Calculus

Problem 32601

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

See Solution

Problem 32602

Find the limit as $x$ approaches 4 for $\frac{x^{3}-64}{x-4}$ and write a simpler function $g(x)$ that matches it except at one point.

See Solution

Problem 32603

Estimate the limit of $\lim _{x \rightarrow 0 \pm} \frac{\sin |x|}{x}$ from both sides and check if it exists.

See Solution

Problem 32604

Find the limit as $x$ approaches 0 for the expression $\frac{x^{2}+5 x}{x}$ .

See Solution

Problem 32605

Find the absolute maximum of the function $f(x)=x^{4}+8 x^{3}+18 x^{2}+4$ on the interval $[-4,1]$ .

See Solution

Problem 32606

Find the intervals where the function $f(x)=\frac{2}{15} x^{6}-5 x^{4}$ is concave up.

See Solution

Problem 32607

Find the derivative $y^{\prime}$ for $y=(4+9x-8x^{2}) e^{x}$ . Simplify your answer.

See Solution

Problem 32608

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{x+6}-\sqrt{6}}{x}$ .

See Solution

Problem 32609

Find the limit as $x$ approaches $\frac{\pi}{2}$ of $\sin x$ .

See Solution

Problem 32610

Find the limit as $x$ approaches -6 for the expression $x^{2}+6x$ .

See Solution

Problem 32611

Find the limit: $\lim _{x \rightarrow -5} \frac{x^{2}-25}{x+5}$ .

See Solution

Problem 32612

Find the limit as $x$ approaches $\pi$ for $\cos(9x)$ .

See Solution

Problem 32613

Find the limit as $\theta$ approaches 0 for the expression $\frac{\cos (5 \theta) \tan (5 \theta)}{\theta}$ .

See Solution

Problem 32614

Find $\frac{d y}{d t}$ when $y=x^{3}+3$ , $\frac{d x}{d t}=5$ , and $x=1$ . Answer: $\frac{d y}{d t}=\square$ .

See Solution

Problem 32615

Find the limit as $x$ approaches 2 for the expression $\frac{x^{2}+6x-16}{x^{2}-4}$ .

See Solution

Problem 32616

1. Analyze the region $R$ between $y=2x$ and $y=x^2$ : (a) Sketch $R$ . (b) Find and label intersection points. (c) Sketch the solid from revolving $R$ around the $x$ -axis. (d) Provide an integral for the volume (no need to solve).
2. Sketch the solid from revolving $R$ around the $y$ -axis.

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Problem 32617

Find $\frac{d y}{d t}$ for $4 x y-6 x+3 y^{3}=-135$ given $\frac{d x}{d t}=-12$ , $x=3$ , $y=-3$ . $\frac{d y}{d t}=\square$

See Solution

Problem 32618

Evaluate the limits given:
1. $\lim _{x \rightarrow c}[5 g(x)]$
2. $\lim _{x \rightarrow c}[f(x)+g(x)]$
3. $\lim _{x \rightarrow c}[f(x) g(x)]$
4. $\lim _{x \rightarrow c} \frac{f(x)}{g(x)}$
with $\lim _{x \rightarrow c} f(x)=\frac{4}{5}$ and $\lim _{x \rightarrow c} g(x)=4$ .

See Solution

Problem 32619

Find the derivative of the function $f(x)=\cos x-5 \tan x$ , i.e., calculate $f^{\prime}(x)$ .

See Solution

Problem 32620

A boat is 2 feet from a dock, and the rope is pulled in at 1 ft/sec. How fast is the distance to the dock decreasing? $\square$ ft/sec.

See Solution

Problem 32621

Find the limits: (a) $\lim _{x \rightarrow 4} (2 x^{2}-3 x+42)$ , (b) $\lim _{x \rightarrow 62} \sqrt[3]{x+2}$ , (c) $\lim _{x \rightarrow 4} g(f(x))$ .

See Solution

Problem 32622

Find the derivative $f'(t)$ of the function $f(t)=(t^2+7t+6)(6t^{-2}+6t^{-3})$ .

See Solution

Problem 32623

A ball is thrown with a velocity of $45 \mathrm{ft/s}$ . Height after $t$ seconds is $y=45t-14t^2$ . Find average velocity from $t=2$ for given intervals.

See Solution

Problem 32624

Evaluate the limit as $x$ approaches infinity for the expression $3 x^{9}+11 x-4$ . What is the limit?

See Solution

Problem 32625

A point moves on $xy=20$ . At $(4,5)$ with $x$ increasing by 4 units/sec, find the rate of change of $y$ .
The $y$ -coordinate is $\square$ at $\square$ units per second.

