Calculus

Problem 401

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

See Solution

Problem 402

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

See Solution

Problem 403

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

See Solution

Problem 404

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

See Solution

Problem 405

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

See Solution

Problem 406

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

See Solution

Problem 407

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

See Solution

Problem 408

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

See Solution

Problem 409

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

See Solution

Problem 410

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

See Solution

Problem 411

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

See Solution

Problem 412

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

See Solution

Problem 413

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

See Solution

Problem 414

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

See Solution

Problem 415

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

See Solution

Problem 416

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

See Solution

Problem 417

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

See Solution

Problem 418

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

See Solution

Problem 419

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

See Solution

Problem 420

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

See Solution

Problem 421

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

See Solution

Problem 422

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

See Solution

Problem 423

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

See Solution

Problem 424

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

See Solution

Problem 425

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

See Solution

Problem 426

Solve the equation $6 \ln (x-4)=30$ for $x$ .

See Solution

Problem 427

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

See Solution

Problem 428

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

See Solution

Problem 429

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

See Solution

Problem 430

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

See Solution

Problem 431

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

See Solution

Problem 432

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

See Solution

Problem 433

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

See Solution

Problem 434

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

See Solution

Problem 435

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

See Solution

Problem 436

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

See Solution

Problem 437

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

See Solution

Problem 438

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

See Solution

Problem 439

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

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Problem 440

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

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Problem 441

Differentiate and simplify these exponential functions: $f(x)=e^{7x}$ , $f(x)=3\cdot e^{2x}$ , $f(x)=e^{2x}$ , $f(x)=0.2\cdot e^{5x-1}$ , $f(x)=0.5\cdot e^{4x}$ , $f(x)=e^{-2x}$ , $f(x)=4\cdot e^{\frac{1}{2}x}$ , $f(x)=e^{0.2x}$ , $f(x)=\frac{1}{3}\cdot e^{-3x}$ , $f(x)=-\frac{1}{10}\cdot e^{5x}$ , $f(x)=3\cdot e^{-4x+2}$ , $f(x)=5\cdot e^{-3x-2}$ . Also differentiate and simplify: $f(x)=2x+e^{2x}$ , $f(x)=e^{2x}-5\cdot e^{\frac{1}{5}x}$ , $f(x)=x^{2}+e^{3x-1}$ , $f(x)=20\cdot e^{0.1x}+x+5$ , $f(x)=\frac{1}{3}x^{3}-2\cdot e^{-0.25x}$ , $f(x)=2x^{2}-e^{-x+3}+x^{3}$ .

See Solution

Problem 442

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

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Problem 443

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

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Problem 444

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

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Problem 445

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

See Solution

Problem 446

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

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Problem 447

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

See Solution

Problem 448

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

See Solution

Problem 449

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

See Solution

Problem 450

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

See Solution

Problem 451

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

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Problem 452

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

See Solution

Problem 453

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

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Problem 454

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

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Problem 455

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

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Problem 456

1a) Solve $2^{x-1}=8$ . 1b) Solve $3^{2x}=243$ . 2) Show $f(x+3)=8f(x)$ for $f(x)=2^x$ . 3) Find $f(4)$ and domain of $g(x)=\log_3(x^2-4x+3)$ . 4) Find inverse of $f(x)=e^{3x-1}$ . 5) Find $g(x)$ s.t. $(f\circ g)(x)=(g\circ f)(x)=x$ for $f(x)=e^{\sqrt{x}}$ . 6) Convert angles a) $315^\circ$ b) $-40^\circ$ c) $330^\circ$ to radians. 7) Sketch $f(x)=\frac{x^3-5x^2+6x}{x^2-4}$ .

See Solution

Problem 457

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

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Problem 458

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

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Problem 459

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

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Problem 460

Find the range of $y=\frac{2 x+1}{3 x-6}$ , 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 $f(x)=x^{2}+\frac{16}{x}$ . Sketch the curve.

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Problem 462

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

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Problem 463

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

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Problem 464

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

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Problem 465

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

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Problem 466

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

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Problem 467

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

See Solution

Problem 468

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

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Problem 469

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

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Problem 470

Find the value of $t$ when $6e^{9t} = 96$ .

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Problem 471

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

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Problem 472

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

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Problem 473

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

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Problem 474

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

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Problem 475

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

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Problem 476

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

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Problem 477

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

See Solution

Problem 478

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

See Solution

Problem 479

Approximate $12^{\frac{2}{3}}$ using a calculator and select the correct answer.

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Problem 480

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

See Solution

Problem 481

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

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Problem 482

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

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Problem 483

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

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Problem 484

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

See Solution

Problem 485

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

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Problem 486

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

See Solution

Problem 487

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

See Solution

Problem 488

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

See Solution

Problem 489

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

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Problem 490

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

See Solution

Problem 491

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

See Solution

Problem 492

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

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Problem 493

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

See Solution

Problem 494

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

See Solution

Problem 495

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

See Solution

Problem 496

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

See Solution

Problem 497

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

See Solution

Problem 498

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

See Solution

Problem 499

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

See Solution

Problem 500

Divide $y^{4}$ by $y^{2}$ using the quotient rule.

See Solution

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Calculus

Problem 401

Find the Cartesian equation x=f(y)x=f(y)x=f(y) from the parametric equations x(t)=2t2x(t)=2t^2x(t)=2t2 and y(t)=−9+4ty(t)=-9+4ty(t)=−9+4t.

