Calculus

Problem 4801

Find the derivative $f^{\prime}(1)$ for the piecewise function $f(x)=\begin{cases} 5-3 x^{2} & x>1 \\ 8-6 x & x \leq 1 \end{cases}$ . If not differentiable, enter DNE.

See Solution

Problem 4802

Find the derivative of $f(x)=\frac{3 x+4}{7 x-5}$ at $x=0$ and provide the exact decimal answer. $f^{\prime}(0)=$

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Problem 4803

Find the derivative of $f(t)=\frac{2}{4-t}$ at $t=-5$ using the limit definition. $f^{\prime}(-5)=$

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Problem 4804

Find the tangent line equation for $f(x)=\frac{5}{9+x^{2}}$ at $x=3$ . Express as $y$ in terms of $x$ .

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Problem 4805

Find the tangent line equation for $f(x)=\frac{5}{9+x^{2}}$ at $x=3$ . Use $y=f(x)$ for the equation.

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Problem 4806

Find the derivative $v^{\prime}(a)$ for the dodecahedron volume $v=\frac{15+7 \sqrt{5}}{4} x^{3}$ at $x=a$ .

See Solution

Problem 4807

Find the derivative at $x=3$ : $f^{\prime}(3)=\lim _{h \rightarrow 0} \frac{(3+h)^{2}+6(3+h)-27}{h}$ .

See Solution

Problem 4808

Profit $P$ (in \ $thousands) from$ x $kg of coffee is given by$ P(x)=\frac{4x-200}{x+400} $. Find average profit, sketch, and rate of change at$ x=1000$.

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Problem 4809

Find the production level $x$ that minimizes the marginal cost $C(x)=x^{2}-120x+8500$ and the minimum cost.

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Problem 4810

Find the derivative $\frac{d y}{d x}$ for the function $y=\sin^{-1}(\sqrt{9 x^{2}})$ .

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Problem 4811

Determine if the function $f(x)=-3 x^{2}+6 x+1$ has a minimum or maximum, its location, and the value.

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Problem 4812

Find $\frac{d y}{d x}$ using the chain rule: $\frac{d y}{d x}=\frac{d y}{d u} \frac{d u}{d x}$ , where $y=5 u^{4}-5$ and $u=3 \sqrt{x}$ .

See Solution

Problem 4813

Find $h^{\prime}(-1)$ for $h(x)=\left(x^{4}+p(x)\right)^{3}$ using given values of $p(x)$ and $p^{\prime}(x)$ .

See Solution

Problem 4814

Find the derivative of $f(x)=0.5^{e^{x}}$ .

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Problem 4815

The Collegiate Investigator sells for \ $90 per copy. The cost to produce$ x $copies is$ C(x)=90+0.10x+0.001x^{2}$.
(a) Find the marginal profit function $P^{\prime}(x)$ . (b) Calculate the marginal profit for 500 copies. Interpret the results.

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Problem 4816

Given cost $C(x)=2x$ and revenue $R(x)=4x-0.01x^2$ , find marginal cost $C'(x)$ , revenue $R'(x)$ , and profit $P'(x)$ at $x=10$ .

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Problem 4817

Find the marginal profit for a car wash with $n=50$ workers using the formula $P=-300n+25n^{2}-0.005n^{4}$ . Interpret the result.

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Problem 4818

Find the 1st, 2nd, and 3rd derivatives of $f(x)=(x^{2}-x) \cdot e^{-0.5 x}$ using factoring.

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Problem 4819

Find the marginal cost function $C'(x)$ for $C(x)=100+43x-0.09x^2$ . At $x=100$ , find $C'(100)$ and average cost $\bar{C}(100)$ .

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Problem 4820

Given cost $C(x)=2x$ and revenue $R(x)=4x-0.01x^2$ , find marginal cost, revenue, profit, and compare actual vs. approximate for the 10th ounce.

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Problem 4821

Given $C(x)=2x$ and $R(x)=4x-0.01x^2$ , find marginal cost $C'(x)$ , revenue $R'(x)$ , and profit $P'(x)$ for the 10th unit.

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Problem 4822

The Collegiate Investigator sells for $90\phi$ per copy. Given cost $C(x)=90+0.10x+0.001x^{2}$ , find:
(a) Marginal profit function $P'(x)=$
(b) Marginal profit for 500 copies: loss from 501st copy is dollars.

See Solution

Problem 4823

Find the marginal cost function $C'(x)$ for $C(x)=150+2,200x-0.08x^2$ . Approximate and find the exact cost for the 5th commercial in dollars.

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Problem 4824

The newspaper sells for \ $904 per copy. The cost for$ x $copies is$ C(x)=90+0.10x+0.001x^{2}$.
(a) Find the marginal profit function $P^{\prime}(x)$ . (b) Calculate the marginal profit for 500 copies. The loss from producing the 501st copy is __ dollars.

See Solution

Problem 4825

Find the marginal cost function $C^{\prime}(x)$ for $C(x)=150+2,200 x-0.08 x^{2}$ . Approximate and find the exact cost for the 5th commercial.

