Calculus

Problem 1701

Find $\lim _{x \rightarrow-11} k(x)$ for $k(x)=\frac{x^{3}+1331}{x+11}$ using values near -11. Complete the table.

See Solution

Problem 1702

Find the limit: $L = \lim _{x \rightarrow 3} \frac{\ln (x)-\ln (3)}{x-3}$ .

See Solution

Problem 1703

Find $\ln(f'(x))$ if $f(x) = e^{x}$ .

See Solution

Problem 1704

Find the derivative of the function $h(x)=\cos x+e^{\sin (2 x)}$ .

See Solution

Problem 1705

Find $f^{\prime}(3)$ for $f(x)=\sqrt{x+2}$ using limits and graphing. Show both algebraic and visual approaches.

See Solution

Problem 1706

Evaluate these integrals: a) $\int_{0}^{1} \frac{x^{3}-4 x-10}{x^{2}-x-6} d x$ b) $\int \arctan \left(\frac{1}{x}\right) d x$ c) $\int_{0}^{\frac{\pi}{6}} \sec ^{3} \theta \tan \theta d \theta$

See Solution

Problem 1707

Evaluate the integral: $\int_{0}^{1} \frac{x^{3}-4 x-10}{x^{2}-x-6} d x$

See Solution

Problem 1708

Evaluate the integral: $\int_{0}^{1} \frac{x^{3}-4 x-10}{x^{2}-x-6} d x$

See Solution

Problem 1709

Find and interpret $\lim _{x \rightarrow \infty} \bar{C}(x)$ for $C(x)=15,000+4x$ . Determine $\bar{C}(x)$ . $\bar{C}(x)=\frac{C(x)}{x}$

See Solution

Problem 1710

Find the limit as $x$ approaches -∞ for $e^{x}$ . What is $\lim _{x \rightarrow-\infty} e^{x}=$ ?

See Solution

Problem 1711

Find $\lim _{x \rightarrow-2} F(x)$ for $F(x)=\frac{4 x}{(x+2)^{5}}$ and identify its vertical asymptote.

See Solution

Problem 1712

Find the limit as $x$ approaches 1 for $\frac{x^{4}+3 x^{3}-6 x^{2}+3 x-1}{x-1}$ by evaluating $f(x)$ near $x=1$ . Fill in values for $f(x)$ at $x = 0.99, 0.999, 0.9999, 1.0001, 1.001, 1.01$ .

See Solution

Problem 1713

Find the limit:
$\lim _{x \rightarrow 1} \frac{x^{4}+3 x^{3}-6 x^{2}+3 x-1}{x-1}$
a. Let $f(x)=\frac{x^{4}+3 x^{3}-6 x^{2}+3 x-1}{x-1}$ . Fill in the values for $f(x)$ near $x=1$ .

See Solution

Problem 1714

Find the limit as $x$ approaches -1 for $f(x) = \frac{x^{1/9} + 1}{x + 1}$ and fill in the values for $f(x)$ at given $x$ .

See Solution

Problem 1715

Find the limit of $f(x)=\frac{\arctan (x-b)}{x-1}$ as $x \to \infty$ , for any real number $b$ .

See Solution

Problem 1716

Graph the function and find the limit:
A. $\lim _{x \rightarrow-\infty} \frac{\sqrt{64 x^{2}+3 x+5}}{2 x}$
B. The limit does not exist.

See Solution

Problem 1717

Find $\bar{C}(x)$ for $C(x)=15,000+4x$ and compute $\lim_{x \rightarrow \infty} \bar{C}(x)$ . A. $\lim_{x \rightarrow \infty} \bar{C}(x)=$ B. limit does not exist.

See Solution

Problem 1718

Calculate sediment depth $D(t)=158(1-e^{-0.0137 t})$ . Find $D(20)$ and $\lim D(t)$ , then interpret results.

See Solution

Problem 1719

Determine the sediment depth $D(t)=158(1-e^{-0.0137 t})$ for $t=20$ and find $\lim D(t)$ .

See Solution

Problem 1720

Find values $x=a$ where $k(x)=e^{\sqrt{x-6}}$ is discontinuous and determine the limits as $x$ approaches $a$ .

See Solution

Problem 1721

Find where $k(x)=3 e^{\sqrt{x-4}}$ is discontinuous and determine the limits as $x$ approaches those points.

See Solution

Problem 1722

Find values of $x=a$ where $k(x)=3 e^{\sqrt{x-4}}$ is discontinuous and determine the limits as $x$ approaches these values.

