Calculus

Problem 12701

Find the number of units $x$ (whole number) that minimizes the unit cost $U(x)=\frac{1}{162 x+1}+2 x$ for $0 \leq x \leq 1$ .

See Solution

Problem 12702

Find the critical numbers of the function $f(x)=x^{3}-48x$ .

See Solution

Problem 12703

Find points on the curve $y=\ln(x^{2}+1)$ where the tangent line is horizontal.

See Solution

Problem 12704

A farmer has 400 m of mesh to build three sides of a rectangular enclosure. What dimensions maximize the area?

See Solution

Problem 12705

Find the tangent line to $y^{2}(y^{2}-4)=x^{2}(x^{2}-5)$ at $(0,-2)$ using implicit differentiation.

See Solution

Problem 12706

Find the critical points of the function $f$ given its derivative $f^{\prime}(x)=5 x^{3}+9 x^{2}-2 x$ .

See Solution

Problem 12707

Evaluate the integral $\int_{0}^{1}(x^{2}-3x+6)dx$ and check if the result matches the given figure.

See Solution

Problem 12708

Calculate the integral from 0 to 1 of the function $2x - 2\sqrt{x}$ .

See Solution

Problem 12709

Evaluate the integral $\int_{-\pi / 4}^{7 \pi / 4}(\sin x+\cos x) dx$ and check consistency with the provided figure.

See Solution

Problem 12710

1. Given marginal profit values for sales levels: Sales: 50, 100, 150, 200, 250, 300, 350, 400; Marginal Profit: 7, 3, 1, 0, -4, -2, 0, 5. (a) Estimate profit change from sales level 100 to 120 using linear approximation. (b) Determine the sign of the second derivative of profit near sales level 100 and its implication on the previous estimate. (c) Identify sales levels for local minimum and maximum profit.

See Solution

Problem 12711

Given $f(x)=2 x^{3}-6$ , find critical points, intervals for $f^{\prime}(x)>0$ , $f^{\prime}(x)<0$ , local maxima, and minima.

See Solution

Problem 12712

Find $R^{\prime}(-2)$ for $R(x)=P\left(x^{2}-5\right)$ given $P(-2)=4$ , $P^{\prime}(-2)=-3$ .

See Solution

Problem 12713

Create a sign diagram for the derivative of the function $f(x)=x^{3}-6 x^{2}-15 x+5$ .

See Solution

Problem 12714

Estimate the profit change from sales of 100 to 120 using linear approximation given marginal profits: 7, 3, 1, 0, -4, -2, 0, 5. Answer: 60.

See Solution

Problem 12715

Determine at which point (Point #1, #2, or #3) a falling puck has maximum kinetic energy, given no info on their heights.

See Solution

Problem 12716

Sneaker price is \ $600, rising \$40/day. Estimate price after 30 days using linear approximation and function$ L(t)$. Explain results.

See Solution

Problem 12717

Find the derivative of the function $\frac{2 x}{x+1}$ with respect to $x$ .

See Solution

Problem 12718

Find $f^{\prime}(1)$ for the function $f(x)=(2 x+1)(x^{7}+3 x)$ . Choices: 0, 12, 38, 1.

See Solution

Problem 12719

Find critical numbers, local minima, and local maxima for $f(x)=\sqrt{x^{4}+3}-\frac{x^{3}}{3}$ .

See Solution

Problem 12720

Find the rate of change of demand $D(p)=\frac{100}{\sqrt{p}}$ at $p=625$ dollars per ton (units not "million tons").

See Solution

Problem 12721

Find the correct derivative of $f(x)=\frac{x^{2}-1}{x^{7}+1}$ . Options include:
1. $f^{\prime}(x)=\frac{2 x}{7 x^{6}}$
2. $f^{\prime}(x)=\frac{2 x(x^{7}+1)-(x^{2}-1)(7 x^{6})}{(x^{7}+1)^{2}}$
3. $f^{\prime}(x)=\frac{2 x(x^{7}+1)-x^{2}(7 x^{6})}{7 x^{6}}$
4. $f^{\prime}(x)=\frac{2 x-1}{7 x^{6}-1}$

See Solution

Problem 12722

Find the rate of change of demand for rice in 2020 using $p(t)=535+9t-5t^2$ . Calculate for $t=0$ and $t=6$ . Include units.