See Solution

Problem 4826

Bestimme die Intervalle, in denen $f$ monoton wachsend oder fallend ist, und untersuche das Krümmungsverhalten. a) $f(x)=x^{2}-x-6$ b) $f(x)=x^{3}+x$ c) $f(x)=\frac{1}{9} x^{3}-x$

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Problem 4827

Bestimme die Ableitungen der Funktionen: a) $f(x)=\frac{1}{x}(x^{4}+3x^{2})$ , b) $f(t)=\sqrt[3]{t}(t^{3}+\frac{1}{t})$ , c) $f(s)=s^{2}(\sqrt{s}+\frac{1}{s^{4}})$ .

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Problem 4828

Find the derivative of $g(x)=\left(\frac{x^{2}+1}{x}\right)^{4}$ . Simplify your answer.

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Problem 4829

Franz investiert in Hasen, deren Population sich nach $h(t)=(240+20 t) \cdot e^{-0,05 t}$ entwickelt.
a) Wie viele Hasen hat er gekauft und wie viele sind es nach einem Jahr? b) Wachstumsrate zu Beginn (in Hasen/Monat)? c) Wann ist das Maximum erreicht? d) Wann verringert sich die Population am stärksten?

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Problem 4830

Find $\frac{d y}{d x}$ for $x=\sqrt{y}$ . Options: (A) $2 \sqrt{x}$ (B) $\sqrt{2} x$ (C) $2 \sqrt{y}$ (D) $\sqrt{2} y$ .

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Problem 4831

Find the value of $k$ given $r=2u^{5}$ and $u=3x^{2}+2$ , with $\frac{dr}{dx}=k(x)(3x^{2}+2)^{4}$ .

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Problem 4832

Nennen Sie zwei Funktionen, die für $x \rightarrow+\infty$ den Grenzwert 1 haben, und analysieren Sie ihr Verhalten für $x \rightarrow-\infty$ .

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Problem 4833

Nennen Sie zwei verschiedene Funktionen, die für $x \rightarrow+\infty$ den Grenzwert 1 haben, und analysieren Sie ihr Verhalten für $x \rightarrow-\infty$ .

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Problem 4834

Find the derivative of the function $f(x)=(x+k) \cdot e^{x}$ . What is $f^{\prime}(x)$ ?

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Problem 4835

Beurteilen Sie Naels Aussage über den Grenzwert von $\mathrm{f}(x) = 0,001 \mathrm{x}^{4}-0,11 \mathrm{x}^{2}+3$ im Unendlichen.

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Problem 4836

Find the derivative, roots, and max/min points of the function $f(x)=450x-3x^{2}$ .

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Problem 4837

Find the partial elasticities of $z(x, y) = 26e^{3x+2y}$ with respect to $x$ and $y$ . Determine coefficients $a, b, c, d, m, n, k$ .

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Problem 4838

Find an approximate value for $F(3.99,0.04)$ given $F(4,0)=-6$ , $F_{1}^{\prime}(4,0)=2$ , $F_{2}^{\prime}(4,0)=3$ . Round to two digits.

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Problem 4839

Find the first and second-order partial derivatives of $G(p, q)=5 p^{5}+20 q^{8}$ w.r.t. $p$ , and determine coefficients.

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Problem 4840

Find partial elasticities of $z=26 \cdot e^{3x+2y}$ with respect to $x$ and $y$ , and determine coefficients $a, b, c, d, m, n, k$ .

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Problem 4841

Find the linear approximation of $h(x_{1}, x_{2})=14 e^{5 x_{1}} \cdot \ln(7+x_{2})$ at $(-0.3,0.5)$ and coefficients $a$ , $b$ , $c$ , $d$ , $k$ .

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Problem 4842

Find the partial elasticities of $z=26 \cdot e^{3x+2y}$ with respect to $x$ and $y$ , and determine the coefficients $a$ , $b$ , $c$ , $d$ , $m$ , $n$ , $k$ .

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Problem 4843

Find an expression for $\frac{2 \cos 3 \theta + 2 \sin 5 \theta + 5 \theta^{2}}{5 \cos 5 \theta}$ when $\theta$ is small.

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Problem 4844

Find the derivative of $f(x) = -6x\sqrt{3x+2}$ and simplify it.

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Problem 4845

A ball is dropped from a height of $4.9 \mathrm{~m}$ . How long until it hits the ground?

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Problem 4846

Find the derivative of $f(x) = 2e^x - e^{-x}$ .

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Problem 4847

Find the derivative of $f(x)=2 e^{x}$ .

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Problem 4848

Find the derivative of $f(x)=2e^x$ .

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Problem 4849

Find the derivative of the function $f(x)=-e^{-x}$ .

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Problem 4850

Find the first three derivatives of $f(x)=2 \cdot e^{x}-e^{-x}$ and simplify each one.

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Problem 4851

Berechnen Sie die Ableitung von $f(x)=2 e^{x}$ . Ist die Produktregel erforderlich? Was geschieht mit der Zahl zwei?

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Problem 4852

Finde den Wert von $b$ (mit $b>0$ und $b \in \mathbb{N}$ ), sodass $\int_{-2}^{b} (4 - x^2) \, dx = 0$ .

