Calculus

Problem 601

Find the derivative of $\left(\frac{e^{2-x}}{x+2}\right)$ .

See Solution

Problem 602

Find the values of $a$ and $b$ in the function $P(t) = a b^t$ that models the number of bacteria in a lab experiment, given the function $P(t) = 47(1.112)^{5t}$ . Round the final values of $a$ and $b$ to 4 decimal places.

See Solution

Problem 603

Evaluate the indefinite integral $\int x \sqrt{29 x-7} d x$ .

See Solution

Problem 604

Soit la fonction $g(x) = e^{x}(1-x)-1$ définie sur $\mathbb{R}$ . Étudier le sens de variation et le signe de $g$ . Soit la fonction $f(x) = \frac{x}{e^{x}-1}+2$ pour $x \neq 0$ , et $f(0)=3$ . Déterminer les limites de $f$ , étudier son sens de variation, et trouver l'équation de la tangente à $(\mathscr{E})$ à l'origine. Soit $h(x) = f(x)-x$ , montrer que $h'(x)<0$ et en déduire qu'il existe une solution unique $\alpha$ à l'équation $f(x)=x$ dans l'intervalle $] 2 ; \frac{5}{2}[$ .

See Solution

Problem 605

Find all values of $t$ where the parametric curve $x=4t^3-2t^2+4t, y=2t^3+4t^2-2$ has a horizontal tangent.

See Solution

Problem 606

Find the equation of the tangent line to $f(x) = \sqrt{5x-1}$ at the point $(1,2)$ in slope-intercept form.

See Solution

Problem 607

Find $dy/dt$ when $x^2 + y^2 = 36$ and $dx/dt = 3$ , for $y > 0$ and (a) $x = 0$ , (b) $x = 2$ , (c) $x = 3$ .

See Solution

Problem 608

Find the Left and Right Riemann sums, then use them to find the Trapezoidal sum on the interval $[-3,7]$ given the table of $x$ and $f(x)$ values.

See Solution

Problem 609

Determine the horizontal asymptote of the function $f(x) = \frac{16 x^{3}-7 x+8}{20 x^{3}+8 x-3}$ . If none exists, state that fact.

See Solution

Problem 610

Identify the critical points of $y=x^{2}e^{x}$ . State the $x$ - and $y$ -coordinates. Determine if they are local max, min, or neither.

See Solution

Problem 611

Find the minimum $y$ -value of $f(x)=x^{2}+\frac{2}{x}$ on $\frac{1}{2} \leq x \leq 2$ .

See Solution

Problem 612

Find the solution, rounded to one decimal place, to the equation $e^{x} = \sqrt{x+8}$ using a graphing calculator.

See Solution

Problem 613

1. Find the antiderivative of $f(x)$ : a) $f(x)=4x^3+3x^4$ b) $f(x)=\frac{1}{10}x^4-\frac{10}{x^4}$ e) $f(x)=2e^x+1$ f) $f(x)=\frac{1}{3}e^x-\frac{x^2}{2}$
2. Find the antiderivative of $f(x)$ : a) $f(x)=(3x-1)^2$ b) $f(x)=\frac{3}{(-5x+1)^4}$ e) $f(x)=e^{4x-1}$ f) $f(x)=\frac{2}{5}e^{2-5x}$ i) $f(x)=\sqrt{6x}$ j) $f(x)=\sqrt{9-\frac{1}{3}x}$

See Solution

Problem 614

Find the degree 4 Maclaurin polynomial for $(1+x)^{1/3}$ and use it to approximate $\sqrt[3]{1.1}$ .

See Solution

Problem 615

Find the inverse function $f^{-1}(x)$ of $f(x)=\sqrt{7x+2}$ where $x \geq -\frac{2}{7}$ .

See Solution

Problem 616

Find the value of a piecewise function $w(t)$ when $t=-7$ .

See Solution

Problem 617

Explain why the graph of $y=-2 \sqrt[3]{x}$ is continuous but not differentiable at $x=0$ .

See Solution

Problem 618

Find the derivative of the function $h(x) = e^{2x^2} - 9x + 3/x$ .

See Solution

Problem 619

Find 4 paths to (1,-1) to show $\lim_{(x,y)\to(1,-1)} f(x,y)$ doesn't exist for $f(x,y)$ . Provide curve equations and graphs.

See Solution

Problem 620

Differentiate $y=(x+2)(x^{3}+2x+1)$ and choose the correct setup.

See Solution

Problem 621

Find the range of the quadratic function $f(x)=-6(x-7)^{2}+5$ .

See Solution

Problem 622

Find the indefinite integral of $5x^3$ .