See Solution

Problem 12723

Evaluate these limits using the definition of derivative:
1. $\lim _{x \rightarrow 1} \frac{x^{2 x+1}+x-2}{x-1}$
2. $\lim _{x \rightarrow 0} \frac{\log _{3}\left(x^{2}+3 x+1\right)}{x}$
Also, find $g^{\prime}(0)$ for $g(x)=f(x)^{3 f(x)}$ if the tangent line at $x=0$ is $y=3 x+2$ .

See Solution

Problem 12724

Find the limit as $x$ approaches 0 from the right for $\frac{4}{x} - \frac{8}{e^{2x}-1}$ . Combine into one fraction first.

See Solution

Problem 12725

Find the rate of revenue increase when 180 units are sold, given $R=1100 x-x^{2}$ and sales increase by 30 units/day.

See Solution

Problem 12726

Find $\frac{d A}{d t}$ for a circle's area $A$ with radius $r$ when $r=2$ and $\frac{d r}{d t}=3$ .

See Solution

Problem 12727

Find the local maximum and minimum of the function $f(x)=x^{3}-12x$ .

See Solution

Problem 12728

Find the critical numbers of the function $f(z)=\frac{5 z+5}{5 z^{2}+5 z+5}$ in increasing order. Use $\mathrm{N}$ for unused blanks.

See Solution

Problem 12729

Calculate the account value after 7 years for an investment of \ $29800 at a 3.5% annual rate, using$ V=P e^{rt}$.

See Solution

Problem 12730

Find if the limit $\lim _{x \rightarrow 1} \frac{x^{8}-1}{x^{3}-1}$ exists and calculate its value.

See Solution

Problem 12731

Find the slope of the tangent line to $y=\tan^{-1}(-x)$ at $x=4$ .

See Solution

Problem 12732

Find the derivative of the inverse of $f(x)=x^{2}-9$ for $x>0$ .

See Solution

Problem 12733

Find the limit as $x$ approaches 1 for $\frac{\ln(4x^2 + 2x - 5)}{3x^2 - 3}$ .

See Solution

Problem 12734

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos (3 x)}{\sin ^{2}(2 x)}$ .

See Solution

Problem 12735

An electron oscillates at 50 Hz with positions +10 and -10. Find the amplitude, period, and key characteristics.

See Solution

Problem 12736

Find if the limit $\lim _{x \rightarrow 0} \frac{4^{\sin (x)}-1}{6 x}$ exists and calculate its value.

See Solution

Problem 12737

Calculate the limit: $\lim _{x \rightarrow 0} \frac{2^{x}-7^{x}}{4 x}$ .

See Solution

Problem 12738

Find values of $x$ for which the series $\sum_{n=0}^{\infty}(-1)^{n} x^{-2 n}$ converges and its sum as a function of $x$ .

See Solution

Problem 12739

Determine if the series $\sum_{n=6}^{\infty} \frac{1}{n^{3}-125}$ converges or diverges using one test.

See Solution

Problem 12740

Find $f(4)$ given that $f^{\prime}(x)=\frac{\sqrt{x}+3}{x^{3}}$ for $x>0$ and $f(1)=0$ .

See Solution

Problem 12741

Evaluate: (a) $\sum_{i=5}^{97}\left(\frac{1}{i}-\frac{1}{i-1}\right)$ ; (b) $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{1}{n}\left[\left(\frac{2 i}{n}\right)^{3}+10\right]$ .

See Solution

Problem 12742

Evaluate $\int_{0}^{5} f(t) d t + 7$ given $\int_{0}^{5} f(t) d t = 10$ .

See Solution

Problem 12743

Find the local minimum and maximum of the function $f(x)=-2 x^{3}+39 x^{2}-216 x+2$ .

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Problem 12744

Evaluate four of these integrals given $\int_{0}^{5} f(t) d t=10$ : (a) $\int_{0}^{5} 7 f(t) d t$ , (b) $\int_{0}^{5}(f(t)+7) d t$ , (c) $\int_{0}^{5} f(t) d t+7$ , (d) $\int_{0}^{5} 7 f(t+7) d t$ .

See Solution

Problem 12745

Evaluate the limit: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{1}{n}\left[\left(\frac{2 i}{n}\right)^{3}+10\right]$

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Problem 12746

Find the time $t$ when the drug concentration $C(t)=\frac{90 t}{6 t^{2}+45}$ is highest. $t=\square$ .