Problem 402

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

Problem 403

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

Problem 404

Express the limit as a definite integral on [1,9][1, 9][1,9]: lim⁡n→∞∑i=1nxi∗(xi∗)2+8Δx\lim_{n \to \infty} \sum_{i=1}^{n} \frac{x_i^*}{(x_i^*)^2 + 8} \Delta xlimn→∞​∑i=1n​(xi∗​)2+8xi∗​​Δx

Problem 405

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

Problem 406

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

Problem 407

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

Problem 408

Problem 409

Find all ccc in the interval (2,7)(2,7)(2,7) such that f′(c)=f(7)−f(2)7−2f'(c) = \frac{f(7) - f(2)}{7 - 2}f′(c)=7−2f(7)−f(2)​ where f(x)=9−14xf(x) = 9 - \frac{14}{x}f(x)=9−x14​.

Problem 410

Find two points, other than the origin, on the graph of the function t(x)=−94x5t(x) = -\frac{9}{4}x^5t(x)=−49​x5 that fit within the [−10,10][-10,10][−10,10] by [−10,10][-10,10][−10,10] grid.

Problem 411

Find the domain and range of the quadratic function y=−3x2y=-3x^2y=−3x2.

Problem 412

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

Problem 413

Find the range of xxx where xe−xx e^{-x}xe−x is positive.

Problem 414

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

Problem 415

Find equivalent expressions for log⁡(107)\log(10^7)log(107). Options: A. 7⋅log⁡107 \cdot \log 107⋅log10, B. 7⋅107 \cdot 107⋅10, C. 1, D. 7.

Problem 416

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

Problem 417

Find the sum of the infinite series ∑n=1∞−7(38)n\sum_{n=1}^{\infty}-7\left(\frac{3}{8}\right)^{n}∑n=1∞​−7(83​)n.

Problem 418

Find the logarithm of 11024\frac{1}{1024}10241​ in base 4.

Problem 419

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

Problem 420

Find the area bounded by the graphs of y=2xy=\frac{2}{x}y=x2​ and y=0y=0y=0 over the interval 1≤x≤4e1 \leq x \leq 4e1≤x≤4e.

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)^xf(x)=4x,8x,(31​)x,(51​)x. Write the inverse functions in exponential and logarithmic form.

Problem 422

Find the ∫sin⁡8(x) dx\int \sin^8(x)\,dx∫sin8(x)dx or ddxsin⁡8(x)\frac{d}{dx}\sin^8(x)dxd​sin8(x).

Problem 423

Find fx(x,y)f_x(x,y)fx​(x,y) and fy(x,y)f_y(x,y)fy​(x,y) for f(x,y)=6x5−xy6−5y3f(x,y)=6x^5-xy^6-5y^3f(x,y)=6x5−xy6−5y3.

Problem 424

Evaluate f(x)=ln⁡x9f(x)=\frac{\ln x}{9}f(x)=9lnx​ when x=2.197225x=2.197225x=2.197225. Round the answer to three decimal places.

Problem 425

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

Problem 426

Solve the equation 6ln⁡(x−4)=306 \ln (x-4)=306ln(x−4)=30 for xxx.

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)f(x)=3x2ln(x) for x>0x > 0x>0.

Problem 428

Find the linear and quadratic approximating polynomials for f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ at x=1x=1x=1 and use them to approximate 11.03\frac{1}{1.03}1.031​.

Problem 429

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

Problem 430

Find the partial derivative of 200x14y34200 x^{\frac{1}{4}} y^{\frac{3}{4}}200x41​y43​ with respect to xxx.

Problem 431

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

Problem 432

Solve the equation x3−4x=2xx^{3}-4 x=2^{x}x3−4x=2x and find the fourth solution, rounded to the nearest tenth.

Problem 433

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

Problem 434

Find the derivative of arctan⁡x2+1\arctan \sqrt{x^{2}+1}arctanx2+1​.

Problem 435

Find the value of xxx that satisfies the equation ln⁡(x−5)=0\ln(x-5) = 0ln(x−5)=0.

Problem 436

Find the first and second derivatives of the linear function y=7x+1y = 7x + 1y=7x+1, where dydx=□\frac{dy}{dx} = \squaredxdy​=□ and d2ydx2=□\frac{d^2y}{dx^2} = \squaredx2d2y​=□.

Problem 437

Suppose f(0)=−3f(0)=-3f(0)=−3 and f′(x)≤9f'(x) \leq 9f′(x)≤9 for all xxx. What is the largest possible value for f(3)f(3)f(3)?The largest possible value for f(3)f(3)f(3) is 000.

Problem 438

Find the partial fraction decomposition of 74x8x2−10x+3\frac{74 x}{8 x^{2}-10 x+3}8x2−10x+374x​ in the form f(x)2x−1+g(x)4x−3\frac{f(x)}{2 x-1}+\frac{g(x)}{4 x-3}2x−1f(x)​+4x−3g(x)​, where f(x)f(x)f(x) and g(x)g(x)g(x) are to be determined.

Problem 439

Find the derivative of y=f(x)=x2−4y = f(x) = x^2 - 4y=f(x)=x2−4 at x=1x = 1x=1, evaluate f(1)f(1)f(1), and graph y=f(x)y = f(x)y=f(x) and the tangent line at (1,f(1))(1, f(1))(1,f(1)).

Problem 440

Use Maclaurin series to estimate eee from exe^xex, rounding to 4 decimal places. e≈2.7183e \approx \boxed{2.7183}e≈2.7183​

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Solve the equation $6 \ln (x-4)=30$ for $x$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Find the value of $t$ when $6e^{9t} = 96$ .

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

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

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

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