See Solution

Problem 623

A snowball's radius decreases at $0.1$ cm/min. Find the rate of volume decrease when radius is $11$ cm. (Round answer to 3 decimal places)
$\frac{d V}{d t} = -4 \pi r^2 \frac{d r}{d t} = -4 \pi (11)^2 (0.1) = -\boxed{12.116} \frac{\mathrm{cm}^3}{\min}$

See Solution

Problem 624

Evaluate the definite integral of $0.25e^{-0.25x}$ .

See Solution

Problem 625

Find the value of $y=a b^{x}$ given $a=1600, b=.25, x=12$ , rounded to 7 decimal places.

See Solution

Problem 626

Evaluate $\frac{\sqrt{x-3}}{9}$ when $x=12$ . Options: A) $\frac{1}{3}$ B) 1 C) 3 D) 81.

See Solution

Problem 627

Prove that the function $f(x)=x^{7}+8 x-2$ has at most one real root using Rolle's Theorem. Show it has exactly one using the Intermediate Value Theorem.

See Solution

Problem 628

Find the average velocity of an object moving along a vertical line with position $L(t)=-3t^2+t+8$ (m) at time $t$ (s) for the given intervals: [3s, 9s], [2s, 7s], [5s, 8s].

See Solution

Problem 629

Find the derivative of $f(x)=\sqrt{5 x-10}$ using the limit definition.

See Solution

Problem 630

At most, how many turning points does the polynomial $P(x)=25 x^{110}-19 x^{37}+43 x^{20}-32 x^{17}$ have?

See Solution

Problem 631

Solve for $y$ in the equation $e^{-8 y}=9$ . Round the answer to the nearest hundredth.

See Solution

Problem 632

Solve the differential equation $\frac{d y}{d x} = -\frac{e^{x}}{8}$ .

See Solution

Problem 633

Find the value of $c$ that satisfies the Mean Value Theorem for $h(x) = \sqrt{3x-3}$ on the interval $4 \leq x \leq 13$ .

See Solution

Problem 634

Find the radius of convergence $R$ of the power series $\sum_{n=1}^{\infty} \frac{(-4)^{n}}{\sqrt{n}}(x+8)^{n}$ . If $R$ is infinite, type "infinity" or "inf". Answer: $R=\frac{1}{4}$ . What is the interval of convergence? Answer (in interval notation): $(-12, -4)$ .

See Solution

Problem 635

Determine the behavior of the function $f(x) = (2x + 1) \cdot e^{-x}$ for very large and very small values of $x$ .

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Problem 636

Evaluate the limit $\lim_{x \to \pi/2} (\sec(x) - \tan(x))$ . Use symbolic notation and fractions.

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Problem 637

Evaluate the natural logarithm of 0.012 and give the answer to 4 decimal places. $\ln (0.012)$

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Problem 638

Analyze the properties of the logarithmic functions $y=\log_2(x+1)$ and $y=\log(x)-3$ , including domain, range, asymptotes, x-intercepts, and transformations.

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Problem 639

Use Newton's method to find the solution to $e^{-2x} = -3x + 9$ starting with $x_0 = -5$ . Provide the first two iterates $x_1$ and $x_2$ , and the final solution accurate to 4 decimal places.

See Solution

Problem 640

Find the limit, continuity, and type of discontinuity of the piecewise function f(x) = \\begin{cases} 2x+1, & x>-1 \\\\ x^2+1, & x \\leq-1 \\end{cases} at $x=a$ .

See Solution

Problem 641

A. $(\mathrm{f}-\mathrm{g})(12)=\sqrt{12-3}-12^{2}+12$

See Solution

Problem 642

Evaluate the integral $2 \cdot \int x^{2} \ln x d x$

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Problem 643

Find the slope of $y=x^3-x$ at $x=a$ . What is the slope at $x=1$ ? Where does the slope equal $11$ ?

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Problem 644

Find the solution to the equation $5 e^{x+2} = 7$ .

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Problem 645

Evaluate the integral $\int_{e^{3 x}}^{e^{4 x}} \frac{1}{x \ln x+x} dx$ .

See Solution

Problem 646

Simplify the expression $\sqrt[4]{\frac{x^{3}}{256}}$ using the quotient rule, assuming all variables are positive real numbers.

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Problem 647

Find the integral that represents the length of the curve $f(x) = \cos x, 0 \leq x \leq \pi$ .

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Problem 648

Find the values of $m$ for which the integral $\int_{0}^{8} x^{m} d x$ converges.

See Solution

Problem 649

Approximate the sum of the series $\sum_{n=1}^{\infty}(-1)^{n} \frac{2}{3 n^{4}}$ with error < 0.001.

See Solution

Problem 650

Determine the limit of the sequence $\left\{\left(\frac{n+n^{2}}{2 n^{2}}\right)^{n}\right\}$ .