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Problem 12747

Find if the limit $\lim _{x \rightarrow 4}\left(\frac{7}{\ln (x-3)}-\frac{2}{x-4}\right)$ exists, and its value.

See Solution

Problem 12748

Find the intervals where the power $P(R)=\frac{120 R}{(0.4+R)^{2}}$ is increasing based on resistance $R$ .

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Problem 12749

Find the derivative of $f(x)=\left(x^{3}-3 x^{2}+1\right)^{-3}$ using the chain rule.

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Problem 12750

Find the remaining amount of technetium $-99 \mathrm{~m}$ from a $40 \mathrm{~g}$ sample after 2 days. Also, calculate the original value of a car sold for \$4200 with a 15% annual depreciation.

See Solution

Problem 12751

Find $g^{\prime}(4)$ given $f(4)=5$ , $f^{\prime}(4)=6$ , and $g(x)=\sqrt{x} f(x)$ .

See Solution

Problem 12752

Find the differential $d y$ of the function $d y=x(1-\cos (x))$ using $d x$ .

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Problem 12753

A rock is thrown up at 13 m/s from a 38 m cliff. When is it 11 m above ground? Round to two decimal places.

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Problem 12754

Calculate the surface area when a 6x16 rectangle rotates around line $z$ . How does it change if $z$ is a side?

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Problem 12755

Calculate the area between the curves $f(x) = x^2 + 4x + 4$ and $g(x) = 4x + 20$ from $x = -4$ to $x = 4$ .

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Problem 12756

Find the differential $d y$ for the function $y=10 x^{4/5}$ . Use " $d x$ " for $d x$ .

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Problem 12757

Sketch the graphs of $f$ and $f''$ given that $f'$ is a parabola with roots at (-2,0) and (2,0) and a minimum at (0,-4).

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Problem 12758

Find the differential $d y$ for the function $y = x \tan(x)$ .

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Problem 12759

Sketch the function $f$ and its second derivative $f''$ from the graph of $f'$ : a parabola with roots at -2 and 2, min at (0,-4).

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Problem 12760

Find the number of people the vaccine prevented from getting sick, given $N_{1}(t)=0.1 t^{2}+0.5 t+150$ for $0 \leq t \leq 25$ and $N_{2}(t)=-0.2 t^{2}+6 t+200$ for $25 < t \leq 50$ .

See Solution

Problem 12761

Approximate the area under $y=x^{2}$ from $x=1$ to $x=4$ using a Right Endpoint method with 6 subdivisions.

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Problem 12762

Find the advertising spend $s$ (in thousands) for max profit given $P=-\frac{1}{10} s^{3}+15 s^{2}+400$ . What is $s$ ?

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Problem 12763

Find $\frac{d y}{d x}$ for the curve $e^{x^{2}-y}=y^{9 x+5}$ using logarithmic differentiation.

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Problem 12764

Find the derivative of $y=e^{-2 e^{-0.02 x}}$ . What is $y^{\prime}=\square$ ?

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Problem 12765

Find the limit: $\lim _{x \rightarrow 1^{-}} \tan ^{-1}\left(\frac{1}{x-1}\right)$ .

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Problem 12766

Find the derivative $f^{\prime}(x)$ for the function $f(x)=e^{\sqrt{x-14}}$ .

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Problem 12767

Find the limit: $\lim _{x \rightarrow \infty} \tan ^{-1} x$ .

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Problem 12768

Find the limit: $\lim _{x \rightarrow \infty} \cos ^{-1}\left(\frac{\sqrt{x-1}}{\sqrt{2 x+1}}\right)$ .

See Solution

Problem 12769

Find the limit: $\lim _{x \rightarrow \infty} \sin ^{-1}\left(\frac{1-x}{2 x+1}\right)$ .

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Problem 12770

Differentiate the function $f(x)=\left(x+\frac{1}{x}\right) e^{4 x}$ . Find $\frac{\mathrm{d}}{\mathrm{dx}}f(x)=\square$ .

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Problem 12771

A well pumps 40000 L on day 1, decreasing by $5\%$ daily. Find total water pumped over its lifetime.

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Problem 12772

Find the vertical asymptote(s) of $g(x)=\frac{1}{(x-4)^{2}}$ . Choose A, B, or C and complete the equations if needed.

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Problem 12773

Find the limit: $\lim _{x \rightarrow 1} \cos ^{-1}\left(x+\frac{1}{2}\right)$ .