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Problem 4853

Differentiate the function $u=\sqrt[3]{t}+6 \sqrt{t^{3}}$ . Find $u'$ .

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Problem 4854

Differentiate $g(x)=3 e^{x} \sqrt{x}$ ; find $g^{\prime}(x)$ .

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Problem 4855

Write the composite function as $f(g(x))$ with $u=g(x)$ and $y=f(u)$ for $y=\sqrt[3]{1+8x}$ . Find $\frac{dy}{dx}$ .

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Problem 4856

Find the derivative of $y=e^{3 x \cos x}$ . What is $y^{\prime}=$ ?

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Problem 4857

Find the light intensity $I$ (in thousands of foot-candles) that maximizes the photosynthesis rate $P=\frac{110 I}{I^{2}+I+9}$ .

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Problem 4858

Find the first and second derivatives, $y^{\prime}$ and $y^{\prime \prime}$ , for $y=e^{\alpha x} \sin \beta x$ .

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Problem 4859

Find the average rate of change of $C(t)$ over these intervals: (i) $[1.0,2.0]$ , (ii) $[1.5,2.0]$ , (iii) $[2.0,2.5]$ , (iv) $[2.0,3.0]$ . Also estimate $C^{\prime}(2)$ .

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Problem 4860

Find the max and min of $f(x)=x e^{-x^{2} / 50}$ on the interval $[-2,10]$ . What are the values?

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Problem 4861

Find the price elasticity of demand for the "Roasted Rooster" at \ $4.00 when$ q=\frac{39}{p^{0.85}}$. What is the % change in demand?

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Problem 4862

Find the derivative of $y=e^{3 x \cos x}$ , where $y'=\square$ .

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Problem 4863

Given $C(x)=3x$ and $R(x)=8x-0.01x^{2}$ , find the marginal cost, revenue, and profit functions.

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Problem 4864

What is the sum of the series $1=\frac{1}{2}+\frac{1}{8}-\frac{1}{48}+\cdots$ ?

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Problem 4865

Substitute $x=\pi / 2$ into the series for $\sin x$ : $1=\frac{\pi}{2}+\frac{\pi^{3}}{3 !}+\frac{\pi^{5}}{5 !}+\cdots$

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Problem 4866

Find the derivative of the function $y=e^{3 x \cos x}$ .

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Problem 4867

Identify the summation notation for the series $\frac{1}{2}-\frac{x}{2^{2}}+\frac{x^{2}}{2^{3} 2 !}-\cdots$ . Options include:
1. $\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{n}}{n ! 2^{n+1}}$
2. $\sum_{n=0}^{\infty} \frac{x^{n}}{n ! 2^{n+1}}$
3. $\sum_{n=1}^{\infty} \frac{x^{n}}{n ! 2^{n}}$
4. $\sum_{n=0}^{\infty} \frac{x^{n}}{n ! 2^{n}}$
5. $\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{n}}{n ! 2^{n}}$
6. $\sum_{n=1}^{\infty}(-1)^{n+1} \frac{x^{n}}{n ! 2^{n}}$

See Solution

Problem 4868

Find the first and second derivatives, $y^{\prime}$ and $y^{\prime \prime}$ , for $y=e^{\alpha x} \sin \beta x$ .

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Problem 4869

Find the first and second derivatives, $y'$ and $y''$ , of the function $y=e^{\alpha x} \sin \beta x$ .

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Problem 4870

Find $f^{\prime}(-1)$ if $f(x)=x^{12} h(x)$ , $h(-1)=2$ , and $h'(-1)=5$ .

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Problem 4871

Find the marginal revenue at 75 units for the revenue function $R(q)=-3 q^{2}+600 q$ . Answer: $M R(75)=$ \$.

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Problem 4872

Find $f^{\prime}(0)$ for $f(x)=3 x^{2}-7 x+7$ and the tangent line at $(0,7)$ in the form $y=m x+b$ .

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Problem 4873

Find the first and second derivatives $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ for the equation $4 \sqrt{y}-2 x=-4 y$ .

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Problem 4874

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan 2 x}{\sin x}$ .

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Problem 4875

Find the limit as $x$ approaches 0 for the expression $\frac{\tan n x}{\sin x}$ .

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Problem 4876

Find the velocity and acceleration vectors using $u_{r}$ and $u_{\theta}$ for $r=2 \sin 3 t$ and $\theta=4 t$ . $v=(\square) u_{r}+(\square) u_{\theta}$

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Problem 4877

Find the marginal profit for a business with 150 customers, given $p=2n^{2}-\sqrt{n}$ .

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Problem 4878

Find the force on a mass $m$ moving at 7 units/sec along $y=6x^2$ at point $(2^{1/2}, 12)$ using $F=ma$ . $F=(\square) i+(\square) j$

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Problem 4879

Find the derivative of $H(x)=\frac{(6 x+1)^{1/2}}{(3 x-2)^{4}}$ .

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Problem 4880

Find the derivative of $f(x)=\log _{4}(5 x+3)+7^{2 x-1}$ .