See Solution

Problem 651

Estimate the total number of plastic widgets sold in the first 6 weeks given the sales function $P(t)=7000 t e^{-0.3 t}$ .

See Solution

Problem 652

Describe the end behavior of $g(x) = -\frac{1}{x^2}$ . As $x \to \pm\infty$ , $g(x)$ goes to 0 or $\pm\infty$ .

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Problem 653

Find the secant line equation $y=mx+b$ passing through points $(-3, f(-3))$ and $(2, f(2))$ where $f(x)=x^{3}-5$ .

See Solution

Problem 654

Find the equation(s) of the vertical asymptote(s) of the rational function $g(t) = \frac{t^{2} - 5}{3t^{2} + 4t - 3}$ .

See Solution

Problem 655

Differentiate the function $g(t) = \frac{2 - 5t}{7t + 8}$ to find $g'(t)$ .

See Solution

Problem 656

Find the linear approximation of $y=e^{5x}\ln(x)$ at $x=1$ . $L(x)=e^5\ln(1)+\frac{5e^5\ln(1)+e^5}{1}(x-1)$

See Solution

Problem 657

The graph $y=500(0.85)^{x}$ represents the amount of drug (in mg) left in a patient's system $x$ hours after administration. Find the $y$ -intercept.

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Problem 658

Derive the function $f(x) = \ln 3x^2$ .

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Problem 659

Find the domain, range, intercepts, and asymptotes of the exponential function $f(x) = 3 \cdot 6^{x}$ . Describe the end behavior.

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Problem 660

Find $\Delta y$ , $dy=f'(x)dx$ , and $dy$ for $y=2x^3$ , $x=4$ , and $\Delta x=0.08$ . a) $\Delta y=\boxed{1.6384}$ b) $dy=f'(x)dx=\boxed{96}dx$ c) $dy=\boxed{7.68}$

See Solution

Problem 661

Find the value of $x$ in the equation $e^{x}=5$ . The correct statements are: $x=\ln 5$ , $x=\log _{5} e$ , and to the nearest hundredth, $x \approx 1.61$ .

See Solution

Problem 662

Find the differential $dy$ for $y=e^{x/4}$ and evaluate $dy$ for $x=0, dx=0.05$ .

See Solution

Problem 663

Identify the hole, vertical asymptote, horizontal asymptote, and domain of the rational function $f(x) = \frac{x^2 + 9x + 20}{x^2 + 3x - 4}$ .

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Problem 664

Evaluate the integral $\int \frac{9 x+2}{x^{2}+x-6} d x$ .

See Solution

Problem 665

Calculate the integral $\int_{2}^{3} 3 x^{2} d t$ .

See Solution

Problem 666

Evaluate the integral $\int_{-2}^{2} \frac{1+x^{2}}{1+2^{x}} d x$ .

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Problem 667

Calculate the integral $\int \tan^{3} x \sec x \, dx$ .

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Problem 668

Find the integral of $-\frac{1}{5} \sin u \, du$ .

See Solution

Problem 669

Differentiate $x^{e^{x}}$ with respect to $x$ .

See Solution

Problem 670

Find the derivative of $x^{2}$ with respect to $x$ .

See Solution

Problem 671

Find the ball's instantaneous velocity at $t=10.0$ s given $x(t)=0.000015 t^{5}-0.004 t^{3}+0.4 t$ .

See Solution

Problem 672

Determine if the series converges or diverges: $\sum_{n=1}^{\infty}\left[\left(\frac{6}{7}\right)^{n}-\frac{3}{2^{n}}\right]$ . If it converges, find the sum.

See Solution

Problem 673

Find the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n(n+2)}$ .

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Problem 674

Find the limit of the sequence $a_{n}=\sqrt{n^{2}+3 n}-n$ . Options: A) 3 B) 2 C) $1 / 2$ D) $3 / 2$ E) 0 F) 1 G) $\infty$ H) does not exist.

See Solution

Problem 675

Determine if the series are convergent or divergent:
(I) $\sum_{n=1}^{\infty}\left(\frac{n+3}{2 n-5}\right)^{n}$
(II) $\sum_{n=1}^{\infty} \sqrt[4]{\frac{2}{n^{3}}}$
(III) $\sum_{n=1}^{\infty} \frac{2^{n}}{n ! \sqrt{n+1}}$
Explain your reasoning.

See Solution

Problem 676

Determine if these series converge or diverge, explaining your reasoning: (I) $\sum_{n=1}^{\infty}\left(\frac{n+3}{2 n-5}\right)^{n}$ , (II) $\sum_{n=1}^{\infty} \sqrt[4]{\frac{2}{n^{3}}}$ , (III) $\sum_{n=1}^{\infty} \frac{2^{n}}{n ! \sqrt{n+1}}$ .