See Solution

Problem 1723

Graph the function $g(x)$ , find discontinuous $x$ values, and determine left/right limits at those points.
$g(x)=\left\{\begin{array}{ll} 2 & \text { if } x<-3 \\ x^{2}+1 & \text { if }-3 \leq x \leq 1 \\ 2 & \text { if } x>1 \end{array}\right.$

See Solution

Problem 1724

Graph the function $g(x)$ , find discontinuous $x$ values, and compute left/right limits at those points.
$g(x)=\left\{\begin{array}{ll} 2 & \text { if } x<-3 \\ x^{2}+1 & \text { if }-3 \leq x \leq 1 \\ 2 & \text { if } x>1 \end{array}\right.$

See Solution

Problem 1725

Find $k$ such that $\lim _{x \rightarrow 6} \frac{2 x^{2}-9 x-18}{x-6}=k(6)-15$ .

See Solution

Problem 1726

Find the limit of postage cost $C(x)$ as $x$ approaches 3 oz from the left: $\lim_{x \rightarrow 3^{-}} C(x) = \$.

See Solution

Problem 1727

Find the limit of postage cost $C(x)$ as $x$ approaches 3 oz from the left: $\lim_{x \rightarrow 3^{-}} C(x) = \$ \square$ .

See Solution

Problem 1728

Postage costs $\$0.36$ for the first ounce and $\$0.29$ for each additional ounce. Find limits and values of $C(x)$ for $x=3$ .

See Solution

Problem 1729

Find the rate of change of demand $D(p)=-4 p^{2}-6 p+700$ with respect to price $p$ and at $p=\$13$ .

See Solution

Problem 1730

Find $\delta$ so that if $|x-2|<\delta$ , then $|4x-8|<0.1$ .

See Solution

Problem 1731

Find the derivative of $f(x)=3 x^{2}+5 x-2$ with respect to $x$ .

See Solution

Problem 1732

Find the limit: $\lim _{x \rightarrow 4}(2 x-5)$ . What is the result?

See Solution

Problem 1733

Find the limit: $\lim _{x \rightarrow a} x=a$ .

See Solution

Problem 1734

Find the average rate of change of $f(x)=\frac{8}{-3 x-5}$ on $[-3,3]$ . Average rate of change $=$ Give exact answer.

See Solution

Problem 1735

Find the derivative $f^{\prime}(x)$ for $f(x)=6x+14$ using the limit definition: $\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ .

See Solution

Problem 1736

Find the slope $m$ of the tangent line $y=3x-4$ at $a=2$ , the intersection point $(x, y)$ , $f(2)$ , and $f'(2)$ .

See Solution

Problem 1737

Find the derivative of the function $f(t)=\sqrt[3]{t}$ at a point $a \neq 0$ : $f^{\prime}(a)=$

See Solution

Problem 1738

Find $\lim _{x \rightarrow 6^{-}} H(x)$ for the function $H(x)=\left\{\begin{array}{ll}x+2, & x<6 \\ x-1, & x \geq 6\end{array}\right.$ . Options: A. $\lim _{x \rightarrow 6^{-}} H(x)=$ B. Limit does not exist.

See Solution

Problem 1739

Find $\lim _{x \rightarrow 12^{-}} H(x)$ for the function $H(x)$ defined as:
$H(x) = \begin{cases} 6, & x=12 \\ \frac{x^{3}-12x^{2}}{16x-192}, & x \neq 12 \end{cases}$

See Solution

Problem 1740

Find the limit as $x$ approaches 3 for $f(x)=\frac{x^{2}+4 x-21}{x^{2}-9}$ . Use a table and verify with a graphing calculator.

See Solution

Problem 1741

Find the limit as $x$ approaches 3 for the function $f(x)=\frac{x^{2}+4x-21}{x^{2}-9}$ . Round to three decimal places.

See Solution

Problem 1742

Find the limit as $x$ approaches 3 for the expression $\frac{x^{2}+4x-21}{x^{2}-9}$ , rounded to three decimal places.

See Solution

Problem 1743

Find the limit: $\lim _{x \rightarrow 2} \frac{x^{2}+3 x-10}{x^{2}-4}$ . Is it a number or does it not exist?

See Solution

Problem 1744

Find the limit: $\lim _{h \rightarrow 0} \frac{\frac{9}{7+h}-\frac{9}{7}}{h}$ . Choose A or B.

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Problem 1745

Find volume $V$ as a function of density $\rho$ using $V = m / \rho$ for mass $m = 8 \mathrm{~kg}$ . Evaluate $\lim_{\rho \rightarrow 0^{+}} V(\rho)$ .

