Calculus

Problem 5801

Calculate the relative rate of change of $f(x)=6 x^{2}-\ln x$ at $x=8$ .

See Solution

Problem 5802

Calculate the relative rate of change of $f(x)=276-3x$ at $x=34$ .

See Solution

Problem 5803

Estimate $f(8)$ using local linear approximation given $f(12)=10$ and $f^{\prime}(12)=-3$ .

See Solution

Problem 5804

Calculate the relative rate of change of the function $f(x)=10+2 e^{-2 x}$ .

See Solution

Problem 5805

Gegeben ist eine differenzierbare Funktion g. Bestimmen Sie die Ableitungen für $f_{1}(x)=g(3 x)$ , $f_{2}(x)=g(1-x)$ und $f_{3}(x)=g\left(\frac{1}{x}\right)$ .

See Solution

Problem 5806

What does $h^{\prime}(12)<0$ indicate about wind speed at 12 km from a hurricane's center?

See Solution

Problem 5807

Calculate the percentage rate of change of $f(x)$ where $f(x)=300+65x$ at $x=7$ .

See Solution

Problem 5808

Find the percentage rate of change of $f(x)$ at $x=45$ for $f(x)=5400-2 x^{2}$ .

See Solution

Problem 5809

Find $f^{\prime}(x)$ for $f(x)=3 x^{2}+2$ using $\frac{f(x+h)-f(x)}{h}$ as $h$ approaches 0.

See Solution

Problem 5810

A plane at 1 mile altitude flies at 500 mph. Find the rate of distance increase from the radar when it's 2 miles away.

See Solution

Problem 5811

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

See Solution

Problem 5812

Find the derivative of $e^{x} \ln x + x^{2} \cos x$ using the product and chain rules.

See Solution

Problem 5813

Find the average velocity of a particle with position $x(t)=(3.5 \mathrm{~m/s}) t-(5.0 \mathrm{~m/s}^2) t^2$ from $t=0.30 \mathrm{~s}$ to $t=0.40 \mathrm{~s}$ .

See Solution

Problem 5814

Find $f^{\prime}(2)$ for $f(x)=\frac{g(x)}{h(x)}$ given $g(2)=5$ , $g^{\prime}(2)=-1$ , $h(2)=4$ , and $h^{\prime}(2)=3$ .

See Solution

Problem 5815

Calculate the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{3 x+2}-\sqrt{x+2}}{4^{x}+5^{x}-2}$ .

See Solution

Problem 5816

Finde die Tangentengleichungen der Funktionen $f$ , die parallel zu den Geraden $g$ sind. a) $f(x)=x^{2}-9 x+13$ , $g: 3 x-5 y=7$ b) $f(x)=-2 x^{2}+\frac{3}{4} x-1$ , $g: 2 x+4 y=-2$ c) $f(x)=\frac{x^{3}}{3}+\frac{3 x^{2}}{2}-7 x+1$ , $g: 6 x-2 y=1$ d) $f(x)=\frac{x^{3}}{3}+\frac{x^{2}}{30}+4$ , $g: 2 x-15 y=5$

See Solution

Problem 5817

Find the derivative of $f(x)=-5 x^{4}+5 x^{3}+2 x^{2}+5 x+5$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 5818

Find the missing part of the derivative for $k(x)=\left(5+5 x^{2}\right)^{-3}$ : $k^{\prime}(x)=\left(-3\left(5+5 x^{2}\right)^{-4} \quad \text { ? }\right)(10 x)$

See Solution

Problem 5819

Leiten Sie die Funktionen ab: (a) $f(x)=\sin (3 x+3)$ , (b) $f(t)=2 \cdot \cos (\pi t-2)$ , (c) $f(t)=-\cos (5 t)$ , (d) $f(x)=-4 \cdot \sin (\pi-x)$ , (e) $f(t)=\sin (t-6)$ , (f) $f(x)=\cos (3 x+\pi)$ .

See Solution

Problem 5820

Bestimme die Steigung von $f(x)=(2x-1)^{3}$ bei $P(1 \mid f(1))$ , finde $a$ mit $g^{\prime}(a)=-2$ für $g(x)=\frac{1}{(x-1)^{2}}$ , und prüfe, ob $h(x)=\sqrt{3x+5}$ waagerechte Tangenten hat.

See Solution

Problem 5821

Find the limit: $\lim _{t \rightarrow \sqrt{2}} \frac{\sqrt{3} t - t}{4^{t^{2}-2} + 5^{t^{2}-2} - 2}$ .

See Solution

Problem 5822

Verify if $\left(\frac{f}{g}\right)^{\prime}(x)=\frac{f^{\prime}(x)}{g^{\prime}(x)}$ for $f(x)=8x+7$ and $g(x)=5x$ .

See Solution

Problem 5823

Calculate the integral from 4 to 9 of the function $\frac{1}{x^{2}}$ .

See Solution

Problem 5824

Verify if the derivative of the quotient $\left(\frac{f}{g}\right)^{\prime}(x)$ equals $\frac{f^{\prime}(x)}{g^{\prime}(x)}$ for $f(x)=7x+3$ and $g(x)=8x$ .

