Calculus

Problem 27601

Differentiate: (i) $3 x^{2}-x$ (ii) $\frac{1}{x}$ . Find points on $y=x^{3}$ where the gradient is 12.

See Solution

Problem 27602

Find the derivative of the curve $y=3 x^{2}-12 x+8$ and the point where the gradient is 18.

See Solution

Problem 27603

Determine if the expression $P(x, y) \mathrm{d} x + Q(x, y) \mathrm{d} y$ is a total differential by testing $P$ and $Q$ .

See Solution

Problem 27604

Analyze a firm's short-run cost function: $TC=2Q^{3}-2Q^{2}+Q+10$ .
a) Find TFC & TVC. b) Derive AFC, AVC, AC, and MC. c) Determine output levels that minimize MC and AVC, and find their minimum values.

See Solution

Problem 27605

Определете вида на ЧдУ: $x^{2} u_{x x}+2 x y u_{x y}+y^{2} u_{y y}=0$ . Направете смяна на променливите: $\xi=\frac{y}{x}, \eta=x$ .

See Solution

Problem 27606

Given the curve $y=x^{2}+3x$ : (a) Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ . (b) Determine the gradient at $x=-4$ . (c) Find the minimum point's coordinates.

See Solution

Problem 27607

Решете ОДУ от I ред: $y=2x-3y$ , $y(0)=1$ с мрежа $x_{i}=ih$ , $h=10^{-1}$ . Напишете 3 стъпки с методите на Ойлер и Хойн.

See Solution

Problem 27608

Намери решението на задачата на Коши за уравнението $u_{tt}=u_{xx}$ с $\varphi(x)=x^{3}$ , $\psi(x)=e^{x}$ .

See Solution

Problem 27609

Graph the functions and find local max/min if they exist: 13. $f(x)=9 x^{6}+20 x$ , 14. $f(x)=x^{3}-4 x^{2}+x+1$ , 15. $f(x)=-x^{2}+6 x-10$ , 16. $f(x)=-5 x^{2}+7$ .

See Solution

Problem 27610

Given $f(1)=6$ , $f^{\prime}(1)=-2$ , find $h^{\prime}(1)$ for $h(x)=\sqrt{x} f(x)$ . Options: a. 0 b. None c. 1 d. -1

See Solution

Problem 27611

Differentiate the function $f(x) = x \cdot e^{-kx}$ with respect to $x$ .

See Solution

Problem 27612

Find the derivative, turning point, and line of symmetry for the curve $y=x^{2}-4 x+1$ . Is the turning point max or min?

See Solution

Problem 27613

Find the gradient of the curve $y=x^{3}-5 x^{2}+8 x-7$ at the point $(2,-3)$ .

See Solution

Problem 27614

Find the gradient of the curve $y=x^{3}-5 x^{2}+8 x-7$ at $(2,-3)$ and explain its significance. Also, for $y=x^{2}+\frac{16}{x}$ , find $\frac{d y}{d x}$ and the turning point coordinates.

See Solution

Problem 27615

Differentiate $y=8 x^{2}+\frac{2}{x}$ and find its turning point coordinates.

See Solution

Problem 27616

What is the balance after 3 years for an investment of \$2,500 at 12\% annual interest, compounded continuously?

See Solution

Problem 27617

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{\sin \sqrt{x}}{\sqrt{x}}$ .

See Solution

Problem 27618

Find $g^{\prime}(2)$ for the function $f(x)=2 x^{4}-x^{3}+x$ where $g(x)=f^{-1}(x)$ and $g(2)=1$ .

See Solution

Problem 27619

Find the partial derivatives $f_{x}(x, y)$ and $f_{y}(x, y)$ for the function $f(x, y)=2 x+3 y$ .

See Solution

Problem 27620

Find $\lim _{x \rightarrow 0} f(x)$ given $3+x \leq f(x) \leq 3 \cos x$ for all $x$ .

See Solution

Problem 27621

Calculate the limit: $\lim _{n \rightarrow \infty} \frac{(e-1) \sum_{k=0}^{\infty} \frac{1}{e^{k}}}{\left(1+\frac{2}{n}\right)^{n}}$

See Solution

Problem 27622

If $\lim _{x \rightarrow-2} \frac{f(x)}{x}=-1$ , find $\lim _{x \rightarrow-2} f(x)$ .

See Solution

Problem 27623

Find the inflection points of the function $f(x)=\frac{x^{5}}{5}-x^{4}+x+1$ . Choose the correct option.

See Solution

Problem 27624

Determine if the series $\sum_{n=1}^{\infty} \frac{n ! \sin (1 / n)}{n^{n}}$ converges or diverges.

See Solution

Problem 27625

Find $\frac{d x}{d t}$ at the point $(2, \sqrt{12})$ on the circle $x^{2}+y^{2}=16$ if $\frac{d y}{d t}=-3$ .

See Solution

Problem 27626

Find the volume of the solid formed by revolving the area between $y=x^{2}$ and $y=2x$ in the first quadrant around the $y$ -axis.