See Solution

Problem 677

Determine if these series are absolutely convergent, conditionally convergent, or divergent:
(I) $\sum_{n=1}^{\infty}\left(\frac{\pi}{2}\right)^{n}$
(II) $\sum_{n=1}^{\infty} \frac{(-1)^{n-1} e}{\sqrt{n}}$
(III) $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}(\sqrt{n}+1)^{3}}$

See Solution

Problem 678

A ball's position is given by $x(t)=0.000015 t^5 - 0.004 t^3 + 0.4 t$ . Find its velocity at $t=10.0$ s.

See Solution

Problem 679

A ball's position is given by $x(t)=0.000015 t^5 - 0.004 t^3 + 0.4 t$ . Find its velocity at $t=10.0$ s.

See Solution

Problem 680

Find the integral of $x^{2}$ with respect to $x$ .

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Problem 681

Find the integral $\int \sin (3 x) d x$ . Choose from: (a) $3 \cos (3 x)+C$ , (b) $\frac{1}{3} \cos (3 x)+C$ , (c) $-3 \cos (3 x)+C$ , (d) $-\frac{1}{3} \cos (3 x)+C$ .

See Solution

Problem 682

Calculate the integral $\int_{1}^{5} \frac{x-1}{x} dx$ and choose the correct answer: (a) $5-\ln 5$ , (b) $4-\ln 5$ , (c) $2-\ln 5$ , (d) $1-\ln 5$ .

See Solution

Problem 683

Find the limit: $\lim _{x \rightarrow 1} \frac{2 \cdot \ln (x)}{\mathrm{e}^{x}-1}$ . Options: (a) $\frac{2}{e}$ (b) 1 (c) 0 (d) nonexistent.

See Solution

Problem 684

Find $f(1)$ given $f^{\prime}(x)=3 x^{2}$ and $f(2)=3$ . Choices: (a) -7 (b) -4 (c) 7 (d) 10

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Problem 685

17) Find the area of the region between $y=2x$ and $y=x^2$ . 18) If a cube's side length $S=2$ in decreases at 5 in³/min, find the rate of change of $S$ in in/min.

See Solution

Problem 686

Find the derivative $\frac{d}{d x}(\tan (\ln (x)))$ . What is $\frac{d y}{d x}$ for $\sqrt{x}+y^{2}=x y+2$ at (4,0)?

See Solution

Problem 687

8) Find the limit: $\lim _{h \rightarrow 0} \frac{\cos \left((x+h)^{2}\right)-\cos \left(x^{2}\right)}{h}$ .
9) Determine $x$ for the maximum of $y=\frac{4}{3} x^{3}-8 x^{2}+15 x$ on $[0,4]$ .

See Solution

Problem 688

A particle's velocity is $v(t)=e^{t-1}-3 \sin (t-1)$ . What is its motion at $t=1$ ? (a) speeding up, (b) slowing down, (c) neither, (d) at rest.

See Solution

Problem 689

At $t=2$ , what does $R^{\prime}(2)=4$ mean? (a) Depth is 4 inches. (b) Rate is 4 in/hr. (c) Rate of change is 4 in/hr². (d) Depth increased by 4 inches.

See Solution

Problem 690

22) Given $\int_{4}^{6} f(x) dx=5$ and $\int_{10}^{4} f(x) dx=8$ , find $\int_{6}^{10}(4 f(x)+10) dx$ .
23) Find the tangent line equation to $y=e^{2x}$ at $x=1$ .

See Solution

Problem 691

Find the limit: $\lim _{x \rightarrow \infty} \frac{4 \cdot \ln (x)+4}{3 x}$ . Choose (a) 2, (b) -2, (c) 0, or (d) nonexistent.

See Solution

Problem 692

Find the $x$ -coordinate of the inflection point of $f(x)=\int_{1}^{x}(3t-6t^{2})dt$ . Options: (a) $-\frac{1}{4}$ (b) $\frac{1}{4}$ (c) 0 (d) $\frac{1}{2}$ . Evaluate $\int \sin(3x)dx$ . Options: (a) $3\cos(3x)+C$ (b) $\frac{1}{3}\cos(3x)+C$ (c) $-3\cos(3x)+C$ (d) $-\frac{1}{3}\cos(3x)+C$ . Count removable discontinuities of $y=\frac{x+2}{x^{4}+16}$ . Options: (a) one (b) two (c) three (d) four.

See Solution

Problem 693

Find $f'(x)$ if $f(x)=\frac{x^{2}+1}{3 x}$ and evaluate $\int(2 x+3)(x^{2}+3 x)^{4} dx$ .