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Problem 12774

Differentiate $y=\frac{x}{-2 \sin x}$ with respect to $x$ .

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Problem 12775

Simplify the derivative: $\frac{d}{d x}\left(\frac{e^{x}+7 e^{6 x}}{e^{x}}\right)=\square$

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Problem 12776

Find the horizontal asymptote of $N(t)=\frac{0.9t+1100}{5t+7}$ as $t \rightarrow \infty$ and explain its meaning.

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Problem 12777

Invest \ $12,367 at 6.4% interest, compounded continuously. Find the function for amount after$ t$ years, balances for 1, 2, 5, 10 years, and doubling time.

See Solution

Problem 12778

Find the horizontal asymptote of $N(t)=\frac{0.9t+1100}{5t+7}$ as $t \to \infty$ and explain its significance.

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Problem 12779

Find the area of the largest rectangle between the $x$ -axis, $y$ -axis, and the curve $y=8-x^{3}$ .

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Problem 12780

Find dimensions of an open-top box with volume $389344 \mathrm{~cm}^{3}$ that minimizes surface area.
1. Surface area formula $A(x)$ in terms of $x$ (base side length).
2. Derivative $A^{\prime}(x)$ .
3. Solve $A^{\prime}(x)=0$ for $x$ .
4. Find second derivative $A^{\prime \prime}(x)$ and evaluate it.

See Solution

Problem 12781

Find a point $c$ in the interval [16,36] such that the Mean Value Theorem applies to $y(x)=\sqrt{x}$ . $c=$

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Problem 12782

Find the cost, average cost, marginal cost, and optimal production level for the cost function $C(x)=57600+700x+x^{2}$ at $x=1850$ .

See Solution

Problem 12783

Bestimmen Sie die Intervalle, in denen $f$ steigt oder fällt, und skizzieren Sie den Verlauf von $f$ aus dem Graphen von $f'$ .

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Problem 12784

Ricardo metabolizes caffeine at $14.7\%$ per hour. Find time to metabolize half of $96 \mathrm{mg}$ , $190 \mathrm{mg}$ , and $c$ mg.

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Problem 12785

Find the derivative $f'(\theta)$ for $f(\theta)=\theta \sin (\theta)$ and evaluate it at $\pi$ .

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Problem 12786

Model Ni-65 decay with a function, find remaining fraction after 10h, and time to decay to $\frac{1}{1024}$ of original mass.

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Problem 12787

Berechne die Extrempunkte der Funktion $f(x)=\frac{1}{3a} x^{3}-x^{2}$ für $a>0$ .

See Solution

Problem 12788

Calculate $\frac{\partial z}{\partial x}$ for $z=f(x, y)$ given $x e^{z}+z e^{y}=x+y$ .

See Solution

Problem 12789

Finde die Extrempunkte der Funktion $f(x)=\frac{1}{3 a} x^{3}-x^{2}$ für $a>0$ .

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Problem 12790

Calculate the limit: $\lim _{x \rightarrow \frac{\pi}{2}}(1+\operatorname{ctg} x)^{\operatorname{tg} x}$ .

See Solution

Problem 12791

Estimate the area under $y=x^{2}$ from $x=0$ to $x=3$ using a Right Endpoint method with 6 subdivisions.

See Solution

Problem 12792

Find the limit: $\lim _{x \rightarrow 0}\left(1+3 x^{4}\right)^{\frac{1}{\sin ^{2} x}}$

See Solution

Problem 12793

Calculate the limit: $\lim _{x \rightarrow 0} \frac{3^{\operatorname{tg} x}-2^{\operatorname{tg} x}}{\operatorname{arctg} \sqrt{x} \cdot \arcsin \sqrt{x}}$

See Solution

Problem 12794

Gegeben ist $f_{t}(x)=x^{3}-12 t^{2} x$ für $t \in \mathbb{R}^{+}$ . Finde Hoch-/Tiefpunkte, Ortskurve, gemeinsame Punkte und Steigung im Ursprung.