See Solution

Problem 32626

Find the limit as $x$ approaches infinity for $3x^{9} + 11x - 4$ . What is the limit?

See Solution

Problem 32627

Find the average velocity for the position function $s(t)=6 t^{2}-5 t-7$ from $t=5$ to $t=5+h$ . Use exact values. $A V_{[5,5+h]}=\frac{s(5+h)-s(5)}{h}$

See Solution

Problem 32628

Find $\frac{d y}{d x}$ for $y=f(1+\sqrt{2-x^{2}})$ at $x=1$ , given $f^{\prime}(2)=-3$ .

See Solution

Problem 32629

Find the limit as $x$ approaches negative infinity for the expression: $-5 x^{4}+3 x^{5}+2$ .

See Solution

Problem 32630

Find removable and nonremovable discontinuities of $f(x)=\frac{7}{x}$ . Identify values of $x$ .

See Solution

Problem 32631

A boat is pulled toward a dock at 3 ft/s. How fast is the distance to the dock decreasing when 2 ft away? Answer: $\square$ ft/s.

See Solution

Problem 32632

Find the derivative of the function $f(t)=(t^{2}+7t+6)(6t^{-2}+6t^{-3})$ . What is $f^{\prime}(t)$ ?

See Solution

Problem 32633

Find the limits:
1. $\lim _{x \rightarrow \infty} \frac{3 x-2}{\sqrt{x^{2}+5}}$
2. $\lim _{x \rightarrow-\infty} \frac{3 x-2}{\sqrt{x^{2}+5}}$

See Solution

Problem 32634

Evaluate the limit: $\lim _{x \rightarrow 4}\left(4 x^{2}-1\right)(\sqrt{6 x+1})$ using basic limit laws.

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Problem 32635

A rock creates ripples in a pond. If the radius increases at 4 ft/s, find the area change rate when the radius is 9 ft. Area change is approximately $\square$ sq ft/s. Round to the nearest thousandth.

See Solution

Problem 32636

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{4}-5 x^{3}+8}}{9 x^{2}+6}$

See Solution

Problem 32637

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

See Solution

Problem 32638

A 10-foot ladder leans against a wall. If the bottom moves away at 3 ft/sec, how fast is the top sliding down when 6 ft away? The rate is $\square \mathrm{ft} / \mathrm{sec}$ .

See Solution

Problem 32639

Evaluate the limit: $\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}$ using direct substitution. Simplify your answer.

See Solution

Problem 32640

Find the derivative $\frac{d y}{d t}$ for $y=\frac{4 t \ln t}{e^{t}}$ . Simplify your answer.

See Solution

Problem 32641

Find the integral: $\int \frac{3}{\sqrt{4-25 x^{2}}} d x$ .

See Solution

Problem 32642

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{10 x \Delta x + 5 \Delta x^{2} - 7 \Delta x}{\Delta x}$ .

See Solution

Problem 32643

Find the limits:
1. $\lim _{x \rightarrow \infty} \frac{3 x-2}{\sqrt{x^{2}+5}}$
2. $\lim _{x \rightarrow-\infty} \frac{3 x-2}{\sqrt{x^{2}+5}}$

See Solution

Problem 32644

Find the derivative of the function $F(x)=\frac{\sqrt{x}-7}{\sqrt{x}+7}$ . What is $F'(x)$ ?

See Solution

Problem 32645

Find the current rate of change of sales given $s=50,000-40,000 e^{-0.00025 x}$ , with $x=\$ 4,000$ and $dx/dt=\$ 300$ .

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Problem 32646

Find the tangent and normal line equations at $(-1,4)$ for the curve $x^{2} y^{2}=16$ . Provide answers in slope-intercept form.

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Problem 32647

Prove that $y=x^3+3x+1$ satisfies $y''' + xy'' - 2y' = 0$ .

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Problem 32648

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{256+h}-16}{h}$ and rewrite it for direct substitution.

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Problem 32649

Evaluate these limits: (a) $\lim _{x \rightarrow \infty} \frac{\sqrt{3+10 x^{2}}}{8+9 x}=$ (b) $\lim _{x \rightarrow-\infty} \frac{\sqrt{3+10 x^{2}}}{8+9 x}=$

See Solution

Problem 32650

Find limits and function values for $a=2$ and $a=0$ : $\lim_{x \rightarrow a^{-}} f(x)$ , $\lim_{x \rightarrow a^{+}} f(x)$ , $\lim_{x \rightarrow a} f(x)$ , and $f(a)$ .