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Problem 4881

Find the limit: $\lim _{h \rightarrow 0} \frac{\ln (1+h)}{h}$ .

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Problem 4882

Find the limit: $\lim _{x \rightarrow 1} \frac{\sqrt{2 x-x^{4}}-\sqrt[3]{x}}{1-x^{\frac{1}{4}}}$ .

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Problem 4883

Find the first and second derivatives of $I(x)=2x-3+c^{-x}$ .

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Problem 4884

Calculate the limit: $\lim _{h \rightarrow 0} \frac{(1+3 h)^{\frac{1}{h}}}{4 e^{3+h}}$ .

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Problem 4885

Bestimme die Wendestellen des Graphen von $f(x)=\frac{1}{5} x^{5}-3 x^{4}+18 x^{3}-54 x^{2}$ . Wo ist $x=$ ?

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Problem 4886

Bestimme die Wendestellen von $f(x)=-\frac{1}{6} x^{4}+9 x^{2}$ . Wo hat der Graph von $f$ Wendepunkte?

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Problem 4887

Differentiate $y=\frac{(2 x+5)^{2 / 3}}{(6 x+7)^{2}(5 x-4)^{5}}$ using logarithmic differentiation.

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Problem 4888

Finde die Nullstellen der zweiten Ableitung $f''(x) = 10x^3 - 36x^2 + 18x$ . Setze $f''(x) = 0$ .

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Problem 4889

Berechne den Flächeninhalt zwischen dem Graphen von $f(x)=2 x^{2}-3 x-5$ und der $x$ -Achse im Intervall $[-1 ; 2,5]$ .

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Problem 4890

Überprüfen Sie, ob $F(x)=0,1 x^{4}-0,1$ und $G(x)=\frac{2}{20} x^{4}$ Stammfunktionen von $h(x)=\frac{2}{5} x^{3}$ sind.

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Problem 4891

Find the third derivative of $f(x)$ if $f''(x) = -x^2 + x + 6$ .

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Problem 4892

Find the inflection point of $f(x)=2 \cdot e^{x}-e^{-x}$ .

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Problem 4893

Find an expression for $\frac{4 \cos \theta + 4 \sin 4\theta + 3 \cos 4\theta}{5 \theta + 5 \cos 5\theta}$ when $\theta$ is small.

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Problem 4894

Find an expression for $\frac{2 \theta^{2}+5 \cos \theta+\cos 2 \theta}{\tan 5 \theta+5 \cos 5 \theta}$ when $\theta$ is small.

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Problem 4895

Find the derivative of $f(x) = \frac{-1}{(5x^{2} - x)^{3}}$ .

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Problem 4896

Calculate the integral from -1 to 4 of the function $3x^{2} - 4x + 1$ .

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Problem 4897

Bestimmen Sie, ob die Folge $a_n = (1/2)^n$ konvergent oder divergent ist und erläutern Sie Ihre Antwort.

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Problem 4898

Find the derivative $\frac{d y}{d x}$ for the equation 9 $x^{3}$ + 9 $y^{3}$ = 8.

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Problem 4899

Find the derivative $\frac{d y}{d x}$ for the equation: $7 x^{3}+y^{3}=4$ .

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Problem 4900

Find the derivative $\frac{d y}{d x}$ for the equation: $8 x^{3} y^{3}=1$ .

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Calculus

Problem 4801

Find the derivative f′(1)f^{\prime}(1)f′(1) for the piecewise function f(x)={5−3x2x>18−6xx≤1f(x)=\begin{cases} 5-3 x^{2} & x>1 \\ 8-6 x & x \leq 1 \end{cases}f(x)={5−3x28−6x​x>1x≤1​. If not differentiable, enter DNE.

Problem 4802

Find the derivative of f(x)=3x+47x−5f(x)=\frac{3 x+4}{7 x-5}f(x)=7x−53x+4​ at x=0x=0x=0 and provide the exact decimal answer. f′(0)=f^{\prime}(0)=f′(0)=

Problem 4803

Find the derivative of f(t)=24−tf(t)=\frac{2}{4-t}f(t)=4−t2​ at t=−5t=-5t=−5 using the limit definition. f′(−5)= f^{\prime}(-5)= f′(−5)=

Problem 4804

Find the tangent line equation for f(x)=59+x2f(x)=\frac{5}{9+x^{2}}f(x)=9+x25​ at x=3x=3x=3. Express as yyy in terms of xxx.

Problem 4805

Find the tangent line equation for f(x)=59+x2f(x)=\frac{5}{9+x^{2}}f(x)=9+x25​ at x=3x=3x=3. Use y=f(x)y=f(x)y=f(x) for the equation.

Problem 4806

Find the derivative v′(a)v^{\prime}(a)v′(a) for the dodecahedron volume v=15+754x3v=\frac{15+7 \sqrt{5}}{4} x^{3}v=415+75​​x3 at x=ax=ax=a.

Problem 4807

Find the derivative at x=3x=3x=3: f′(3)=lim⁡h→0(3+h)2+6(3+h)−27hf^{\prime}(3)=\lim _{h \rightarrow 0} \frac{(3+h)^{2}+6(3+h)-27}{h}f′(3)=limh→0​h(3+h)2+6(3+h)−27​.