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Problem 12795

Gegeben ist die Funktion $f_{k}(x)=x^{3}-3 k x^{2}$ .
a) Zeichne den Graphen für $k=-1, -0,5, 0, 0,5, 1$ . b) Finde die Nullstellen von $f_{k}$ . c) Berechne die ersten drei Ableitungen von $f_{k}$ . d) Bestimme die Extrempunkte und deren Art. e) Finde den Wendepunkt und zeige, dass es sich um eine Wendestelle handelt. f) Bestimme die Wendetangente. g) Finde $k$ , damit die Wendetangente durch $P(0, 8)$ verläuft.

See Solution

Problem 12796

Find the limit: $\lim _{x \rightarrow \frac{\pi}{2}}(\sin x)^{\operatorname{tg}^{2} x}$ .

See Solution

Problem 12797

Calculate $u_{8} = \frac{1}{8} \sum_{k=0}^{7} \left(\frac{k}{8}\right)^{2}$ . What is $u_{8}$ ?

See Solution

Problem 12798

Find the third derivative of the function $f(x)=x^{2}+\frac{3}{x}$ .

See Solution

Problem 12799

Determine the radius and interval of convergence for the series $\sum_{k=0}^{\infty} \frac{(-2)^{k} x^{k+1}}{k+1}$ .

See Solution

Problem 12800

Calculate $u_{8} = \frac{1}{8} \cdot \sum_{k=0}^{7} \left(\frac{k}{8}\right)^{2}$ and find its approximate value.

See Solution

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Calculus

Problem 12701

Find the number of units xxx (whole number) that minimizes the unit cost U(x)=1162x+1+2xU(x)=\frac{1}{162 x+1}+2 xU(x)=162x+11​+2x for 0≤x≤10 \leq x \leq 10≤x≤1.

Problem 12702

Find the critical numbers of the function f(x)=x3−48xf(x)=x^{3}-48xf(x)=x3−48x.

Problem 12703

Find points on the curve y=ln⁡(x2+1)y=\ln(x^{2}+1)y=ln(x2+1) where the tangent line is horizontal.

Problem 12704

A farmer has 400 m of mesh to build three sides of a rectangular enclosure. What dimensions maximize the area?

Problem 12705

Find the tangent line to y2(y2−4)=x2(x2−5)y^{2}(y^{2}-4)=x^{2}(x^{2}-5)y2(y2−4)=x2(x2−5) at (0,−2)(0,-2)(0,−2) using implicit differentiation.

Problem 12706

Find the critical points of the function fff given its derivative f′(x)=5x3+9x2−2xf^{\prime}(x)=5 x^{3}+9 x^{2}-2 xf′(x)=5x3+9x2−2x.

Problem 12707

Evaluate the integral ∫01(x2−3x+6)dx\int_{0}^{1}(x^{2}-3x+6)dx∫01​(x2−3x+6)dx and check if the result matches the given figure.

Problem 12708

Calculate the integral from 0 to 1 of the function 2x−2x2x - 2\sqrt{x}2x−2x​.

Problem 12709

Evaluate the integral ∫−π/47π/4(sin⁡x+cos⁡x)dx\int_{-\pi / 4}^{7 \pi / 4}(\sin x+\cos x) dx∫−π/47π/4​(sinx+cosx)dx and check consistency with the provided figure.

Problem 12710

Problem 12711

Given f(x)=2x3−6f(x)=2 x^{3}-6f(x)=2x3−6, find critical points, intervals for f′(x)>0f^{\prime}(x)>0f′(x)>0, f′(x)<0f^{\prime}(x)<0f′(x)<0, local maxima, and minima.

Problem 12712

Find R′(−2)R^{\prime}(-2)R′(−2) for R(x)=P(x2−5)R(x)=P\left(x^{2}-5\right)R(x)=P(x2−5) given P(−2)=4P(-2)=4P(−2)=4, P′(−2)=−3P^{\prime}(-2)=-3P′(−2)=−3.

Problem 12713

Create a sign diagram for the derivative of the function f(x)=x3−6x2−15x+5f(x)=x^{3}-6 x^{2}-15 x+5f(x)=x3−6x2−15x+5.

Problem 12714

Estimate the profit change from sales of 100 to 120 using linear approximation given marginal profits: 7, 3, 1, 0, -4, -2, 0, 5. Answer: 60.

Problem 12715

Determine at which point (Point #1, #2, or #3) a falling puck has maximum kinetic energy, given no info on their heights.

Problem 12716

Sneaker price is \600,rising$40/day.Estimatepriceafter30daysusinglinearapproximationandfunction600, rising \$40/day. Estimate price after 30 days using linear approximation and function 600,rising$40/day.Estimatepriceafter30daysusinglinearapproximationandfunctionL(t)$. Explain results.