See Solution

Problem 32651

Find the limit as $t$ approaches $\infty$ for $\frac{-3t-9}{\sqrt{t^2+3t+5}}$ . Enter I, -I, or DNE. Limit =

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Problem 32652

Evaluate the limit: $\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}$ and simplify to find the exact answer.

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Problem 32653

Determine if the function $f(x)$ has a jump discontinuity at $x=8$ by finding the limits: $\lim _{x \rightarrow 8^{-}} f(x) = \square, \quad \lim _{x \rightarrow 8^{+}} f(x) = \square.$

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Problem 32654

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}=\square$

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Problem 32655

Evaluate these limits: 1. $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{4}-6 x^{3}-2}}{9 x^{2}-1}=\square$ 2. $\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{4}-6 x^{3}-2}}{9 x^{2}-1}=\square$

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Problem 32656

Differentiate the function $g(x)=x^{5}+\frac{1}{x^{5}}$ . Find $g^{\prime}(x)=$ .

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Problem 32657

Find $f'(2)$ and $f'(-3)$ for the function $f(x)=\frac{7}{x^{7}}$ .

See Solution

Problem 32658

Calculate the relative rate of change of $f(x)=3 x^{2}-\ln x$ at $x=4$ . Answer: $\square$ .

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Problem 32659

Find $f'(x)$ for $f(x)=12x(\sin(x)+\cos(x))$ and evaluate $f'(-\frac{\pi}{6})$ .

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Problem 32660

Simplify the integral $\int \frac{1}{x^{2} \sqrt{9 x^{2}+16}} d x$ with a trigonometric substitution.

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Problem 32661

Find the limit: $\lim _{x \rightarrow \frac{\pi}{2}} \frac{9 \sec (x)+3}{\tan (x)}$ .

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Problem 32662

Find the slope of the tangent line to $y=3 x^{3}$ at $(-5,-375)$ and write the equation in $y=m x+b$ form.

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Problem 32663

Find the derivative of the function $f(x)=4 x^{2}-4 x+8$ . What is $f^{\prime}(x)=\square$ ?

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Problem 32664

Differentiate the function $f(x)=x^{2}+8x+30$ . Find $f^{\prime}(x)$ .

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Problem 32665

Find $\frac{d y}{d x}$ for $\sqrt[3]{x^{2}}+\sqrt[3]{y^{2}}=1$ and $\frac{d^{2} y}{d x^{2}}$ at $x=\frac{1}{125}$ .

See Solution

Problem 32666

Calculate the relative rate of change of $f(x)=264-3x$ at $x=33$ . Answer: $\square$ (round to three decimals).

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Problem 32667

Find the derivative $f^{\prime}(x)$ for the function $f(x)=7 x^{13}$ . What is $f^{\prime}(x)=?$

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Problem 32668

Find the derivative of the function $f(x) = 8x^3 + 9x^2 + 3x - 7$ . What is $f'(x) = \square$ ?

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Problem 32669

Given $f(t)=(t^{2}+2 t+2)(3 t^{2}+6)$ , find $f^{\prime}(t)$ and then calculate $f^{\prime}(5)$ .

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Problem 32670

Find the derivative $\frac{\mathrm{dy}}{\mathrm{dx}}$ for the equation $x^{3}+2 x^{2} y-x y^{2}+y^{3}+3=0$ at the point (1,-1).

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Problem 32671

Find $F'(x)$ for $F(x)=12x(\sin x+\cos x)$ and evaluate $F'(-\frac{\pi}{6})$ .

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Problem 32672

Find the percentage rate of change of $f(x)=5250-5 x^{2}$ at $x=25$ . Answer: $\square \%$ .

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Problem 32673

Find the derivative of $f(x)=7 \frac{1}{x^{9}}+x^{2}+14$ . What is $f^{\prime}(x)=\square$ ?

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Problem 32674

Simplify the integral $\int \frac{1}{x^{2} \sqrt{9 x^{2}+16}} d x$ using trigonometric substitution.

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Problem 32675

Compute the derivative using the Product Rule: Find $\frac{d}{d t}\left((t^{2}+1)(t+9)\right)$ at $t=-3$ .

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Problem 32676

Evaluate if you can use substitution for the limit: $\lim _{x \rightarrow \frac{\pi}{2}}\left(\sin x+x^{2}\right)$ and find the limit.

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Problem 32677

Find the derivative of $\frac{1}{x+2}$ at $x=4$ .

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Problem 32678

Find the derivative of the function $f(x)=\frac{2 x}{x^{3}+1}$ . What is $f^{\prime}(x)=\square$ ?