See Solution

Problem 5825

Check if the derivative of a quotient is the quotient of the derivatives for $f(x)=7x+3$ and $g(x)=8x$ .

See Solution

Problem 5826

Find the derivative of $f(x)=\frac{\sin(3x)}{4x^{2}-6x}$ .

See Solution

Problem 5827

Find the limit: $\lim _{x \rightarrow-2} \frac{\sqrt[3]{4 x^{2}-15}-1}{\sqrt{x+3}-1}$ .

See Solution

Problem 5828

Bestimmen Sie die Stammfunktion $F$ von $f(x)=3 x^{2}-2 x$ , die durch den Punkt $P(2, -1)$ verläuft. Wählen Sie $C$ passend.

See Solution

Problem 5829

Ein Kreuzfahrtschiff hat 2500 Passagiere. Am Tag 1 erkranken 22 Passagiere. Zeigen Sie, dass $f^{\prime}(t)=2498 \cdot 0,008 \cdot e^{-0,008 \cdot t}$ gilt und bestimmen Sie, wie lange medizinisches Personal benötigt wird.

See Solution

Problem 5830

Verify if $\left(\frac{f}{g}\right)^{\prime}(x)=\frac{f^{\prime}(x)}{g^{\prime}(x)}$ for $f(x)=9x+5$ , $g(x)=2x$ .

See Solution

Problem 5831

Verify if $\left(\frac{f}{g}\right)^{\prime}(x)=\frac{f^{\prime}(x)}{g^{\prime}(x)}$ is true for $f(x)=3x+4$ , $g(x)=7x$ .

See Solution

Problem 5832

Find the limit: $\lim _{x \rightarrow \infty} \frac{12 x^{5}+6 x-1}{5 x^{4}-4 x^{3}+3}$ .

See Solution

Problem 5833

Find the derivative $f'(x)$ of $f(x)=4x(\sin x+\cos x)$ and calculate $f'(2)$ rounded to the nearest hundredth.

See Solution

Problem 5834

Calculate the limit: $\lim _{x \rightarrow 1} \frac{\sqrt[3]{7+x^{3}}-\sqrt[3]{9-x}}{x-1}$ .

See Solution

Problem 5835

Check if the derivative of a product $(f g)^{\prime}(x)$ equals the product of derivatives $f^{\prime}(x) g^{\prime}(x)$ for $f(x)=5x+4$ , $g(x)=3x+2$ .

See Solution

Problem 5836

Find the derivative $f'(x)$ for $f(x)=3x(\sin x+\cos x)$ and calculate $f'(3)$ rounded to the nearest hundredth.

See Solution

Problem 5837

Find the selling price $x$ that maximizes profit $P(x) = R(x) - C(x)$ , where $C(x)$ and $R(x)$ are defined.

See Solution

Problem 5838

Finde die Ableitung von $f(x) = (x-1)(x-3)^{2}$ und vereinfache $f'(x)$ .

See Solution

Problem 5839

Find the degree 3 Taylor expansion of $f(x) = \ln(x^2 + 8x + 17)$ around $x = -4$ , including the best big- $O$ term.

See Solution

Problem 5840

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

See Solution

Problem 5841

Sketch a function graph and check if $\mathrm{f}$ is continuous at $\mathrm{x}=2$ : $f(2)=-3$ , $\lim _{x \rightarrow 2^{-}} f(x)=3$ , $\lim _{x \rightarrow 2^{+}} f(x)=-3$ .

See Solution

Problem 5842

Given the production function $Y=Z K+B N$ , find output per worker $y$ in terms of capital per worker $k$ . Determine the marginal and average product of $k$ . Then derive the dynamic equation for $k$ . Discuss conditions for steady state and endogenous growth. Analyze growth rates of capital, output, and consumption per worker. Finally, calculate long-run growth rate of output per worker for $s=0.4, Z=1, B=2, d=0.08, n=0.02$ and compare with $B=5$ .

See Solution

Problem 5843

Find $f^{\prime}(0)$ for the piecewise function: $f(x)=-2x^{2}+7x$ for $x<0$ and $f(x)=3x^{2}-2$ for $x\geq0$ .

See Solution

Problem 5844

Find the limits of the function $f(x)$ at $x=0$ :
1. $\lim _{x \rightarrow 0^{-}} f(x)$
2. $\lim _{x \rightarrow 0^{+}} f(x)$
Choose the correct answer:
A. $2 ;-1$ B. 2; Does not exist C. $-1 ; 2$ D. Does not exist; does not exist

See Solution

Problem 5845

Identify vertical asymptotes for $g(x)=\frac{x}{6-x}$ and describe limits at $x=6$ .

See Solution

Problem 5846

Sketch a function graph that meets these conditions: $f(2)=1$ and $\lim_{x \rightarrow 2} f(x)=1$ . Is $f$ continuous at $x=2$ ?