Problem 12717

Find the derivative of the function 2xx+1\frac{2 x}{x+1}x+12x​ with respect to xxx.

Problem 12718

Find f′(1)f^{\prime}(1)f′(1) for the function f(x)=(2x+1)(x7+3x)f(x)=(2 x+1)(x^{7}+3 x)f(x)=(2x+1)(x7+3x). Choices: 0, 12, 38, 1.

Problem 12719

Find critical numbers, local minima, and local maxima for f(x)=x4+3−x33f(x)=\sqrt{x^{4}+3}-\frac{x^{3}}{3}f(x)=x4+3​−3x3​.

Problem 12720

Find the rate of change of demand D(p)=100pD(p)=\frac{100}{\sqrt{p}}D(p)=p​100​ at p=625p=625p=625 dollars per ton (units not "million tons").

Problem 12721

Problem 12722

Find the rate of change of demand for rice in 2020 using p(t)=535+9t−5t2p(t)=535+9t-5t^2p(t)=535+9t−5t2. Calculate for t=0t=0t=0 and t=6t=6t=6. Include units.

Problem 12723

Problem 12724

Find the limit as xxx approaches 0 from the right for 4x−8e2x−1\frac{4}{x} - \frac{8}{e^{2x}-1}x4​−e2x−18​. Combine into one fraction first.

Problem 12725

Find the rate of revenue increase when 180 units are sold, given R=1100x−x2R=1100 x-x^{2}R=1100x−x2 and sales increase by 30 units/day.

Problem 12726

Find dAdt\frac{d A}{d t}dtdA​ for a circle's area AAA with radius rrr when r=2r=2r=2 and drdt=3\frac{d r}{d t}=3dtdr​=3.

Problem 12727

Find the local maximum and minimum of the function f(x)=x3−12xf(x)=x^{3}-12xf(x)=x3−12x.

Problem 12728

Find the critical numbers of the function f(z)=5z+55z2+5z+5f(z)=\frac{5 z+5}{5 z^{2}+5 z+5}f(z)=5z2+5z+55z+5​ in increasing order. Use N\mathrm{N}N for unused blanks.

Problem 12729

Calculate the account value after 7 years for an investment of \29800ata3.529800 at a 3.5% annual rate, using 29800ata3.5V=P e^{rt}$.

Problem 12730

Find if the limit lim⁡x→1x8−1x3−1\lim _{x \rightarrow 1} \frac{x^{8}-1}{x^{3}-1}limx→1​x3−1x8−1​ exists and calculate its value.

Problem 12731

Find the slope of the tangent line to y=tan⁡−1(−x)y=\tan^{-1}(-x)y=tan−1(−x) at x=4x=4x=4.

Problem 12732

Find the derivative of the inverse of f(x)=x2−9f(x)=x^{2}-9f(x)=x2−9 for x>0x>0x>0.

Problem 12733

Find the limit as xxx approaches 1 for ln⁡(4x2+2x−5)3x2−3\frac{\ln(4x^2 + 2x - 5)}{3x^2 - 3}3x2−3ln(4x2+2x−5)​.

Problem 12734

Find the limit: lim⁡x→01−cos⁡(3x)sin⁡2(2x)\lim _{x \rightarrow 0} \frac{1-\cos (3 x)}{\sin ^{2}(2 x)}limx→0​sin2(2x)1−cos(3x)​.

Problem 12735

An electron oscillates at 50 Hz with positions +10 and -10. Find the amplitude, period, and key characteristics.

Problem 12736

Find if the limit lim⁡x→04sin⁡(x)−16x\lim _{x \rightarrow 0} \frac{4^{\sin (x)}-1}{6 x}limx→0​6x4sin(x)−1​ exists and calculate its value.

Problem 12737

Calculate the limit: lim⁡x→02x−7x4x \lim _{x \rightarrow 0} \frac{2^{x}-7^{x}}{4 x} limx→0​4x2x−7x​.

Problem 12738

Find values of xxx for which the series ∑n=0∞(−1)nx−2n\sum_{n=0}^{\infty}(-1)^{n} x^{-2 n}∑n=0∞​(−1)nx−2n converges and its sum as a function of xxx.

