Calculus

Problem 701

Find the derivative of $y$ with respect to $x$ for the expression $x+\ln |x-3|+\frac{2}{x-3}$ .

See Solution

Problem 702

Find the formula for the marginal average profit of the profit function $P(x)=100 \ln (2 x+1)-5 x-10$ . Options are provided.

See Solution

Problem 703

Find the derivatives of these functions: 1. $f(x)=6 x^{3}-9 x+4$ , 2. $y=2 t^{4}- t^{2}+13 t$ , 3. $g(z)=4 z^{7}-3 z^{-7}+9 z$ .

See Solution

Problem 704

Un point $\mathrm{P}$ de masse $\mathrm{m}$ se déplace en coordonnées polaires. Trouvez les équations du mouvement et les expressions pour $\mathrm{r}(t)$ et $\omega$ quand $\omega$ est constant.

See Solution

Problem 705

Find the integral of $\frac{2}{3 x^{2}} - 2 \sec^{2}(5x)$ with respect to $x$ .

See Solution

Problem 706

Evaluate the integral: $\int_{2}\left(\frac{2}{3 x^{2}}-2 \sec ^{2} 5 x\right) d x$

See Solution

Problem 707

Find the integral: $\int 6 \operatorname{cosec} 2 x \cot 2 x \, dx$

See Solution

Problem 708

Find the integral of $10^{3 x}$ with respect to $x$ .

See Solution

Problem 709

Calculate the integral: $\int_{0}^{\frac{\pi}{4}} 4 \sin 2 x \, dx$

See Solution

Problem 710

Find the integral: $\int \frac{5-x-3 x^{2}}{4 \sqrt{x}} \, dx$

See Solution

Problem 711

Find the limit of $f(x)=\frac{\sin 4x}{4 \sin 3x}$ as $x$ approaches 0.

See Solution

Problem 712

Find the limit of $f(x)=\frac{\sin 4x}{4\sin 3x}+\frac{\sin (x/3)}{9x}$ as $x \to 0$ .

See Solution

Problem 713

Find the limit: $\lim_{x \rightarrow 0} f(x)$ where $f(x)=2 x \cos \left(\frac{1}{x^{2}}\right)$ .

See Solution

Problem 714

Find $\frac{d}{d x} f^{-1}(0)$ for $f(x)=x+\sin x$ . At $x=0$ , $f$ has slope 2, so $f^{-1}$ slope is $\frac{1}{2}$ .

See Solution

Problem 715

Find the derivative of $f(x) = x^2$ using the limit definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ .

See Solution

Problem 716

Find the limit: If $f(z)=\frac{z+\bar{z}}{z}$ , then $\lim _{x \rightarrow 0} f(x+0 i)=$ .

See Solution

Problem 717

Find $\lim _{x \rightarrow 0} f(x+0 i)$ and $\lim _{y \rightarrow 0} f(0+i y)$ for $f(z)=\frac{z+\bar{z}}{z}$ .

See Solution

Problem 718

Evaluate the limit as $x$ approaches 0: $\lim _{x \rightarrow 0} \frac{x^{2}-25}{x^{2}-4 x-5}$ .

See Solution

Problem 719

Find $\frac{d y}{d x}$ for the equation $2 \cdot x \cdot \ln (y) + y^{2} = 5$ .

See Solution

Problem 720

Find $\frac{d y}{d x}$ if $e^{x \cdot y}+5 \cdot y^{4}=2 \cdot y$ .

See Solution

Problem 721

Find the marginal profit $P^{\prime}(x)$ for $P(x)=3 \cdot x-8 \cdot \sqrt{x}$ and calculate $\frac{P^{\prime}(x)}{x}$ .

See Solution

Problem 722

Find the marginal cost for businesses A and B with $C_A(x)=350+40x+0.12x^2$ and $C_B(x)=280+30x+0.09x^2$ . For $x=500$ , which is cheaper?

See Solution

Problem 723

Find the greatest integer less than or equal to $\int_{1}^{2} \log _{2}(x^{3}+1) dx + \int_{1}^{\log _{2} 9}(2^{x}-1)^{\frac{1}{3}} dx$ .

See Solution

Problem 724

Find the cost increase when producing 60 items instead of 50, given $C=0.001 x^{3}-0.3 x^{2}+40 x+1000$ .

See Solution

Problem 725

Find the cost increase when producing from 50 to 60 items with $C=0.001 x^{3}-0.3 x^{2}+40 x+1000$ . Also, calculate the average cost per additional unit.

See Solution

Problem 726

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 727

A company sells $Q(x)=60x-x^{2}$ units after spending $\$x$ thousand on ads. Find $Q(7)$ and $Q^{\prime}(7)$ .