See Solution

Problem 5847

Find the limit as $x$ approaches -10 from the left for $f(x)=\frac{1}{x+10}$ . Options: A. -1 B. $\infty$ C. 0 D. $-\infty$

See Solution

Problem 5848

Find $\lim _{x \rightarrow 0} f(x)$ and $f(0)$ from the graph. Options: A, B, C, D based on the graph's behavior.

See Solution

Problem 5849

A bucket catches water leaking at $0.5 \mathrm{~cm}^{3}$ per minute. Find $\frac{dh}{dt}$ when $h=10 \mathrm{~cm}$ .

See Solution

Problem 5850

Find the derivative of $f(x)=\frac{\sin 3 x}{\cos 2 x}$ at $x=\frac{\pi}{6}$ .

See Solution

Problem 5851

Find the local linear approximation of $f(x)=\frac{5 x+7}{x-4}$ near $x=5$ . Answer: $f(x) \approx$

See Solution

Problem 5852

Find the vertical asymptotes of $f(x)=\frac{x-7}{x^{2}-49}$ and evaluate the limits: a. $\lim _{x \rightarrow 7} f(x)$ , b. $\lim _{x \rightarrow-7^{-}} f(x)$ , c. $\lim _{x \rightarrow-7^{+}} f(x)$ .

See Solution

Problem 5853

Determine vertical asymptotes $x=a$ for $f(x)=\frac{x^{3}-11 x^{2}+28 x}{x^{2}-7 x}$ . Evaluate limits as $x$ approaches $a$ .

See Solution

Problem 5854

Estimate $p^{\prime \prime}(1997)$ using the percentages of seniors using marijuana from 1991 to 1999. Choose A, B, C, D, or E.

See Solution

Problem 5855

Find $f'(x)$ for $f(x)=\frac{\sqrt{x}-6}{\sqrt{x}+6}$ and calculate $f'(4)$ (round to the nearest hundredth).

See Solution

Problem 5856

Find the derivative $f'(x)$ of $f(x)=5x(\sin x+\cos x)$ and calculate $f'(3)$ rounded to the nearest hundredth.

See Solution

Problem 5857

Prove that $y=e^{x}(\sin 2 x+\cos 2 x)$ satisfies $y^{\prime \prime}-2 y^{\prime}+5 y=0$ .

See Solution

Problem 5858

Find the derivative of the function $f(x)=\sec(g(x))$ , i.e., compute $f'(x)$ .

See Solution

Problem 5859

A steak at $25^{\circ} \mathrm{F}$ thaws in a $71^{\circ} \mathrm{F}$ room. After 9 mins it's $36^{\circ} \mathrm{F}$ . When is it $60^{\circ} \mathrm{F}$ ?

See Solution

Problem 5860

Déterminez si la suite $a_{n}=\frac{3 n^{4}}{\sqrt{5 n^{8}+4 n^{5}}}$ converge et trouvez sa limite ou écrivez diverge.

See Solution

Problem 5861

A particle moves on the circle $x^{2}+y^{2}=100$ . At $(8,6)$ , if $\frac{d x}{d t}=5$ , find $\frac{d y}{d t}$ .

See Solution

Problem 5862

Determine the intervals where the function $f(x)=3x^{2}-18x+24$ is increasing.

See Solution

Problem 5863

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(-11 x^{-4}\right)$ .

See Solution

Problem 5864

Find the limit as $x$ approaches infinity for the expression $-11 x^{-4}$ .

See Solution

Problem 5865

Find the rate of change of gravitational force $F(r)=\frac{2.99 \times 10^{16}}{r^{2}}$ at $r=6.77 \times 10^{6}$ m.

See Solution

Problem 5866

Find the limit: $\lim _{x \rightarrow \infty} \frac{8+8 x+9 x^{3}}{x^{3}}.$

See Solution

Problem 5867

The logistic growth function $f(t)=\frac{107,000}{1+5100 e^{-t}}$ models flu cases.
a. Find initial cases. b. Find cases after 4 weeks. c. Determine maximum cases.

See Solution

Problem 5868

Find the limit as $x$ approaches infinity of $e^{x}$ .

See Solution

Problem 5869

Find the tangent plane and normal line equations for $f(x, y)=x e^{x y}$ at $(1,0,1)$ .

See Solution

Problem 5870

Find $\partial f / \partial r$ for $f(x, y)=y^{2}+6 x^{3}$ , where $x(r, s)=r s$ and $y(r, s)=r^{2}+s^{2}$ .

See Solution

Problem 5871

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sin 15 x}{13 x}$ .

See Solution

Problem 5872

Find the rate of change of power $P(R)=\frac{2.1 R}{(R+0.5)^{2}}$ W at $R=3 \Omega$ . Round to four decimal places.

See Solution

Problem 5873

Find when the object moves right on the $x$ -axis given $x(t)=\frac{1}{2} t^{4}-\frac{10}{3} t^{3}+4 t^{2}$ . Options: $(0,1) \cup(4, \infty)$ , $(0,1) \cup(6, \infty)$ , $\left(0, \frac{2}{3}\right) \cup\left(\frac{8}{3}, \infty\right)$ , $(1,4)$ , $\left(0, \frac{1}{2}\right) \cup(2, \infty)$ .