Problem 12739

Determine if the series ∑n=6∞1n3−125\sum_{n=6}^{\infty} \frac{1}{n^{3}-125}n=6∑∞​n3−1251​ converges or diverges using one test.

Problem 12740

Find f(4)f(4)f(4) given that f′(x)=x+3x3f^{\prime}(x)=\frac{\sqrt{x}+3}{x^{3}}f′(x)=x3x​+3​ for x>0x>0x>0 and f(1)=0f(1)=0f(1)=0.

Problem 12741

Problem 12742

Find the number of units $x$ (whole number) that minimizes the unit cost $U(x)=\frac{1}{162 x+1}+2 x$ for $0 \leq x \leq 1$ .

Find the critical numbers of the function $f(x)=x^{3}-48x$ .

Find points on the curve $y=\ln(x^{2}+1)$ where the tangent line is horizontal.

Find the tangent line to $y^{2}(y^{2}-4)=x^{2}(x^{2}-5)$ at $(0,-2)$ using implicit differentiation.

Find the critical points of the function $f$ given its derivative $f^{\prime}(x)=5 x^{3}+9 x^{2}-2 x$ .

Evaluate the integral $\int_{0}^{1}(x^{2}-3x+6)dx$ and check if the result matches the given figure.

Calculate the integral from 0 to 1 of the function $2x - 2\sqrt{x}$ .

Evaluate the integral $\int_{-\pi / 4}^{7 \pi / 4}(\sin x+\cos x) dx$ and check consistency with the provided figure.

Given $f(x)=2 x^{3}-6$ , find critical points, intervals for $f^{\prime}(x)>0$ , $f^{\prime}(x)<0$ , local maxima, and minima.

Find $R^{\prime}(-2)$ for $R(x)=P\left(x^{2}-5\right)$ given $P(-2)=4$ , $P^{\prime}(-2)=-3$ .

Create a sign diagram for the derivative of the function $f(x)=x^{3}-6 x^{2}-15 x+5$ .

Sneaker price is \ $600, rising \$40/day. Estimate price after 30 days using linear approximation and function$ L(t)$. Explain results.

Find the derivative of the function $\frac{2 x}{x+1}$ with respect to $x$ .

Find $f^{\prime}(1)$ for the function $f(x)=(2 x+1)(x^{7}+3 x)$ . Choices: 0, 12, 38, 1.

Find critical numbers, local minima, and local maxima for $f(x)=\sqrt{x^{4}+3}-\frac{x^{3}}{3}$ .

Find the rate of change of demand $D(p)=\frac{100}{\sqrt{p}}$ at $p=625$ dollars per ton (units not "million tons").

Find the rate of change of demand for rice in 2020 using $p(t)=535+9t-5t^2$ . Calculate for $t=0$ and $t=6$ . Include units.

Find the limit as $x$ approaches 0 from the right for $\frac{4}{x} - \frac{8}{e^{2x}-1}$ . Combine into one fraction first.

Find the rate of revenue increase when 180 units are sold, given $R=1100 x-x^{2}$ and sales increase by 30 units/day.

Find $\frac{d A}{d t}$ for a circle's area $A$ with radius $r$ when $r=2$ and $\frac{d r}{d t}=3$ .

Find the local maximum and minimum of the function $f(x)=x^{3}-12x$ .

Find the critical numbers of the function $f(z)=\frac{5 z+5}{5 z^{2}+5 z+5}$ in increasing order. Use $\mathrm{N}$ for unused blanks.

Calculate the account value after 7 years for an investment of \ $29800 at a 3.5% annual rate, using$ V=P e^{rt}$.

Find if the limit $\lim _{x \rightarrow 1} \frac{x^{8}-1}{x^{3}-1}$ exists and calculate its value.

Find the slope of the tangent line to $y=\tan^{-1}(-x)$ at $x=4$ .

Find the derivative of the inverse of $f(x)=x^{2}-9$ for $x>0$ .

Find the limit as $x$ approaches 1 for $\frac{\ln(4x^2 + 2x - 5)}{3x^2 - 3}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos (3 x)}{\sin ^{2}(2 x)}$ .

Find if the limit $\lim _{x \rightarrow 0} \frac{4^{\sin (x)}-1}{6 x}$ exists and calculate its value.

Calculate the limit: $\lim _{x \rightarrow 0} \frac{2^{x}-7^{x}}{4 x}$ .