See Solution

Problem 728

Find the change in demand $x$ when price $p$ drops from \ $2.25 to \$1 using$ x=\frac{1000}{\sqrt{p}+1} $. Also, calculate$ \frac{\Delta X}{\Delta P}$.

See Solution

Problem 729

Find the derivative of $(x+1) \sin x$ . What is it?

See Solution

Problem 730

Find the integral of the constant function $f(x)=k$ . What is it?

See Solution

Problem 731

Find the derivative of $f(x)=4 \cos 2 x+\log x+x$ . Options: $8 \sin 2 x+1 / x+x$ , $-8 \sin 2 x+1 / x+x$ , $-8 \sin 2 x+1 / x+1$ , $8 \sin 2 x+1 / x+1$ .

See Solution

Problem 732

Find the rate of price change when $x=100$ and supply decreases at 8 crates/day from $p x - 20 p - 6 x + 40 = 0$ .

See Solution

Problem 733

Differentiate the function: $y = \sqrt{\frac{\sqrt{x}+\frac{1}{\sqrt{x}}}{\sqrt{x}-\frac{1}{\sqrt{x}}}}$

See Solution

Problem 734

Find the derivative of $2^{x+2} + 3^{y+1} = 18$ with respect to $x$ .

See Solution

Problem 735

Evaluate the integral: $\int x^{3}\left(4+x^{2}\right)^{\frac{3}{2}} d x$

See Solution

Problem 736

Sketch the curves $y=x^{2}+x$ and $y=-2 \cdot x^{2}+4$ from $x=0$ to $x=2$ . Find the area between them using integrals.

See Solution

Problem 737

Find a formula for $\int \frac{d x}{(x-a)(x-b)}$ and compute the length of $y=\log (\sin x)$ from $\left(\frac{\pi}{4}, \log \frac{\sqrt{2}}{2}\right)$ .

See Solution

Problem 738

Find the derivative $f^{\prime}(x)$ of $f(x)=|x|$ for $x>0$ , $x<0$ , and at $x=0$ . Explain why it's undefined at $0$ .

See Solution

Problem 739

Find the $x$ values where the function $f$ has a relative maximum given $f^{\prime}(x)=x^{2}(x+1)^{3}(x-4)^{2}$ .

See Solution

Problem 740

Evaluate $\int_{\pi / 4}^{\pi / 2} \sin ^{3} \alpha \cos \alpha d \alpha$ . What is the result?

See Solution

Problem 741

Find the value of $c$ such that $f(x)=x+\frac{c}{x}$ has a local minimum at $x=3$ .

See Solution

Problem 742

Given values of $f^{\prime \prime}(x)$ : at $x=-1$ is -4, $x=0$ is -1, $x=1$ is 2, $x=2$ is 5, $x=3$ is 8. What type of function is $f$ ? (A) linear (B) quadratic (C) cubic (D) fourth-degree (E) exponential

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Problem 743

Find the time $t$ (from 0 to 10) when the object with velocity $v(t)=t \cos t-\ln (t+2)$ reaches maximum speed.

See Solution

Problem 744

Find the area under the curve $y=\frac{4}{1+x^{2}}$ from $x=-1$ to $x=0$ . Choices: (A) $4-\frac{\pi}{4}$ , (B) $8-2 \pi$ , (C) $8-\pi$ , (D) $8-\frac{\pi}{2}$ , (E) $2 \pi-4$ .

See Solution

Problem 745

Calculate the area under the curve $y=\frac{4}{1+x^{2}}$ from $x=-1$ to $x=1$ .

See Solution

Problem 746

Find the function $f(x)$ given that $f^{\prime}(x)=2 f(x)$ and $f(2)=1$ .

See Solution

Problem 747

Find the time $t$ in $[0, 10]$ when the velocity $v(t)=t \cos t - \ln(t+2)$ is maximized. Options: A. 9.5 B. 5.1 C. 6.4 D. 7.6

See Solution

Problem 748

Estimate $y$ at $x=3.1$ using the tangent line of the curve $x^{3}+x \tan y=27$ at $(3,0)$ . Choose from (A) -2.7 (B) -0.9 (C) 0 (D) 0.1 (E) 3.0.

See Solution

Problem 749

Find the rate of change of sales $S(t)=10,000+2000 t-200 t^{2}$ and interpret $S^{\prime}(t)$ .

See Solution

Problem 750

Analyze the stability of equilibrium points for $\ddot{x}=(x-a)(x^{2}-a)$ across all real values of $a$ .

See Solution

Problem 751

Find the derivative of the function $f(x)=6 \cdot \sqrt{x} \cdot(x-5)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 752

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-3 \cdot \ln (4 \cdot x)$ and calculate $f^{\prime}(2)$ .

See Solution

Problem 753

Find the derivative of $f(x)=\frac{\sqrt{x}-7}{\sqrt{x}+7}$ and calculate $f^{\prime}(5)$ .

