \to \infty} f(x), limxf(x)\lim_{x \to -\infty} f(x), and the horizontal asymptotes of f(x)f(x).

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Problem 481

Find the slope of the normal line to the curve f(x)=x+9xf(x)=x+\frac{9}{x} at x=1x=-1.

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Problem 482

Find the derivative of y=xln(ln(x))y=x \ln (\ln (x)) evaluated at x=ex=e.

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Problem 483

Calculate f(2+h)f(2)h\frac{f(2+h)-f(2)}{h} for h=.1,.01,.01,.1h=.1, .01, -.01, -.1 where f(x)=x37xf(x)=x^{3}-7x. If the derivative of f(x)f(x) at x=2x=2 is an integer, what would you expect it to be?

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Problem 484

Find the value of the definite integral 132sin(x5)x45dx\int_{1}^{32} \frac{\sin (\sqrt[5]{x})}{\sqrt[5]{x^{4}}} dx.

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Problem 485

Find the value of tt in the equation 2.2=60(10)t2.2=60(10)^{t}.

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Problem 486

Kira's distance from Tomsville after tt hours is D(t)=10.44tD(t) = 10.4 - 4t km. Find the inverse function D1(x)D^{-1}(x) and the time when she is 5.2 km from Tomsville.
(a) The amount of time she has walked (in hours) when she is xx kilometers from Tomsville. (b) D1(x)=10.4x4D^{-1}(x) = \frac{10.4 - x}{4} (c) D1(5.2)=1.3D^{-1}(5.2) = 1.3 hours

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Problem 487

Find the derivative of x+y=5\sqrt{x} + \sqrt{y} = 5 at the point (x,y)=(9,4)(x, y) = (9, 4).

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Problem 488

Find the value of f(x)=4xf(x) = 4^x when x=12x = \frac{1}{2}.

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Problem 489

Determine the number of critical points for the function f(x)=ex1+exf(x) = \frac{e^{x}}{1+e^{x}}.

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Problem 490

Find the derivative of 2xy+eycos(x)=e2 x y + e^y \cos(x) = e at the point (0,1)(0, 1). a. 0 b. e1-e^{-1} c. 2e1-2 e^{-1} d. e2\frac{e}{2}

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Problem 491

Find the value of f(1)f(1) for the function f(x)=10(12)x+7f(x)=10(12)^{x}+7.

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Problem 492

If the function f(x)f(x) has a critical point at x=cx=c, but its derivative f(c)f'(c) does not exist, then the statement is True.

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Problem 493

Find the open interval(s) where the function f(x)=x34x2+5xf(x) = x^3 - 4x^2 + 5x is concave down.

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Problem 494

Solve the equation 4e2x7=14 e^{2 x}-7=1 and round the solution to the hundredths place.

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Problem 495

Find the asymptote of the function f(x)=exf(x) = e^x. Options: a. y=1y = 1 b. x=0x = 0 c. y=0y = 0 d. y=xy = x

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Problem 496

Find the derivative dydx\frac{dy}{dx} of a curve with parametric equations x(t)=9cos(10t)x(t)=9\cos(10t) and y(t)=8e9ty(t)=-8e^{9t}.

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Problem 497

Find the derivative of y=2lnx+5log2xy=2 \ln x+5 \log _{2} x with respect to xx.

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Problem 498

Find the value of 60e(6)60 e^{-(6)}.

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Problem 499

Use implicit differentiation to find the derivative of yy, then evaluate the derivative at (3,0) for the equation 9ey=x2+y59e^y = x^2 + y^5.

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Problem 500

Divide y4y^{4} by y2y^{2} using the quotient rule.

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