Find values of $x$ for which the series $\sum_{n=0}^{\infty}(-1)^{n} x^{-2 n}$ converges and its sum as a function of $x$ .

Determine if the series $\sum_{n=6}^{\infty} \frac{1}{n^{3}-125}$ converges or diverges using one test.

Find $f(4)$ given that $f^{\prime}(x)=\frac{\sqrt{x}+3}{x^{3}}$ for $x>0$ and $f(1)=0$ .

Evaluate $\int_{0}^{5} f(t) d t + 7$ given $\int_{0}^{5} f(t) d t = 10$ .

Find the local minimum and maximum of the function $f(x)=-2 x^{3}+39 x^{2}-216 x+2$ .

Evaluate the limit: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{1}{n}\left[\left(\frac{2 i}{n}\right)^{3}+10\right]$

Find the time $t$ when the drug concentration $C(t)=\frac{90 t}{6 t^{2}+45}$ is highest. $t=\square$ .

Find if the limit $\lim _{x \rightarrow 4}\left(\frac{7}{\ln (x-3)}-\frac{2}{x-4}\right)$ exists, and its value.

Find the intervals where the power $P(R)=\frac{120 R}{(0.4+R)^{2}}$ is increasing based on resistance $R$ .

Find the derivative of $f(x)=\left(x^{3}-3 x^{2}+1\right)^{-3}$ using the chain rule.

Find the remaining amount of technetium $-99 \mathrm{~m}$ from a $40 \mathrm{~g}$ sample after 2 days. Also, calculate the original value of a car sold for \$4200 with a 15% annual depreciation.

Find $g^{\prime}(4)$ given $f(4)=5$ , $f^{\prime}(4)=6$ , and $g(x)=\sqrt{x} f(x)$ .

Find the differential $d y$ of the function $d y=x(1-\cos (x))$ using $d x$ .

Calculate the surface area when a 6x16 rectangle rotates around line $z$ . How does it change if $z$ is a side?

Calculate the area between the curves $f(x) = x^2 + 4x + 4$ and $g(x) = 4x + 20$ from $x = -4$ to $x = 4$ .

Find the differential $d y$ for the function $y=10 x^{4/5}$ . Use " $d x$ " for $d x$ .

Sketch the graphs of $f$ and $f''$ given that $f'$ is a parabola with roots at (-2,0) and (2,0) and a minimum at (0,-4).

Find the differential $d y$ for the function $y = x \tan(x)$ .

Sketch the function $f$ and its second derivative $f''$ from the graph of $f'$ : a parabola with roots at -2 and 2, min at (0,-4).

Approximate the area under $y=x^{2}$ from $x=1$ to $x=4$ using a Right Endpoint method with 6 subdivisions.

Find the advertising spend $s$ (in thousands) for max profit given $P=-\frac{1}{10} s^{3}+15 s^{2}+400$ . What is $s$ ?

Find $\frac{d y}{d x}$ for the curve $e^{x^{2}-y}=y^{9 x+5}$ using logarithmic differentiation.

Find the derivative of $y=e^{-2 e^{-0.02 x}}$ . What is $y^{\prime}=\square$ ?

Find the limit: $\lim _{x \rightarrow 1^{-}} \tan ^{-1}\left(\frac{1}{x-1}\right)$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=e^{\sqrt{x-14}}$ .

Find the limit: $\lim _{x \rightarrow \infty} \tan ^{-1} x$ .

Find the limit: $\lim _{x \rightarrow \infty} \cos ^{-1}\left(\frac{\sqrt{x-1}}{\sqrt{2 x+1}}\right)$ .

Find the limit: $\lim _{x \rightarrow \infty} \sin ^{-1}\left(\frac{1-x}{2 x+1}\right)$ .

Differentiate the function $f(x)=\left(x+\frac{1}{x}\right) e^{4 x}$ . Find $\frac{\mathrm{d}}{\mathrm{dx}}f(x)=\square$ .

A well pumps 40000 L on day 1, decreasing by $5\%$ daily. Find total water pumped over its lifetime.

Find the vertical asymptote(s) of $g(x)=\frac{1}{(x-4)^{2}}$ . Choose A, B, or C and complete the equations if needed.

Find the limit: $\lim _{x \rightarrow 1} \cos ^{-1}\left(x+\frac{1}{2}\right)$ .

Differentiate $y=\frac{x}{-2 \sin x}$ with respect to $x$ .