See Solution

Problem 754

Find the derivative $f'(x)$ of the function $f(x)=\sqrt{4x^2 + 4x + 4}$ and evaluate it at $x=4$ .

See Solution

Problem 755

Calculate the derivative of $g(x)=(4x^{2}+5x) \cdot e^{x}$ .

See Solution

Problem 756

Find $y^{\prime}(4)$ using implicit differentiation for $4 \cdot x^{2}+3 \cdot x+x \cdot y=4$ and $y(4)=-18$ .

See Solution

Problem 757

Find discontinuities of $f(x)=\left\{\begin{array}{ll}x^{2}-1, & x<1 \\ x, & x \geq 1\end{array}\right.$ and explain why.

See Solution

Problem 758

Find the derivative $f^{\prime}(x)$ of the function $f(x)=2+\frac{7}{x}+\frac{7}{x^{2}}$ .

See Solution

Problem 759

Find the derivative of $R(j)=\left(\ln \left(j^{2}\right)\right)^{2}$ at $j=e$ .

See Solution

Problem 760

Find the derivative $f^{\prime}(t)$ of the function $f(t)=(t^{2}+5t+3)(2t^{-2}+6t^{-5})$ .

See Solution

Problem 761

Find the slope of the tangent line to the curve $4xy^{3}+3xy=7$ at the point $(1,1)$ using implicit differentiation.

See Solution

Problem 762

Find the marginal cost for both businesses given their cost functions: $C_A(x)=200+25x+0.1x^2$ and $C_B(x)=400+80x+0.06x^2$ . For $x=500$ , which business has the lowest cost for the next tire?

See Solution

Problem 763

Find the critical number of the function $f(x)=(8 \cdot x-7) \cdot e^{5 \cdot x}$ .

See Solution

Problem 764

The demand function is $D(x)=967-25-x$ .
(a) Find the elasticity of demand. (b) Find the price where elasticity equals 1. (c) Determine the price for maximum revenue.

See Solution

Problem 765

Find the value of $x$ that satisfies the equation $\frac{d}{d x}(x \cdot(967-25-x))=0$ .

See Solution

Problem 766

Evaluate the integral: $\int(3x^{2}+5x-4) \, dx$

See Solution

Problem 767

Projectile height is given by $h(t)=-16 \cdot t^{2}+256-t$ . Find: (a) average velocity for 3s, (b) speed & height at 6s, (c) max height & time, (d) acceleration at $t=5$ .

See Solution

Problem 768

Find the antiderivative of the function: $\int(5 - 6x^{2} - x^{3}) \, dx$

See Solution

Problem 769

Helium fills a balloon at 2 ft³/s. Find the radius increase rate (in ft/s) after 7 minutes. Use $V=\frac{4}{3} \cdot \pi \cdot r^{3}$ .

See Solution

Problem 770

Evaluate the integral $\int_{0}^{3} \frac{y}{y+1} dy$ using the substitution $t=y+1$ .

See Solution

Problem 771

Substitute $t=y+1$ in the integral $\int_{0}^{3} \frac{y}{y+1} dy$ .

See Solution

Problem 772

Approximate the integral $\int_{-1}^{2}(x^{4}-3 \cdot x^{2}+1) d x$ using left Riemann sums with 8 subdivisions. Round to three decimal places.

See Solution

Problem 773

Solve the integral $\int_{0}^{3} \frac{y}{y+1} d y$ using $t=y+1$ . Which option is correct? 1-5.

See Solution

Problem 774

Find the total displacement of a ball with velocity $v(t)=90-32 \cdot t$ from $t=2$ to $t=5$ seconds.

See Solution

Problem 775

Calculate the integral using integration by parts: $\int x \cdot \ln (x) \, dx =$

See Solution

Problem 776

Calculate the area between $y=\sqrt{x}$ and $y=0$ over the interval $[-1,1]$ .

See Solution

Problem 777

Calculate the daily consumer surplus for sales at price $p=15$ using the demand equation $p=30 e^{-401}$ .

See Solution

Problem 778

Find the average rate of change of profit $P(x)=-400x^2+6800x-12000$ over $[6,6+h]$ for $h=1, 0.1, 0.01, 0.001, 0.0001$ . What do the results indicate?

See Solution

Problem 779

Evaluate the integral $\int_{4}^{\infty} e^{-\frac{x}{4}} dx$ . Enter "undefined" if it diverges.

See Solution

Problem 780

Evaluate the integral: $\int \frac{1}{\sqrt{1-4 x^{2}}} d x$ .

See Solution

Problem 781

A particle's position is given by $s=2 \cdot t^{3}-7 \cdot t^{2}-9 \cdot t+12$ . Find velocity, acceleration, and specific values.