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Problem 32679

Find the derivative of $f(x)=e^{9x}$ . What is $f'(x)$ ?

See Solution

Problem 32680

Find the derivative using the Quotient Rule: $\frac{d}{d t}\left(\frac{t^{2}+1}{t^{2}-1}\right)$ at $t=-4$ .

See Solution

Problem 32681

Find the derivative of $f(x)=x e^{6 x}$ . What is $f^{\prime}(x)=\square$ ?

See Solution

Problem 32682

Find the tangent line equation for $y=4 \ln (x)$ at $x=2$ . Tangent Line: $y=$

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Problem 32683

Find the derivative of $f(x)=e^{-3 x^{3}}$ . What is $f^{\prime}(x)=\square$ ?

See Solution

Problem 32684

Differentiate the function $f(x)=\frac{5\left(x^{4}-3\right)}{2}$ .

See Solution

Problem 32685

Find the average velocity of a stone tossed with height $h(t)=15t-4.9t^2$ over intervals: a. $[1,1.05]$ , b. $[1,1.01]$ , c. $[1,1.005]$ , d. $[1,1.001]$ .

See Solution

Problem 32686

Find the derivative $s^{\prime}(t)$ of the function $s(t) = (8 - 7t)(t^2 - 2)$ .

See Solution

Problem 32687

Find the slope of the function $4x + y^3 + xy^2 = 16$ at the point $(1,2)$ .

See Solution

Problem 32688

Differentiate $y=\ln(3x+2)^{2}$ .

See Solution

Problem 32689

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

See Solution

Problem 32690

Find if the critical point (1, 12) of the function $f(x) = 2x^3 - 9x^2 + 12x + 7$ is a max, min, or neither.

See Solution

Problem 32691

Find the limit as $x$ approaches 0 for the expression $\frac{x^{2}-2 x}{x}$ .

See Solution

Problem 32692

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

See Solution

Problem 32693

Find values of $x$ for which demand is elastic or inelastic for $p=g(x)=420-0.7 x$ . Use $E(x)=-\frac{g(x)}{x g^{\prime}(x)}$ .

See Solution

Problem 32694

Find all values of $p$ for which demand is elastic in the equation $p + 0.02x = 50$ , where $0 \leq p \leq 50. (Interval)$

See Solution

Problem 32695

Evaluate the limit: $\lim _{x \rightarrow 7} \frac{x-7}{x^{2}-49}$ .

See Solution

Problem 32696

Find values of $x$ for which demand is elastic or inelastic using $p=g(x)=640-0.8x$ . Elasticity is $E(x)=-\frac{g(x)}{x g^{\prime}(x)}$ .

See Solution

Problem 32697

Find the limit as $x$ approaches infinity for $\frac{5 x^{3}-2 x^{5}+2}{x^{6}+3 x^{2}}$ . Options: (a) $\infty$ (b) (c) $\frac{1}{2}$

See Solution

Problem 32698

At the maximum point $x=a$ of $y=f(x)$ , which conditions can be true? (a) $dy/dx>0$ , $d^2y/dx^2=0$ (b) $dy/dx=0$ , $d^2y/dx^2<0$ (c) $dy/dx=0$ , $d^2y/dx^2>0$ (d) $dy/dx<0$ , $d^2y/dx^2=0$

See Solution

Problem 32699

Find the derivative of $f(x) = e^{2x}$ and the tangent line equation at point $(x_0, f(x_0))$ . $x_0$ is unspecified.

See Solution

Problem 32700

Find $\int_{1}^{6} 2 f(x) d x$ given $\int_{-1}^{1} f(x) d x=3$ and $\int_{-1}^{6} f(x) d x=7$ .

See Solution

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Calculus

Problem 32601

Find the derivative of f(x)=44x+4f(x)=\frac{4}{4x+4}f(x)=4x+44​. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 32602

Find the limit as xxx approaches 4 for x3−64x−4\frac{x^{3}-64}{x-4}x−4x3−64​ and write a simpler function g(x)g(x)g(x) that matches it except at one point.

Problem 32603

Estimate the limit of lim⁡x→0±sin⁡∣x∣x\lim _{x \rightarrow 0 \pm} \frac{\sin |x|}{x}limx→0±​xsin∣x∣​ from both sides and check if it exists.

Problem 32604

Find the limit as xxx approaches 0 for the expression x2+5xx\frac{x^{2}+5 x}{x}xx2+5x​.

