Calculus

Problem 1801

Find the derivative of $f(t)=1.1 t^{2}-0.2 t^{0.4}+27$ .

See Solution

Problem 1802

Find the derivative of $f(t)=1.1 t^{2}-0.2 t^{0.4}+27$ .

See Solution

Problem 1803

Find the derivative of $f(x) = 4x^2 + 11x - 2$ at $x=3$ using $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .

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Problem 1804

Find the slope of the tangent line for the function $f(x)=3 x^{2}+9 x^{3}-5 x$ at $x=-4$ .

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Problem 1805

Find the tangent line equation for $f(x)=3 x^{2}+9 x^{3}-5 x$ at $x=-4$ .

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Problem 1806

Find the limit as $x$ approaches -9 for the expression $\frac{x^{2}+10 x+9}{x+9}$ .

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Problem 1807

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{3 x+81}-9}{x}$ . What is the exact value?

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Problem 1808

Find the limits: $\lim _{m \rightarrow 5^{-}} \sqrt{m-5},$ $\lim _{m \rightarrow 5^{+}} \sqrt{m-5},$ $\lim _{m \rightarrow 5} \sqrt{m-5}.$ Enter DNE if a limit does not exist.

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Problem 1809

Find the limits of the piecewise function $f(x)=\left\{\begin{array}{ll}2-x-x^{2} & \text { if } x \leq 4 \\ 2 x-9 & \text { if } x>4\end{array}\right.$ at $x=4$ :
1. $\lim _{x \rightarrow 4^{-}} f(x)=$
2. $\lim _{x \rightarrow 4^{+}} f(x)=$
3. $\lim _{x \rightarrow 4} f(x)=$

See Solution

Problem 1810

Find the limits of the piecewise function $f(x)=\left\{\begin{array}{lll}4-x-x^{2} & \text { if } & x \leq 4 \\ 2 x-9 & \text { if } & x>4\end{array}\right.$ at $x=4$ :
1. $\lim _{x \rightarrow 4^{-}} f(x)$
2. $\lim _{x \rightarrow 4^{+}} f(x)$
3. $\lim _{x \rightarrow 4} f(x)$
Enter "DNE" if a limit does not exist.

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Problem 1811

Find the velocity and acceleration of a particle with position $s(t)=7 t^{2}+6 t+9$ at $t=3$ seconds using limits.

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Problem 1812

Estimate the jumper's height change $h^{\prime}(5.5)$ , $h^{\prime}(6)$ , $h^{\prime}(6.5)$ and identify the fastest rise time.

See Solution

Problem 1813

Find the velocity of a particle at time $t$ from its position $s(t)=7 t^{2}+6 t+9$ using the limit definition of the derivative. $v(t)=\frac{m}{s}$

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Problem 1814

Estimate the bungee jumper's height change $h'(5.5)$ , $h'(6)$ , $h'(6.5)$ , and find when the jumper rises fastest. Also, approximate $h''(6)$ .

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Problem 1815

A bungee jumper's height $h(t)$ is given by data points. Estimate the jumper's speed and find when they rise fastest. Approximate $h^{\prime \prime}(6)$ .

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Problem 1816

Find the limits:
1. $\lim _{x \rightarrow \frac{6}{7}} x^{4}=$
2. $\lim _{x \rightarrow 64} x^{3 / 2}=$

See Solution

Problem 1817

Find the limit: $\lim _{x \rightarrow-7} 18=$ . Use Basic Limit Laws.

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Problem 1818

Find the limit: $\lim _{x \rightarrow 125} x^{\frac{4}{3}}$ .

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Problem 1819

Find the limit: $\lim _{x \rightarrow 2}(6 x+5)$ . Provide your answer as a whole number.

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Problem 1820

Find the limit using Basic Limit Laws: $\lim _{x \rightarrow 8}\left(2 x^{2 / 3}-3 x^{-1}\right)=$

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Problem 1821

Find the limit using Basic Limit Laws: $\lim _{t \rightarrow 8} \frac{3 t-2}{t+6} =$

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Problem 1822

Find the derivative of the function $f(x)=\frac{1}{2 \sqrt[6]{x^{5}}}$ and express it in radical form.

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Problem 1823

Find the limit as $x$ approaches 7 for the expression $\frac{\sqrt{x+2}+1}{\sqrt{x+9}-1}$ .

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Problem 1824

Find the limit as $t$ approaches 2 for the expression $t^{-1}$ .

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Problem 1825

Find the limit: $\lim _{x \rightarrow 4} \frac{8 \sqrt{x}-\frac{1}{2} x}{(x-25)^{2}}$ .

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Problem 1826

Find the limit as $x$ approaches $\frac{13}{2}$ for $(4 x^{2}+10 x-7)^{3 / 2}$ .

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Problem 1827

Find the derivative of the function $y=-\frac{1}{2 x}$ , expressed in simplest form without negative exponents.

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Problem 1828

Evaluate the limit: $\lim _{x \rightarrow \frac{13}{2}}\left(4 x^{2}+10 x-7\right)^{3 / 2}$ .