See Solution

Problem 782

Projectile height is given by $h(t)=-16 \cdot t^{2}+256 \cdot t$ . Find average velocity, speed, max height, and acceleration at $t=5$ .

See Solution

Problem 783

Show that the limit $\operatorname{limit}_{x \rightarrow 2} \frac{|x-2|}{x-2}$ does not exist.

See Solution

Problem 784

Calculate the integral $\int\left(2+5 x-\frac{1}{(x-2)^{2}}\right) d x$ .

See Solution

Problem 785

Calculate the integral of the function $2x$ with respect to $x$ : $\int 2 x \, dx$ .

See Solution

Problem 786

Solve the differential equation: $y^{\prime \prime}-7 y^{\prime}+12 y=0$ .

See Solution

Problem 787

Calculate the integral $\int 2 x \cos \left(x^{2}-5\right) \, dx$ .

See Solution

Problem 788

Differentiate $y=\frac{1}{\sqrt{a+x^{2}}}$ with respect to $x$ .

See Solution

Problem 789

Differentiate $y=\sqrt{a+x}$ .

See Solution

Problem 790

Calculate the integral: $\int_{-\infty}^{\infty} x(4 x+1) e^{-4 x^{2}} d x$ given that $\int_{-\infty}^{\infty} e^{-x^{2}} d x=\sqrt{\pi}$ .

See Solution

Problem 791

Calculate the integral $\int_{-\infty}^{\infty} x^{2} e^{-x^{2}} d x$ given that $\int_{-\infty}^{\infty} e^{-x^{2}} d x=\sqrt{\pi}$ .

See Solution

Problem 792

Calculate the integral: $\int \frac{\cos (\ln (x))}{x} d x$ .

See Solution

Problem 793

Find the derivative of $y=x^{3} \ln (x)$ with respect to $x$ : $\frac{d y}{d x}$ .

See Solution

Problem 794

Evaluate $\frac{J_{29}-J_{58}}{I_{15}-I_{14}}$ and $\frac{1}{15 \pi} \sum_{k=1}^{29} I_{k}+(-1)^{k} J_{2 k}$ .

See Solution

Problem 795

Evaluate the integral $\int d \theta$ .

See Solution

Problem 796

Calculate the integral: $\int -5 \cos \pi x \, dx$

See Solution

Problem 797

Find the area between $y=e^{x}$ , $y=x^{2}$ , from $x=0$ to $x=1$ .

See Solution

Problem 798

Find the slant asymptotes of the curve given by the equation $y=\sqrt{x^{2}+4 x}$ .

See Solution

Problem 799

Evaluate the integral: $\int \frac{1}{5 x^{2}-30 x+65} dx$

See Solution

Problem 800

Find the integral $\int \frac{1}{x^{2}-8 x+65} d x$ . Choose the correct answer: (A), (B), (C), or (D).

See Solution

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Calculus

Problem 701

Find the derivative of yyy with respect to xxx for the expression x+ln⁡∣x−3∣+2x−3x+\ln |x-3|+\frac{2}{x-3}x+ln∣x−3∣+x−32​.

Problem 702

Find the formula for the marginal average profit of the profit function P(x)=100ln⁡(2x+1)−5x−10 P(x)=100 \ln (2 x+1)-5 x-10 P(x)=100ln(2x+1)−5x−10. Options are provided.

Problem 703

Find the derivatives of these functions: 1. f(x)=6x3−9x+4f(x)=6 x^{3}-9 x+4f(x)=6x3−9x+4, 2. y=2t4−t2+13ty=2 t^{4}- t^{2}+13 ty=2t4−t2+13t, 3. g(z)=4z7−3z−7+9zg(z)=4 z^{7}-3 z^{-7}+9 zg(z)=4z7−3z−7+9z.

Problem 704

Un point P\mathrm{P}P de masse m\mathrm{m}m se déplace en coordonnées polaires. Trouvez les équations du mouvement et les expressions pour r(t)\mathrm{r}(t)r(t) et ω\omegaω quand ω\omegaω est constant.

Problem 705

Find the integral of 23x2−2sec⁡2(5x)\frac{2}{3 x^{2}} - 2 \sec^{2}(5x)3x22​−2sec2(5x) with respect to xxx.

Problem 706

Evaluate the integral: ∫2(23x2−2sec⁡25x)dx\int_{2}\left(\frac{2}{3 x^{2}}-2 \sec ^{2} 5 x\right) d x∫2​(3x22​−2sec25x)dx

Problem 707

Find the integral: ∫6cosec⁡2xcot⁡2x dx\int 6 \operatorname{cosec} 2 x \cot 2 x \, dx∫6cosec2xcot2xdx

Problem 708

Find the integral of 103x10^{3 x}103x with respect to xxx.