Problem 32605

Find the absolute maximum of the function f(x)=x4+8x3+18x2+4f(x)=x^{4}+8 x^{3}+18 x^{2}+4f(x)=x4+8x3+18x2+4 on the interval [−4,1][-4,1][−4,1].

Problem 32606

Find the intervals where the function f(x)=215x6−5x4f(x)=\frac{2}{15} x^{6}-5 x^{4}f(x)=152​x6−5x4 is concave up.

Problem 32607

Find the derivative y′y^{\prime}y′ for y=(4+9x−8x2)exy=(4+9x-8x^{2}) e^{x}y=(4+9x−8x2)ex. Simplify your answer.

Problem 32608

Find the limit: lim⁡x→0x+6−6x\lim _{x \rightarrow 0} \frac{\sqrt{x+6}-\sqrt{6}}{x}limx→0​xx+6​−6​​.

Problem 32609

Find the limit as xxx approaches π2\frac{\pi}{2}2π​ of sin⁡x\sin xsinx.

Problem 32610

Find the limit as xxx approaches -6 for the expression x2+6xx^{2}+6xx2+6x.

Problem 32611

Find the limit: lim⁡x→−5x2−25x+5\lim _{x \rightarrow -5} \frac{x^{2}-25}{x+5}limx→−5​x+5x2−25​.

Problem 32612

Find the limit as xxx approaches π\piπ for cos⁡(9x)\cos(9x)cos(9x).

Problem 32613

Find the limit as θ\thetaθ approaches 0 for the expression cos⁡(5θ)tan⁡(5θ)θ\frac{\cos (5 \theta) \tan (5 \theta)}{\theta}θcos(5θ)tan(5θ)​.

Problem 32614

Find dydt\frac{d y}{d t}dtdy​ when y=x3+3y=x^{3}+3y=x3+3, dxdt=5\frac{d x}{d t}=5dtdx​=5, and x=1x=1x=1. Answer: dydt=□\frac{d y}{d t}=\squaredtdy​=□.

Problem 32615

Find the limit as xxx approaches 2 for the expression x2+6x−16x2−4\frac{x^{2}+6x-16}{x^{2}-4}x2−4x2+6x−16​.

Problem 32616

Problem 32617

Find dydt\frac{d y}{d t}dtdy​ for 4xy−6x+3y3=−1354 x y-6 x+3 y^{3}=-1354xy−6x+3y3=−135 given dxdt=−12\frac{d x}{d t}=-12dtdx​=−12, x=3x=3x=3, y=−3y=-3y=−3. dydt=□\frac{d y}{d t}=\squaredtdy​=□

Problem 32618

Problem 32619

Find the derivative of the function f(x)=cos⁡x−5tan⁡xf(x)=\cos x-5 \tan xf(x)=cosx−5tanx, i.e., calculate f′(x)f^{\prime}(x)f′(x).

Problem 32620

A boat is 2 feet from a dock, and the rope is pulled in at 1 ft/sec. How fast is the distance to the dock decreasing? □\square□ ft/sec.

Problem 32621

Find the limits: (a) lim⁡x→4(2x2−3x+42)\lim _{x \rightarrow 4} (2 x^{2}-3 x+42)limx→4​(2x2−3x+42), (b) lim⁡x→62x+23\lim _{x \rightarrow 62} \sqrt[3]{x+2}limx→62​3x+2​, (c) lim⁡x→4g(f(x))\lim _{x \rightarrow 4} g(f(x))limx→4​g(f(x)).

Problem 32622

Find the derivative f′(t)f'(t)f′(t) of the function f(t)=(t2+7t+6)(6t−2+6t−3)f(t)=(t^2+7t+6)(6t^{-2}+6t^{-3})f(t)=(t2+7t+6)(6t−2+6t−3).

Problem 32623

A ball is thrown with a velocity of 45ft/s45 \mathrm{ft/s}45ft/s. Height after ttt seconds is y=45t−14t2y=45t-14t^2y=45t−14t2. Find average velocity from t=2t=2t=2 for given intervals.

Problem 32624

Evaluate the limit as xxx approaches infinity for the expression 3x9+11x−43 x^{9}+11 x-43x9+11x−4. What is the limit?

Problem 32625

A point moves on xy=20xy=20xy=20. At (4,5)(4,5)(4,5) with xxx increasing by 4 units/sec, find the rate of change of yyy. The yyy-coordinate is □\square□ at □\square□ units per second.

Problem 32626

Find the limit as xxx approaches infinity for 3x9+11x−43x^{9} + 11x - 43x9+11x−4. What is the limit?