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Problem 1829

Evaluate the limit: $\lim _{x \rightarrow-3}(5 f(x)+3 g(x))$ given $\lim _{x \rightarrow-3} f(x)=5$ and $\lim _{x \rightarrow-3} g(x)=1$ .

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Problem 1830

Evaluate the limit given that $\lim _{x \rightarrow 2} g(x)=-3$ : $\lim _{x \rightarrow 2} \frac{g(x)}{x^{2}}=$

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Problem 1831

Find the derivative $f'(1)$ for the function $f(x)=\frac{5 \sqrt{x}}{3}+2 \sqrt{x^{3}}$ . Simplify to a single fraction.

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Problem 1832

Evaluate the limit: $\lim _{x \rightarrow-3} \frac{f(x)+1}{4 g(x)-9}$ given $\lim _{x \rightarrow-3} f(x)=3$ and $\lim _{x \rightarrow-3} g(x)=5$ .

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Problem 1833

Find the derivative $f^{\prime}(3)$ for the function $f(x)=-\sqrt{x^{3}}-\frac{\sqrt{x}}{2}$ as a simplified fraction.

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Problem 1834

Given $\lim _{x \rightarrow c} f(x) g(x)=I$ and $\lim _{x \rightarrow c} g(x)=M \neq 0$ , find $\lim _{x \rightarrow c} f(x)$ . Is the proof valid? Select all that apply.

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Problem 1835

Given $\lim _{x \rightarrow c} f(x) g(x)=L$ and $\lim _{x \rightarrow c} g(x)=M \neq 0$ , find $\lim _{x \rightarrow c} f(x)$ . Is the proof valid?

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Problem 1836

Given $\lim _{x \rightarrow c} f(x) g(x)=L$ and $\lim _{x \rightarrow c} g(x)=M \neq 0$ , find $\lim _{x \rightarrow c} f(x)$ . Is the proof valid? Select reasons.

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Problem 1837

Given $\lim _{x \rightarrow c} f(x) g(x)=L$ and $\lim _{x \rightarrow c} g(x)=M \neq 0$ , find $\lim _{x \rightarrow c} f(x)$ . Is the proof valid? Select reasons if not.

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Problem 1838

Find functions $f$ and $g$ where $\lim_{x \to 0} f(x)$ and $\lim_{x \to 0} g(x)$ don't exist, but $\lim_{x \to 0}(f(x)+g(x))$ does.

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Problem 1839

Given $\lim _{x \rightarrow 6} f(x)=3$ , find these limits:
1. $\lim _{x \rightarrow 6}(f(x))^{2}=$
2. $\lim _{x \rightarrow 6} \frac{1}{f(x)}=$
3. $\lim _{x \rightarrow 6} x \sqrt{f(x)}=$

See Solution

Problem 1840

Cho hàm số $y=f(x)$ với $f^{\prime}(x)=x(x-1)(x-2)$ . Số khẳng định đúng từ (I) đến (IV) là bao nhiêu? A. 4, B. 3, C. 2, D. 1.

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Problem 1841

Evaluate the limit: $\lim _{x \rightarrow-1}(f(x) \cdot g(x)+2)$ given that $\lim _{x \rightarrow-1} f(x)=2$ and $\lim _{x \rightarrow-1} g(x)=10$ .

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Problem 1842

Evaluate the limit: $\lim _{x \rightarrow 7} \frac{\sqrt{x+2}+1}{\sqrt{x+9}-1}.$

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Problem 1843

Find the limit: $\lim _{x \rightarrow 7} \frac{\sqrt{x+2}+1}{\sqrt{x+9}-1}$

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Problem 1844

Evaluate the limit: $\lim _{x \rightarrow 7} \frac{3 f(x)+x}{g(x)+1}$ given that $\lim _{x \rightarrow 7} f(x)=7$ and $\lim _{x \rightarrow 7} g(x)=-12$ .

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Problem 1845

Evaluate the limit: $\lim _{x \rightarrow 64}\left(4 x^{2 / 3}-11 x^{-1}\right)=?$

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Problem 1846

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x^{4}-4 x^{3}+1}{2-2 x^{2}-7 x^{4}}$ .

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Problem 1847

Calculate the limit: $\lim _{x \rightarrow-\infty} \frac{x^{4}-4 x^{3}+1}{2-2 x^{2}-2 x^{4}}$ .

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Problem 1848

Evaluate the integral $\int (x^{2}-1) \sqrt{x} \, dx$ . Choose the correct answer from the options provided.

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Problem 1849

Evaluate the integral $\int\left(x^{2}-1\right) \sqrt{x} d x$ . Choose the correct answer from the options provided.

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Problem 1850

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

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Problem 1851

Evaluate the integral $\int e^{x} e^{2 x} dx$ . What is the result? (A) $\frac{1}{3} e^{3 x}+c$ (B) $3 e^{3 x}+c$ (C) $3 x-e^{-3 x}+c$ (D) none of the choices

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Problem 1852

Evaluate $\int \frac{e^{2 x} d x}{e^{x}+1}$ . Choose the correct answer from the options given.

