Calculus

Problem 301

Evaluate the quotient (5.44×1018)÷(6.8×109)\left(5.44 \times 10^{-18}\right) \div\left(6.8 \times 10^{-9}\right) and express the result in scientific notation.

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Problem 302

Find the regions of increasing and decreasing for the function y=2x+72xy=\frac{2x+7}{2x} where x0x \neq 0.

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Problem 303

Solve the equation 7e3x4=147 e^{3 x}-4=14 using natural logarithms. Round the solution to the nearest thousandth.

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Problem 304

Find the quadratic equation of a rocket's trajectory given its maximum height of 12801280 feet and landing point at (16,0)(16,0) in 88 seconds.

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Problem 305

Find the derivative of the functions f(x)=2x2f(x)=2x^2, f(x)=2xf(x)=2x, f(x)=5f(x)=5, f(x)=x2f(x)=-x^2, f(x)=2x+3f(x)=2x+3, and f(x)=ax2f(x)=ax^2.

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Problem 306

Find the absolute extreme values of f(x)=12x33+16x212xf(x) = \frac{12x^3}{3} + 16x^2 - 12x on the interval [4,1][-4, 1]. Select the correct choice: A. The absolute maximum is \square at x=x = \square. B. There is no absolute maximum on the given interval.

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Problem 307

Find the derivative of the inverse function f1(x)f^{-1}(x) where f(x)=1(1x)2f(x)=\frac{1}{(1-x)^{2}} and x>1x>1, evaluated at x=14x=\frac{1}{4}.

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Problem 308

Calculate the specified partial derivatives for the following functions: (a) f(x,y)=xy3ey,fy2=2fy2f(x, y)=x y^{3} e^{y}, \quad f_{y^{2}}^{\prime \prime}=\frac{\partial^{2} f}{\partial y^{2}} (b) f(p,q)=3p3q2,fp2=2fp2f(p, q)=3 p^{3} q^{2}, \quad f_{p^{2}}^{\prime \prime}=\frac{\partial^{2} f}{\partial p^{2}} (c) f(k,l)=5kl3,fkl=2fklf(k, l)=5 \sqrt{k} l^{3}, \quad f_{kl}^{\prime \prime}=\frac{\partial^{2} f}{\partial k \partial l}

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Problem 309

Find the approximate value of y=4+sinxy=\sqrt{4+\sin x} at x=0.12x=0.12.

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Problem 310

Find the limit of the given rational expression as xx approaches 3.

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Problem 311

Determine constant cc so that f(x)=ce(2x+3)f(x) = c e^{-(2x+3)} is a valid probability distribution over 0.621>x>0.43790.621 > x > 0.4379.

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Problem 312

Find the indefinite integral of cos23x\cos^2 3x.

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Problem 313

Evaluate the integral xsec1xdx\int x \sec^{-1} x dx for x1x \leq -1.

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Problem 314

Find the area of the region between y=3sin(x)y=3\sin(x) and y=4cos(x)y=4\cos(x) from x=0x=0 to x=0.6πx=0.6\pi. The region consists of two parts.

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Problem 315

Calculate the integral xlnxdx\int x \ln x d x. (3 marks for calculation, 3 marks for explanations, 6 marks total.)

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Problem 316

Find the second derivative f(1)f''(1) of a differentiable function ff satisfying f(x)+3f(x2)=f2(x)f'(x) + 3f(x^2) = f^2(x) for x>0x > 0 and f(1)=5f(1) = 5.

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Problem 317

Find the function k(x)k(-x) given k(x)=8x97xk(x)=-8 x^{9}-7 x.

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Problem 318

Find the range of f(x)=x2+3f(x)=x^2+3 when x>0x>0.

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Problem 319

Find the derivatives of h(x)=f(x)g(x)h(x) = f(x)g(x) at x=3x = 3 and h(x)=f(x)/g(x)h(x) = f(x)/g(x) at x=2x = 2, given a table of f(x),f(x),g(x),g(x)f(x), f'(x), g(x), g'(x). Also, find the points where the tangent line of f(x)=(2x2+10)/(x+2)f(x) = (2x^2 + 10)/(x + 2) is horizontal.

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Problem 320

Find the point(s) on the graph of f(x)=3x25xf(x)=3x^2-5x with tangent lines parallel to y=7x+5y=7x+5. The point(s) is/are ((x, y)).

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Problem 321

(1 point) Evaluate the definite integral 1316x2dx\int_{1}^{3} 16 x^{2} d x using: a) Trapezoid Rule with n=4n=4, b) Simpson's Rule with n=4n=4, and c) find the exact value.