Problem 709

Calculate the integral: ∫0π44sin⁡2x dx\int_{0}^{\frac{\pi}{4}} 4 \sin 2 x \, dx∫04π​​4sin2xdx

Problem 710

Find the integral: ∫5−x−3x24x dx\int \frac{5-x-3 x^{2}}{4 \sqrt{x}} \, dx∫4x​5−x−3x2​dx

Problem 711

Find the limit of f(x)=sin⁡4x4sin⁡3xf(x)=\frac{\sin 4x}{4 \sin 3x}f(x)=4sin3xsin4x​ as xxx approaches 0.

Problem 712

Find the limit of f(x)=sin⁡4x4sin⁡3x+sin⁡(x/3)9xf(x)=\frac{\sin 4x}{4\sin 3x}+\frac{\sin (x/3)}{9x}f(x)=4sin3xsin4x​+9xsin(x/3)​ as x→0x \to 0x→0.

Problem 713

Find the limit: lim⁡x→0f(x)\lim_{x \rightarrow 0} f(x)limx→0​f(x) where f(x)=2xcos⁡(1x2)f(x)=2 x \cos \left(\frac{1}{x^{2}}\right)f(x)=2xcos(x21​).

Problem 714

Find ddxf−1(0)\frac{d}{d x} f^{-1}(0)dxd​f−1(0) for f(x)=x+sin⁡xf(x)=x+\sin xf(x)=x+sinx. At x=0x=0x=0, fff has slope 2, so f−1f^{-1}f−1 slope is 12\frac{1}{2}21​.

Problem 715

Find the derivative of f(x)=x2f(x) = x^2f(x)=x2 using the limit definition: f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}f′(x)=limh→0​hf(x+h)−f(x)​.

Problem 716

Find the limit: If f(z)=z+zˉzf(z)=\frac{z+\bar{z}}{z}f(z)=zz+zˉ​, then lim⁡x→0f(x+0i)=\lim _{x \rightarrow 0} f(x+0 i)=limx→0​f(x+0i)=.

Problem 717

Find lim⁡x→0f(x+0i)\lim _{x \rightarrow 0} f(x+0 i)limx→0​f(x+0i) and lim⁡y→0f(0+iy)\lim _{y \rightarrow 0} f(0+i y)limy→0​f(0+iy) for f(z)=z+zˉzf(z)=\frac{z+\bar{z}}{z}f(z)=zz+zˉ​.

Problem 718

Evaluate the limit as xxx approaches 0: lim⁡x→0x2−25x2−4x−5\lim _{x \rightarrow 0} \frac{x^{2}-25}{x^{2}-4 x-5}limx→0​x2−4x−5x2−25​.

Problem 719

Find dydx\frac{d y}{d x}dxdy​ for the equation 2⋅x⋅ln⁡(y)+y2=52 \cdot x \cdot \ln (y) + y^{2} = 52⋅x⋅ln(y)+y2=5.

Problem 720

Find dydx\frac{d y}{d x}dxdy​ if ex⋅y+5⋅y4=2⋅ye^{x \cdot y}+5 \cdot y^{4}=2 \cdot yex⋅y+5⋅y4=2⋅y.

Problem 721

Find the marginal profit P′(x)P^{\prime}(x)P′(x) for P(x)=3⋅x−8⋅xP(x)=3 \cdot x-8 \cdot \sqrt{x}P(x)=3⋅x−8⋅x​ and calculate P′(x)x\frac{P^{\prime}(x)}{x}xP′(x)​.

Problem 722

Find the marginal cost for businesses A and B with CA(x)=350+40x+0.12x2C_A(x)=350+40x+0.12x^2CA​(x)=350+40x+0.12x2 and CB(x)=280+30x+0.09x2C_B(x)=280+30x+0.09x^2CB​(x)=280+30x+0.09x2. For x=500x=500x=500, which is cheaper?

Problem 723

Find the greatest integer less than or equal to ∫12log⁡2(x3+1)dx+∫1log⁡29(2x−1)13dx\int_{1}^{2} \log _{2}(x^{3}+1) dx + \int_{1}^{\log _{2} 9}(2^{x}-1)^{\frac{1}{3}} dx∫12​log2​(x3+1)dx+∫1log2​9​(2x−1)31​dx.

Problem 724

Find the cost increase when producing 60 items instead of 50, given C=0.001x3−0.3x2+40x+1000C=0.001 x^{3}-0.3 x^{2}+40 x+1000C=0.001x3−0.3x2+40x+1000.

Problem 725

Find the cost increase when producing from 50 to 60 items with C=0.001x3−0.3x2+40x+1000C=0.001 x^{3}-0.3 x^{2}+40 x+1000C=0.001x3−0.3x2+40x+1000. Also, calculate the average cost per additional unit.