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Problem 694

26) Find the integral for the volume of a solid with base $R$ and height 5 times the base: (a) $25 \int_{a}^{b}(g(x)-f(x))^{2} dx$ (b) $5 \int_{a}^{b}(g(x)-f(x)) dx$ (c) $5 \int_{a}^{b}(f(x)-g(x)) dx$ (d) $5 \int_{a}^{b}(f(x)-g(x))^{2} dx$
27) Solve $\frac{dy}{dx}=\frac{x}{y}$ with $y=4$ at $x=2$ : (a) $\sqrt{\frac{1}{2} x^{2}+14}$ (b) $\sqrt{2 x^{2}+8}$ (c) $\sqrt{x^{2}+6}$ (d) $\sqrt{x^{2}+12}$

See Solution

Problem 695

Find the integral of $f(x) = \sin 2x$ . Choose from: $(-1/2) \cos 2x + C$ or others.

See Solution

Problem 696

Evaluate the integral: $\int \frac{3 a x}{b^{2}+c^{2} x^{2}} \, dx$

See Solution

Problem 697

Find the ball's instantaneous velocity at $t=10.0$ s given its position $x(t)=0.000015 t^5 - 0.004 t^3 + 0.4 t$ .

See Solution

Problem 698

Find the limit: $\lim _{x \rightarrow \pi} \frac{\cos x+\sin (2 x)+1}{x^{2}-\pi^{2}}$ . What is it?

See Solution

Problem 699

Solve the differential equation: $x y \, dx - (x+2) \, dy = 0$ .

See Solution

Problem 700

Find the derivative of the function $100+(t-3)^{4}(t-5)^{2}$ with respect to $t$ .

See Solution

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Calculus

Problem 601

Find the derivative of (e2−xx+2)\left(\frac{e^{2-x}}{x+2}\right)(x+2e2−x​).

Problem 602

Find the values of aaa and bbb in the function P(t)=abtP(t) = a b^tP(t)=abt that models the number of bacteria in a lab experiment, given the function P(t)=47(1.112)5tP(t) = 47(1.112)^{5t}P(t)=47(1.112)5t. Round the final values of aaa and bbb to 4 decimal places.

Problem 603

Evaluate the indefinite integral ∫x29x−7dx\int x \sqrt{29 x-7} d x∫x29x−7​dx.

Problem 604

Problem 605

Find all values of ttt where the parametric curve x=4t3−2t2+4t,y=2t3+4t2−2x=4t^3-2t^2+4t, y=2t^3+4t^2-2x=4t3−2t2+4t,y=2t3+4t2−2 has a horizontal tangent.

Problem 606

Find the equation of the tangent line to f(x)=5x−1f(x) = \sqrt{5x-1}f(x)=5x−1​ at the point (1,2)(1,2)(1,2) in slope-intercept form.

Problem 607

Find dy/dtdy/dtdy/dt when x2+y2=36x^2 + y^2 = 36x2+y2=36 and dx/dt=3dx/dt = 3dx/dt=3, for y>0y > 0y>0 and (a) x=0x = 0x=0, (b) x=2x = 2x=2, (c) x=3x = 3x=3.

Problem 608

Find the Left and Right Riemann sums, then use them to find the Trapezoidal sum on the interval [−3,7][-3,7][−3,7] given the table of xxx and f(x)f(x)f(x) values.

Problem 609

Determine the horizontal asymptote of the function f(x)=16x3−7x+820x3+8x−3f(x) = \frac{16 x^{3}-7 x+8}{20 x^{3}+8 x-3}f(x)=20x3+8x−316x3−7x+8​. If none exists, state that fact.

Problem 610

Identify the critical points of y=x2exy=x^{2}e^{x}y=x2ex. State the xxx- and yyy-coordinates. Determine if they are local max, min, or neither.

Problem 611

Find the minimum yyy-value of f(x)=x2+2xf(x)=x^{2}+\frac{2}{x}f(x)=x2+x2​ on 12≤x≤2\frac{1}{2} \leq x \leq 221​≤x≤2.

Problem 612

Find the solution, rounded to one decimal place, to the equation ex=x+8e^{x} = \sqrt{x+8}ex=x+8​ using a graphing calculator.

Problem 613

Problem 614

Find the degree 4 Maclaurin polynomial for (1+x)1/3(1+x)^{1/3}(1+x)1/3 and use it to approximate 1.13\sqrt[3]{1.1}31.1​.

Problem 615

Find the inverse function f−1(x)f^{-1}(x)f−1(x) of f(x)=7x+2f(x)=\sqrt{7x+2}f(x)=7x+2​ where x≥−27x \geq -\frac{2}{7}x≥−72​.

Problem 616

Find the value of a piecewise function w(t)w(t)w(t) when t=−7t=-7t=−7.

