Calculus

Problem 501

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

See Solution

Problem 502

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

See Solution

Problem 503

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

See Solution

Problem 504

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

See Solution

Problem 505

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

See Solution

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.
$V = (4/3) \pi r^3$ , where $r$ is the radius.

See Solution

Problem 507

Find the indefinite integral of $5 dx$ .

See Solution

Problem 508

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

See Solution

Problem 509

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

See Solution

Problem 510

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

See Solution

Problem 511

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

See Solution

Problem 512

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

See Solution

Problem 513

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

See Solution

Problem 514

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

See Solution

Problem 515

Find the value of $e^{-0.5}$ .

See Solution

Problem 516

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

See Solution

Problem 517

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

See Solution

Problem 518

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

See Solution

Problem 519

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

See Solution

Problem 520

Complete the exponential decay table: Original Amount: $19,000$ , Decay Rate: $13\%$ per year, Years: $18$ , Final Amount after $18$ years of decay.

See Solution

Problem 521

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

See Solution

Problem 522

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

See Solution

Problem 523

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

See Solution

Problem 524

Solve for time $t$ in the equation $r = r_0 + \frac{1}{2}at^2$ , where $r_0$ and $a$ are known constants.

See Solution

Problem 525

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

See Solution

Problem 526

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

See Solution

Problem 527

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

See Solution

Problem 528

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

See Solution

Problem 529

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

See Solution

Problem 530

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

See Solution

Problem 531

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

See Solution

Problem 532

Show that for positive real numbers $u$ and $v$ , and a positive real number $a \neq 1$ , $\log_{a}\left(\frac{u}{v}\right) = \log_{a}u - \log_{a}v$ .

See Solution

Problem 533

Find the logarithm of 1.6.

See Solution

Problem 534

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

See Solution

Problem 535

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

See Solution

Problem 536

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

See Solution

Problem 537

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

See Solution

Problem 538

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

See Solution

Problem 539

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

See Solution

Problem 540

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

See Solution

Problem 541

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

See Solution

Problem 542

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

See Solution

Problem 543

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

See Solution

Problem 544

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

See Solution

Problem 545

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

See Solution

Problem 546

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

See Solution

Problem 547

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

See Solution

Problem 548

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

See Solution

Problem 549

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

See Solution

Problem 550

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

See Solution

Problem 551

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

See Solution

Problem 552

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

See Solution

Problem 553

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

See Solution

Problem 554

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

See Solution

Problem 555

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

See Solution

Problem 556

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

See Solution

Problem 557

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

See Solution

Problem 558

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

See Solution

Problem 559

Find the second derivative of the secant function with respect to $x$ .

See Solution

Problem 560

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

See Solution

Problem 561

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

See Solution

Problem 562

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

See Solution

Problem 563

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

See Solution

Problem 564

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

See Solution

Problem 565

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

See Solution

Problem 566

Calculate $R=\ln(h_1/h_2)$ given $h_1=24.1\pm0.1\mathrm{~cm}$ and $h_2=12.8\pm0.1\mathrm{~cm}$ . Select the correct value of $R$ .

See Solution

Problem 567

Sami measured physical quantities $X \pm \Delta X, Y \pm \Delta Y, Z \pm \Delta Z$ . To calculate $R=4 (Z^{7} Y^{4})/(X^{7})$ , $\Delta R/R$ is given by option a: $14 \Delta Z/Z + 16 \Delta Y/Y - 14 \Delta X/X$ .

See Solution

Problem 568

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

See Solution

Problem 569

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

See Solution

Problem 570

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

See Solution

Problem 571

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

See Solution

Problem 572

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

See Solution

Problem 573

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

See Solution

Problem 574

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

See Solution

Problem 575

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

See Solution

Problem 576

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

See Solution

Problem 577

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

See Solution

Problem 578

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

See Solution

Problem 579

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

See Solution

Problem 580

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

See Solution

Problem 581

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

See Solution

Problem 582

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

See Solution

Problem 583

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

See Solution

Problem 584

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

See Solution

Problem 585

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

See Solution

Problem 586

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

See Solution

Problem 587

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

See Solution

Problem 588

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

See Solution

Problem 589

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

See Solution

Problem 590

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

See Solution

Problem 591

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

See Solution

Problem 592

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

See Solution

Problem 593

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

See Solution

Problem 594

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

See Solution

Problem 595

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

See Solution

Problem 596

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

See Solution

Problem 597

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

See Solution

Problem 598

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

See Solution

Problem 599

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

See Solution

Problem 600

Find the unit tangent vector and length of the curve $\mathbf{r}(t) = 6t^3\mathbf{i} + 2t^3\mathbf{j} - 3t^3\mathbf{k}$ for $1 \leq t \leq 2$ . The unit tangent vector is $(\frac{6}{7})\mathbf{i} + (\frac{2}{7})\mathbf{j} + (-\frac{3}{7})\mathbf{k}$ , and the length is $9\sqrt{7}$ units.

See Solution

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Calculus

Problem 501

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

Problem 502

Differentiate the function f(x)=x7−4e8xf(x) = x^7 - 4e^{8x}f(x)=x7−4e8x to find f′(x)f'(x)f′(x).

Problem 503

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

Problem 504

Find the derivative y′y'y′ of y=cot⁡−1(t)y=\cot^{-1}(\sqrt{t})y=cot−1(t​).

Problem 505

Find the derivative of y=ln⁡tan⁡xy=\ln \sqrt{\tan x}y=lntanx​ evaluated at x=π4x=\frac{\pi}{4}x=4π​.

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^3V=(4/3)πr3, where rrr is the radius.