Problem 32627

Find the average velocity for the position function s(t)=6t2−5t−7s(t)=6 t^{2}-5 t-7s(t)=6t2−5t−7 from t=5t=5t=5 to t=5+ht=5+ht=5+h. Use exact values. AV[5,5+h]=s(5+h)−s(5)h A V_{[5,5+h]}=\frac{s(5+h)-s(5)}{h} AV[5,5+h]​=hs(5+h)−s(5)​

Problem 32628

Find dydx\frac{d y}{d x}dxdy​ for y=f(1+2−x2)y=f(1+\sqrt{2-x^{2}})y=f(1+2−x2​) at x=1x=1x=1, given f′(2)=−3f^{\prime}(2)=-3f′(2)=−3.

Problem 32629

Find the limit as x x x approaches negative infinity for the expression: −5x4+3x5+2 -5 x^{4}+3 x^{5}+2 −5x4+3x5+2.

Problem 32630

Find removable and nonremovable discontinuities of f(x)=7xf(x)=\frac{7}{x}f(x)=x7​. Identify values of xxx.

Problem 32631

A boat is pulled toward a dock at 3 ft/s. How fast is the distance to the dock decreasing when 2 ft away? Answer: □\square□ ft/s.

Problem 32632

Find the derivative of the function f(t)=(t2+7t+6)(6t−2+6t−3)f(t)=(t^{2}+7t+6)(6t^{-2}+6t^{-3})f(t)=(t2+7t+6)(6t−2+6t−3). What is f′(t)f^{\prime}(t)f′(t)?

Problem 32633

Find the limits: 1. lim⁡x→∞3x−2x2+5\lim _{x \rightarrow \infty} \frac{3 x-2}{\sqrt{x^{2}+5}}limx→∞​x2+5​3x−2​ 2. lim⁡x→−∞3x−2x2+5\lim _{x \rightarrow-\infty} \frac{3 x-2}{\sqrt{x^{2}+5}}limx→−∞​x2+5​3x−2​

Problem 32634

Evaluate the limit: lim⁡x→4(4x2−1)(6x+1)\lim _{x \rightarrow 4}\left(4 x^{2}-1\right)(\sqrt{6 x+1})x→4lim​(4x2−1)(6x+1​) using basic limit laws.

Problem 32635

A rock creates ripples in a pond. If the radius increases at 4 ft/s, find the area change rate when the radius is 9 ft. Area change is approximately □\square□ sq ft/s. Round to the nearest thousandth.

Problem 32636

Evaluate the limit: lim⁡x→∞x4−5x3+89x2+6 \lim _{x \rightarrow \infty} \frac{\sqrt{x^{4}-5 x^{3}+8}}{9 x^{2}+6} limx→∞​9x2+6x4−5x3+8​​

Problem 32637

Find the derivative of the function f(x)=2−x23+x2f(x)=\frac{2-x^{2}}{3+x^{2}}f(x)=3+x22−x2​.

Problem 32638

A 10-foot ladder leans against a wall. If the bottom moves away at 3 ft/sec, how fast is the top sliding down when 6 ft away? The rate is □ft/sec\square \mathrm{ft} / \mathrm{sec}□ft/sec.

Problem 32639

Evaluate the limit: lim⁡x→64x−8x−64\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}x→64lim​x−64x​−8​ using direct substitution. Simplify your answer.

Problem 32640

Find the derivative dydt\frac{d y}{d t}dtdy​ for y=4tln⁡tety=\frac{4 t \ln t}{e^{t}}y=et4tlnt​. Simplify your answer.

Problem 32641

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

Find the limit as $x$ approaches 4 for $\frac{x^{3}-64}{x-4}$ and write a simpler function $g(x)$ that matches it except at one point.

Estimate the limit of $\lim _{x \rightarrow 0 \pm} \frac{\sin |x|}{x}$ from both sides and check if it exists.

Find the limit as $x$ approaches 0 for the expression $\frac{x^{2}+5 x}{x}$ .

Find the absolute maximum of the function $f(x)=x^{4}+8 x^{3}+18 x^{2}+4$ on the interval $[-4,1]$ .

Find the intervals where the function $f(x)=\frac{2}{15} x^{6}-5 x^{4}$ is concave up.

Find the derivative $y^{\prime}$ for $y=(4+9x-8x^{2}) e^{x}$ . Simplify your answer.

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{x+6}-\sqrt{6}}{x}$ .

Find the limit as $x$ approaches $\frac{\pi}{2}$ of $\sin x$ .

Find the limit as $x$ approaches -6 for the expression $x^{2}+6x$ .