See Solution

Problem 5874

Find the time intervals when the object moves right given $x(t)=\frac{1}{2} t^{4}-\frac{10}{3} t^{3}+4 t^{2}$ .

See Solution

Problem 5875

Find the limit: $\lim _{x \rightarrow 7} \frac{x^{2}-6 x-7}{-2 x+14}$ .

See Solution

Problem 5876

Find the limit: $\lim _{x \rightarrow 0} \frac{x^{2}-7 x}{x^{2}-x}$ in simplest form.

See Solution

Problem 5877

Find the limit: $\lim _{x \rightarrow-3} \frac{x^{2}+11 x+24}{x^{2}+3 x}$ .

See Solution

Problem 5878

Find the limit as $x$ approaches infinity: $\lim_{x \rightarrow \infty} \frac{14x^{3}+10-49x^{4}-35x}{x+9x^{4}-4-36x^{3}}$ . State DNE if infinite.

See Solution

Problem 5879

Find the limit as $x$ approaches infinity for $\frac{(8+x^{2})(2x-7)}{(2x^{3}+1)(3x^{3}+7)}$ . State DNE if infinite.

See Solution

Problem 5880

Find the limit: $\lim _{x \rightarrow \infty} \frac{(3 x-2)(6+5 x^{3})}{(1-7 x^{2})(5 x+6)}$ . State if it DNE.

See Solution

Problem 5881

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

See Solution

Problem 5882

Find the limit: $\lim _{x \rightarrow \infty} \frac{-2(4 x^{3}-1)}{(2 x^{3}-9)(3-2 x)}$ . State if DNE.

See Solution

Problem 5883

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{8 x^{12}+49 x^{6}-23 x^{4}}}{2 x+4 x^{4}+6}$ . State if DNE.

See Solution

Problem 5884

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sqrt{-47 x^{7}+x^{8}}}{5+4 x^{2}+9 x^{4}}$ . If infinite, state DNE.

See Solution

Problem 5885

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{-20 x^{3}+x^{4}}}{x+10 x^{3}+3 x^{2}}$ . State if it DNE.

See Solution

Problem 5886

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{27 x^{9}-30 x^{3}}}{x^{3}+7+7 x^{4}}$ . State if it DNE.

See Solution

Problem 5887

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{40+64 x^{13}}}{9 x^{4}+4 x}$ . State DNE if infinite.

See Solution

Problem 5888

Find the time $t$ for maximum rabbit population given $P(t)=140t-0.3t^4+1000$ and when it disappears.

See Solution

Problem 5889

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{27 x^{9}-30 x^{3}}}{x^{3}+7+7 x^{4}}$ . State DNE if infinite.

See Solution

Problem 5890

Find the velocity of a ball dropped from $150 \mathrm{~m}$ when it hits the ground, using $g=9.8 \mathrm{~m/s}^2$ .

See Solution

Problem 5891

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sqrt{49 x^{6}+46 x^{2}+42 x^{5}}}{8 x^{2}+5+5 x^{3}}$ . State DNE if infinite.

See Solution

Problem 5892

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{36 x^{4}-45 x-41}}{9 x+5 x^{2}+2}$ . State if it DNE.

See Solution

Problem 5893

Find the maximum rabbit population and when it occurs for $P(t)=140t-0.3t^4+1000$ . Also, determine when it disappears.

See Solution

Problem 5894

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{-16 x^{5}-31+25 x^{10}}}{x^{2}+7 x^{4}+1}$ . State DNE if infinite.

See Solution

Problem 5895

Find the tangent line to $f(x)=-2 \arccos (x)$ at $x=-\frac{\sqrt{3}}{2}$ .

See Solution

Problem 5896

Determine the limit as $x$ approaches infinity for $\frac{x^{72}}{2^{x}}$ . What does it equal?

See Solution

Problem 5897

Find the horizontal asymptotes for the function $f(x)=\frac{\sqrt{4 x^{4}-25 x^{3}}}{2 x-5 x^{2}}$ .

See Solution

Problem 5898

Determine the limit: $\lim _{x \rightarrow \infty} \frac{8 \log _{5} x-6 x^{36}}{x^{43}}$ . What does it equal?

See Solution

Problem 5899

Evaluate the limit: $L = \lim _{x \rightarrow 0} \frac{7-\cos (x)-10 x}{3 x-1}$

See Solution

Problem 5900

Find the half-life of Technetium-99 if a 250 g sample decays to 62.5 g in 319,500 years.

See Solution

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Calculus

Problem 5801

Calculate the relative rate of change of f(x)=6x2−ln⁡xf(x)=6 x^{2}-\ln xf(x)=6x2−lnx at x=8x=8x=8.

Problem 5802

Calculate the relative rate of change of f(x)=276−3xf(x)=276-3xf(x)=276−3x at x=34x=34x=34.

Problem 5803

Estimate f(8)f(8)f(8) using local linear approximation given f(12)=10f(12)=10f(12)=10 and f′(12)=−3f^{\prime}(12)=-3f′(12)=−3.