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Problem 322

Find the left Riemann sum LnL_n, right Riemann sum RnR_n, and their average for 15(x2+4)dx\int_1^5 (x^2 + 4) dx with n=4n=4.

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Problem 323

Find the integral of 2xf(x2)+82x f(x^2) + 8 from x=2x=2 to x=4x=4 given that f(x)f(x) is continuous and 416f(x)dx=19\int_{4}^{16} f(x) dx = 19.

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Problem 324

Find the derivative dydx\frac{d y}{d x} of the equation 6y2(x1)y=56 y^{2} - (x - 1) y = 5 in simplified factored form.

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Problem 325

Find the range of the function f(x)=x2f(x) = \sqrt{x-2} on its domain.

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Problem 326

Find the indefinite integral of 3xsinx23x\sin x^2.

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Problem 327

Approximate e4e^{4} and round the answer to the nearest tenth.

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Problem 328

Find the maximum height of a vertically moving body with s=12gt2+v0t+s0,g>0s=-\frac{1}{2}gt^2+v_0t+s_0, g>0. Which expression gives the correct max height: v022g+s0\frac{v_0^2}{2g}+s_0, v0g+s0\frac{v_0}{g}+s_0, v02g+s0\frac{v_0^2}{g}+s_0, v02g+s0\frac{v_0}{2g}+s_0?

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Problem 329

Find the derivative of y=3secxtanxy=3 \sec x \tan x with respect to xx.

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Problem 330

Find the integral of ex/xe^{x}/x from 1 to 2, given F(x)=0xeetdtF(x)=\int_{0}^{x} e^{e^{t}} dt.

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Problem 331

Find the value of the third derivative of f(x)f(x) at x=1x=-1, given the degree 3 Taylor polynomial P(x)=8910(x+1)+87(x+1)2103(x+1)3P(x)=8-\frac{9}{10}(x+1)+\frac{8}{7}(x+1)^{2}-\frac{10}{3}(x+1)^{3}.

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Problem 332

Evaluate the natural logarithm of e7e^{7}.

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Problem 333

Approximate the integral of f(x)=2x2f(x) = 2 - x^2 from 0 to 1 using Riemann sums. Lisa calculates the upper sum using the maximum function value, while Theo calculates the lower sum using the minimum function value. Compare the results for 5 and 500 rectangles.

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Problem 334

Determine the intersection point of the tangents at points AA and BB on the graph of the function f(x)=12x22xf(x) = -\frac{1}{2}x^{2} - 2x, where A(1/1.5)A(-1/1.5) and B(1/2.5)B(1/2.5).

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Problem 335

Determine if y=14e3xy=\frac{1}{4} e^{-3 x} represents exponential growth or decay. Identify the graph.

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Problem 336

Find the derivative dydx\frac{d y}{d x} when x=yx=\sqrt{y}.

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Problem 337

Find the critical points of the function f(θ)=8secθ+4tanθf(\theta) = 8 \sec \theta + 4 \tan \theta for 0<θ<2π0 < \theta < 2\pi.

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Problem 338

Find a function f(x)f(x) with derivative f(x)=2f'(x) = 2 that passes through the point (0,3)(0, 3).

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Problem 339

Find the derivative of f(x)=7x+4f(x)=\sqrt{7x+4} using the limit definition. Then find the equation of the tangent line at x=3x=3.

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Problem 340

Find the limit of the piecewise function g(x)g(x) as xx approaches 2 from the right.

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Problem 341

Find the values of ln35.2\ln 35.2 and log45\log \frac{4}{5}.

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Problem 342

Find the term independent of xx in the expansion of (1x+x2)9\left(\frac{1}{x}+x^{2}\right)^{9}.

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Problem 343

Find the value of s(r(4))s(r(4)) where r(x)=2x2r(x) = 2x - 2 and s(x)=x2s(x) = -x^2.

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Problem 344

Use leading coefficient test to determine if yy \to \infty or yy \to -\infty as xx \to -\infty for y=7x56x4y = 7x^5 - 6x^4.

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Problem 345

Expand the expression exlnxe^{x-\ln x}.

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Problem 346

Determine if the graphs of the function family fa(x)=x4ax2f_a(x) = x^4 - ax^2 have no local maximum for a0a \leq 0.

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Problem 347

Find the slope of the function y=x+1y = |x| + 1.