Find the limit: $\lim _{x \rightarrow -5} \frac{x^{2}-25}{x+5}$ .

Find the limit as $x$ approaches $\pi$ for $\cos(9x)$ .

Find the limit as $\theta$ approaches 0 for the expression $\frac{\cos (5 \theta) \tan (5 \theta)}{\theta}$ .

Find $\frac{d y}{d t}$ when $y=x^{3}+3$ , $\frac{d x}{d t}=5$ , and $x=1$ . Answer: $\frac{d y}{d t}=\square$ .

Find the limit as $x$ approaches 2 for the expression $\frac{x^{2}+6x-16}{x^{2}-4}$ .

Find $\frac{d y}{d t}$ for $4 x y-6 x+3 y^{3}=-135$ given $\frac{d x}{d t}=-12$ , $x=3$ , $y=-3$ . $\frac{d y}{d t}=\square$

Find the derivative of the function $f(x)=\cos x-5 \tan x$ , i.e., calculate $f^{\prime}(x)$ .

A boat is 2 feet from a dock, and the rope is pulled in at 1 ft/sec. How fast is the distance to the dock decreasing? $\square$ ft/sec.

Find the limits: (a) $\lim _{x \rightarrow 4} (2 x^{2}-3 x+42)$ , (b) $\lim _{x \rightarrow 62} \sqrt[3]{x+2}$ , (c) $\lim _{x \rightarrow 4} g(f(x))$ .

Find the derivative $f'(t)$ of the function $f(t)=(t^2+7t+6)(6t^{-2}+6t^{-3})$ .

A ball is thrown with a velocity of $45 \mathrm{ft/s}$ . Height after $t$ seconds is $y=45t-14t^2$ . Find average velocity from $t=2$ for given intervals.

Evaluate the limit as $x$ approaches infinity for the expression $3 x^{9}+11 x-4$ . What is the limit?

A point moves on $xy=20$ . At $(4,5)$ with $x$ increasing by 4 units/sec, find the rate of change of $y$ .
The $y$ -coordinate is $\square$ at $\square$ units per second.

Find the limit as $x$ approaches infinity for $3x^{9} + 11x - 4$ . What is the limit?

Find the average velocity for the position function $s(t)=6 t^{2}-5 t-7$ from $t=5$ to $t=5+h$ . Use exact values. $A V_{[5,5+h]}=\frac{s(5+h)-s(5)}{h}$

Find $\frac{d y}{d x}$ for $y=f(1+\sqrt{2-x^{2}})$ at $x=1$ , given $f^{\prime}(2)=-3$ .

Find the limit as $x$ approaches negative infinity for the expression: $-5 x^{4}+3 x^{5}+2$ .

Find removable and nonremovable discontinuities of $f(x)=\frac{7}{x}$ . Identify values of $x$ .

A boat is pulled toward a dock at 3 ft/s. How fast is the distance to the dock decreasing when 2 ft away? Answer: $\square$ ft/s.

Find the derivative of the function $f(t)=(t^{2}+7t+6)(6t^{-2}+6t^{-3})$ . What is $f^{\prime}(t)$ ?

Find the limits:
1. $\lim _{x \rightarrow \infty} \frac{3 x-2}{\sqrt{x^{2}+5}}$
2. $\lim _{x \rightarrow-\infty} \frac{3 x-2}{\sqrt{x^{2}+5}}$

Evaluate the limit: $\lim _{x \rightarrow 4}\left(4 x^{2}-1\right)(\sqrt{6 x+1})$ using basic limit laws.

A rock creates ripples in a pond. If the radius increases at 4 ft/s, find the area change rate when the radius is 9 ft. Area change is approximately $\square$ sq ft/s. Round to the nearest thousandth.

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{4}-5 x^{3}+8}}{9 x^{2}+6}$

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

A 10-foot ladder leans against a wall. If the bottom moves away at 3 ft/sec, how fast is the top sliding down when 6 ft away? The rate is $\square \mathrm{ft} / \mathrm{sec}$ .

Evaluate the limit: $\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}$ using direct substitution. Simplify your answer.

Find the derivative $\frac{d y}{d t}$ for $y=\frac{4 t \ln t}{e^{t}}$ . Simplify your answer.

Find the integral: $\int \frac{3}{\sqrt{4-25 x^{2}}} d x$ .

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{10 x \Delta x + 5 \Delta x^{2} - 7 \Delta x}{\Delta x}$ .