See Solution

Problem 27627

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ given that $x^{2}+y^{2}=1$ . Options: a. $\frac{-x}{y^{2}}$ b. $\frac{-1}{y^{3}}$ c. None d. $\frac{x^{2}+y^{2}}{y^{3}}$

See Solution

Problem 27628

Which statement about the function $f(x)=\left\{\begin{array}{ll}2 x & 0<x<1 \\ 4 & x=1 \\ -x+3 & 1<x<2\end{array}\right.$ is false? A) $\lim _{x \rightarrow 2^{-}} f(x)$ exists B) $\lim _{x \rightarrow 0^{+}} f(x)$ exists C) $\lim _{x \rightarrow 1} f(x)$ exists D) $f(1)$ is not defined

See Solution

Problem 27629

Gegeben sind die Kostenfunktion $K(x) = 2x^3 - 18x^2 + 96x + 80$ und die Preisabsatzfunktion $p(x) = 120e^{-0,05x}$ . Bestimmen Sie den Gewinnbereich und den maximalen Gewinn sowie den Preis für eine ME.

See Solution

Problem 27630

Which function has an absolute maximum on $[0,4]$ by the Extreme Value Theorem? Options: $y=\sin x$ , $y=\sin^{-1} x$ , $y=\tan x$ , $y=\frac{x^{2}-16}{x^{2}+x-20}$ .

See Solution

Problem 27631

Compare the rates of change for $y=2x-1$ . Choose: A. Equal, B. II > I, C. I > II.

See Solution

Problem 27632

Which function can't use the Second Derivative Test at $x=4$ for relative extrema?
1. $y=\frac{1}{3} x^{3}-4 x^{2}+16 x$
2. $y=x e^{\frac{x}{4}}$
3. $y=-\cos (x-4)$
4. $y=-x^{2}+8 x$

See Solution

Problem 27633

Given $f^{\prime}(x)=-4 x^{3}+12 x^{2}$ , find critical points, inflection points, and analyze $f$ on $[2,4]$ .

See Solution

Problem 27634

Find the value of $\mathbf{c}$ in Rolle's Theorem for $f(x)=e^{x} \cos x$ on $[0, \pi]$ . Options: a. $\frac{\pi}{4}$ b. $\frac{\pi}{2}$ c. $\frac{3 \pi}{4}$ d. None.

See Solution

Problem 27635

A 1 kg block on a spring (100 N/m) is displaced 0.2 m. Find its velocity at the equilibrium point.

See Solution

Problem 27636

Find the volume of the solid formed by revolving the curve $y=\sqrt{x}$ from $0$ to $4$ around the x-axis.

See Solution

Problem 27637

Find the $x$ -values for points of inflection given $f^{\prime}(x)=10 x^{4}+16 x^{3}$ .

See Solution

Problem 27638

Find intervals where $f$ is increasing and concave up given $f^{\prime}(x)=4 x^{3}+12 x^{2}$ .

See Solution

Problem 27639

A pumpkin is dropped from a height of $440 \mathrm{~m}$ . Find the free fall distance, max speed, and total time to hit the ground.

See Solution

Problem 27640

A student runs for 12s starting at 0m. Find position $x(t)$ and acceleration $a(t)$ from velocity $v(t)=0.033 t^{2}+1.4 t$ .

See Solution

Problem 27641

Find the intervals where $f$ is increasing and concave up given $f'(x)=4x^3+12x^2$ .

See Solution

Problem 27642

A cannonball is fired up at $25.0 \mathrm{~m/s}$ from $40.0 \mathrm{~m}$ . Find max height, impact speed, and total flight time.

See Solution

Problem 27643

Find the number of inflection points of $f$ where $f^{\prime \prime}(x)=x \cos x$ in the interval $(-2 \pi, \pi)$ .

See Solution

Problem 27644

Find $p$ so that $\lim_{x \rightarrow 0} f(x)$ exists for $f(x)=\left\{\begin{array}{ll}2-x^{2}, & x \geq 0 \\ x+p, & x<0\end{array}\right.$ .

See Solution

Problem 27645

Find the value of $k$ so that the limit $\lim _{x \rightarrow 4} h(x)$ exists for the piecewise function:
$h(x)=\begin{cases} 20 x^{2}-k, & x<4 \\ k x^{3}-5, & x \geq 4 \end{cases}$

See Solution

Problem 27646

Find the slope of the tangent line to $\cos (\pi x) = x^{7} y^{2}$ at the point $(-1, 1)$ .

See Solution

Problem 27647

Given the temperature function $f(t)$ , answer these:
(a) What is the average rate of change of temperature from $t=0$ to $t=12$ ? (b) Approximate $f^{\prime}(10)$ using the table data. (c) Is $f$ continuous on $[0, 24]$ ? Justify. (d) Find $f^{\prime}(20)$ .

See Solution

Problem 27648

The temperature function $f(t)$ is defined in two parts.
(a) What is the average rate of change of $f$ from $t=0$ to $t=12$ ? (b) Estimate $f^{\prime}(10)$ using the table data. (c) Is $f$ continuous on $[0, 24]$ ? Justify. (d) Calculate $f^{\prime}(20)$ .

See Solution

Problem 27649

Find the average rate of change of $f(t)$ from $t=0$ to $t=12$ , approximate $f'(10)$ , and check continuity for $0 \leq t \leq 24$ .

See Solution

Problem 27650

Find the average rate of change of $f(t)$ from $t=0$ to $t=12$ and approximate $f'(10)$ using $g(t)$ values.

See Solution

Problem 27651

Given the piecewise function $f(x)$ , determine:
a) The value of $k$ for continuity at $x=-2$ . b) The type of discontinuity at $x=0$ and why. c) All horizontal asymptotes of $f$ with justification.