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Problem 348

Find two functions ff and gg such that (fg)(x)=(6x+7)7(f \circ g)(x) = (6x+7)^7, where neither function is the identity function.

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Problem 349

Compute Δy\Delta y and dydy for y=x1y=\sqrt{x-1}, x=2x=2, Δx=0.8\Delta x=0.8. Round answers to three decimal places. Sketch a diagram showing dxdx, dydy, and Δy\Delta y.

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Problem 350

Find bb in ln(25b)=\ln(25b) = \square.

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Problem 351

Find the values of xx where the function f(x)f(x) is negative, based on the graph.

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Problem 352

Find the rate of change of the base of a triangle with altitude 12 cm and area 97 sq cm, given the altitude is increasing at 2.5 cm/min and the area at 3.5 sq cm/min.

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Problem 353

Find the exponential function g(t)=abtg(t) = ab^t given g(10)=95g(10) = 95 and g(30)=27g(30) = 27. Round the solution to four decimal places.

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Problem 354

Find the derivative of f(x)=x3x2f(x) = -\sqrt{x^3} - \frac{\sqrt{x}}{2} and evaluate it at x=3x = 3. Express the answer as a simplified radical fraction.

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Problem 355

Find the limit of (f(x+h)f(x))/h(f(x+h) - f(x))/h as hh approaches 0, given f(x)=x28f(x) = x^2 - 8.

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Problem 356

Determine the xx- and yy-coordinates of the inflection point of the function f(x)=ex1+exf(x)=\frac{e^{x}}{1+e^{x}}.

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Problem 357

Given functions f(x)=0.5x2f(x) = 0.5x^2 and g(x)=3x3+1g(x) = 3x^3 + 1. Find the difference quotient for each interval: a) [0;2][0; 2], b) [1;3][-1; 3], c) [1;1][-1; 1], d) [2;1][-2; -1].

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Problem 358

Determine if E=mc2E=mc^2 is a power function. If so, state the power and constant of variation.

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Problem 359

Describe the end behavior of the function y=2x+2+3y=-|2x+2|+3. As x,y+x \rightarrow-\infty, y \rightarrow+\infty; As x+,yx \rightarrow+\infty, y \rightarrow-\infty.

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Problem 360

Given the function V(x)=(x4)2+6,x4V(x) = (x - 4)^2 + 6, x \geq 4, find the minimum value of the function.

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Problem 361

Differentiate the function y=1x+1x+1y=\frac{1}{x+\frac{1}{x+1}} with respect to xx.

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Problem 362

Finde kritische Stelle und Grenzwerte einer Funktion ff mit vertikaler Asymptote bei x=1x=1.

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Problem 363

Evaluate N(0)N(0) and N(18)N(18) for N(t)=12.800.02+0.53tN(t) = \frac{12.80}{0.02 + 0.53^{t}}. Round answers to 2 decimal places.

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Problem 364

Find the derivative of f(x)=xx2f(x) = x - x^2 using the limit definition of the derivative.

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Problem 365

Find the absolute extrema and their xx-values for f(x)=2x3+8x2+8x2f(x)=2x^3+8x^2+8x-2 on [5,0][-5,0].

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Problem 366

Find the critical point and determine the behavior of the function y=x7/24x2y=x^{7/2}-4x^2 for x>0x>0. Critical point: c=c=

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Problem 367

Simplify ex+7e6xex\frac{e^{x} + 7e^{6x}}{e^{x}} before finding its derivative.

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Problem 368

Find the domain, vertical asymptote(s), and horizontal asymptote of f(x)=1x+2f(x)=\frac{1}{x+2}. Domain: x2x\neq -2, Vertical Asymptote(s): x=2x=-2, Horizontal Asymptote: y=0y=0.

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Problem 369

Find the yy-values of the function f(x)=3exf(x) = -3e^x to two decimal places when x=2,1,0,1,2x = -2, -1, 0, 1, 2.

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Problem 370

Find the time when the drug concentration C(t)=0.7tt2+49C(t)=\frac{0.7 t}{t^{2}+49} is maximized, accurate to at least 2 decimal places.

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Problem 371

Find the logarithmic function f(t)=a+logb(t)f(t) = a + \log_b(t) passing through points (4,0.5)(4, -0.5) and (16,0)(16, 0).

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Problem 372

Determine the function resulting from horizontally compressing y=f(x)y=f(x) by 1/31/3 and vertically translating the graph 6 units down.

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Problem 373

Determine the derivative of the function f(x)=12x3f(x) = \frac{1}{2} x^{3}.