Find the limits:
1. $\lim _{x \rightarrow \infty} \frac{3 x-2}{\sqrt{x^{2}+5}}$
2. $\lim _{x \rightarrow-\infty} \frac{3 x-2}{\sqrt{x^{2}+5}}$

Find the derivative of the function $F(x)=\frac{\sqrt{x}-7}{\sqrt{x}+7}$ . What is $F'(x)$ ?

Find the current rate of change of sales given $s=50,000-40,000 e^{-0.00025 x}$ , with $x=\$ 4,000$ and $dx/dt=\$ 300$ .

Find the tangent and normal line equations at $(-1,4)$ for the curve $x^{2} y^{2}=16$ . Provide answers in slope-intercept form.

Prove that $y=x^3+3x+1$ satisfies $y''' + xy'' - 2y' = 0$ .

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{256+h}-16}{h}$ and rewrite it for direct substitution.

Evaluate these limits: (a) $\lim _{x \rightarrow \infty} \frac{\sqrt{3+10 x^{2}}}{8+9 x}=$ (b) $\lim _{x \rightarrow-\infty} \frac{\sqrt{3+10 x^{2}}}{8+9 x}=$

Find limits and function values for $a=2$ and $a=0$ : $\lim_{x \rightarrow a^{-}} f(x)$ , $\lim_{x \rightarrow a^{+}} f(x)$ , $\lim_{x \rightarrow a} f(x)$ , and $f(a)$ .

Find the limit as $t$ approaches $\infty$ for $\frac{-3t-9}{\sqrt{t^2+3t+5}}$ . Enter I, -I, or DNE. Limit =

Evaluate the limit: $\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}$ and simplify to find the exact answer.

Determine if the function $f(x)$ has a jump discontinuity at $x=8$ by finding the limits: $\lim _{x \rightarrow 8^{-}} f(x) = \square, \quad \lim _{x \rightarrow 8^{+}} f(x) = \square.$

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}=\square$

Differentiate the function $g(x)=x^{5}+\frac{1}{x^{5}}$ . Find $g^{\prime}(x)=$ .

Find $f'(2)$ and $f'(-3)$ for the function $f(x)=\frac{7}{x^{7}}$ .

Calculate the relative rate of change of $f(x)=3 x^{2}-\ln x$ at $x=4$ . Answer: $\square$ .

Find $f'(x)$ for $f(x)=12x(\sin(x)+\cos(x))$ and evaluate $f'(-\frac{\pi}{6})$ .

Simplify the integral $\int \frac{1}{x^{2} \sqrt{9 x^{2}+16}} d x$ with a trigonometric substitution.

Find the limit: $\lim _{x \rightarrow \frac{\pi}{2}} \frac{9 \sec (x)+3}{\tan (x)}$ .

Find the slope of the tangent line to $y=3 x^{3}$ at $(-5,-375)$ and write the equation in $y=m x+b$ form.

Find the derivative of the function $f(x)=4 x^{2}-4 x+8$ . What is $f^{\prime}(x)=\square$ ?

Differentiate the function $f(x)=x^{2}+8x+30$ . Find $f^{\prime}(x)$ .

Find $\frac{d y}{d x}$ for $\sqrt[3]{x^{2}}+\sqrt[3]{y^{2}}=1$ and $\frac{d^{2} y}{d x^{2}}$ at $x=\frac{1}{125}$ .

Calculate the relative rate of change of $f(x)=264-3x$ at $x=33$ . Answer: $\square$ (round to three decimals).

Find the derivative $f^{\prime}(x)$ for the function $f(x)=7 x^{13}$ . What is $f^{\prime}(x)=?$

Find the derivative of the function $f(x) = 8x^3 + 9x^2 + 3x - 7$ . What is $f'(x) = \square$ ?

Given $f(t)=(t^{2}+2 t+2)(3 t^{2}+6)$ , find $f^{\prime}(t)$ and then calculate $f^{\prime}(5)$ .

Find the derivative $\frac{\mathrm{dy}}{\mathrm{dx}}$ for the equation $x^{3}+2 x^{2} y-x y^{2}+y^{3}+3=0$ at the point (1,-1).

Find $F'(x)$ for $F(x)=12x(\sin x+\cos x)$ and evaluate $F'(-\frac{\pi}{6})$ .

Find the percentage rate of change of $f(x)=5250-5 x^{2}$ at $x=25$ . Answer: $\square \%$ .

Find the derivative of $f(x)=7 \frac{1}{x^{9}}+x^{2}+14$ . What is $f^{\prime}(x)=\square$ ?

Simplify the integral $\int \frac{1}{x^{2} \sqrt{9 x^{2}+16}} d x$ using trigonometric substitution.