See Solution

Problem 27652

Show that $\frac{d}{d x}\left(\frac{x}{\sqrt{4-x^{2}}}\right)=\frac{4}{\left(4-x^{2}\right)^{\frac{3}{2}}}$ . Then evaluate $\int_{0}^{1} \frac{8}{\left(4-x^{2}\right)^{3}} d x$ . For part (b), find the volume generated by rotating the region under $y=\frac{x}{\sqrt{4-x^{2}}}$ from $x=-1$ to $x=1$ about the $x$ -axis.

See Solution

Problem 27653

Find the limit: $\lim _{x \rightarrow 0} \frac{x^{3}-2 x^{2}+x}{2 x^{3}+x^{2}-2 x}$ .

See Solution

Problem 27654

Find the integral of $\frac{\sin ^{5} x}{\cos ^{7} x} d x$ .

See Solution

Problem 27655

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ given that $x^{2}+y^{2}=1$ .

See Solution

Problem 27656

Find $c$ in Rolle's Theorem for $f(x)=e^{x} \cos x$ on $x \in [0, \pi]$ .

See Solution

Problem 27657

Find the value of $c$ for the mean value theorem of $f(x)=\frac{1}{2}+1$ on $[1,6]$ .

See Solution

Problem 27658

Find the volume of the solid formed by revolving the area between $y=\sqrt{x}$ , $0 \leq x \leq 4$ , and the x-axis around the x-axis.

See Solution

Problem 27659

Find the value of $c$ for the function $f(x)=x^{3}-3x^{2}$ in the interval $[0,3]$ that meets the Mean Value Theorem.

See Solution

Problem 27660

Find the critical points of the function $f(x)=\frac{x^{2}-2}{x^{2}-1}$ .

See Solution

Problem 27661

Find the inflection point of the function $f(x)=\frac{x^{5}}{5}-x^{4}+x+1$ .

See Solution

Problem 27662

Find all $x$ where the curve $y=4 x^{4}-2 x^{2}-4$ has a horizontal tangent.

See Solution

Problem 27663

Find the value of $c$ in Rolle's Theorem for $f(x)=e^{x} \cos x$ on $[0, \pi]$ . Options: a. $\frac{\pi}{4}$ b. $\frac{\pi}{2}$ c. $\frac{3 \pi}{4}$ d. None.

See Solution

Problem 27664

Find the value of $c$ in Rolle's Theorem for $f(x)=e^{x} \cos x$ on $[0, 2\pi]$ . Options: a) $\frac{\pi}{4}$ b) $\frac{3\pi}{4}$ c) $\frac{5\pi}{4}$ d) Two are true.

See Solution

Problem 27665

Find the limit: $\lim _{x \rightarrow 0} \frac{x^{3}-4 x}{2 x^{2}+3 x}$ .

See Solution

Problem 27666

Find the volume of the solid of revolution for the region in the first quadrant between $y=x^{3}+3$ , $y=4$ , and the $y$ -axis for different axes: (a) $y$ -axis (disk), (b) $x$ -axis (washer), (c) $y=1$ (shell), (d) $x=-1$ (shell).

See Solution

Problem 27667

In the WKB approximation, analyze two potentials $V_A(x)$ and $V_B(x)$ for a quantum particle. Answer the following:
(a) State boundary conditions for $V_A(x)$ at $x=0$ and $x=2L$ . (b) Determine the energy spectrum for $E > 2V_0L$ in $V_A(x)$ , leaving the equation for $E_n$ . (c) Explain how to compute tunneling probability for $V_B(x)$ using WKB. (d) Calculate tunneling probability for a particle with energy $E < V_0$ approaching $V_B(x)$ from the left.

See Solution

Problem 27668

Calculate the integral: $\int x^{3} \sqrt{x^{2}+1} \, dx$

See Solution

Problem 27669

Calculate the integral: $\int \sqrt{\frac{x-1}{x^{5}}} d x$

See Solution

Problem 27670

Evaluate the integral: $\int \frac{d x}{\sqrt{x}(1+\sqrt{x})^{2}}$

See Solution

Problem 27671

Find where the graph of $y=x^{3}+x^{2}-27 x$ is concave down. Options: a. $x \leq \frac{-1}{3}$ , b. $-6 \leq x \leq 2$ , c. $x \backslash \lg a$ , d. $x \leq 0$ , e. None.

See Solution

Problem 27672

A function $f$ is defined piecewise. Find $p$ for limit at $x=3$ and check limit existence at $x=5$ .

See Solution

Problem 27673

Find the value of $\int_{c}^{b}(f(x)-g(x)) \, dx$ given the area shown, options: a) 6, b) -2, c) 2, d) -5.

See Solution

Problem 27674

Determine the interval where the function $f(x)=x \ln x-x, x>0$ is increasing: a. decreasing for all $x>0$ b. $(0, \infty)$ c. $[1, \infty)$ d. $(0,1]$

See Solution

Problem 27675

Calculate the integral: $\int_{0}^{\pi / 6}(\sec x+\tan x)^{2} d x$

See Solution

Problem 27676

Given $f(x)$ is decreasing on $[3,8]$ and increasing on $[-3,2]$ , which statement is false? a) $f^{\prime}(4)<f^{\prime}(2)$ b) $f^{\prime}(6)>f^{\prime}(-1)$ c) $\frac{f(5)-f(7)}{f^{\prime}(0)}>0$ d) $\frac{f(7)-f(5)}{f^{\prime}(-1)}<0$

See Solution

Problem 27677

Consider the function $f(x)=\frac{x-1}{x^{2}(x-2)}$ . Which statement about its extrema is true?