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Problem 374

Solve for xx to two decimal places using the exponential equation 700=500(1.04)x700=500(1.04)^{x}.

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Problem 375

Drug concentration function K(t)=8tt2+4K(t) = \frac{8t}{t^2 + 4} for 0<t<0 < t < \infty. a. Find intervals where K(t)K(t) is increasing. b. Find intervals where K(t)K(t) is decreasing.

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Problem 376

Solve the equation 8xlnx+x=08 x \ln x + x = 0 and round the answer to 3 decimal places.

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Problem 377

Find the roots of the derivative fa(t)=1.5t23(a+1)t+6af'_a(t) = 1.5t^2 - 3(a+1)t + 6a for a general 'a'. Solve 1.5t23(a+1)t+6a=01.5t^2 - 3(a+1)t + 6a = 0.

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Problem 378

Use implicit differentiation to find the derivative dpdx\frac{d p}{d x} for the demand equation p2+p+7x=200p^{2} + p + 7x = 200.

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Problem 379

Evaluate e2e^2, e2e^{-2}, and e1/2e^{1/2} using a calculator, rounding to three decimal places.

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Problem 380

Estimate Δy\Delta y for y=sin(5x)y=\sin(5x) using linear approximation when Δx=0.3\Delta x=0.3 at x=0x=0. Find the percentage error.

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Problem 381

Find the domain and range of y=log4(4+2x)y=\log_{4}(4+2x). Explain how to solve without graphing.

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Problem 382

Find the values of a, b, and c such that 79/4=abc7^{9/4} = \sqrt[c]{a^{b}}.

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Problem 383

The exponential function f(x)=561(1.026)xf(x)=561(1.026)^{x} models a country's population f(x)f(x) in millions, xx years after 1974. Find the population in 1974, 2001, 2028, and 2055, and describe the population growth pattern.

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Problem 384

Find the function with a different asymptote from the others: q(x)=6log5(x2)+3q(x)=6 \cdot \log _{5}(x-2)+3, b(x)=4log5(x2)+3b(x)=4 \cdot \log _{5}(x-2)+3, c(x)=4logs(x+1)+3c(x)=4 \cdot \log _{s}(x+1)+3, d(x)=4log5(x2)7d(x)=4 \cdot \log _{5}(x-2)-7.

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Problem 385

Find the natural logarithm of the 10th root of e without using a calculator. Solve for the 10th root of e.

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Problem 386

Find the range of the function f(x)=log2(x+2)+1f(x) = \log_2(x+2) + 1 where x>2x > -2 and 2<y<4-2 < y < 4.

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Problem 387

Find variables in V=πr2hV=\pi r^{2} h to create linear or quadratic relationships. Explain your choices.

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Problem 388

Estimate 85\sqrt{85} using linear approximation. Choose a value of aa to minimize the error. Round the approximation to three decimal places.

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Problem 389

Approximate ee by substituting large nn into (1+1n)n\left(1+\frac{1}{n}\right)^{n}.

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Problem 390

Expand (12x)1/2(1-2x)^{1/2} as a power series in xx up to and including x3x^3, simplifying coefficients, where 1/2<x<1/2-1/2<x<1/2.

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Problem 391

Sketch the graph of y=x48x3+16x22y=x^{4}-8x^{3}+16x^{2}-2 and summarize key points in a table.

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Problem 392

Find the value of aa if 7a=7387^{a}=\sqrt[8]{7^{3}}.

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Problem 393

Solve the equation 4x+6=74 \sqrt{x} + 6 = 7 for xx.

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Problem 394

Use the quotient rule to simplify p2q3pq2\frac{p^{2} q^{3}}{p q^{2}}, where all bases are not equal to 0.

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Problem 395

What operation undoes X2X^{2}? a) Divide by 2 b) Logit? c) Square root? d) Square it.

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Problem 396

Solve for xx in the equation ln(9x+4)=4.2\ln (9 x+4)=4.2, rounding the final solution to 3 decimal places.

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Problem 397

Bestimme die Ableitung und Nullstellen der Funktion y=x2+1y=x^{2}+1 und skizziere den Graphen.

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Problem 398

Graph the equation y=log3(x+3)+3y=\log_{3}(x+3)+3, then find its domain, range, and vertical asymptote.

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Problem 399

Evaluate f(x)=x+8f(x)=\sqrt{x+8} at x=4x=-4 and simplify.

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Problem 400

Find the limit of the sequence limn7n3nn\lim _{n \rightarrow \infty} \sqrt[n]{7^{n}-3^{n}}.

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