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Problem 1853

Evaluate $\int \frac{\ln x}{x} d x$ . Choose the correct answer: (A) $\frac{1}{2}(\ln |x|)^{2}+c$ , (B) $-2(\ln |x|)^{2}+c$ , (C) $\frac{x}{2} \ln |x|^{2}+c$ , (D) none.

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Problem 1854

Find the numerical coefficient of the integral $\int \sin x \cos^{2} x \, dx$ rounded to 2 decimal places.

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Problem 1855

Evaluate $\int \frac{1}{3 x} d x$ and choose the correct answer: (A) $\frac{1}{3} \ln |x|+C$ , (B) $\frac{1}{3} \ln |x|$ , (C) $-3 \ln |x|+C$ , (D) $3 \ln |x|-x+C$ .

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Problem 1856

Rotate the ellipse around the $y$ -axis. Use the shell method to find the volume: $\frac{4}{3} \pi a^{2} b$ .

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Problem 1857

Find the 2 positive terms from the integral of $\cos^{4} x \sin^{2} x \, dx$ . Options: (A), (B), (C), (D).

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Problem 1858

Evaluate the limit using limit laws: $\lim _{x \rightarrow \frac{15}{2}}\left(4 x^{2}+6 x-6\right)^{3 / 2}=$

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Problem 1859

What is the negative term in the integral of $\tan^{4} x \, dx$ ? Answer format: "term".

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Problem 1860

Evaluate the limit: $\lim _{x \rightarrow-1}(f(x) \cdot g(x)+2)$ given $\lim _{x \rightarrow-1} f(x)=2$ and $\lim _{x \rightarrow-1} g(x)=10$ .

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Problem 1861

Air is pumped into a balloon at 10 cm³/s. Find the rate of diameter increase when the radius is $5 \mathrm{~cm}$ .

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Problem 1862

What is the numerical coefficient of the inverse trigonometric function in the integral $\frac{d x}{2+7 x^{2}}$ ? Round to 2 decimal places.

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Problem 1863

Evaluate the integral $\int \frac{d x}{\sqrt{3+2 x-x^{2}}}$ using trigonometric substitution. What is the result?

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Problem 1864

Evaluate $\int \frac{d x}{(x-3) \sqrt{x^{2}-6 x+8}}$ . Choose the correct answer: (A), (B), (C), or (D).

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Problem 1865

What is the term inside the inverse trigonometric function in the integral of $\frac{d x}{\sqrt{25-x^{2}}}$ ?

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Problem 1866

Integrate $\int \frac{d x}{(4 x^{2}-9)^{3/2}}$ . Choose the correct answer from the options provided.

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Problem 1867

Evaluate $\int \frac{d x}{x^{2}-4 x+5}$ using trigonometric substitution. Choose the correct answer: (A), (B), (C), or (D).

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Problem 1868

Find the integral of $e^{x} \sin x$ with respect to $x$ : $\int e^{x} \sin x \, dx$ . Choose the correct answer.

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Problem 1869

Find the integral of $\int(\ln x)^{2} d x$ . Choose the correct answer from the options provided.

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Problem 1870

Integrate $\int \tan^{-1}(3x) \, dx$ . Choose the correct answer from the options provided.

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Problem 1871

Evaluate $\int \sec ^{2}(4 x+1) d x$ using $u=4 x+1$ . What is the result?

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Problem 1872

Find the marginal cost function for the production cost $C(x)=130+6x-0.8x^{2}$ .

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Problem 1873

Find the marginal cost function for the total cost $C(x)=600+17 x-0.1 x^{2}+0.4 x^{3}$ of producing $x$ magazines.

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Problem 1874

Find the marginal average cost for the weekly hat production cost function $C(x)=2800+26x+0.3x^{2}$ .

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Problem 1875

Find the marginal profit function given price $p=D(x)=8.00-0.004 x$ and cost $C(x)=0.007 x^{2}+0.55 x+270$ .

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Problem 1876

Find the marginal profit for selling 145 toasters, given price $p=D(x)=81.7-0.01 x$ and cost $C(x)=0.03 x^{2}+4.7 x+6100$ .

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Problem 1877

Is the statement true or false? The derivative of $f(x)$ is the instantaneous rate of change of $y=f(x)$ with respect to $x$ . Choose A, B, C, or D.

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Problem 1878

Find the limit as $x$ approaches 2 for the function $f(x)=\frac{x^{2}+4 x-12}{x^{2}-4}$ using a table and verify with a graphing calculator.

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Problem 1879

Find the limit as $x$ approaches 2 for the expression $\frac{x^{2}+4x-12}{x^{2}-4}$ , rounded to three decimal places.

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Problem 1880

Find values of $x=a$ where $k(x)=4 e^{\sqrt{x-4}}$ is discontinuous and state the limits as $x$ approaches $a$ .

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Problem 1881

Find values of $x=a$ where $k(x)=4 e^{\sqrt{x-4}}$ is discontinuous. State the limits or if they don't exist.

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Problem 1882

Find values of $x=a$ where $k(x)=4 e^{\sqrt{x-4}}$ is discontinuous and determine the limits as $x$ approaches $a$ .