See Solution

Problem 27678

Calculate the value of $\int_{0}^{\frac{\pi}{3}}(\sec x+\tan x)^{2} d x$ . Choose the correct answer.

See Solution

Problem 27679

Evaluate the integral $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}}\left(4 \sec ^{2} t+\frac{\pi}{t^{2}}\right) dt$ .

See Solution

Problem 27680

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=0$ , $x=-\frac{\pi}{4}$ , $x=\frac{\pi}{4}$ around $y=-2$ . Select the correct integral:
1) $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \pi\left((\sec x+2)^{2}-4\right) d x$ 2) $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \pi\left(\sec ^{2} x-2\right) d x$ 3) $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \pi(\sec x+4)^{2} d x$ 4) $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \pi(\sec x-2)^{2} d x$

See Solution

Problem 27681

Consider the function $f(x)=\frac{x-1}{x^{2}(x-2)}$ . Which statement about its extrema is true?

See Solution

Problem 27682

Given $f'(3)=0$ , $f'(5)=0$ , $f''(3)=-4$ , and $f''(5)=-5$ , determine the nature of $f$ at $x=3$ and $x=5$ .

See Solution

Problem 27683

Analyze the function $f(x)=\frac{x^{4}+1}{x^{2}}$ and select the correct statement about its critical points and asymptotes.

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Problem 27684

Find the area between the curves $y=5 \sqrt{x}$ and $y=5 x^{2}$ . Choices: $\frac{5}{3}, \frac{1}{2}, \frac{5}{2}, \frac{1}{3},$ none.

See Solution

Problem 27685

Estimate the profit from the $101^{st}$ unit sold using marginal analysis for $P(x)=-0.04 x^{2}+27 x-1000$ .

See Solution

Problem 27686

Find the area between the curves $y=2 \sqrt{x}$ and $y=2 x^{2}$ . Choose: $\frac{2}{15}$ , $\frac{2}{5}$ , none.

See Solution

Problem 27687

Find the value of $\int_{0}^{\frac{\pi}{3}}(\cos x+\sec x)^{2} d x$ .

See Solution

Problem 27688

Find the derivative $f^{\prime}(x)$ if $f(x)=x^{2} g(x)$ . Choose from the options provided.

See Solution

Problem 27689

Find the volume of a solid with a base under $y=4-x^{2}$ , where cross-sections perpendicular to the $y$ -axis are squares.

See Solution

Problem 27690

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=0$ , $x=-\frac{\pi}{4}$ , and $x=\frac{\pi}{4}$ around the $x$ -axis.

See Solution

Problem 27691

Find the area between the curves $y=\sqrt{x}$ and $y=x$ . Choose from: $\frac{1}{6}, \frac{1}{3}, \frac{1}{4}, \frac{1}{2}$ .

See Solution

Problem 27692

Find the area between the curves $y=\sqrt{x}$ and $y=\sqrt{x^{3}}$ . What is the area?

See Solution

Problem 27693

If $f'(2)=0$ , which statement about $f$ is true?
1. The tangent line at $(2, f(2))$ is $x=2$ .
2. At $x=2$ , $f$ changes direction.
3. $(2, f(2))$ is a local max or min.
4. The tangent line at $(2, f(2))$ is horizontal.

See Solution

Problem 27694

Find the volume of a solid with a square cross-section above the $x$ -axis and under the parabola $y=9-x^{2}$ .

See Solution

Problem 27695

Analyze the function $f(x)=\frac{x^{4}+1}{x^{2}}$ and determine which statement is true about it.

See Solution

Problem 27696

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=0$ , $x=-\frac{\pi}{4}$ , $x=\frac{\pi}{4}$ around $y=-1$ .

See Solution

Problem 27697

Analyze the function $f(x)=\frac{4}{3} x-\tan x$ for concavity in specified intervals. Which statement is true?

See Solution

Problem 27698

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=\sqrt{2}$ , $x=-\frac{\pi}{4}$ , $x=\frac{\pi}{4}$ around $y=-\sqrt{2}$ .

See Solution

Problem 27699

Evaluate the integral $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}}\left(4 \sec ^{2} t+\frac{\pi}{t^{2}}\right) dt$ .

See Solution

Problem 27700

Analyze the function $f(x)=\frac{4}{3} x-\tan x$ for $-\frac{\pi}{2}<x<\frac{\pi}{2}$ . Which interval is $f$ increasing?

See Solution

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Calculus

Problem 27601

Differentiate: (i) 3x2−x3 x^{2}-x3x2−x (ii) 1x\frac{1}{x}x1​. Find points on y=x3y=x^{3}y=x3 where the gradient is 12.

Problem 27602

Find the derivative of the curve y=3x2−12x+8y=3 x^{2}-12 x+8y=3x2−12x+8 and the point where the gradient is 18.