Problem 726

Find the ball's instantaneous velocity at t=10.0t=10.0t=10.0 s given x(t)=0.000015t5−0.004t3+0.4tx(t)=0.000015 t^5 - 0.004 t^3 + 0.4 tx(t)=0.000015t5−0.004t3+0.4t.

Problem 727

A company sells Q(x)=60x−x2Q(x)=60x-x^{2}Q(x)=60x−x2 units after spending $x\$x$x thousand on ads. Find Q(7)Q(7)Q(7) and Q′(7)Q^{\prime}(7)Q′(7).

Problem 728

Find the change in demand xxx when price ppp drops from \2.25to$1using2.25 to \$1 using 2.25to$1usingx=\frac{1000}{\sqrt{p}+1}.Also,calculate. Also, calculate .Also,calculate\frac{\Delta X}{\Delta P}$.

Problem 729

Find the derivative of (x+1)sin⁡x(x+1) \sin x(x+1)sinx. What is it?

Problem 730

Find the integral of the constant function f(x)=kf(x)=kf(x)=k. What is it?

Problem 731

Problem 732

Find the rate of price change when x=100x=100x=100 and supply decreases at 8 crates/day from px−20p−6x+40=0p x - 20 p - 6 x + 40 = 0px−20p−6x+40=0.

Problem 733

Differentiate the function: y=x+1xx−1xy = \sqrt{\frac{\sqrt{x}+\frac{1}{\sqrt{x}}}{\sqrt{x}-\frac{1}{\sqrt{x}}}}y=x​−x​1​x​+x​1​​​

Problem 734

Find the derivative of 2x+2+3y+1=182^{x+2} + 3^{y+1} = 182x+2+3y+1=18 with respect to xxx.

Problem 735

Evaluate the integral: ∫x3(4+x2)32dx\int x^{3}\left(4+x^{2}\right)^{\frac{3}{2}} d x∫x3(4+x2)23​dx

Problem 736

Sketch the curves y=x2+xy=x^{2}+xy=x2+x and y=−2⋅x2+4y=-2 \cdot x^{2}+4y=−2⋅x2+4 from x=0x=0x=0 to x=2x=2x=2. Find the area between them using integrals.

Problem 737

Find a formula for ∫dx(x−a)(x−b)\int \frac{d x}{(x-a)(x-b)}∫(x−a)(x−b)dx​ and compute the length of y=log⁡(sin⁡x)y=\log (\sin x)y=log(sinx) from (π4,log⁡22)\left(\frac{\pi}{4}, \log \frac{\sqrt{2}}{2}\right)(4π​,log22​​).

Problem 738

Find the derivative f′(x)f^{\prime}(x)f′(x) of f(x)=∣x∣f(x)=|x|f(x)=∣x∣ for x>0x>0x>0, x<0x<0x<0, and at x=0x=0x=0. Explain why it's undefined at 000.

Problem 739

Find the xxx values where the function fff has a relative maximum given f′(x)=x2(x+1)3(x−4)2f^{\prime}(x)=x^{2}(x+1)^{3}(x-4)^{2}f′(x)=x2(x+1)3(x−4)2.

Problem 740

Evaluate ∫π/4π/2sin⁡3αcos⁡αdα\int_{\pi / 4}^{\pi / 2} \sin ^{3} \alpha \cos \alpha d \alpha∫π/4π/2​sin3αcosαdα. What is the result?

Find the derivative of $y$ with respect to $x$ for the expression $x+\ln |x-3|+\frac{2}{x-3}$ .

Find the formula for the marginal average profit of the profit function $P(x)=100 \ln (2 x+1)-5 x-10$ . Options are provided.

Find the derivatives of these functions: 1. $f(x)=6 x^{3}-9 x+4$ , 2. $y=2 t^{4}- t^{2}+13 t$ , 3. $g(z)=4 z^{7}-3 z^{-7}+9 z$ .

Un point $\mathrm{P}$ de masse $\mathrm{m}$ se déplace en coordonnées polaires. Trouvez les équations du mouvement et les expressions pour $\mathrm{r}(t)$ et $\omega$ quand $\omega$ est constant.

Find the integral of $\frac{2}{3 x^{2}} - 2 \sec^{2}(5x)$ with respect to $x$ .

Evaluate the integral: $\int_{2}\left(\frac{2}{3 x^{2}}-2 \sec ^{2} 5 x\right) d x$

Find the integral: $\int 6 \operatorname{cosec} 2 x \cot 2 x \, dx$

Find the integral of $10^{3 x}$ with respect to $x$ .

Calculate the integral: $\int_{0}^{\frac{\pi}{4}} 4 \sin 2 x \, dx$

Find the integral: $\int \frac{5-x-3 x^{2}}{4 \sqrt{x}} \, dx$

Find the limit of $f(x)=\frac{\sin 4x}{4 \sin 3x}$ as $x$ approaches 0.