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Problem 1746

Calculate the average velocity of $s(t) = \ln(t)$ in mm over $1 \leq t \leq 9$ . Round to three decimal places.

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Problem 1747

Calculate the average velocity of $s(t)$ in $\mathrm{mm}$ from $t=1$ to $t=3$ given $s(t)$ values: $s(1)=3$ , $s(3)=11$ .

See Solution

Problem 1748

Calculate the average velocity for $s(t)$ from $t=1$ to $t=3$ , given $s(1)=4$ mm and $s(3)=4$ mm.

See Solution

Problem 1749

Determine if the rate of change of $h(x)=x^{4}+x^{3}-12 x^{2}+10 x+30$ is increasing or decreasing for $(-\infty,-1.686)$ .

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Problem 1750

A baseball is thrown up. (a) Sketch a velocity vs. time graph until it's caught. (b) Describe velocity changes. (c) Describe acceleration changes. (d) Label upward, peak, and downward motion on the graph.

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Problem 1751

Find the slope of the tangent line for $f(x)=3 x^{3}+x^{5}$ at $x=4$ using the limit method. No need to simplify.

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Problem 1752

Find the derivative of the function $f(x)=e^{x-1}$ using the limit definition without simplifying.

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Problem 1753

Hitung hasil dari $\int(3 x^{2}+2 x+3) \, dx$ .

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Problem 1754

Find the derivative of $f(x)=\sqrt{x+5}$ at $x=3$ using the limit definition. No need to simplify.

See Solution

Problem 1755

Find the derivative of $f(x)=\sqrt{x+5}$ at $x=3$ using the limit definition. No need to simplify.

See Solution

Problem 1756

Find the derivative of $f(x)=\sqrt{x+5}$ at $x=3$ using the limit definition. No need to simplify.

See Solution

Problem 1757

Find the tangent line equation for $f(x)=-12 x^{2}-3 x+1$ at $x=5$ .

See Solution

Problem 1758

Hitung $\int\left(\frac{4}{x^{3}}+2 x^{2}-\frac{3}{x^{2}}\right) d x$ .

See Solution

Problem 1759

Hitung $\int\left(2 x^{-3 / 2}+x^{-2 / 3}\right) d x$ dan pilih hasil yang benar.

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Problem 1760

Hitung $\int\left(\frac{4}{3 x^{5}}+\frac{x^{2}}{2}-\frac{3}{4}\right) d x$ .

See Solution

Problem 1761

Calculate the integral of the function $e^{-x^{2}}$ from $-\infty$ to $\infty$ .

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Problem 1762

A girl throws a stone at $60^{\circ}$ with a speed of $12 \mathrm{~m/s}$ from $30 \mathrm{~m}$ high. Find:
1. Max height from sea level.
2. Impact speed.
3. Graph horizontal & vertical velocity vs. time.

See Solution

Problem 1763

A coffee cup has $140 \mathrm{mg}$ caffeine. If it decreases by $10 \%$ hourly, how long to eliminate half?

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Problem 1764

Find $f^{\prime}(3)$ for the function $f(x)=2x+5$ .

See Solution

Problem 1765

Find $f \prime(3)$ for $f(x)=\sqrt{x+1}$ ; options: $1/2$ , $1/4$ .

See Solution

Problem 1766

Find $\frac{f(1+h)-f(1)}{h}$ for $f(x)=-x^{2}+3x$ , where $h \neq 0$ .

See Solution

Problem 1767

Graph a function with a horizontal tangent line at $x=5$ .

See Solution

Problem 1768

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

See Solution

Problem 1769

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

See Solution

Problem 1770

Find the fuel consumption rate at 9 AM using $f(t)=0.6 t^{2}-0.3 t^{0.3}+21$ . Round to 2 decimal places.

See Solution

Problem 1771

Find the average fuel consumption rate from 6 AM to 2 PM using $f(t)=0.6 t^{2}-0.3 t^{0.3}+21$ . Round to 2 decimal places.

See Solution

Problem 1772

Find the second derivative $f^{\prime \prime}(x)$ of the function $f(x)=2 \sqrt{x}+6 x^{\frac{1}{3}}+\frac{3}{x^{2}}$ .

See Solution

Problem 1773

Calculate the average fuel consumption rate from 6 AM to 2 PM using $f(t)=0.6 t^{2}-0.3 t^{0.3}+21$ . Round to 2 decimal places.

See Solution

Problem 1774

Find the second derivative $f^{\prime \prime}\left(\frac{\pi}{6}\right)$ for $f(x)=10 \sin (x)+8 \cos (x)$ .