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Problem 1883

Is it true or false that a runner's mile time can be used to find average speed? Choose the correct answer: A. True, average rate is time over distance. B. False, instantaneous speed is unknown. C. True, average rate is distance over time. D. False, only instantaneous speed can be found.

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Problem 1884

Find the instantaneous velocity of the object at $t=4$ for the position function $s(t)=4 t^{2}+5 t+2$ .

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Problem 1885

Approximate the derivative of $f(x)=5 x^{x}$ at $a=2$ using small $h$ . Find the slope of the line near that point.

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Problem 1886

Approximate $e^{-0.6}$ to three decimal places. What is the value?

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Problem 1887

Calculate the average rate of change of $f(x)=x^{2}+5x$ from $x=0$ to $x=8$ . Simplify your answer.

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Problem 1888

Find the limit: If $h(x)=\frac{\sqrt{x}+2}{x-7}$ , what is $\lim _{x \rightarrow 7} h(x)$ ? A. $=$ B. DNE.

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Problem 1889

Find the limit: If $h(x)=\frac{\sqrt{x}+2}{x-7}$ , then determine $\lim _{x \rightarrow 7} h(x)$ . A. $\lim _{x \rightarrow 7} h(x)=$ B. Limit does not exist.

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Problem 1890

Find the limit: If $h(x)=\frac{\sqrt{x}+2}{x-7}$ , then calculate $\lim _{x \rightarrow 7} h(x)$ .

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Problem 1891

Find $\lim _{x \rightarrow 7} h(x)$ for $h(x)=\frac{\sqrt{x}+2}{x-7}$ . Does it exist?

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Problem 1892

Find the limit: If $h(x)=\frac{\sqrt{x}+2}{x-7}$ , then determine $\lim _{x \rightarrow 7} h(x)$ . Options: A. Limit value or B. Limit does not exist.

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Problem 1893

Find the integral of $x^{2} \cos x$ with respect to $x$ .

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Problem 1894

Find the marginal revenue $R'(x)$ for $R(x)=55x-\frac{x^2}{100}$ , estimate $R'(2000)$ , actual revenue $R(2001)$ , and compare.

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Problem 1895

Calculate the integral: $\int \sqrt{x^{2}+16} \, dx$

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Problem 1896

Find $f^{\prime}(2)$ , $f^{\prime}(16)$ , and $f^{\prime}(-3)$ for $f(x)=\frac{-2}{x}$ . Does the derivative exist?

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Problem 1897

Find the instantaneous velocity of the object at time $t=3$ for the position function $s(t)=3 t^{2}-7 t-6$ .

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Problem 1898

Find the secant line for $f(x)=\frac{8}{x}$ between $x=2$ and $x=8$ , and the tangent line at $x=2$ .

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Problem 1899

Untersuchen Sie das Verhalten von $f(x)$ für $x \rightarrow \infty$ und $x \rightarrow -\infty$ für die folgenden Funktionen: a) $f(x)=x \cdot e^{x}$ , b) $f(x)=x^{2} \cdot e^{-x}$ , c) $f(x)=-x^{3} \cdot e^{x}$ , d) $f(x)=5+x^{4} \cdot e^{x}$ , e) $f(x)=4+x \cdot e^{-x}$ , f) $f(x)=(x^{2}+x+7) \cdot e^{x}$ , g) $f(x)=3 x-e^{-x}$ , h) $f(x)=x-e^{x}+2$ , i) $f(x)=\frac{x}{e^{x}}-\frac{x}{2}$ .

See Solution

Problem 1900

Find the instantaneous rate of change of $g(t)=5-t^{2}$ at $t=-6$ . What is the value?

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Calculus

Problem 1801

Find the derivative of f(t)=1.1t2−0.2t0.4+27f(t)=1.1 t^{2}-0.2 t^{0.4}+27f(t)=1.1t2−0.2t0.4+27.

Problem 1802

Find the derivative of f(t)=1.1t2−0.2t0.4+27f(t)=1.1 t^{2}-0.2 t^{0.4}+27f(t)=1.1t2−0.2t0.4+27.

Problem 1803

Find the derivative of f(x)=4x2+11x−2f(x) = 4x^2 + 11x - 2f(x)=4x2+11x−2 at x=3x=3x=3 using f′(x)=lim⁡h→0f(x+h)−f(x)hf^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}f′(x)=limh→0​hf(x+h)−f(x)​.

Problem 1804

Find the slope of the tangent line for the function f(x)=3x2+9x3−5xf(x)=3 x^{2}+9 x^{3}-5 xf(x)=3x2+9x3−5x at x=−4x=-4x=−4.

Problem 1805

Find the tangent line equation for f(x)=3x2+9x3−5xf(x)=3 x^{2}+9 x^{3}-5 xf(x)=3x2+9x3−5x at x=−4x=-4x=−4.

Problem 1806

Find the limit as xxx approaches -9 for the expression x2+10x+9x+9\frac{x^{2}+10 x+9}{x+9}x+9x2+10x+9​.

Problem 1807

Find the limit: lim⁡x→03x+81−9x\lim _{x \rightarrow 0} \frac{\sqrt{3 x+81}-9}{x}limx→0​x3x+81​−9​. What is the exact value?