Problem 4808

Profit PPP (in \thousands)from thousands) from thousands)fromxkgofcoffeeisgivenby kg of coffee is given by kgofcoffeeisgivenbyP(x)=\frac{4x-200}{x+400}.Findaverageprofit,sketch,andrateofchangeat. Find average profit, sketch, and rate of change at .Findaverageprofit,sketch,andrateofchangeatx=1000$.

Problem 4809

Find the production level xxx that minimizes the marginal cost C(x)=x2−120x+8500C(x)=x^{2}-120x+8500C(x)=x2−120x+8500 and the minimum cost.

Problem 4810

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=sin⁡−1(9x2)y=\sin^{-1}(\sqrt{9 x^{2}})y=sin−1(9x2​).

Problem 4811

Determine if the function f(x)=−3x2+6x+1f(x)=-3 x^{2}+6 x+1f(x)=−3x2+6x+1 has a minimum or maximum, its location, and the value.

Problem 4812

Find dydx\frac{d y}{d x}dxdy​ using the chain rule: dydx=dydududx\frac{d y}{d x}=\frac{d y}{d u} \frac{d u}{d x}dxdy​=dudy​dxdu​, where y=5u4−5y=5 u^{4}-5y=5u4−5 and u=3xu=3 \sqrt{x}u=3x​.

Problem 4813

Find h′(−1)h^{\prime}(-1)h′(−1) for h(x)=(x4+p(x))3h(x)=\left(x^{4}+p(x)\right)^{3}h(x)=(x4+p(x))3 using given values of p(x)p(x)p(x) and p′(x)p^{\prime}(x)p′(x).

Problem 4814

Find the derivative of f(x)=0.5exf(x)=0.5^{e^{x}}f(x)=0.5ex.

Problem 4815

Problem 4816

Given cost C(x)=2xC(x)=2xC(x)=2x and revenue R(x)=4x−0.01x2R(x)=4x-0.01x^2R(x)=4x−0.01x2, find marginal cost C′(x)C'(x)C′(x), revenue R′(x)R'(x)R′(x), and profit P′(x)P'(x)P′(x) at x=10x=10x=10.

Problem 4817

Find the marginal profit for a car wash with n=50n=50n=50 workers using the formula P=−300n+25n2−0.005n4P=-300n+25n^{2}-0.005n^{4}P=−300n+25n2−0.005n4. Interpret the result.

Problem 4818

Find the 1st, 2nd, and 3rd derivatives of f(x)=(x2−x)⋅e−0.5xf(x)=(x^{2}-x) \cdot e^{-0.5 x}f(x)=(x2−x)⋅e−0.5x using factoring.

Problem 4819

Find the marginal cost function C′(x)C'(x)C′(x) for C(x)=100+43x−0.09x2C(x)=100+43x-0.09x^2C(x)=100+43x−0.09x2. At x=100x=100x=100, find C′(100)C'(100)C′(100) and average cost Cˉ(100)\bar{C}(100)Cˉ(100).

Problem 4820

Given cost C(x)=2xC(x)=2xC(x)=2x and revenue R(x)=4x−0.01x2R(x)=4x-0.01x^2R(x)=4x−0.01x2, find marginal cost, revenue, profit, and compare actual vs. approximate for the 10th ounce.

Problem 4821

Given C(x)=2xC(x)=2xC(x)=2x and R(x)=4x−0.01x2R(x)=4x-0.01x^2R(x)=4x−0.01x2, find marginal cost C′(x)C'(x)C′(x), revenue R′(x)R'(x)R′(x), and profit P′(x)P'(x)P′(x) for the 10th unit.

Problem 4822

The Collegiate Investigator sells for 90ϕ90\phi90ϕ per copy. Given cost C(x)=90+0.10x+0.001x2C(x)=90+0.10x+0.001x^{2}C(x)=90+0.10x+0.001x2, find:(a) Marginal profit function P′(x)=P'(x)=P′(x)=(b) Marginal profit for 500 copies: loss from 501st copy is dollars.

Problem 4823

Find the marginal cost function C′(x)C'(x)C′(x) for C(x)=150+2,200x−0.08x2C(x)=150+2,200x-0.08x^2C(x)=150+2,200x−0.08x2. Approximate and find the exact cost for the 5th commercial in dollars.

Problem 4824

Problem 4825

Find the marginal cost function C′(x)C^{\prime}(x)C′(x) for C(x)=150+2,200x−0.08x2C(x)=150+2,200 x-0.08 x^{2}C(x)=150+2,200x−0.08x2. Approximate and find the exact cost for the 5th commercial.