See Solution

Problem 1775

Find the second derivative $f^{\prime \prime}(x)$ for the function $f(x)=2 \sqrt{x}+6 x^{\frac{1}{3}}+\frac{3}{x^{2}}$ .

See Solution

Problem 1776

Find the derivative of the function $g(x)=8 x^{4}+6 x^{3}-3.7$ .

See Solution

Problem 1777

Find the derivative of $g(x)=-7 x^{2}-2 x^{5}$ .

See Solution

Problem 1778

Find the derivative of $g(x) = -5 x^{3} + 3 + 2 x^{5}$ .

See Solution

Problem 1779

Find the slopes of the tangent lines for $f(x)=6+5x-3x^{2}$ at $x=0$ , $x=1$ , and $x=3$ .

See Solution

Problem 1780

Find the slopes of the tangent lines for $f(x)=6+5x-3x^{2}$ at $x=0$ , $x=1$ , and $x=3$ .

See Solution

Problem 1781

Find the velocity of a particle at $t=9$ given its position $s(t)=3.2 t^{2}+22 t$ for $0 \leq t \leq 18$ .

See Solution

Problem 1782

A pitcher throws a baseball at 98 mph over a $2.8 \mathrm{~m}$ distance. Find the acceleration without time given.

See Solution

Problem 1783

Find the derivative of the function $f(x)=-2 x^{\frac{3}{4}}-5-8 x^{\frac{1}{2}}$ .

See Solution

Problem 1784

Rewrite $f(x)=(3 x+2)(2 x+3)$ as a sum and find $f^{\prime}(x)$ .

See Solution

Problem 1785

Find the fuel consumption rate at noon using $f(t)=0.4 t^{2}-0.2 t^{0.4}+19$ . Round to 2 decimal places.

See Solution

Problem 1786

Find the fuel consumption rate at 9 AM using $f(t)=0.4 t^{2}-0.1 t^{0.5}+21$ . Round to 2 decimal places.

See Solution

Problem 1787

Find the fuel consumption rate at 10 AM using $f(t)=0.8 t^{3}-0.2 t^{0.3}+27$ . Round to 2 decimal places.

See Solution

Problem 1788

Find the derivative of $f(t)=0.8 t^{3}-0.2 t^{0.3}+27$ .

See Solution

Problem 1789

Find the average fuel consumption rate from 6 AM to 1 PM using $f(t)=0.8 t^{3}-0.2 t^{0.3}+27$ . Round to 2 decimal places.

See Solution

Problem 1790

Find the value of $a$ that makes the function $f(t)=\frac{\sqrt{4+t^{2}}-2}{t^{2}}$ for $t \neq 0$ and $f(0)=a$ continuous.

See Solution

Problem 1791

Find the value of $a$ for which the function $f(t)=\frac{\sqrt{4+t^{2}}-2}{t^{2}}$ (for $t \neq 0$ ) is continuous at $t=0$ .

See Solution

Problem 1792

Find the derivative of the function $f(x) = (2 + x^4)^{2/3}$ .

See Solution

Problem 1793

Annie wants \$2300 in 7 years with 5.2% continuous interest. How much must she deposit? Round to the nearest dollar.

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Problem 1794

Find the derivative of the function $f(x)=(2+x^{4})^{2/3}$ .

See Solution

Problem 1795

Find the slope of the secant line for $f(x)=2 x^{2}-12 x+10$ on $[4,4+h]$ for given $h$ values: 1, 0.1, 0.01, 0.001.

See Solution

Problem 1796

Find $f^{\prime}(x)$ and $f^{\prime}(-5)$ if $f(x)=14$ . What are the values?

See Solution

Problem 1797

Find the average fuel consumption rate from 6 AM to 1 PM using $f(t)=0.8 t^{3}-0.2 t^{0.3}+27$ . Round to 2 decimal places.

See Solution

Problem 1798

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

See Solution

Problem 1799

Expand and simplify: $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{1}{2 x+3}$ .

See Solution

Problem 1800

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

See Solution

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Calculus

Problem 1701

Find lim⁡x→−11k(x)\lim _{x \rightarrow-11} k(x)limx→−11​k(x) for k(x)=x3+1331x+11k(x)=\frac{x^{3}+1331}{x+11}k(x)=x+11x3+1331​ using values near -11. Complete the table.

Problem 1702

Find the limit: L=lim⁡x→3ln⁡(x)−ln⁡(3)x−3L = \lim _{x \rightarrow 3} \frac{\ln (x)-\ln (3)}{x-3}L=limx→3​x−3ln(x)−ln(3)​.