Problem 1808

Find the limits: lim⁡m→5−m−5,\lim _{m \rightarrow 5^{-}} \sqrt{m-5},m→5−lim​m−5​, lim⁡m→5+m−5,\lim _{m \rightarrow 5^{+}} \sqrt{m-5},m→5+lim​m−5​, lim⁡m→5m−5.\lim _{m \rightarrow 5} \sqrt{m-5}.m→5lim​m−5​. Enter DNE if a limit does not exist.

Problem 1809

Problem 1810

Problem 1811

Find the velocity and acceleration of a particle with position s(t)=7t2+6t+9s(t)=7 t^{2}+6 t+9s(t)=7t2+6t+9 at t=3t=3t=3 seconds using limits.

Problem 1812

Estimate the jumper's height change h′(5.5)h^{\prime}(5.5)h′(5.5), h′(6)h^{\prime}(6)h′(6), h′(6.5)h^{\prime}(6.5)h′(6.5) and identify the fastest rise time.

Problem 1813

Find the velocity of a particle at time ttt from its position s(t)=7t2+6t+9s(t)=7 t^{2}+6 t+9s(t)=7t2+6t+9 using the limit definition of the derivative. v(t)=msv(t)=\frac{m}{s}v(t)=sm​

Problem 1814

Estimate the bungee jumper's height change h′(5.5)h'(5.5)h′(5.5), h′(6)h'(6)h′(6), h′(6.5)h'(6.5)h′(6.5), and find when the jumper rises fastest. Also, approximate h′′(6)h''(6)h′′(6).

Problem 1815

A bungee jumper's height h(t)h(t)h(t) is given by data points. Estimate the jumper's speed and find when they rise fastest. Approximate h′′(6)h^{\prime \prime}(6)h′′(6).

Problem 1816

Find the limits: 1. lim⁡x→67x4=\lim _{x \rightarrow \frac{6}{7}} x^{4}=limx→76​​x4= 2. lim⁡x→64x3/2=\lim _{x \rightarrow 64} x^{3 / 2}=limx→64​x3/2=

Problem 1817

Find the limit: lim⁡x→−718=\lim _{x \rightarrow-7} 18=limx→−7​18=. Use Basic Limit Laws.

Problem 1818

Find the limit: lim⁡x→125x43\lim _{x \rightarrow 125} x^{\frac{4}{3}}x→125lim​x34​.

Problem 1819

Find the limit: lim⁡x→2(6x+5)\lim _{x \rightarrow 2}(6 x+5)limx→2​(6x+5). Provide your answer as a whole number.

Problem 1820

Find the limit using Basic Limit Laws: lim⁡x→8(2x2/3−3x−1)=\lim _{x \rightarrow 8}\left(2 x^{2 / 3}-3 x^{-1}\right)= x→8lim​(2x2/3−3x−1)=

Problem 1821

Find the limit using Basic Limit Laws: lim⁡t→83t−2t+6=\lim _{t \rightarrow 8} \frac{3 t-2}{t+6} = t→8lim​t+63t−2​=

Problem 1822

Find the derivative of the function f(x)=12x56f(x)=\frac{1}{2 \sqrt[6]{x^{5}}}f(x)=26x5​1​ and express it in radical form.

Problem 1823

Find the limit as xxx approaches 7 for the expression x+2+1x+9−1\frac{\sqrt{x+2}+1}{\sqrt{x+9}-1}x+9​−1x+2​+1​.

Problem 1824

Find the limit as ttt approaches 2 for the expression t−1t^{-1}t−1.

Problem 1825

Find the limit: lim⁡x→48x−12x(x−25)2\lim _{x \rightarrow 4} \frac{8 \sqrt{x}-\frac{1}{2} x}{(x-25)^{2}}limx→4​(x−25)28x​−21​x​.

Problem 1826

Find the limit as xxx approaches 132\frac{13}{2}213​ for (4x2+10x−7)3/2(4 x^{2}+10 x-7)^{3 / 2}(4x2+10x−7)3/2.

Problem 1827

Find the derivative of the function y=−12xy=-\frac{1}{2 x}y=−2x1​, expressed in simplest form without negative exponents.

Problem 1828

Evaluate the limit: lim⁡x→132(4x2+10x−7)3/2\lim _{x \rightarrow \frac{13}{2}}\left(4 x^{2}+10 x-7\right)^{3 / 2}x→213​lim​(4x2+10x−7)3/2.

Problem 1829

Evaluate the limit: lim⁡x→−3(5f(x)+3g(x))\lim _{x \rightarrow-3}(5 f(x)+3 g(x))limx→−3​(5f(x)+3g(x)) given lim⁡x→−3f(x)=5\lim _{x \rightarrow-3} f(x)=5limx→−3​f(x)=5 and lim⁡x→−3g(x)=1\lim _{x \rightarrow-3} g(x)=1limx→−3​g(x)=1.