Problem 27603

Determine if the expression P(x,y)dx+Q(x,y)dyP(x, y) \mathrm{d} x + Q(x, y) \mathrm{d} yP(x,y)dx+Q(x,y)dy is a total differential by testing PPP and QQQ.

Problem 27604

Analyze a firm's short-run cost function: TC=2Q3−2Q2+Q+10TC=2Q^{3}-2Q^{2}+Q+10TC=2Q3−2Q2+Q+10. a) Find TFC & TVC. b) Derive AFC, AVC, AC, and MC. c) Determine output levels that minimize MC and AVC, and find their minimum values.

Problem 27605

Определете вида на ЧдУ: x2uxx+2xyuxy+y2uyy=0x^{2} u_{x x}+2 x y u_{x y}+y^{2} u_{y y}=0x2uxx​+2xyuxy​+y2uyy​=0. Направете смяна на променливите: ξ=yx,η=x\xi=\frac{y}{x}, \eta=xξ=xy​,η=x.

Problem 27606

Given the curve y=x2+3xy=x^{2}+3xy=x2+3x: (a) Find dy dx\frac{\mathrm{d} y}{\mathrm{~d} x} dxdy​. (b) Determine the gradient at x=−4x=-4x=−4. (c) Find the minimum point's coordinates.

Problem 27607

Решете ОДУ от I ред: y=2x−3yy=2x-3yy=2x−3y, y(0)=1y(0)=1y(0)=1 с мрежа xi=ihx_{i}=ihxi​=ih, h=10−1h=10^{-1}h=10−1. Напишете 3 стъпки с методите на Ойлер и Хойн.

Problem 27608

Намери решението на задачата на Коши за уравнението utt=uxxu_{tt}=u_{xx}utt​=uxx​ с φ(x)=x3\varphi(x)=x^{3}φ(x)=x3, ψ(x)=ex\psi(x)=e^{x}ψ(x)=ex.

Problem 27609

Graph the functions and find local max/min if they exist: 13. f(x)=9x6+20xf(x)=9 x^{6}+20 xf(x)=9x6+20x, 14. f(x)=x3−4x2+x+1f(x)=x^{3}-4 x^{2}+x+1f(x)=x3−4x2+x+1, 15. f(x)=−x2+6x−10f(x)=-x^{2}+6 x-10f(x)=−x2+6x−10, 16. f(x)=−5x2+7f(x)=-5 x^{2}+7f(x)=−5x2+7.

Problem 27610

Given f(1)=6f(1)=6f(1)=6, f′(1)=−2f^{\prime}(1)=-2f′(1)=−2, find h′(1)h^{\prime}(1)h′(1) for h(x)=xf(x)h(x)=\sqrt{x} f(x)h(x)=x​f(x). Options: a. 0 b. None c. 1 d. -1

Problem 27611

Differentiate the function f(x)=x⋅e−kxf(x) = x \cdot e^{-kx}f(x)=x⋅e−kx with respect to xxx.

Problem 27612

Find the derivative, turning point, and line of symmetry for the curve y=x2−4x+1y=x^{2}-4 x+1y=x2−4x+1. Is the turning point max or min?

Problem 27613

Find the gradient of the curve y=x3−5x2+8x−7y=x^{3}-5 x^{2}+8 x-7y=x3−5x2+8x−7 at the point (2,−3)(2,-3)(2,−3).

Problem 27614

Find the gradient of the curve y=x3−5x2+8x−7y=x^{3}-5 x^{2}+8 x-7y=x3−5x2+8x−7 at (2,−3)(2,-3)(2,−3) and explain its significance. Also, for y=x2+16xy=x^{2}+\frac{16}{x}y=x2+x16​, find dydx\frac{d y}{d x}dxdy​ and the turning point coordinates.

Problem 27615

Differentiate y=8x2+2xy=8 x^{2}+\frac{2}{x}y=8x2+x2​ and find its turning point coordinates.

Problem 27616

What is the balance after 3 years for an investment of \$2,500 at 12\% annual interest, compounded continuously?

Problem 27617

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=sin⁡xxf(x)=\frac{\sin \sqrt{x}}{\sqrt{x}}f(x)=x​sinx​​.

Problem 27618

Find g′(2)g^{\prime}(2)g′(2) for the function f(x)=2x4−x3+xf(x)=2 x^{4}-x^{3}+xf(x)=2x4−x3+x where g(x)=f−1(x)g(x)=f^{-1}(x)g(x)=f−1(x) and g(2)=1g(2)=1g(2)=1.

Problem 27619

Find the partial derivatives fx(x,y)f_{x}(x, y)fx​(x,y) and fy(x,y)f_{y}(x, y)fy​(x,y) for the function f(x,y)=2x+3yf(x, y)=2 x+3 yf(x,y)=2x+3y.

Problem 27620

Find lim⁡x→0f(x)\lim _{x \rightarrow 0} f(x)limx→0​f(x) given 3+x≤f(x)≤3cos⁡x3+x \leq f(x) \leq 3 \cos x3+x≤f(x)≤3cosx for all xxx.

Problem 27621

Calculate the limit: lim⁡n→∞(e−1)∑k=0∞1ek(1+2n)n\lim _{n \rightarrow \infty} \frac{(e-1) \sum_{k=0}^{\infty} \frac{1}{e^{k}}}{\left(1+\frac{2}{n}\right)^{n}}limn→∞​(1+n2​)n(e−1)∑k=0∞​ek1​​

Problem 27622

If lim⁡x→−2f(x)x=−1\lim _{x \rightarrow-2} \frac{f(x)}{x}=-1limx→−2​xf(x)​=−1, find lim⁡x→−2f(x)\lim _{x \rightarrow-2} f(x)limx→−2​f(x).