Problem 5804

Calculate the relative rate of change of the function f(x)=10+2e−2xf(x)=10+2 e^{-2 x}f(x)=10+2e−2x.

Problem 5805

Gegeben ist eine differenzierbare Funktion g. Bestimmen Sie die Ableitungen für f1(x)=g(3x)f_{1}(x)=g(3 x)f1​(x)=g(3x), f2(x)=g(1−x)f_{2}(x)=g(1-x)f2​(x)=g(1−x) und f3(x)=g(1x)f_{3}(x)=g\left(\frac{1}{x}\right)f3​(x)=g(x1​).

Problem 5806

What does h′(12)<0h^{\prime}(12)<0h′(12)<0 indicate about wind speed at 12 km from a hurricane's center?

Problem 5807

Calculate the percentage rate of change of f(x)f(x)f(x) where f(x)=300+65xf(x)=300+65xf(x)=300+65x at x=7x=7x=7.

Problem 5808

Find the percentage rate of change of f(x)f(x)f(x) at x=45x=45x=45 for f(x)=5400−2x2f(x)=5400-2 x^{2}f(x)=5400−2x2.

Problem 5809

Find f′(x)f^{\prime}(x)f′(x) for f(x)=3x2+2f(x)=3 x^{2}+2f(x)=3x2+2 using f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ as hhh approaches 0.

Problem 5810

A plane at 1 mile altitude flies at 500 mph. Find the rate of distance increase from the radar when it's 2 miles away.

Problem 5811

Find the limit: lim⁡x→−∞x2+13x−1\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}+1}}{3 x-1}limx→−∞​3x−1x2+1​​.

Problem 5812

Find the derivative of exln⁡x+x2cos⁡xe^{x} \ln x + x^{2} \cos xexlnx+x2cosx using the product and chain rules.

Problem 5813

Find the average velocity of a particle with position x(t)=(3.5 m/s)t−(5.0 m/s2)t2x(t)=(3.5 \mathrm{~m/s}) t-(5.0 \mathrm{~m/s}^2) t^2x(t)=(3.5 m/s)t−(5.0 m/s2)t2 from t=0.30 st=0.30 \mathrm{~s}t=0.30 s to t=0.40 st=0.40 \mathrm{~s}t=0.40 s.

Problem 5814

Find f′(2)f^{\prime}(2)f′(2) for f(x)=g(x)h(x)f(x)=\frac{g(x)}{h(x)}f(x)=h(x)g(x)​ given g(2)=5g(2)=5g(2)=5, g′(2)=−1g^{\prime}(2)=-1g′(2)=−1, h(2)=4h(2)=4h(2)=4, and h′(2)=3h^{\prime}(2)=3h′(2)=3.

Problem 5815

Calculate the limit: lim⁡x→03x+2−x+24x+5x−2\lim _{x \rightarrow 0} \frac{\sqrt{3 x+2}-\sqrt{x+2}}{4^{x}+5^{x}-2}limx→0​4x+5x−23x+2​−x+2​​.

Problem 5816

Problem 5817

Find the derivative of f(x)=−5x4+5x3+2x2+5x+5f(x)=-5 x^{4}+5 x^{3}+2 x^{2}+5 x+5f(x)=−5x4+5x3+2x2+5x+5. What is f′(x)f^{\prime}(x)f′(x)?

Problem 5818

Find the missing part of the derivative for k(x)=(5+5x2)−3k(x)=\left(5+5 x^{2}\right)^{-3}k(x)=(5+5x2)−3: k′(x)=(−3(5+5x2)−4 ? )(10x)k^{\prime}(x)=\left(-3\left(5+5 x^{2}\right)^{-4} \quad \text { ? }\right)(10 x)k′(x)=(−3(5+5x2)−4 ? )(10x)

Problem 5819

Problem 5820

Problem 5821

Find the limit: lim⁡t→23t−t4t2−2+5t2−2−2\lim _{t \rightarrow \sqrt{2}} \frac{\sqrt{3} t - t}{4^{t^{2}-2} + 5^{t^{2}-2} - 2}limt→2​​4t2−2+5t2−2−23​t−t​.

Problem 5822

Verify if (fg)′(x)=f′(x)g′(x)\left(\frac{f}{g}\right)^{\prime}(x)=\frac{f^{\prime}(x)}{g^{\prime}(x)}(gf​)′(x)=g′(x)f′(x)​ for f(x)=8x+7f(x)=8x+7f(x)=8x+7 and g(x)=5xg(x)=5xg(x)=5x.

Problem 5823

Calculate the integral from 4 to 9 of the function 1x2\frac{1}{x^{2}}x21​.

Problem 5824

Verify if the derivative of the quotient (fg)′(x)\left(\frac{f}{g}\right)^{\prime}(x)(gf​)′(x) equals f′(x)g′(x)\frac{f^{\prime}(x)}{g^{\prime}(x)}g′(x)f′(x)​ for f(x)=7x+3f(x)=7x+3f(x)=7x+3 and g(x)=8xg(x)=8xg(x)=8x.