Problem 1703

Find ln⁡(f′(x))\ln(f'(x))ln(f′(x)) if f(x)=exf(x) = e^{x}f(x)=ex.

Problem 1704

Find the derivative of the function h(x)=cos⁡x+esin⁡(2x)h(x)=\cos x+e^{\sin (2 x)}h(x)=cosx+esin(2x).

Problem 1705

Find f′(3)f^{\prime}(3)f′(3) for f(x)=x+2f(x)=\sqrt{x+2}f(x)=x+2​ using limits and graphing. Show both algebraic and visual approaches.

Problem 1706

Problem 1707

Evaluate the integral: ∫01x3−4x−10x2−x−6dx\int_{0}^{1} \frac{x^{3}-4 x-10}{x^{2}-x-6} d x∫01​x2−x−6x3−4x−10​dx

Problem 1708

Evaluate the integral: ∫01x3−4x−10x2−x−6dx\int_{0}^{1} \frac{x^{3}-4 x-10}{x^{2}-x-6} d x∫01​x2−x−6x3−4x−10​dx

Problem 1709

Find and interpret lim⁡x→∞Cˉ(x)\lim _{x \rightarrow \infty} \bar{C}(x)limx→∞​Cˉ(x) for C(x)=15,000+4xC(x)=15,000+4xC(x)=15,000+4x. Determine Cˉ(x)\bar{C}(x)Cˉ(x). Cˉ(x)=C(x)x\bar{C}(x)=\frac{C(x)}{x}Cˉ(x)=xC(x)​

Problem 1710

Find the limit as xxx approaches -∞ for exe^{x}ex. What is lim⁡x→−∞ex=\lim _{x \rightarrow-\infty} e^{x}=limx→−∞​ex=?

Problem 1711

Find lim⁡x→−2F(x)\lim _{x \rightarrow-2} F(x)limx→−2​F(x) for F(x)=4x(x+2)5F(x)=\frac{4 x}{(x+2)^{5}}F(x)=(x+2)54x​ and identify its vertical asymptote.

Problem 1712

Problem 1713

Problem 1714

Find the limit as xxx approaches -1 for f(x)=x1/9+1x+1f(x) = \frac{x^{1/9} + 1}{x + 1}f(x)=x+1x1/9+1​ and fill in the values for f(x)f(x)f(x) at given xxx.

Problem 1715

Find the limit of f(x)=arctan⁡(x−b)x−1f(x)=\frac{\arctan (x-b)}{x-1}f(x)=x−1arctan(x−b)​ as x→∞x \to \inftyx→∞, for any real number bbb.

Problem 1716

Graph the function and find the limit: A. lim⁡x→−∞64x2+3x+52x\lim _{x \rightarrow-\infty} \frac{\sqrt{64 x^{2}+3 x+5}}{2 x}x→−∞lim​2x64x2+3x+5​​ B. The limit does not exist.

Problem 1717

Find Cˉ(x)\bar{C}(x)Cˉ(x) for C(x)=15,000+4xC(x)=15,000+4xC(x)=15,000+4x and compute lim⁡x→∞Cˉ(x)\lim_{x \rightarrow \infty} \bar{C}(x)limx→∞​Cˉ(x). A. lim⁡x→∞Cˉ(x)=\lim_{x \rightarrow \infty} \bar{C}(x)=limx→∞​Cˉ(x)= B. limit does not exist.

Problem 1718

Calculate sediment depth D(t)=158(1−e−0.0137t)D(t)=158(1-e^{-0.0137 t})D(t)=158(1−e−0.0137t). Find D(20)D(20)D(20) and lim⁡D(t)\lim D(t)limD(t), then interpret results.

Problem 1719

Determine the sediment depth D(t)=158(1−e−0.0137t)D(t)=158(1-e^{-0.0137 t})D(t)=158(1−e−0.0137t) for t=20t=20t=20 and find lim⁡D(t)\lim D(t)limD(t).

Problem 1720

Find values x=ax=ax=a where k(x)=ex−6k(x)=e^{\sqrt{x-6}}k(x)=ex−6​ is discontinuous and determine the limits as xxx approaches aaa.

Problem 1721

Find where k(x)=3ex−4k(x)=3 e^{\sqrt{x-4}}k(x)=3ex−4​ is discontinuous and determine the limits as xxx approaches those points.

Problem 1722

Find values of x=ax=ax=a where k(x)=3ex−4k(x)=3 e^{\sqrt{x-4}}k(x)=3ex−4​ is discontinuous and determine the limits as xxx approaches these values.