Problem 27623

Find the inflection points of the function f(x)=x55−x4+x+1f(x)=\frac{x^{5}}{5}-x^{4}+x+1f(x)=5x5​−x4+x+1. Choose the correct option.

Problem 27624

Determine if the series ∑n=1∞n!sin⁡(1/n)nn\sum_{n=1}^{\infty} \frac{n ! \sin (1 / n)}{n^{n}}n=1∑∞​nnn!sin(1/n)​ converges or diverges.

Problem 27625

Find dxdt\frac{d x}{d t}dtdx​ at the point (2,12)(2, \sqrt{12})(2,12​) on the circle x2+y2=16x^{2}+y^{2}=16x2+y2=16 if dydt=−3\frac{d y}{d t}=-3dtdy​=−3.

Problem 27626

Find the volume of the solid formed by revolving the area between y=x2y=x^{2}y=x2 and y=2xy=2xy=2x in the first quadrant around the yyy-axis.

Problem 27627

Find the second derivative d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ given that x2+y2=1x^{2}+y^{2}=1x2+y2=1. Options: a. −xy2\frac{-x}{y^{2}}y2−x​ b. −1y3\frac{-1}{y^{3}}y3−1​ c. None d. x2+y2y3\frac{x^{2}+y^{2}}{y^{3}}y3x2+y2​

Problem 27628

Problem 27629

Gegeben sind die Kostenfunktion K(x)=2x3−18x2+96x+80K(x) = 2x^3 - 18x^2 + 96x + 80K(x)=2x3−18x2+96x+80 und die Preisabsatzfunktion p(x)=120e−0,05xp(x) = 120e^{-0,05x}p(x)=120e−0,05x. Bestimmen Sie den Gewinnbereich und den maximalen Gewinn sowie den Preis für eine ME.

Problem 27630

Which function has an absolute maximum on [0,4][0,4][0,4] by the Extreme Value Theorem? Options: y=sin⁡xy=\sin xy=sinx, y=sin⁡−1xy=\sin^{-1} xy=sin−1x, y=tan⁡xy=\tan xy=tanx, y=x2−16x2+x−20y=\frac{x^{2}-16}{x^{2}+x-20}y=x2+x−20x2−16​.

Problem 27631

Compare the rates of change for y=2x−1y=2x-1y=2x−1. Choose: A. Equal, B. II > I, C. I > II.

Problem 27632

Which function can't use the Second Derivative Test at x=4x=4x=4 for relative extrema? 1. y=13x3−4x2+16xy=\frac{1}{3} x^{3}-4 x^{2}+16 xy=31​x3−4x2+16x 2. y=xex4y=x e^{\frac{x}{4}}y=xe4x​ 3. y=−cos⁡(x−4)y=-\cos (x-4)y=−cos(x−4) 4. y=−x2+8xy=-x^{2}+8 xy=−x2+8x

Problem 27633

Given f′(x)=−4x3+12x2f^{\prime}(x)=-4 x^{3}+12 x^{2}f′(x)=−4x3+12x2, find critical points, inflection points, and analyze fff on [2,4][2,4][2,4].

Problem 27634

Find the value of c\mathbf{c}c in Rolle's Theorem for f(x)=excos⁡xf(x)=e^{x} \cos xf(x)=excosx on [0,π][0, \pi][0,π]. Options: a. π4\frac{\pi}{4}4π​ b. π2\frac{\pi}{2}2π​ c. 3π4\frac{3 \pi}{4}43π​ d. None.

Problem 27635

A 1 kg block on a spring (100 N/m) is displaced 0.2 m. Find its velocity at the equilibrium point.

Problem 27636

Find the volume of the solid formed by revolving the curve y=xy=\sqrt{x}y=x​ from 000 to 444 around the x-axis.

Problem 27637

Find the xxx-values for points of inflection given f′(x)=10x4+16x3f^{\prime}(x)=10 x^{4}+16 x^{3}f′(x)=10x4+16x3.

Problem 27638

Find intervals where fff is increasing and concave up given f′(x)=4x3+12x2f^{\prime}(x)=4 x^{3}+12 x^{2}f′(x)=4x3+12x2.

Problem 27639

A pumpkin is dropped from a height of 440 m440 \mathrm{~m}440 m. Find the free fall distance, max speed, and total time to hit the ground.

Problem 27640

A student runs for 12s starting at 0m. Find position x(t)x(t)x(t) and acceleration a(t)a(t)a(t) from velocity v(t)=0.033t2+1.4tv(t)=0.033 t^{2}+1.4 tv(t)=0.033t2+1.4t.

Differentiate: (i) $3 x^{2}-x$ (ii) $\frac{1}{x}$ . Find points on $y=x^{3}$ where the gradient is 12.

Find the derivative of the curve $y=3 x^{2}-12 x+8$ and the point where the gradient is 18.

Determine if the expression $P(x, y) \mathrm{d} x + Q(x, y) \mathrm{d} y$ is a total differential by testing $P$ and $Q$ .