Find the limit of $f(x)=\frac{\sin 4x}{4\sin 3x}+\frac{\sin (x/3)}{9x}$ as $x \to 0$ .

Find the limit: $\lim_{x \rightarrow 0} f(x)$ where $f(x)=2 x \cos \left(\frac{1}{x^{2}}\right)$ .

Find $\frac{d}{d x} f^{-1}(0)$ for $f(x)=x+\sin x$ . At $x=0$ , $f$ has slope 2, so $f^{-1}$ slope is $\frac{1}{2}$ .

Find the derivative of $f(x) = x^2$ using the limit definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ .

Find the limit: If $f(z)=\frac{z+\bar{z}}{z}$ , then $\lim _{x \rightarrow 0} f(x+0 i)=$ .

Find $\lim _{x \rightarrow 0} f(x+0 i)$ and $\lim _{y \rightarrow 0} f(0+i y)$ for $f(z)=\frac{z+\bar{z}}{z}$ .

Evaluate the limit as $x$ approaches 0: $\lim _{x \rightarrow 0} \frac{x^{2}-25}{x^{2}-4 x-5}$ .

Find $\frac{d y}{d x}$ for the equation $2 \cdot x \cdot \ln (y) + y^{2} = 5$ .

Find $\frac{d y}{d x}$ if $e^{x \cdot y}+5 \cdot y^{4}=2 \cdot y$ .

Find the marginal profit $P^{\prime}(x)$ for $P(x)=3 \cdot x-8 \cdot \sqrt{x}$ and calculate $\frac{P^{\prime}(x)}{x}$ .

Find the marginal cost for businesses A and B with $C_A(x)=350+40x+0.12x^2$ and $C_B(x)=280+30x+0.09x^2$ . For $x=500$ , which is cheaper?

Find the greatest integer less than or equal to $\int_{1}^{2} \log _{2}(x^{3}+1) dx + \int_{1}^{\log _{2} 9}(2^{x}-1)^{\frac{1}{3}} dx$ .

Find the cost increase when producing 60 items instead of 50, given $C=0.001 x^{3}-0.3 x^{2}+40 x+1000$ .

Find the cost increase when producing from 50 to 60 items with $C=0.001 x^{3}-0.3 x^{2}+40 x+1000$ . Also, calculate the average cost per additional unit.

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

A company sells $Q(x)=60x-x^{2}$ units after spending $\$x$ thousand on ads. Find $Q(7)$ and $Q^{\prime}(7)$ .

Find the change in demand $x$ when price $p$ drops from \ $2.25 to \$1 using$ x=\frac{1000}{\sqrt{p}+1} $. Also, calculate$ \frac{\Delta X}{\Delta P}$.

Find the derivative of $(x+1) \sin x$ . What is it?

Find the integral of the constant function $f(x)=k$ . What is it?

Find the rate of price change when $x=100$ and supply decreases at 8 crates/day from $p x - 20 p - 6 x + 40 = 0$ .

Differentiate the function: $y = \sqrt{\frac{\sqrt{x}+\frac{1}{\sqrt{x}}}{\sqrt{x}-\frac{1}{\sqrt{x}}}}$

Find the derivative of $2^{x+2} + 3^{y+1} = 18$ with respect to $x$ .

Evaluate the integral: $\int x^{3}\left(4+x^{2}\right)^{\frac{3}{2}} d x$

Sketch the curves $y=x^{2}+x$ and $y=-2 \cdot x^{2}+4$ from $x=0$ to $x=2$ . Find the area between them using integrals.

Find a formula for $\int \frac{d x}{(x-a)(x-b)}$ and compute the length of $y=\log (\sin x)$ from $\left(\frac{\pi}{4}, \log \frac{\sqrt{2}}{2}\right)$ .

Find the derivative $f^{\prime}(x)$ of $f(x)=|x|$ for $x>0$ , $x<0$ , and at $x=0$ . Explain why it's undefined at $0$ .

Find the $x$ values where the function $f$ has a relative maximum given $f^{\prime}(x)=x^{2}(x+1)^{3}(x-4)^{2}$ .

Evaluate $\int_{\pi / 4}^{\pi / 2} \sin ^{3} \alpha \cos \alpha d \alpha$ . What is the result?

Find the value of $c$ such that $f(x)=x+\frac{c}{x}$ has a local minimum at $x=3$ .

Given values of $f^{\prime \prime}(x)$ : at $x=-1$ is -4, $x=0$ is -1, $x=1$ is 2, $x=2$ is 5, $x=3$ is 8. What type of function is $f$ ? (A) linear (B) quadratic (C) cubic (D) fourth-degree (E) exponential

Find the time $t$ (from 0 to 10) when the object with velocity $v(t)=t \cos t-\ln (t+2)$ reaches maximum speed.