Problem 1830

Evaluate the limit given that lim⁡x→2g(x)=−3\lim _{x \rightarrow 2} g(x)=-3limx→2​g(x)=−3: lim⁡x→2g(x)x2=\lim _{x \rightarrow 2} \frac{g(x)}{x^{2}}= x→2lim​x2g(x)​=

Problem 1831

Find the derivative f′(1)f'(1)f′(1) for the function f(x)=5x3+2x3f(x)=\frac{5 \sqrt{x}}{3}+2 \sqrt{x^{3}}f(x)=35x​​+2x3​. Simplify to a single fraction.

Problem 1832

Evaluate the limit: lim⁡x→−3f(x)+14g(x)−9\lim _{x \rightarrow-3} \frac{f(x)+1}{4 g(x)-9}limx→−3​4g(x)−9f(x)+1​ given lim⁡x→−3f(x)=3\lim _{x \rightarrow-3} f(x)=3limx→−3​f(x)=3 and lim⁡x→−3g(x)=5\lim _{x \rightarrow-3} g(x)=5limx→−3​g(x)=5.

Problem 1833

Find the derivative f′(3)f^{\prime}(3)f′(3) for the function f(x)=−x3−x2f(x)=-\sqrt{x^{3}}-\frac{\sqrt{x}}{2}f(x)=−x3​−2x​​ as a simplified fraction.

Problem 1834

Given lim⁡x→cf(x)g(x)=I\lim _{x \rightarrow c} f(x) g(x)=Ilimx→c​f(x)g(x)=I and lim⁡x→cg(x)=M≠0\lim _{x \rightarrow c} g(x)=M \neq 0limx→c​g(x)=M=0, find lim⁡x→cf(x)\lim _{x \rightarrow c} f(x)limx→c​f(x). Is the proof valid? Select all that apply.

Problem 1835

Given lim⁡x→cf(x)g(x)=L\lim _{x \rightarrow c} f(x) g(x)=Llimx→c​f(x)g(x)=L and lim⁡x→cg(x)=M≠0\lim _{x \rightarrow c} g(x)=M \neq 0limx→c​g(x)=M=0, find lim⁡x→cf(x)\lim _{x \rightarrow c} f(x)limx→c​f(x). Is the proof valid?

Problem 1836

Given lim⁡x→cf(x)g(x)=L\lim _{x \rightarrow c} f(x) g(x)=Llimx→c​f(x)g(x)=L and lim⁡x→cg(x)=M≠0\lim _{x \rightarrow c} g(x)=M \neq 0limx→c​g(x)=M=0, find lim⁡x→cf(x)\lim _{x \rightarrow c} f(x)limx→c​f(x). Is the proof valid? Select reasons.

Problem 1837

Given lim⁡x→cf(x)g(x)=L\lim _{x \rightarrow c} f(x) g(x)=Llimx→c​f(x)g(x)=L and lim⁡x→cg(x)=M≠0\lim _{x \rightarrow c} g(x)=M \neq 0limx→c​g(x)=M=0, find lim⁡x→cf(x)\lim _{x \rightarrow c} f(x)limx→c​f(x). Is the proof valid? Select reasons if not.

Problem 1838

Find functions fff and ggg where lim⁡x→0f(x)\lim_{x \to 0} f(x)limx→0​f(x) and lim⁡x→0g(x)\lim_{x \to 0} g(x)limx→0​g(x) don't exist, but lim⁡x→0(f(x)+g(x))\lim_{x \to 0}(f(x)+g(x))limx→0​(f(x)+g(x)) does.

Problem 1839

Problem 1840

Cho hàm số y=f(x)y=f(x)y=f(x) với f′(x)=x(x−1)(x−2)f^{\prime}(x)=x(x-1)(x-2)f′(x)=x(x−1)(x−2). Số khẳng định đúng từ (I) đến (IV) là bao nhiêu? A. 4, B. 3, C. 2, D. 1.

Problem 1841

Evaluate the limit: lim⁡x→−1(f(x)⋅g(x)+2)\lim _{x \rightarrow-1}(f(x) \cdot g(x)+2)x→−1lim​(f(x)⋅g(x)+2) given that lim⁡x→−1f(x)=2\lim _{x \rightarrow-1} f(x)=2limx→−1​f(x)=2 and lim⁡x→−1g(x)=10\lim _{x \rightarrow-1} g(x)=10limx→−1​g(x)=10.

Find the derivative of $f(t)=1.1 t^{2}-0.2 t^{0.4}+27$ .

Find the derivative of $f(t)=1.1 t^{2}-0.2 t^{0.4}+27$ .

Find the derivative of $f(x) = 4x^2 + 11x - 2$ at $x=3$ using $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .

Find the slope of the tangent line for the function $f(x)=3 x^{2}+9 x^{3}-5 x$ at $x=-4$ .

Find the tangent line equation for $f(x)=3 x^{2}+9 x^{3}-5 x$ at $x=-4$ .

Find the limit as $x$ approaches -9 for the expression $\frac{x^{2}+10 x+9}{x+9}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{3 x+81}-9}{x}$ . What is the exact value?