Analyze a firm's short-run cost function: $TC=2Q^{3}-2Q^{2}+Q+10$ .
a) Find TFC & TVC. b) Derive AFC, AVC, AC, and MC. c) Determine output levels that minimize MC and AVC, and find their minimum values.

Определете вида на ЧдУ: $x^{2} u_{x x}+2 x y u_{x y}+y^{2} u_{y y}=0$ . Направете смяна на променливите: $\xi=\frac{y}{x}, \eta=x$ .

Given the curve $y=x^{2}+3x$ : (a) Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ . (b) Determine the gradient at $x=-4$ . (c) Find the minimum point's coordinates.

Решете ОДУ от I ред: $y=2x-3y$ , $y(0)=1$ с мрежа $x_{i}=ih$ , $h=10^{-1}$ . Напишете 3 стъпки с методите на Ойлер и Хойн.

Намери решението на задачата на Коши за уравнението $u_{tt}=u_{xx}$ с $\varphi(x)=x^{3}$ , $\psi(x)=e^{x}$ .

Graph the functions and find local max/min if they exist: 13. $f(x)=9 x^{6}+20 x$ , 14. $f(x)=x^{3}-4 x^{2}+x+1$ , 15. $f(x)=-x^{2}+6 x-10$ , 16. $f(x)=-5 x^{2}+7$ .

Given $f(1)=6$ , $f^{\prime}(1)=-2$ , find $h^{\prime}(1)$ for $h(x)=\sqrt{x} f(x)$ . Options: a. 0 b. None c. 1 d. -1

Differentiate the function $f(x) = x \cdot e^{-kx}$ with respect to $x$ .

Find the derivative, turning point, and line of symmetry for the curve $y=x^{2}-4 x+1$ . Is the turning point max or min?

Find the gradient of the curve $y=x^{3}-5 x^{2}+8 x-7$ at the point $(2,-3)$ .

Find the gradient of the curve $y=x^{3}-5 x^{2}+8 x-7$ at $(2,-3)$ and explain its significance. Also, for $y=x^{2}+\frac{16}{x}$ , find $\frac{d y}{d x}$ and the turning point coordinates.

Differentiate $y=8 x^{2}+\frac{2}{x}$ and find its turning point coordinates.

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{\sin \sqrt{x}}{\sqrt{x}}$ .

Find $g^{\prime}(2)$ for the function $f(x)=2 x^{4}-x^{3}+x$ where $g(x)=f^{-1}(x)$ and $g(2)=1$ .

Find the partial derivatives $f_{x}(x, y)$ and $f_{y}(x, y)$ for the function $f(x, y)=2 x+3 y$ .

Find $\lim _{x \rightarrow 0} f(x)$ given $3+x \leq f(x) \leq 3 \cos x$ for all $x$ .

Calculate the limit: $\lim _{n \rightarrow \infty} \frac{(e-1) \sum_{k=0}^{\infty} \frac{1}{e^{k}}}{\left(1+\frac{2}{n}\right)^{n}}$

If $\lim _{x \rightarrow-2} \frac{f(x)}{x}=-1$ , find $\lim _{x \rightarrow-2} f(x)$ .

Find the inflection points of the function $f(x)=\frac{x^{5}}{5}-x^{4}+x+1$ . Choose the correct option.

Determine if the series $\sum_{n=1}^{\infty} \frac{n ! \sin (1 / n)}{n^{n}}$ converges or diverges.

Find $\frac{d x}{d t}$ at the point $(2, \sqrt{12})$ on the circle $x^{2}+y^{2}=16$ if $\frac{d y}{d t}=-3$ .

Find the volume of the solid formed by revolving the area between $y=x^{2}$ and $y=2x$ in the first quadrant around the $y$ -axis.

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ given that $x^{2}+y^{2}=1$ . Options: a. $\frac{-x}{y^{2}}$ b. $\frac{-1}{y^{3}}$ c. None d. $\frac{x^{2}+y^{2}}{y^{3}}$

Gegeben sind die Kostenfunktion $K(x) = 2x^3 - 18x^2 + 96x + 80$ und die Preisabsatzfunktion $p(x) = 120e^{-0,05x}$ . Bestimmen Sie den Gewinnbereich und den maximalen Gewinn sowie den Preis für eine ME.

Which function has an absolute maximum on $[0,4]$ by the Extreme Value Theorem? Options: $y=\sin x$ , $y=\sin^{-1} x$ , $y=\tan x$ , $y=\frac{x^{2}-16}{x^{2}+x-20}$ .

Compare the rates of change for $y=2x-1$ . Choose: A. Equal, B. II > I, C. I > II.

Which function can't use the Second Derivative Test at $x=4$ for relative extrema?
1. $y=\frac{1}{3} x^{3}-4 x^{2}+16 x$
2. $y=x e^{\frac{x}{4}}$
3. $y=-\cos (x-4)$
4. $y=-x^{2}+8 x$

Given $f^{\prime}(x)=-4 x^{3}+12 x^{2}$ , find critical points, inflection points, and analyze $f$ on $[2,4]$ .

Find the value of $\mathbf{c}$ in Rolle's Theorem for $f(x)=e^{x} \cos x$ on $[0, \pi]$ . Options: a. $\frac{\pi}{4}$ b. $\frac{\pi}{2}$ c. $\frac{3 \pi}{4}$ d. None.