Problem 617

Explain why the graph of y=−2x3y=-2 \sqrt[3]{x}y=−23x​ is continuous but not differentiable at x=0x=0x=0.

Problem 618

Find the derivative of the function h(x)=e2x2−9x+3/xh(x) = e^{2x^2} - 9x + 3/xh(x)=e2x2−9x+3/x.

Problem 619

Find 4 paths to (1,-1) to show lim⁡(x,y)→(1,−1)f(x,y)\lim_{(x,y)\to(1,-1)} f(x,y)lim(x,y)→(1,−1)​f(x,y) doesn't exist for f(x,y)f(x,y)f(x,y). Provide curve equations and graphs.

Problem 620

Differentiate y=(x+2)(x3+2x+1)y=(x+2)(x^{3}+2x+1)y=(x+2)(x3+2x+1) and choose the correct setup.

Problem 621

Find the range of the quadratic function f(x)=−6(x−7)2+5f(x)=-6(x-7)^{2}+5f(x)=−6(x−7)2+5.

Problem 622

Find the indefinite integral of 5x35x^35x3.

Problem 623

Problem 624

Evaluate the definite integral of 0.25e−0.25x0.25e^{-0.25x}0.25e−0.25x.

Problem 625

Find the value of y=abxy=a b^{x}y=abx given a=1600,b=.25,x=12a=1600, b=.25, x=12a=1600,b=.25,x=12, rounded to 7 decimal places.

Problem 626

Evaluate x−39\frac{\sqrt{x-3}}{9}9x−3​​ when x=12x=12x=12. Options: A) 13\frac{1}{3}31​ B) 1 C) 3 D) 81.

Problem 627

Prove that the function f(x)=x7+8x−2f(x)=x^{7}+8 x-2f(x)=x7+8x−2 has at most one real root using Rolle's Theorem. Show it has exactly one using the Intermediate Value Theorem.

Problem 628

Find the average velocity of an object moving along a vertical line with position L(t)=−3t2+t+8L(t)=-3t^2+t+8L(t)=−3t2+t+8 (m) at time ttt (s) for the given intervals: [3s, 9s], [2s, 7s], [5s, 8s].

Problem 629

Find the derivative of f(x)=5x−10f(x)=\sqrt{5 x-10}f(x)=5x−10​ using the limit definition.

Problem 630

At most, how many turning points does the polynomial P(x)=25x110−19x37+43x20−32x17P(x)=25 x^{110}-19 x^{37}+43 x^{20}-32 x^{17}P(x)=25x110−19x37+43x20−32x17 have?

Problem 631

Solve for yyy in the equation e−8y=9e^{-8 y}=9e−8y=9. Round the answer to the nearest hundredth.

Problem 632

Solve the differential equation dydx=−ex8\frac{d y}{d x} = -\frac{e^{x}}{8}dxdy​=−8ex​.

Problem 633

Find the value of ccc that satisfies the Mean Value Theorem for h(x)=3x−3h(x) = \sqrt{3x-3}h(x)=3x−3​ on the interval 4≤x≤134 \leq x \leq 134≤x≤13.

Problem 634

Problem 635

Determine the behavior of the function f(x)=(2x+1)⋅e−xf(x) = (2x + 1) \cdot e^{-x}f(x)=(2x+1)⋅e−x for very large and very small values of xxx.

Problem 636

Evaluate the limit lim⁡x→π/2(sec⁡(x)−tan⁡(x))\lim_{x \to \pi/2} (\sec(x) - \tan(x))limx→π/2​(sec(x)−tan(x)). Use symbolic notation and fractions.

Problem 637

Evaluate the natural logarithm of 0.012 and give the answer to 4 decimal places. ln⁡(0.012)\ln (0.012)ln(0.012)

Problem 638

Analyze the properties of the logarithmic functions y=log⁡2(x+1)y=\log_2(x+1)y=log2​(x+1) and y=log⁡(x)−3y=\log(x)-3y=log(x)−3, including domain, range, asymptotes, x-intercepts, and transformations.

Problem 639

Use Newton's method to find the solution to e−2x=−3x+9e^{-2x} = -3x + 9e−2x=−3x+9 starting with x0=−5x_0 = -5x0​=−5. Provide the first two iterates x1x_1x1​ and x2x_2x2​, and the final solution accurate to 4 decimal places.

Problem 640

Find the limit, continuity, and type of discontinuity of the piecewise function f(x) = \\begin{cases} 2x+1, & x>-1 \\\\ x^2+1, & x \\leq-1 \\end{cases} at x=ax=ax=a.

Problem 641

A. (f−g)(12)=12−3−122+12(\mathrm{f}-\mathrm{g})(12)=\sqrt{12-3}-12^{2}+12(f−g)(12)=12−3​−122+12

Problem 642

Find the derivative of $\left(\frac{e^{2-x}}{x+2}\right)$ .