Find the limits: $\lim _{m \rightarrow 5^{-}} \sqrt{m-5},$ $\lim _{m \rightarrow 5^{+}} \sqrt{m-5},$ $\lim _{m \rightarrow 5} \sqrt{m-5}.$ Enter DNE if a limit does not exist.

Find the velocity and acceleration of a particle with position $s(t)=7 t^{2}+6 t+9$ at $t=3$ seconds using limits.

Estimate the jumper's height change $h^{\prime}(5.5)$ , $h^{\prime}(6)$ , $h^{\prime}(6.5)$ and identify the fastest rise time.

Find the velocity of a particle at time $t$ from its position $s(t)=7 t^{2}+6 t+9$ using the limit definition of the derivative. $v(t)=\frac{m}{s}$

Estimate the bungee jumper's height change $h'(5.5)$ , $h'(6)$ , $h'(6.5)$ , and find when the jumper rises fastest. Also, approximate $h''(6)$ .

A bungee jumper's height $h(t)$ is given by data points. Estimate the jumper's speed and find when they rise fastest. Approximate $h^{\prime \prime}(6)$ .

Find the limits:
1. $\lim _{x \rightarrow \frac{6}{7}} x^{4}=$
2. $\lim _{x \rightarrow 64} x^{3 / 2}=$

Find the limit: $\lim _{x \rightarrow-7} 18=$ . Use Basic Limit Laws.

Find the limit: $\lim _{x \rightarrow 125} x^{\frac{4}{3}}$ .

Find the limit: $\lim _{x \rightarrow 2}(6 x+5)$ . Provide your answer as a whole number.

Find the limit using Basic Limit Laws: $\lim _{x \rightarrow 8}\left(2 x^{2 / 3}-3 x^{-1}\right)=$

Find the limit using Basic Limit Laws: $\lim _{t \rightarrow 8} \frac{3 t-2}{t+6} =$

Find the derivative of the function $f(x)=\frac{1}{2 \sqrt[6]{x^{5}}}$ and express it in radical form.

Find the limit as $x$ approaches 7 for the expression $\frac{\sqrt{x+2}+1}{\sqrt{x+9}-1}$ .

Find the limit as $t$ approaches 2 for the expression $t^{-1}$ .

Find the limit: $\lim _{x \rightarrow 4} \frac{8 \sqrt{x}-\frac{1}{2} x}{(x-25)^{2}}$ .

Find the limit as $x$ approaches $\frac{13}{2}$ for $(4 x^{2}+10 x-7)^{3 / 2}$ .

Find the derivative of the function $y=-\frac{1}{2 x}$ , expressed in simplest form without negative exponents.

Evaluate the limit: $\lim _{x \rightarrow \frac{13}{2}}\left(4 x^{2}+10 x-7\right)^{3 / 2}$ .

Evaluate the limit: $\lim _{x \rightarrow-3}(5 f(x)+3 g(x))$ given $\lim _{x \rightarrow-3} f(x)=5$ and $\lim _{x \rightarrow-3} g(x)=1$ .

Evaluate the limit given that $\lim _{x \rightarrow 2} g(x)=-3$ : $\lim _{x \rightarrow 2} \frac{g(x)}{x^{2}}=$

Find the derivative $f'(1)$ for the function $f(x)=\frac{5 \sqrt{x}}{3}+2 \sqrt{x^{3}}$ . Simplify to a single fraction.

Evaluate the limit: $\lim _{x \rightarrow-3} \frac{f(x)+1}{4 g(x)-9}$ given $\lim _{x \rightarrow-3} f(x)=3$ and $\lim _{x \rightarrow-3} g(x)=5$ .

Find the derivative $f^{\prime}(3)$ for the function $f(x)=-\sqrt{x^{3}}-\frac{\sqrt{x}}{2}$ as a simplified fraction.

Given $\lim _{x \rightarrow c} f(x) g(x)=I$ and $\lim _{x \rightarrow c} g(x)=M \neq 0$ , find $\lim _{x \rightarrow c} f(x)$ . Is the proof valid? Select all that apply.

Given $\lim _{x \rightarrow c} f(x) g(x)=L$ and $\lim _{x \rightarrow c} g(x)=M \neq 0$ , find $\lim _{x \rightarrow c} f(x)$ . Is the proof valid?

Given $\lim _{x \rightarrow c} f(x) g(x)=L$ and $\lim _{x \rightarrow c} g(x)=M \neq 0$ , find $\lim _{x \rightarrow c} f(x)$ . Is the proof valid? Select reasons.

Given $\lim _{x \rightarrow c} f(x) g(x)=L$ and $\lim _{x \rightarrow c} g(x)=M \neq 0$ , find $\lim _{x \rightarrow c} f(x)$ . Is the proof valid? Select reasons if not.

Find functions $f$ and $g$ where $\lim_{x \to 0} f(x)$ and $\lim_{x \to 0} g(x)$ don't exist, but $\lim_{x \to 0}(f(x)+g(x))$ does.

Cho hàm số $y=f(x)$ với $f^{\prime}(x)=x(x-1)(x-2)$ . Số khẳng định đúng từ (I) đến (IV) là bao nhiêu? A. 4, B. 3, C. 2, D. 1.