Problem 1723

Problem 1724

Problem 1725

Find kkk such that lim⁡x→62x2−9x−18x−6=k(6)−15\lim _{x \rightarrow 6} \frac{2 x^{2}-9 x-18}{x-6}=k(6)-15limx→6​x−62x2−9x−18​=k(6)−15.

Problem 1726

Find the limit of postage cost C(x)C(x)C(x) as xxx approaches 3 oz from the left: $\lim_{x \rightarrow 3^{-}} C(x) = \$.

Problem 1727

Find the limit of postage cost C(x)C(x)C(x) as xxx approaches 3 oz from the left: lim⁡x→3−C(x)=$□\lim_{x \rightarrow 3^{-}} C(x) = \$ \squarelimx→3−​C(x)=$□.

Problem 1728

Postage costs $0.36\$0.36$0.36 for the first ounce and $0.29\$0.29$0.29 for each additional ounce. Find limits and values of C(x)C(x)C(x) for x=3x=3x=3.

Problem 1729

Find the rate of change of demand D(p)=−4p2−6p+700D(p)=-4 p^{2}-6 p+700D(p)=−4p2−6p+700 with respect to price ppp and at p=$13p=\$13p=$13.

Problem 1730

Find δ\deltaδ so that if ∣x−2∣<δ|x-2|<\delta∣x−2∣<δ, then ∣4x−8∣<0.1|4x-8|<0.1∣4x−8∣<0.1.

Problem 1731

Find the derivative of f(x)=3x2+5x−2f(x)=3 x^{2}+5 x-2f(x)=3x2+5x−2 with respect to xxx.

Problem 1732

Find the limit: lim⁡x→4(2x−5)\lim _{x \rightarrow 4}(2 x-5)limx→4​(2x−5). What is the result?

Problem 1733

Find the limit: lim⁡x→ax=a\lim _{x \rightarrow a} x=alimx→a​x=a.

Problem 1734

Find the average rate of change of f(x)=8−3x−5f(x)=\frac{8}{-3 x-5}f(x)=−3x−58​ on [−3,3][-3,3][−3,3]. Average rate of change === Give exact answer.

Problem 1735

Find the derivative f′(x)f^{\prime}(x)f′(x) for f(x)=6x+14f(x)=6x+14f(x)=6x+14 using the limit definition: lim⁡h→0f(x+h)−f(x)h\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}limh→0​hf(x+h)−f(x)​.

Problem 1736

Find the slope mmm of the tangent line y=3x−4y=3x-4y=3x−4 at a=2a=2a=2, the intersection point (x,y)(x, y)(x,y), f(2)f(2)f(2), and f′(2)f'(2)f′(2).

Problem 1737

Find the derivative of the function f(t)=t3f(t)=\sqrt[3]{t}f(t)=3t​ at a point a≠0a \neq 0a=0: f′(a)=f^{\prime}(a)= f′(a)=

Problem 1738

Problem 1739

Find lim⁡x→12−H(x)\lim _{x \rightarrow 12^{-}} H(x)limx→12−​H(x) for the function H(x)H(x)H(x) defined as: H(x)={6,x=12x3−12x216x−192,x≠12H(x) = \begin{cases} 6, & x=12 \\ \frac{x^{3}-12x^{2}}{16x-192}, & x \neq 12 \end{cases}H(x)={6,16x−192x3−12x2​,​x=12x=12​

Problem 1740

Find the limit as xxx approaches 3 for f(x)=x2+4x−21x2−9f(x)=\frac{x^{2}+4 x-21}{x^{2}-9}f(x)=x2−9x2+4x−21​. Use a table and verify with a graphing calculator.

Problem 1741

Find the limit as xxx approaches 3 for the function f(x)=x2+4x−21x2−9f(x)=\frac{x^{2}+4x-21}{x^{2}-9}f(x)=x2−9x2+4x−21​. Round to three decimal places.

Problem 1742

Find the limit as xxx approaches 3 for the expression x2+4x−21x2−9\frac{x^{2}+4x-21}{x^{2}-9}x2−9x2+4x−21​, rounded to three decimal places.

Problem 1743

Find $\lim _{x \rightarrow-11} k(x)$ for $k(x)=\frac{x^{3}+1331}{x+11}$ using values near -11. Complete the table.

Find the limit: $L = \lim _{x \rightarrow 3} \frac{\ln (x)-\ln (3)}{x-3}$ .

Find $\ln(f'(x))$ if $f(x) = e^{x}$ .