Problem 5825

Check if the derivative of a quotient is the quotient of the derivatives for f(x)=7x+3f(x)=7x+3f(x)=7x+3 and g(x)=8xg(x)=8xg(x)=8x.

Problem 5826

Find the derivative of f(x)=sin⁡(3x)4x2−6xf(x)=\frac{\sin(3x)}{4x^{2}-6x}f(x)=4x2−6xsin(3x)​.

Problem 5827

Find the limit: lim⁡x→−24x2−153−1x+3−1\lim _{x \rightarrow-2} \frac{\sqrt[3]{4 x^{2}-15}-1}{\sqrt{x+3}-1}limx→−2​x+3​−134x2−15​−1​.

Problem 5828

Bestimmen Sie die Stammfunktion FFF von f(x)=3x2−2xf(x)=3 x^{2}-2 xf(x)=3x2−2x, die durch den Punkt P(2,−1)P(2, -1)P(2,−1) verläuft. Wählen Sie CCC passend.

Problem 5829

Ein Kreuzfahrtschiff hat 2500 Passagiere. Am Tag 1 erkranken 22 Passagiere. Zeigen Sie, dass f′(t)=2498⋅0,008⋅e−0,008⋅tf^{\prime}(t)=2498 \cdot 0,008 \cdot e^{-0,008 \cdot t}f′(t)=2498⋅0,008⋅e−0,008⋅t gilt und bestimmen Sie, wie lange medizinisches Personal benötigt wird.

Problem 5830

Verify if (fg)′(x)=f′(x)g′(x)\left(\frac{f}{g}\right)^{\prime}(x)=\frac{f^{\prime}(x)}{g^{\prime}(x)}(gf​)′(x)=g′(x)f′(x)​ for f(x)=9x+5f(x)=9x+5f(x)=9x+5, g(x)=2xg(x)=2xg(x)=2x.

Problem 5831

Verify if (fg)′(x)=f′(x)g′(x)\left(\frac{f}{g}\right)^{\prime}(x)=\frac{f^{\prime}(x)}{g^{\prime}(x)}(gf​)′(x)=g′(x)f′(x)​ is true for f(x)=3x+4f(x)=3x+4f(x)=3x+4, g(x)=7xg(x)=7xg(x)=7x.

Problem 5832

Find the limit: lim⁡x→∞12x5+6x−15x4−4x3+3\lim _{x \rightarrow \infty} \frac{12 x^{5}+6 x-1}{5 x^{4}-4 x^{3}+3}limx→∞​5x4−4x3+312x5+6x−1​.

Problem 5833

Find the derivative f′(x)f'(x)f′(x) of f(x)=4x(sin⁡x+cos⁡x)f(x)=4x(\sin x+\cos x)f(x)=4x(sinx+cosx) and calculate f′(2)f'(2)f′(2) rounded to the nearest hundredth.

Problem 5834

Calculate the limit: lim⁡x→17+x33−9−x3x−1\lim _{x \rightarrow 1} \frac{\sqrt[3]{7+x^{3}}-\sqrt[3]{9-x}}{x-1}limx→1​x−137+x3​−39−x​​.

Problem 5835

Check if the derivative of a product (fg)′(x)(f g)^{\prime}(x)(fg)′(x) equals the product of derivatives f′(x)g′(x)f^{\prime}(x) g^{\prime}(x)f′(x)g′(x) for f(x)=5x+4f(x)=5x+4f(x)=5x+4, g(x)=3x+2g(x)=3x+2g(x)=3x+2.

Problem 5836

Find the derivative f′(x)f'(x)f′(x) for f(x)=3x(sin⁡x+cos⁡x)f(x)=3x(\sin x+\cos x)f(x)=3x(sinx+cosx) and calculate f′(3)f'(3)f′(3) rounded to the nearest hundredth.

Problem 5837

Find the selling price xxx that maximizes profit P(x)=R(x)−C(x)P(x) = R(x) - C(x)P(x)=R(x)−C(x), where C(x)C(x)C(x) and R(x)R(x)R(x) are defined.

Problem 5838

Finde die Ableitung von f(x)=(x−1)(x−3)2f(x) = (x-1)(x-3)^{2}f(x)=(x−1)(x−3)2 und vereinfache f′(x)f'(x)f′(x).

Problem 5839

Find the degree 3 Taylor expansion of f(x)=ln⁡(x2+8x+17)f(x) = \ln(x^2 + 8x + 17)f(x)=ln(x2+8x+17) around x=−4x = -4x=−4, including the best big-OOO term.

Problem 5840

Find the integral of the function f(x)=x−2−ex−2f(x) = x - 2 - e^{x - 2}f(x)=x−2−ex−2.

Problem 5841

Sketch a function graph and check if f\mathrm{f}f is continuous at x=2\mathrm{x}=2x=2: f(2)=−3f(2)=-3f(2)=−3, lim⁡x→2−f(x)=3\lim _{x \rightarrow 2^{-}} f(x)=3limx→2−​f(x)=3, lim⁡x→2+f(x)=−3\lim _{x \rightarrow 2^{+}} f(x)=-3limx→2+​f(x)=−3.