Find the values of $a$ and $b$ in the function $P(t) = a b^t$ that models the number of bacteria in a lab experiment, given the function $P(t) = 47(1.112)^{5t}$ . Round the final values of $a$ and $b$ to 4 decimal places.

Evaluate the indefinite integral $\int x \sqrt{29 x-7} d x$ .

Find all values of $t$ where the parametric curve $x=4t^3-2t^2+4t, y=2t^3+4t^2-2$ has a horizontal tangent.

Find the equation of the tangent line to $f(x) = \sqrt{5x-1}$ at the point $(1,2)$ in slope-intercept form.

Find $dy/dt$ when $x^2 + y^2 = 36$ and $dx/dt = 3$ , for $y > 0$ and (a) $x = 0$ , (b) $x = 2$ , (c) $x = 3$ .

Find the Left and Right Riemann sums, then use them to find the Trapezoidal sum on the interval $[-3,7]$ given the table of $x$ and $f(x)$ values.

Determine the horizontal asymptote of the function $f(x) = \frac{16 x^{3}-7 x+8}{20 x^{3}+8 x-3}$ . If none exists, state that fact.

Identify the critical points of $y=x^{2}e^{x}$ . State the $x$ - and $y$ -coordinates. Determine if they are local max, min, or neither.

Find the minimum $y$ -value of $f(x)=x^{2}+\frac{2}{x}$ on $\frac{1}{2} \leq x \leq 2$ .

Find the solution, rounded to one decimal place, to the equation $e^{x} = \sqrt{x+8}$ using a graphing calculator.

Find the degree 4 Maclaurin polynomial for $(1+x)^{1/3}$ and use it to approximate $\sqrt[3]{1.1}$ .

Find the inverse function $f^{-1}(x)$ of $f(x)=\sqrt{7x+2}$ where $x \geq -\frac{2}{7}$ .

Find the value of a piecewise function $w(t)$ when $t=-7$ .

Explain why the graph of $y=-2 \sqrt[3]{x}$ is continuous but not differentiable at $x=0$ .

Find the derivative of the function $h(x) = e^{2x^2} - 9x + 3/x$ .

Find 4 paths to (1,-1) to show $\lim_{(x,y)\to(1,-1)} f(x,y)$ doesn't exist for $f(x,y)$ . Provide curve equations and graphs.

Differentiate $y=(x+2)(x^{3}+2x+1)$ and choose the correct setup.

Find the range of the quadratic function $f(x)=-6(x-7)^{2}+5$ .

Find the indefinite integral of $5x^3$ .

Evaluate the definite integral of $0.25e^{-0.25x}$ .

Find the value of $y=a b^{x}$ given $a=1600, b=.25, x=12$ , rounded to 7 decimal places.

Evaluate $\frac{\sqrt{x-3}}{9}$ when $x=12$ . Options: A) $\frac{1}{3}$ B) 1 C) 3 D) 81.

Prove that the function $f(x)=x^{7}+8 x-2$ has at most one real root using Rolle's Theorem. Show it has exactly one using the Intermediate Value Theorem.

Find the average velocity of an object moving along a vertical line with position $L(t)=-3t^2+t+8$ (m) at time $t$ (s) for the given intervals: [3s, 9s], [2s, 7s], [5s, 8s].

Find the derivative of $f(x)=\sqrt{5 x-10}$ using the limit definition.

At most, how many turning points does the polynomial $P(x)=25 x^{110}-19 x^{37}+43 x^{20}-32 x^{17}$ have?

Solve for $y$ in the equation $e^{-8 y}=9$ . Round the answer to the nearest hundredth.

Solve the differential equation $\frac{d y}{d x} = -\frac{e^{x}}{8}$ .

Find the value of $c$ that satisfies the Mean Value Theorem for $h(x) = \sqrt{3x-3}$ on the interval $4 \leq x \leq 13$ .

Determine the behavior of the function $f(x) = (2x + 1) \cdot e^{-x}$ for very large and very small values of $x$ .

Evaluate the limit $\lim_{x \to \pi/2} (\sec(x) - \tan(x))$ . Use symbolic notation and fractions.

Evaluate the natural logarithm of 0.012 and give the answer to 4 decimal places. $\ln (0.012)$

Analyze the properties of the logarithmic functions $y=\log_2(x+1)$ and $y=\log(x)-3$ , including domain, range, asymptotes, x-intercepts, and transformations.

Use Newton's method to find the solution to $e^{-2x} = -3x + 9$ starting with $x_0 = -5$ . Provide the first two iterates $x_1$ and $x_2$ , and the final solution accurate to 4 decimal places.