Problem 507

Find the indefinite integral of 5dx5 dx5dx.

Problem 508

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

Problem 509

Find the vertical asymptote of the logarithmic function g(x)=log⁡4(x+7)g(x) = \log_4(x+7)g(x)=log4​(x+7).

Problem 510

Find the limit of g(x)g(x)g(x) as xxx approaches 1, given that 2x≤g(x)≤x4−x2+22x \leq g(x) \leq x^4 - x^2 + 22x≤g(x)≤x4−x2+2 for all x≥0x \geq 0x≥0.

Problem 511

Find the end behavior of the quadratic function D(x)=49−x2D(x) = 49 - x^2D(x)=49−x2.

Problem 512

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

Problem 513

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

Problem 514

Find the antiderivative of the rational function 2x5+10x2+7x4\frac{2 x^{5}+10 x^{2}+7}{x^{4}}x42x5+10x2+7​.

Problem 515

Find the value of e−0.5e^{-0.5}e−0.5.

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)v(t)=7sin(t)+7cos(t), between t=3t = 3t=3 and t=8t = 8t=8 seconds. Give the answer to at least three decimal places.

Problem 517

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

Problem 518

Find the value of ttt where the logistic function y=M1+ce−kMty=\frac{M}{1+c e^{-k M t}}y=1+ce−kMtM​ equals M2\frac{M}{2}2M​.

Problem 519

Find the value of xxx when y=a+bln⁡xy = a + b \ln xy=a+blnx, given a=12.3301088919a = 12.3301088919a=12.3301088919 and b=−8.22097130319b = -8.22097130319b=−8.22097130319, and y=−16y = -16y=−16.

Problem 520

Complete the exponential decay table: Original Amount: 19,00019,00019,000, Decay Rate: 13%13\%13% per year, Years: 181818, Final Amount after 181818 years of decay.

Problem 521

Find the demand function D(p)=−2p2−6p+300D(p) = -2p^2 - 6p + 300D(p)=−2p2−6p+300 and its rate of change w.r.t. price ppp. Interpret the rate of change when p=$11p = \$11p=$11.

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)(x−3)h(x) = -5(x+1)(x-3)h(x)=−5(x+1)(x−3).

Problem 523

Evaluate lim⁡x→1x−1x−1\lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1}limx→1​x−1x​−1​ using the provided table of xxx and f(x)f(x)f(x) values.

Problem 524

Solve for time ttt in the equation r=r0+12at2r = r_0 + \frac{1}{2}at^2r=r0​+21​at2, where r0r_0r0​ and aaa are known constants.

Problem 525

Differentiate the function f(θ)=sec⁡θ5+sec⁡θf(\theta) = \frac{\sec \theta}{5 + \sec \theta}f(θ)=5+secθsecθ​.

Problem 526

Find the derivative of 5x2−4\sqrt{5 x^{2}-4}5x2−4​ with respect to xxx.

Problem 527

Evaluate the definite integral of 4ye2y\frac{4y}{e^{2y}}e2y4y​ from 0 to 1.

Problem 528

Solve the following exponential equations: a. e−0.2t=9e^{-0.2 t}=9e−0.2t=9, b. ekt=15e^{k t}=\frac{1}{5}ekt=51​, c. e(ln⁡0.4)t=0.7e^{(\ln 0.4) t}=0.7e(ln0.4)t=0.7.

Problem 529

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

Problem 530

Find the derivative of f(x)=5x4(x3−2)f(x) = 5x^4(x^3 - 2)f(x)=5x4(x3−2) and identify the correct application of the product rule.

Problem 531

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

Problem 532

Show that for positive real numbers uuu and vvv, and a positive real number a≠1a \neq 1a=1, log⁡a(uv)=log⁡au−log⁡av\log_{a}\left(\frac{u}{v}\right) = \log_{a}u - \log_{a}vloga​(vu​)=loga​u−loga​v.

Problem 533

Find the logarithm of 1.6.

Problem 534

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

Problem 535

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

Problem 536

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

Problem 537

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

Problem 538

Find the function y(x)y(x)y(x) that satisfies the first-order differential equation y′=ysin⁡xy' = y \sin xy′=ysinx, where y=πe−cos⁡xy = \pi e^{-\cos x}y=πe−cosx is a solution.

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}R(x)=(1+ex+5)2100ex+5​, where xxx is the natural log of a histamine dose in mM. Evaluate the antiderivative at x=−8.9x=-8.9x=−8.9 and round to 3 decimal places.

Problem 540

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

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

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

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

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

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.
$V = (4/3) \pi r^3$ , where $r$ is the radius.

Find the indefinite integral of $5 dx$ .

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

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

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

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

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

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

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

Find the value of $e^{-0.5}$ .

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

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

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

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

Complete the exponential decay table: Original Amount: $19,000$ , Decay Rate: $13\%$ per year, Years: $18$ , Final Amount after $18$ years of decay.

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

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

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

Solve for time $t$ in the equation $r = r_0 + \frac{1}{2}at^2$ , where $r_0$ and $a$ are known constants.

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

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

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

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

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

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

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

Show that for positive real numbers $u$ and $v$ , and a positive real number $a \neq 1$ , $\log_{a}\left(\frac{u}{v}\right) = \log_{a}u - \log_{a}v$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Find the second derivative of the secant function with respect to $x$ .

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

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

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

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

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

Calculate $R=\ln(h_1/h_2)$ given $h_1=24.1\pm0.1\mathrm{~cm}$ and $h_2=12.8\pm0.1\mathrm{~cm}$ . Select the correct value of $R$ .

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

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

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

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

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

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

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