Calculate the relative rate of change of $f(x)=6 x^{2}-\ln x$ at $x=8$ .

Calculate the relative rate of change of $f(x)=276-3x$ at $x=34$ .

Estimate $f(8)$ using local linear approximation given $f(12)=10$ and $f^{\prime}(12)=-3$ .

Calculate the relative rate of change of the function $f(x)=10+2 e^{-2 x}$ .

Gegeben ist eine differenzierbare Funktion g. Bestimmen Sie die Ableitungen für $f_{1}(x)=g(3 x)$ , $f_{2}(x)=g(1-x)$ und $f_{3}(x)=g\left(\frac{1}{x}\right)$ .

What does $h^{\prime}(12)<0$ indicate about wind speed at 12 km from a hurricane's center?

Calculate the percentage rate of change of $f(x)$ where $f(x)=300+65x$ at $x=7$ .

Find the percentage rate of change of $f(x)$ at $x=45$ for $f(x)=5400-2 x^{2}$ .

Find $f^{\prime}(x)$ for $f(x)=3 x^{2}+2$ using $\frac{f(x+h)-f(x)}{h}$ as $h$ approaches 0.

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

Find the derivative of $e^{x} \ln x + x^{2} \cos x$ using the product and chain rules.

Find the average velocity of a particle with position $x(t)=(3.5 \mathrm{~m/s}) t-(5.0 \mathrm{~m/s}^2) t^2$ from $t=0.30 \mathrm{~s}$ to $t=0.40 \mathrm{~s}$ .

Find $f^{\prime}(2)$ for $f(x)=\frac{g(x)}{h(x)}$ given $g(2)=5$ , $g^{\prime}(2)=-1$ , $h(2)=4$ , and $h^{\prime}(2)=3$ .

Calculate the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{3 x+2}-\sqrt{x+2}}{4^{x}+5^{x}-2}$ .

Find the derivative of $f(x)=-5 x^{4}+5 x^{3}+2 x^{2}+5 x+5$ . What is $f^{\prime}(x)$ ?

Find the missing part of the derivative for $k(x)=\left(5+5 x^{2}\right)^{-3}$ : $k^{\prime}(x)=\left(-3\left(5+5 x^{2}\right)^{-4} \quad \text { ? }\right)(10 x)$

Find the limit: $\lim _{t \rightarrow \sqrt{2}} \frac{\sqrt{3} t - t}{4^{t^{2}-2} + 5^{t^{2}-2} - 2}$ .

Verify if $\left(\frac{f}{g}\right)^{\prime}(x)=\frac{f^{\prime}(x)}{g^{\prime}(x)}$ for $f(x)=8x+7$ and $g(x)=5x$ .

Calculate the integral from 4 to 9 of the function $\frac{1}{x^{2}}$ .

Verify if the derivative of the quotient $\left(\frac{f}{g}\right)^{\prime}(x)$ equals $\frac{f^{\prime}(x)}{g^{\prime}(x)}$ for $f(x)=7x+3$ and $g(x)=8x$ .

Check if the derivative of a quotient is the quotient of the derivatives for $f(x)=7x+3$ and $g(x)=8x$ .

Find the derivative of $f(x)=\frac{\sin(3x)}{4x^{2}-6x}$ .

Find the limit: $\lim _{x \rightarrow-2} \frac{\sqrt[3]{4 x^{2}-15}-1}{\sqrt{x+3}-1}$ .

Bestimmen Sie die Stammfunktion $F$ von $f(x)=3 x^{2}-2 x$ , die durch den Punkt $P(2, -1)$ verläuft. Wählen Sie $C$ passend.

Ein Kreuzfahrtschiff hat 2500 Passagiere. Am Tag 1 erkranken 22 Passagiere. Zeigen Sie, dass $f^{\prime}(t)=2498 \cdot 0,008 \cdot e^{-0,008 \cdot t}$ gilt und bestimmen Sie, wie lange medizinisches Personal benötigt wird.

Verify if $\left(\frac{f}{g}\right)^{\prime}(x)=\frac{f^{\prime}(x)}{g^{\prime}(x)}$ for $f(x)=9x+5$ , $g(x)=2x$ .

Verify if $\left(\frac{f}{g}\right)^{\prime}(x)=\frac{f^{\prime}(x)}{g^{\prime}(x)}$ is true for $f(x)=3x+4$ , $g(x)=7x$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{12 x^{5}+6 x-1}{5 x^{4}-4 x^{3}+3}$ .

Find the derivative $f'(x)$ of $f(x)=4x(\sin x+\cos x)$ and calculate $f'(2)$ rounded to the nearest hundredth.

Calculate the limit: $\lim _{x \rightarrow 1} \frac{\sqrt[3]{7+x^{3}}-\sqrt[3]{9-x}}{x-1}$ .

Check if the derivative of a product $(f g)^{\prime}(x)$ equals the product of derivatives $f^{\prime}(x) g^{\prime}(x)$ for $f(x)=5x+4$ , $g(x)=3x+2$ .