Find the derivative of the function $h(x)=\cos x+e^{\sin (2 x)}$ .

Find $f^{\prime}(3)$ for $f(x)=\sqrt{x+2}$ using limits and graphing. Show both algebraic and visual approaches.

Evaluate the integral: $\int_{0}^{1} \frac{x^{3}-4 x-10}{x^{2}-x-6} d x$

Evaluate the integral: $\int_{0}^{1} \frac{x^{3}-4 x-10}{x^{2}-x-6} d x$

Find and interpret $\lim _{x \rightarrow \infty} \bar{C}(x)$ for $C(x)=15,000+4x$ . Determine $\bar{C}(x)$ . $\bar{C}(x)=\frac{C(x)}{x}$

Find the limit as $x$ approaches -∞ for $e^{x}$ . What is $\lim _{x \rightarrow-\infty} e^{x}=$ ?

Find $\lim _{x \rightarrow-2} F(x)$ for $F(x)=\frac{4 x}{(x+2)^{5}}$ and identify its vertical asymptote.

Find the limit as $x$ approaches -1 for $f(x) = \frac{x^{1/9} + 1}{x + 1}$ and fill in the values for $f(x)$ at given $x$ .

Find the limit of $f(x)=\frac{\arctan (x-b)}{x-1}$ as $x \to \infty$ , for any real number $b$ .

Graph the function and find the limit:
A. $\lim _{x \rightarrow-\infty} \frac{\sqrt{64 x^{2}+3 x+5}}{2 x}$
B. The limit does not exist.

Find $\bar{C}(x)$ for $C(x)=15,000+4x$ and compute $\lim_{x \rightarrow \infty} \bar{C}(x)$ . A. $\lim_{x \rightarrow \infty} \bar{C}(x)=$ B. limit does not exist.

Calculate sediment depth $D(t)=158(1-e^{-0.0137 t})$ . Find $D(20)$ and $\lim D(t)$ , then interpret results.

Determine the sediment depth $D(t)=158(1-e^{-0.0137 t})$ for $t=20$ and find $\lim D(t)$ .

Find values $x=a$ where $k(x)=e^{\sqrt{x-6}}$ is discontinuous and determine the limits as $x$ approaches $a$ .

Find where $k(x)=3 e^{\sqrt{x-4}}$ is discontinuous and determine the limits as $x$ approaches those points.

Find values of $x=a$ where $k(x)=3 e^{\sqrt{x-4}}$ is discontinuous and determine the limits as $x$ approaches these values.

Find $k$ such that $\lim _{x \rightarrow 6} \frac{2 x^{2}-9 x-18}{x-6}=k(6)-15$ .

Find the limit of postage cost $C(x)$ as $x$ approaches 3 oz from the left: $\lim_{x \rightarrow 3^{-}} C(x) = \$.

Find the limit of postage cost $C(x)$ as $x$ approaches 3 oz from the left: $\lim_{x \rightarrow 3^{-}} C(x) = \$ \square$ .

Postage costs $\$0.36$ for the first ounce and $\$0.29$ for each additional ounce. Find limits and values of $C(x)$ for $x=3$ .

Find the rate of change of demand $D(p)=-4 p^{2}-6 p+700$ with respect to price $p$ and at $p=\$13$ .

Find $\delta$ so that if $|x-2|<\delta$ , then $|4x-8|<0.1$ .

Find the derivative of $f(x)=3 x^{2}+5 x-2$ with respect to $x$ .

Find the limit: $\lim _{x \rightarrow 4}(2 x-5)$ . What is the result?

Find the limit: $\lim _{x \rightarrow a} x=a$ .

Find the average rate of change of $f(x)=\frac{8}{-3 x-5}$ on $[-3,3]$ . Average rate of change $=$ Give exact answer.

Find the derivative $f^{\prime}(x)$ for $f(x)=6x+14$ using the limit definition: $\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ .

Find the slope $m$ of the tangent line $y=3x-4$ at $a=2$ , the intersection point $(x, y)$ , $f(2)$ , and $f'(2)$ .

Find the derivative of the function $f(t)=\sqrt[3]{t}$ at a point $a \neq 0$ : $f^{\prime}(a)=$

Find $\lim _{x \rightarrow 12^{-}} H(x)$ for the function $H(x)$ defined as:
$H(x) = \begin{cases} 6, & x=12 \\ \frac{x^{3}-12x^{2}}{16x-192}, & x \neq 12 \end{cases}$

Find the limit as $x$ approaches 3 for $f(x)=\frac{x^{2}+4 x-21}{x^{2}-9}$ . Use a table and verify with a graphing calculator.