Find the area under the curve $y=\frac{4}{1+x^{2}}$ from $x=-1$ to $x=0$ . Choices: (A) $4-\frac{\pi}{4}$ , (B) $8-2 \pi$ , (C) $8-\pi$ , (D) $8-\frac{\pi}{2}$ , (E) $2 \pi-4$ .

Calculate the area under the curve $y=\frac{4}{1+x^{2}}$ from $x=-1$ to $x=1$ .

Find the function $f(x)$ given that $f^{\prime}(x)=2 f(x)$ and $f(2)=1$ .

Find the time $t$ in $[0, 10]$ when the velocity $v(t)=t \cos t - \ln(t+2)$ is maximized. Options: A. 9.5 B. 5.1 C. 6.4 D. 7.6

Estimate $y$ at $x=3.1$ using the tangent line of the curve $x^{3}+x \tan y=27$ at $(3,0)$ . Choose from (A) -2.7 (B) -0.9 (C) 0 (D) 0.1 (E) 3.0.

Find the rate of change of sales $S(t)=10,000+2000 t-200 t^{2}$ and interpret $S^{\prime}(t)$ .

Analyze the stability of equilibrium points for $\ddot{x}=(x-a)(x^{2}-a)$ across all real values of $a$ .

Find the derivative of the function $f(x)=6 \cdot \sqrt{x} \cdot(x-5)$ . What is $f^{\prime}(x)$ ?

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-3 \cdot \ln (4 \cdot x)$ and calculate $f^{\prime}(2)$ .

Find the derivative of $f(x)=\frac{\sqrt{x}-7}{\sqrt{x}+7}$ and calculate $f^{\prime}(5)$ .

Find the derivative $f'(x)$ of the function $f(x)=\sqrt{4x^2 + 4x + 4}$ and evaluate it at $x=4$ .

Calculate the derivative of $g(x)=(4x^{2}+5x) \cdot e^{x}$ .

Find $y^{\prime}(4)$ using implicit differentiation for $4 \cdot x^{2}+3 \cdot x+x \cdot y=4$ and $y(4)=-18$ .

Find discontinuities of $f(x)=\left\{\begin{array}{ll}x^{2}-1, & x<1 \\ x, & x \geq 1\end{array}\right.$ and explain why.

Find the derivative $f^{\prime}(x)$ of the function $f(x)=2+\frac{7}{x}+\frac{7}{x^{2}}$ .

Find the derivative of $R(j)=\left(\ln \left(j^{2}\right)\right)^{2}$ at $j=e$ .

Find the derivative $f^{\prime}(t)$ of the function $f(t)=(t^{2}+5t+3)(2t^{-2}+6t^{-5})$ .

Find the slope of the tangent line to the curve $4xy^{3}+3xy=7$ at the point $(1,1)$ using implicit differentiation.

Find the marginal cost for both businesses given their cost functions: $C_A(x)=200+25x+0.1x^2$ and $C_B(x)=400+80x+0.06x^2$ . For $x=500$ , which business has the lowest cost for the next tire?

Find the critical number of the function $f(x)=(8 \cdot x-7) \cdot e^{5 \cdot x}$ .

The demand function is $D(x)=967-25-x$ .
(a) Find the elasticity of demand. (b) Find the price where elasticity equals 1. (c) Determine the price for maximum revenue.

Find the value of $x$ that satisfies the equation $\frac{d}{d x}(x \cdot(967-25-x))=0$ .

Evaluate the integral: $\int(3x^{2}+5x-4) \, dx$

Projectile height is given by $h(t)=-16 \cdot t^{2}+256-t$ . Find: (a) average velocity for 3s, (b) speed & height at 6s, (c) max height & time, (d) acceleration at $t=5$ .

Find the antiderivative of the function: $\int(5 - 6x^{2} - x^{3}) \, dx$

Helium fills a balloon at 2 ft³/s. Find the radius increase rate (in ft/s) after 7 minutes. Use $V=\frac{4}{3} \cdot \pi \cdot r^{3}$ .

Evaluate the integral $\int_{0}^{3} \frac{y}{y+1} dy$ using the substitution $t=y+1$ .

Substitute $t=y+1$ in the integral $\int_{0}^{3} \frac{y}{y+1} dy$ .

Approximate the integral $\int_{-1}^{2}(x^{4}-3 \cdot x^{2}+1) d x$ using left Riemann sums with 8 subdivisions. Round to three decimal places.

Solve the integral $\int_{0}^{3} \frac{y}{y+1} d y$ using $t=y+1$ . Which option is correct? 1-5.

Find the total displacement of a ball with velocity $v(t)=90-32 \cdot t$ from $t=2$ to $t=5$ seconds.