Problem 4826

Bestimme die Intervalle, in denen fff monoton wachsend oder fallend ist, und untersuche das Krümmungsverhalten. a) f(x)=x2−x−6f(x)=x^{2}-x-6f(x)=x2−x−6 b) f(x)=x3+xf(x)=x^{3}+xf(x)=x3+x c) f(x)=19x3−xf(x)=\frac{1}{9} x^{3}-xf(x)=91​x3−x

Problem 4827

Bestimme die Ableitungen der Funktionen: a) f(x)=1x(x4+3x2)f(x)=\frac{1}{x}(x^{4}+3x^{2})f(x)=x1​(x4+3x2), b) f(t)=t3(t3+1t)f(t)=\sqrt[3]{t}(t^{3}+\frac{1}{t})f(t)=3t​(t3+t1​), c) f(s)=s2(s+1s4)f(s)=s^{2}(\sqrt{s}+\frac{1}{s^{4}})f(s)=s2(s​+s41​).

Problem 4828

Find the derivative of g(x)=(x2+1x)4g(x)=\left(\frac{x^{2}+1}{x}\right)^{4}g(x)=(xx2+1​)4. Simplify your answer.

Problem 4829

Problem 4830

Find dydx\frac{d y}{d x}dxdy​ for x=yx=\sqrt{y}x=y​. Options: (A) 2x2 \sqrt{x}2x​ (B) 2x\sqrt{2} x2​x (C) 2y2 \sqrt{y}2y​ (D) 2y\sqrt{2} y2​y.

Problem 4831

Find the value of kkk given r=2u5r=2u^{5}r=2u5 and u=3x2+2u=3x^{2}+2u=3x2+2, with drdx=k(x)(3x2+2)4\frac{dr}{dx}=k(x)(3x^{2}+2)^{4}dxdr​=k(x)(3x2+2)4.

Problem 4832

Nennen Sie zwei Funktionen, die für x→+∞x \rightarrow+\inftyx→+∞ den Grenzwert 1 haben, und analysieren Sie ihr Verhalten für x→−∞x \rightarrow-\inftyx→−∞.

Problem 4833

Nennen Sie zwei verschiedene Funktionen, die für x→+∞x \rightarrow+\inftyx→+∞ den Grenzwert 1 haben, und analysieren Sie ihr Verhalten für x→−∞x \rightarrow-\inftyx→−∞.

Problem 4834

Find the derivative of the function f(x)=(x+k)⋅exf(x)=(x+k) \cdot e^{x}f(x)=(x+k)⋅ex. What is f′(x)f^{\prime}(x)f′(x)?

Problem 4835

Beurteilen Sie Naels Aussage über den Grenzwert von f(x)=0,001x4−0,11x2+3\mathrm{f}(x) = 0,001 \mathrm{x}^{4}-0,11 \mathrm{x}^{2}+3f(x)=0,001x4−0,11x2+3 im Unendlichen.

Problem 4836

Find the derivative, roots, and max/min points of the function f(x)=450x−3x2f(x)=450x-3x^{2}f(x)=450x−3x2.

Problem 4837

Find the partial elasticities of z(x,y)=26e3x+2yz(x, y) = 26e^{3x+2y}z(x,y)=26e3x+2y with respect to xxx and yyy. Determine coefficients a,b,c,d,m,n,ka, b, c, d, m, n, ka,b,c,d,m,n,k.

Problem 4838

Find an approximate value for F(3.99,0.04)F(3.99,0.04)F(3.99,0.04) given F(4,0)=−6F(4,0)=-6F(4,0)=−6, F1′(4,0)=2F_{1}^{\prime}(4,0)=2F1′​(4,0)=2, F2′(4,0)=3F_{2}^{\prime}(4,0)=3F2′​(4,0)=3. Round to two digits.

Problem 4839

Find the first and second-order partial derivatives of G(p,q)=5p5+20q8G(p, q)=5 p^{5}+20 q^{8}G(p,q)=5p5+20q8 w.r.t. ppp, and determine coefficients.

Problem 4840

Find partial elasticities of z=26⋅e3x+2yz=26 \cdot e^{3x+2y}z=26⋅e3x+2y with respect to xxx and yyy, and determine coefficients a,b,c,d,m,n,ka, b, c, d, m, n, ka,b,c,d,m,n,k.

Problem 4841

Find the linear approximation of h(x1,x2)=14e5x1⋅ln⁡(7+x2)h(x_{1}, x_{2})=14 e^{5 x_{1}} \cdot \ln(7+x_{2})h(x1​,x2​)=14e5x1​⋅ln(7+x2​) at (−0.3,0.5)(-0.3,0.5)(−0.3,0.5) and coefficients aaa, bbb, ccc, ddd, kkk.

Find the derivative $f^{\prime}(1)$ for the piecewise function $f(x)=\begin{cases} 5-3 x^{2} & x>1 \\ 8-6 x & x \leq 1 \end{cases}$ . If not differentiable, enter DNE.

Find the derivative of $f(x)=\frac{3 x+4}{7 x-5}$ at $x=0$ and provide the exact decimal answer. $f^{\prime}(0)=$

Find the derivative of $f(t)=\frac{2}{4-t}$ at $t=-5$ using the limit definition. $f^{\prime}(-5)=$

Find the tangent line equation for $f(x)=\frac{5}{9+x^{2}}$ at $x=3$ . Express as $y$ in terms of $x$ .

Find the tangent line equation for $f(x)=\frac{5}{9+x^{2}}$ at $x=3$ . Use $y=f(x)$ for the equation.