Find the limit as $x$ approaches 3 for the function $f(x)=\frac{x^{2}+4x-21}{x^{2}-9}$ . Round to three decimal places.

Find the limit as $x$ approaches 3 for the expression $\frac{x^{2}+4x-21}{x^{2}-9}$ , rounded to three decimal places.

Find the limit: $\lim _{x \rightarrow 2} \frac{x^{2}+3 x-10}{x^{2}-4}$ . Is it a number or does it not exist?

Find the limit: $\lim _{h \rightarrow 0} \frac{\frac{9}{7+h}-\frac{9}{7}}{h}$ . Choose A or B.

Find volume $V$ as a function of density $\rho$ using $V = m / \rho$ for mass $m = 8 \mathrm{~kg}$ . Evaluate $\lim_{\rho \rightarrow 0^{+}} V(\rho)$ .

Calculate the average velocity of $s(t) = \ln(t)$ in mm over $1 \leq t \leq 9$ . Round to three decimal places.

Calculate the average velocity of $s(t)$ in $\mathrm{mm}$ from $t=1$ to $t=3$ given $s(t)$ values: $s(1)=3$ , $s(3)=11$ .

Calculate the average velocity for $s(t)$ from $t=1$ to $t=3$ , given $s(1)=4$ mm and $s(3)=4$ mm.

Determine if the rate of change of $h(x)=x^{4}+x^{3}-12 x^{2}+10 x+30$ is increasing or decreasing for $(-\infty,-1.686)$ .

Find the slope of the tangent line for $f(x)=3 x^{3}+x^{5}$ at $x=4$ using the limit method. No need to simplify.

Find the derivative of the function $f(x)=e^{x-1}$ using the limit definition without simplifying.

Hitung hasil dari $\int(3 x^{2}+2 x+3) \, dx$ .

Find the derivative of $f(x)=\sqrt{x+5}$ at $x=3$ using the limit definition. No need to simplify.

Find the derivative of $f(x)=\sqrt{x+5}$ at $x=3$ using the limit definition. No need to simplify.

Find the derivative of $f(x)=\sqrt{x+5}$ at $x=3$ using the limit definition. No need to simplify.

Find the tangent line equation for $f(x)=-12 x^{2}-3 x+1$ at $x=5$ .

Hitung $\int\left(\frac{4}{x^{3}}+2 x^{2}-\frac{3}{x^{2}}\right) d x$ .

Hitung $\int\left(2 x^{-3 / 2}+x^{-2 / 3}\right) d x$ dan pilih hasil yang benar.

Hitung $\int\left(\frac{4}{3 x^{5}}+\frac{x^{2}}{2}-\frac{3}{4}\right) d x$ .

Calculate the integral of the function $e^{-x^{2}}$ from $-\infty$ to $\infty$ .

A girl throws a stone at $60^{\circ}$ with a speed of $12 \mathrm{~m/s}$ from $30 \mathrm{~m}$ high. Find:
1. Max height from sea level.
2. Impact speed.
3. Graph horizontal & vertical velocity vs. time.

A coffee cup has $140 \mathrm{mg}$ caffeine. If it decreases by $10 \%$ hourly, how long to eliminate half?

Find $f^{\prime}(3)$ for the function $f(x)=2x+5$ .

Find $f \prime(3)$ for $f(x)=\sqrt{x+1}$ ; options: $1/2$ , $1/4$ .

Find $\frac{f(1+h)-f(1)}{h}$ for $f(x)=-x^{2}+3x$ , where $h \neq 0$ .

Graph a function with a horizontal tangent line at $x=5$ .

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

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

Find the fuel consumption rate at 9 AM using $f(t)=0.6 t^{2}-0.3 t^{0.3}+21$ . Round to 2 decimal places.

Find the average fuel consumption rate from 6 AM to 2 PM using $f(t)=0.6 t^{2}-0.3 t^{0.3}+21$ . Round to 2 decimal places.

Find the second derivative $f^{\prime \prime}(x)$ of the function $f(x)=2 \sqrt{x}+6 x^{\frac{1}{3}}+\frac{3}{x^{2}}$ .

Calculate the average fuel consumption rate from 6 AM to 2 PM using $f(t)=0.6 t^{2}-0.3 t^{0.3}+21$ . Round to 2 decimal places.

Find the second derivative $f^{\prime \prime}\left(\frac{\pi}{6}\right)$ for $f(x)=10 \sin (x)+8 \cos (x)$ .