Evaluate the limit: $\lim _{x \rightarrow-1}(f(x) \cdot g(x)+2)$ given that $\lim _{x \rightarrow-1} f(x)=2$ and $\lim _{x \rightarrow-1} g(x)=10$ .

Evaluate the limit: $\lim _{x \rightarrow 7} \frac{\sqrt{x+2}+1}{\sqrt{x+9}-1}.$

Find the limit: $\lim _{x \rightarrow 7} \frac{\sqrt{x+2}+1}{\sqrt{x+9}-1}$

Evaluate the limit: $\lim _{x \rightarrow 7} \frac{3 f(x)+x}{g(x)+1}$ given that $\lim _{x \rightarrow 7} f(x)=7$ and $\lim _{x \rightarrow 7} g(x)=-12$ .

Evaluate the limit: $\lim _{x \rightarrow 64}\left(4 x^{2 / 3}-11 x^{-1}\right)=?$

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x^{4}-4 x^{3}+1}{2-2 x^{2}-7 x^{4}}$ .

Calculate the limit: $\lim _{x \rightarrow-\infty} \frac{x^{4}-4 x^{3}+1}{2-2 x^{2}-2 x^{4}}$ .

Evaluate the integral $\int (x^{2}-1) \sqrt{x} \, dx$ . Choose the correct answer from the options provided.

Evaluate the integral $\int\left(x^{2}-1\right) \sqrt{x} d x$ . Choose the correct answer from the options provided.

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

Evaluate the integral $\int e^{x} e^{2 x} dx$ . What is the result? (A) $\frac{1}{3} e^{3 x}+c$ (B) $3 e^{3 x}+c$ (C) $3 x-e^{-3 x}+c$ (D) none of the choices

Evaluate $\int \frac{e^{2 x} d x}{e^{x}+1}$ . Choose the correct answer from the options given.

Evaluate $\int \frac{\ln x}{x} d x$ . Choose the correct answer: (A) $\frac{1}{2}(\ln |x|)^{2}+c$ , (B) $-2(\ln |x|)^{2}+c$ , (C) $\frac{x}{2} \ln |x|^{2}+c$ , (D) none.

Find the numerical coefficient of the integral $\int \sin x \cos^{2} x \, dx$ rounded to 2 decimal places.

Evaluate $\int \frac{1}{3 x} d x$ and choose the correct answer: (A) $\frac{1}{3} \ln |x|+C$ , (B) $\frac{1}{3} \ln |x|$ , (C) $-3 \ln |x|+C$ , (D) $3 \ln |x|-x+C$ .

Rotate the ellipse around the $y$ -axis. Use the shell method to find the volume: $\frac{4}{3} \pi a^{2} b$ .

Find the 2 positive terms from the integral of $\cos^{4} x \sin^{2} x \, dx$ . Options: (A), (B), (C), (D).

Evaluate the limit using limit laws: $\lim _{x \rightarrow \frac{15}{2}}\left(4 x^{2}+6 x-6\right)^{3 / 2}=$

What is the negative term in the integral of $\tan^{4} x \, dx$ ? Answer format: "term".

Evaluate the limit: $\lim _{x \rightarrow-1}(f(x) \cdot g(x)+2)$ given $\lim _{x \rightarrow-1} f(x)=2$ and $\lim _{x \rightarrow-1} g(x)=10$ .

Air is pumped into a balloon at 10 cm³/s. Find the rate of diameter increase when the radius is $5 \mathrm{~cm}$ .

What is the numerical coefficient of the inverse trigonometric function in the integral $\frac{d x}{2+7 x^{2}}$ ? Round to 2 decimal places.

Evaluate the integral $\int \frac{d x}{\sqrt{3+2 x-x^{2}}}$ using trigonometric substitution. What is the result?

Evaluate $\int \frac{d x}{(x-3) \sqrt{x^{2}-6 x+8}}$ . Choose the correct answer: (A), (B), (C), or (D).

What is the term inside the inverse trigonometric function in the integral of $\frac{d x}{\sqrt{25-x^{2}}}$ ?

Integrate $\int \frac{d x}{(4 x^{2}-9)^{3/2}}$ . Choose the correct answer from the options provided.

Evaluate $\int \frac{d x}{x^{2}-4 x+5}$ using trigonometric substitution. Choose the correct answer: (A), (B), (C), or (D).

Find the integral of $e^{x} \sin x$ with respect to $x$ : $\int e^{x} \sin x \, dx$ . Choose the correct answer.

Find the integral of $\int(\ln x)^{2} d x$ . Choose the correct answer from the options provided.

Integrate $\int \tan^{-1}(3x) \, dx$ . Choose the correct answer from the options provided.

Evaluate $\int \sec ^{2}(4 x+1) d x$ using $u=4 x+1$ . What is the result?

Find the marginal cost function for the production cost $C(x)=130+6x-0.8x^{2}$ .

Find the marginal cost function for the total cost $C(x)=600+17 x-0.1 x^{2}+0.4 x^{3}$ of producing $x$ magazines.

Find the marginal average cost for the weekly hat production cost function $C(x)=2800+26x+0.3x^{2}$ .