Find the derivative $f'(x)$ for $f(x)=3x(\sin x+\cos x)$ and calculate $f'(3)$ rounded to the nearest hundredth.

Find the selling price $x$ that maximizes profit $P(x) = R(x) - C(x)$ , where $C(x)$ and $R(x)$ are defined.

Finde die Ableitung von $f(x) = (x-1)(x-3)^{2}$ und vereinfache $f'(x)$ .

Find the degree 3 Taylor expansion of $f(x) = \ln(x^2 + 8x + 17)$ around $x = -4$ , including the best big- $O$ term.

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

Sketch a function graph and check if $\mathrm{f}$ is continuous at $\mathrm{x}=2$ : $f(2)=-3$ , $\lim _{x \rightarrow 2^{-}} f(x)=3$ , $\lim _{x \rightarrow 2^{+}} f(x)=-3$ .

Find $f^{\prime}(0)$ for the piecewise function: $f(x)=-2x^{2}+7x$ for $x<0$ and $f(x)=3x^{2}-2$ for $x\geq0$ .

Identify vertical asymptotes for $g(x)=\frac{x}{6-x}$ and describe limits at $x=6$ .

Sketch a function graph that meets these conditions: $f(2)=1$ and $\lim_{x \rightarrow 2} f(x)=1$ . Is $f$ continuous at $x=2$ ?

Find the limit as $x$ approaches -10 from the left for $f(x)=\frac{1}{x+10}$ . Options: A. -1 B. $\infty$ C. 0 D. $-\infty$

Find $\lim _{x \rightarrow 0} f(x)$ and $f(0)$ from the graph. Options: A, B, C, D based on the graph's behavior.

A bucket catches water leaking at $0.5 \mathrm{~cm}^{3}$ per minute. Find $\frac{dh}{dt}$ when $h=10 \mathrm{~cm}$ .

Find the derivative of $f(x)=\frac{\sin 3 x}{\cos 2 x}$ at $x=\frac{\pi}{6}$ .

Find the local linear approximation of $f(x)=\frac{5 x+7}{x-4}$ near $x=5$ . Answer: $f(x) \approx$

Determine vertical asymptotes $x=a$ for $f(x)=\frac{x^{3}-11 x^{2}+28 x}{x^{2}-7 x}$ . Evaluate limits as $x$ approaches $a$ .

Estimate $p^{\prime \prime}(1997)$ using the percentages of seniors using marijuana from 1991 to 1999. Choose A, B, C, D, or E.

Find $f'(x)$ for $f(x)=\frac{\sqrt{x}-6}{\sqrt{x}+6}$ and calculate $f'(4)$ (round to the nearest hundredth).

Find the derivative $f'(x)$ of $f(x)=5x(\sin x+\cos x)$ and calculate $f'(3)$ rounded to the nearest hundredth.

Prove that $y=e^{x}(\sin 2 x+\cos 2 x)$ satisfies $y^{\prime \prime}-2 y^{\prime}+5 y=0$ .

Find the derivative of the function $f(x)=\sec(g(x))$ , i.e., compute $f'(x)$ .

A steak at $25^{\circ} \mathrm{F}$ thaws in a $71^{\circ} \mathrm{F}$ room. After 9 mins it's $36^{\circ} \mathrm{F}$ . When is it $60^{\circ} \mathrm{F}$ ?

Déterminez si la suite $a_{n}=\frac{3 n^{4}}{\sqrt{5 n^{8}+4 n^{5}}}$ converge et trouvez sa limite ou écrivez diverge.

A particle moves on the circle $x^{2}+y^{2}=100$ . At $(8,6)$ , if $\frac{d x}{d t}=5$ , find $\frac{d y}{d t}$ .

Determine the intervals where the function $f(x)=3x^{2}-18x+24$ is increasing.

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(-11 x^{-4}\right)$ .

Find the limit as $x$ approaches infinity for the expression $-11 x^{-4}$ .

Find the rate of change of gravitational force $F(r)=\frac{2.99 \times 10^{16}}{r^{2}}$ at $r=6.77 \times 10^{6}$ m.

Find the limit: $\lim _{x \rightarrow \infty} \frac{8+8 x+9 x^{3}}{x^{3}}.$

The logistic growth function $f(t)=\frac{107,000}{1+5100 e^{-t}}$ models flu cases.
a. Find initial cases. b. Find cases after 4 weeks. c. Determine maximum cases.

Find the limit as $x$ approaches infinity of $e^{x}$ .

Find the tangent plane and normal line equations for $f(x, y)=x e^{x y}$ at $(1,0,1)$ .

Find $\partial f / \partial r$ for $f(x, y)=y^{2}+6 x^{3}$ , where $x(r, s)=r s$ and $y(r, s)=r^{2}+s^{2}$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sin 15 x}{13 x}$ .

Find the rate of change of power $P(R)=\frac{2.1 R}{(R+0.5)^{2}}$ W at $R=3 \Omega$ . Round to four decimal places.

Find the time intervals when the object moves right given $x(t)=\frac{1}{2} t^{4}-\frac{10}{3} t^{3}+4 t^{2}$ .