Find the derivative $v^{\prime}(a)$ for the dodecahedron volume $v=\frac{15+7 \sqrt{5}}{4} x^{3}$ at $x=a$ .

Find the derivative at $x=3$ : $f^{\prime}(3)=\lim _{h \rightarrow 0} \frac{(3+h)^{2}+6(3+h)-27}{h}$ .

Profit $P$ (in \ $thousands) from$ x $kg of coffee is given by$ P(x)=\frac{4x-200}{x+400} $. Find average profit, sketch, and rate of change at$ x=1000$.

Find the production level $x$ that minimizes the marginal cost $C(x)=x^{2}-120x+8500$ and the minimum cost.

Find the derivative $\frac{d y}{d x}$ for the function $y=\sin^{-1}(\sqrt{9 x^{2}})$ .

Determine if the function $f(x)=-3 x^{2}+6 x+1$ has a minimum or maximum, its location, and the value.

Find $\frac{d y}{d x}$ using the chain rule: $\frac{d y}{d x}=\frac{d y}{d u} \frac{d u}{d x}$ , where $y=5 u^{4}-5$ and $u=3 \sqrt{x}$ .

Find $h^{\prime}(-1)$ for $h(x)=\left(x^{4}+p(x)\right)^{3}$ using given values of $p(x)$ and $p^{\prime}(x)$ .

Find the derivative of $f(x)=0.5^{e^{x}}$ .

Given cost $C(x)=2x$ and revenue $R(x)=4x-0.01x^2$ , find marginal cost $C'(x)$ , revenue $R'(x)$ , and profit $P'(x)$ at $x=10$ .

Find the marginal profit for a car wash with $n=50$ workers using the formula $P=-300n+25n^{2}-0.005n^{4}$ . Interpret the result.

Find the 1st, 2nd, and 3rd derivatives of $f(x)=(x^{2}-x) \cdot e^{-0.5 x}$ using factoring.

Find the marginal cost function $C'(x)$ for $C(x)=100+43x-0.09x^2$ . At $x=100$ , find $C'(100)$ and average cost $\bar{C}(100)$ .

Given cost $C(x)=2x$ and revenue $R(x)=4x-0.01x^2$ , find marginal cost, revenue, profit, and compare actual vs. approximate for the 10th ounce.

Given $C(x)=2x$ and $R(x)=4x-0.01x^2$ , find marginal cost $C'(x)$ , revenue $R'(x)$ , and profit $P'(x)$ for the 10th unit.

The Collegiate Investigator sells for $90\phi$ per copy. Given cost $C(x)=90+0.10x+0.001x^{2}$ , find:
(a) Marginal profit function $P'(x)=$
(b) Marginal profit for 500 copies: loss from 501st copy is dollars.

Find the marginal cost function $C'(x)$ for $C(x)=150+2,200x-0.08x^2$ . Approximate and find the exact cost for the 5th commercial in dollars.

Find the marginal cost function $C^{\prime}(x)$ for $C(x)=150+2,200 x-0.08 x^{2}$ . Approximate and find the exact cost for the 5th commercial.

Bestimme die Intervalle, in denen $f$ monoton wachsend oder fallend ist, und untersuche das Krümmungsverhalten. a) $f(x)=x^{2}-x-6$ b) $f(x)=x^{3}+x$ c) $f(x)=\frac{1}{9} x^{3}-x$

Bestimme die Ableitungen der Funktionen: a) $f(x)=\frac{1}{x}(x^{4}+3x^{2})$ , b) $f(t)=\sqrt[3]{t}(t^{3}+\frac{1}{t})$ , c) $f(s)=s^{2}(\sqrt{s}+\frac{1}{s^{4}})$ .

Find the derivative of $g(x)=\left(\frac{x^{2}+1}{x}\right)^{4}$ . Simplify your answer.

Find $\frac{d y}{d x}$ for $x=\sqrt{y}$ . Options: (A) $2 \sqrt{x}$ (B) $\sqrt{2} x$ (C) $2 \sqrt{y}$ (D) $\sqrt{2} y$ .

Find the value of $k$ given $r=2u^{5}$ and $u=3x^{2}+2$ , with $\frac{dr}{dx}=k(x)(3x^{2}+2)^{4}$ .

Nennen Sie zwei Funktionen, die für $x \rightarrow+\infty$ den Grenzwert 1 haben, und analysieren Sie ihr Verhalten für $x \rightarrow-\infty$ .

Nennen Sie zwei verschiedene Funktionen, die für $x \rightarrow+\infty$ den Grenzwert 1 haben, und analysieren Sie ihr Verhalten für $x \rightarrow-\infty$ .

Find the derivative of the function $f(x)=(x+k) \cdot e^{x}$ . What is $f^{\prime}(x)$ ?

Beurteilen Sie Naels Aussage über den Grenzwert von $\mathrm{f}(x) = 0,001 \mathrm{x}^{4}-0,11 \mathrm{x}^{2}+3$ im Unendlichen.

Find the derivative, roots, and max/min points of the function $f(x)=450x-3x^{2}$ .