Find the limit, continuity, and type of discontinuity of the piecewise function f(x) = \\begin{cases} 2x+1, & x>-1 \\\\ x^2+1, & x \\leq-1 \\end{cases} at $x=a$ .

A. $(\mathrm{f}-\mathrm{g})(12)=\sqrt{12-3}-12^{2}+12$

Evaluate the integral $2 \cdot \int x^{2} \ln x d x$

Find the slope of $y=x^3-x$ at $x=a$ . What is the slope at $x=1$ ? Where does the slope equal $11$ ?

Find the solution to the equation $5 e^{x+2} = 7$ .

Evaluate the integral $\int_{e^{3 x}}^{e^{4 x}} \frac{1}{x \ln x+x} dx$ .

Simplify the expression $\sqrt[4]{\frac{x^{3}}{256}}$ using the quotient rule, assuming all variables are positive real numbers.

Find the integral that represents the length of the curve $f(x) = \cos x, 0 \leq x \leq \pi$ .

Find the values of $m$ for which the integral $\int_{0}^{8} x^{m} d x$ converges.

Approximate the sum of the series $\sum_{n=1}^{\infty}(-1)^{n} \frac{2}{3 n^{4}}$ with error < 0.001.

Determine the limit of the sequence $\left\{\left(\frac{n+n^{2}}{2 n^{2}}\right)^{n}\right\}$ .

Estimate the total number of plastic widgets sold in the first 6 weeks given the sales function $P(t)=7000 t e^{-0.3 t}$ .

Describe the end behavior of $g(x) = -\frac{1}{x^2}$ . As $x \to \pm\infty$ , $g(x)$ goes to 0 or $\pm\infty$ .

Find the secant line equation $y=mx+b$ passing through points $(-3, f(-3))$ and $(2, f(2))$ where $f(x)=x^{3}-5$ .

Find the equation(s) of the vertical asymptote(s) of the rational function $g(t) = \frac{t^{2} - 5}{3t^{2} + 4t - 3}$ .

Differentiate the function $g(t) = \frac{2 - 5t}{7t + 8}$ to find $g'(t)$ .

Find the linear approximation of $y=e^{5x}\ln(x)$ at $x=1$ . $L(x)=e^5\ln(1)+\frac{5e^5\ln(1)+e^5}{1}(x-1)$

The graph $y=500(0.85)^{x}$ represents the amount of drug (in mg) left in a patient's system $x$ hours after administration. Find the $y$ -intercept.

Derive the function $f(x) = \ln 3x^2$ .

Find the domain, range, intercepts, and asymptotes of the exponential function $f(x) = 3 \cdot 6^{x}$ . Describe the end behavior.

Find $\Delta y$ , $dy=f'(x)dx$ , and $dy$ for $y=2x^3$ , $x=4$ , and $\Delta x=0.08$ . a) $\Delta y=\boxed{1.6384}$ b) $dy=f'(x)dx=\boxed{96}dx$ c) $dy=\boxed{7.68}$

Find the value of $x$ in the equation $e^{x}=5$ . The correct statements are: $x=\ln 5$ , $x=\log _{5} e$ , and to the nearest hundredth, $x \approx 1.61$ .

Find the differential $dy$ for $y=e^{x/4}$ and evaluate $dy$ for $x=0, dx=0.05$ .

Identify the hole, vertical asymptote, horizontal asymptote, and domain of the rational function $f(x) = \frac{x^2 + 9x + 20}{x^2 + 3x - 4}$ .

Evaluate the integral $\int \frac{9 x+2}{x^{2}+x-6} d x$ .

Calculate the integral $\int_{2}^{3} 3 x^{2} d t$ .

Evaluate the integral $\int_{-2}^{2} \frac{1+x^{2}}{1+2^{x}} d x$ .

Calculate the integral $\int \tan^{3} x \sec x \, dx$ .

Find the integral of $-\frac{1}{5} \sin u \, du$ .

Differentiate $x^{e^{x}}$ with respect to $x$ .

Find the derivative of $x^{2}$ with respect to $x$ .

Find the ball's instantaneous velocity at $t=10.0$ s given $x(t)=0.000015 t^{5}-0.004 t^{3}+0.4 t$ .

Determine if the series converges or diverges: $\sum_{n=1}^{\infty}\left[\left(\frac{6}{7}\right)^{n}-\frac{3}{2^{n}}\right]$ . If it converges, find the sum.

Find the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n(n+2)}$ .

Find the limit of the sequence $a_{n}=\sqrt{n^{2}+3 n}-n$ . Options: A) 3 B) 2 C) $1 / 2$ D) $3 / 2$ E) 0 F) 1 G) $\infty$ H) does not exist.