Find the volume of the solid formed by revolving the curve $y=\sqrt{x}$ from $0$ to $4$ around the x-axis.

Find the $x$ -values for points of inflection given $f^{\prime}(x)=10 x^{4}+16 x^{3}$ .

Find intervals where $f$ is increasing and concave up given $f^{\prime}(x)=4 x^{3}+12 x^{2}$ .

A pumpkin is dropped from a height of $440 \mathrm{~m}$ . Find the free fall distance, max speed, and total time to hit the ground.

A student runs for 12s starting at 0m. Find position $x(t)$ and acceleration $a(t)$ from velocity $v(t)=0.033 t^{2}+1.4 t$ .

Find the intervals where $f$ is increasing and concave up given $f'(x)=4x^3+12x^2$ .

A cannonball is fired up at $25.0 \mathrm{~m/s}$ from $40.0 \mathrm{~m}$ . Find max height, impact speed, and total flight time.

Find the number of inflection points of $f$ where $f^{\prime \prime}(x)=x \cos x$ in the interval $(-2 \pi, \pi)$ .

Find $p$ so that $\lim_{x \rightarrow 0} f(x)$ exists for $f(x)=\left\{\begin{array}{ll}2-x^{2}, & x \geq 0 \\ x+p, & x<0\end{array}\right.$ .

Find the value of $k$ so that the limit $\lim _{x \rightarrow 4} h(x)$ exists for the piecewise function:
$h(x)=\begin{cases} 20 x^{2}-k, & x<4 \\ k x^{3}-5, & x \geq 4 \end{cases}$

Find the slope of the tangent line to $\cos (\pi x) = x^{7} y^{2}$ at the point $(-1, 1)$ .

Find the average rate of change of $f(t)$ from $t=0$ to $t=12$ , approximate $f'(10)$ , and check continuity for $0 \leq t \leq 24$ .

Find the average rate of change of $f(t)$ from $t=0$ to $t=12$ and approximate $f'(10)$ using $g(t)$ values.

Given the piecewise function $f(x)$ , determine:
a) The value of $k$ for continuity at $x=-2$ . b) The type of discontinuity at $x=0$ and why. c) All horizontal asymptotes of $f$ with justification.

Find the limit: $\lim _{x \rightarrow 0} \frac{x^{3}-2 x^{2}+x}{2 x^{3}+x^{2}-2 x}$ .

Find the integral of $\frac{\sin ^{5} x}{\cos ^{7} x} d x$ .

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ given that $x^{2}+y^{2}=1$ .

Find $c$ in Rolle's Theorem for $f(x)=e^{x} \cos x$ on $x \in [0, \pi]$ .

Find the value of $c$ for the mean value theorem of $f(x)=\frac{1}{2}+1$ on $[1,6]$ .

Find the volume of the solid formed by revolving the area between $y=\sqrt{x}$ , $0 \leq x \leq 4$ , and the x-axis around the x-axis.

Find the value of $c$ for the function $f(x)=x^{3}-3x^{2}$ in the interval $[0,3]$ that meets the Mean Value Theorem.

Find the critical points of the function $f(x)=\frac{x^{2}-2}{x^{2}-1}$ .

Find the inflection point of the function $f(x)=\frac{x^{5}}{5}-x^{4}+x+1$ .

Find all $x$ where the curve $y=4 x^{4}-2 x^{2}-4$ has a horizontal tangent.

Find the value of $c$ in Rolle's Theorem for $f(x)=e^{x} \cos x$ on $[0, \pi]$ . Options: a. $\frac{\pi}{4}$ b. $\frac{\pi}{2}$ c. $\frac{3 \pi}{4}$ d. None.

Find the value of $c$ in Rolle's Theorem for $f(x)=e^{x} \cos x$ on $[0, 2\pi]$ . Options: a) $\frac{\pi}{4}$ b) $\frac{3\pi}{4}$ c) $\frac{5\pi}{4}$ d) Two are true.

Find the limit: $\lim _{x \rightarrow 0} \frac{x^{3}-4 x}{2 x^{2}+3 x}$ .

Find the volume of the solid of revolution for the region in the first quadrant between $y=x^{3}+3$ , $y=4$ , and the $y$ -axis for different axes: (a) $y$ -axis (disk), (b) $x$ -axis (washer), (c) $y=1$ (shell), (d) $x=-1$ (shell).

Calculate the integral: $\int x^{3} \sqrt{x^{2}+1} \, dx$

Calculate the integral: $\int \sqrt{\frac{x-1}{x^{5}}} d x$

Evaluate the integral: $\int \frac{d x}{\sqrt{x}(1+\sqrt{x})^{2}}$

Find where the graph of $y=x^{3}+x^{2}-27 x$ is concave down. Options: a. $x \leq \frac{-1}{3}$ , b. $-6 \leq x \leq 2$ , c. $x \backslash \lg a$ , d. $x \leq 0$ , e. None.

A function $f$ is defined piecewise. Find $p$ for limit at $x=3$ and check limit existence at $x=5$ .

Find the value of $\int_{c}^{b}(f(x)-g(x)) \, dx$ given the area shown, options: a) 6, b) -2, c) 2, d) -5.

Determine the interval where the function $f(x)=x \ln x-x, x>0$ is increasing: a. decreasing for all $x>0$ b. $(0, \infty)$ c. $[1, \infty)$ d. $(0,1]$