Find the partial elasticities of $z(x, y) = 26e^{3x+2y}$ with respect to $x$ and $y$ . Determine coefficients $a, b, c, d, m, n, k$ .

Find an approximate value for $F(3.99,0.04)$ given $F(4,0)=-6$ , $F_{1}^{\prime}(4,0)=2$ , $F_{2}^{\prime}(4,0)=3$ . Round to two digits.

Find the first and second-order partial derivatives of $G(p, q)=5 p^{5}+20 q^{8}$ w.r.t. $p$ , and determine coefficients.

Find partial elasticities of $z=26 \cdot e^{3x+2y}$ with respect to $x$ and $y$ , and determine coefficients $a, b, c, d, m, n, k$ .

Find the linear approximation of $h(x_{1}, x_{2})=14 e^{5 x_{1}} \cdot \ln(7+x_{2})$ at $(-0.3,0.5)$ and coefficients $a$ , $b$ , $c$ , $d$ , $k$ .

Find the partial elasticities of $z=26 \cdot e^{3x+2y}$ with respect to $x$ and $y$ , and determine the coefficients $a$ , $b$ , $c$ , $d$ , $m$ , $n$ , $k$ .

Find an expression for $\frac{2 \cos 3 \theta + 2 \sin 5 \theta + 5 \theta^{2}}{5 \cos 5 \theta}$ when $\theta$ is small.

Find the derivative of $f(x) = -6x\sqrt{3x+2}$ and simplify it.

A ball is dropped from a height of $4.9 \mathrm{~m}$ . How long until it hits the ground?

Find the derivative of $f(x) = 2e^x - e^{-x}$ .

Find the derivative of $f(x)=2 e^{x}$ .

Find the derivative of $f(x)=2e^x$ .

Find the derivative of the function $f(x)=-e^{-x}$ .

Find the first three derivatives of $f(x)=2 \cdot e^{x}-e^{-x}$ and simplify each one.

Berechnen Sie die Ableitung von $f(x)=2 e^{x}$ . Ist die Produktregel erforderlich? Was geschieht mit der Zahl zwei?

Finde den Wert von $b$ (mit $b>0$ und $b \in \mathbb{N}$ ), sodass $\int_{-2}^{b} (4 - x^2) \, dx = 0$ .

Differentiate the function $u=\sqrt[3]{t}+6 \sqrt{t^{3}}$ . Find $u'$ .

Differentiate $g(x)=3 e^{x} \sqrt{x}$ ; find $g^{\prime}(x)$ .

Write the composite function as $f(g(x))$ with $u=g(x)$ and $y=f(u)$ for $y=\sqrt[3]{1+8x}$ . Find $\frac{dy}{dx}$ .

Find the derivative of $y=e^{3 x \cos x}$ . What is $y^{\prime}=$ ?

Find the light intensity $I$ (in thousands of foot-candles) that maximizes the photosynthesis rate $P=\frac{110 I}{I^{2}+I+9}$ .

Find the first and second derivatives, $y^{\prime}$ and $y^{\prime \prime}$ , for $y=e^{\alpha x} \sin \beta x$ .

Find the average rate of change of $C(t)$ over these intervals: (i) $[1.0,2.0]$ , (ii) $[1.5,2.0]$ , (iii) $[2.0,2.5]$ , (iv) $[2.0,3.0]$ . Also estimate $C^{\prime}(2)$ .

Find the max and min of $f(x)=x e^{-x^{2} / 50}$ on the interval $[-2,10]$ . What are the values?

Find the price elasticity of demand for the "Roasted Rooster" at \ $4.00 when$ q=\frac{39}{p^{0.85}}$. What is the % change in demand?

Find the derivative of $y=e^{3 x \cos x}$ , where $y'=\square$ .

Given $C(x)=3x$ and $R(x)=8x-0.01x^{2}$ , find the marginal cost, revenue, and profit functions.

What is the sum of the series $1=\frac{1}{2}+\frac{1}{8}-\frac{1}{48}+\cdots$ ?

Substitute $x=\pi / 2$ into the series for $\sin x$ : $1=\frac{\pi}{2}+\frac{\pi^{3}}{3 !}+\frac{\pi^{5}}{5 !}+\cdots$

Find the derivative of the function $y=e^{3 x \cos x}$ .

Find the first and second derivatives, $y^{\prime}$ and $y^{\prime \prime}$ , for $y=e^{\alpha x} \sin \beta x$ .

Find the first and second derivatives, $y'$ and $y''$ , of the function $y=e^{\alpha x} \sin \beta x$ .

Find $f^{\prime}(-1)$ if $f(x)=x^{12} h(x)$ , $h(-1)=2$ , and $h'(-1)=5$ .

Find the marginal revenue at 75 units for the revenue function $R(q)=-3 q^{2}+600 q$ . Answer: $M R(75)=$ \$.

Find $f^{\prime}(0)$ for $f(x)=3 x^{2}-7 x+7$ and the tangent line at $(0,7)$ in the form $y=m x+b$ .

Find the first and second derivatives $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ for the equation $4 \sqrt{y}-2 x=-4 y$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan 2 x}{\sin x}$ .