Calculus

Problem 10401

Use the substitution $x=\sqrt{48} \sec \theta$ to solve the integral $\int \frac{24}{x^{2} \sqrt{x^{2}-48}} dx$ .

See Solution

Problem 10402

Find the critical number $A$ for the function $f(x)=-4x^{2}+4x-4$ . Is it a local min (LMIN), local max (LMAX), or neither (NEITHER)?

See Solution

Problem 10403

Find the local minimum and maximum of the function $f(x)=-2 x^{3}+36 x^{2}-192 x+9$ .

See Solution

Problem 10404

Compute the integral of $\frac{3}{(3+x^{2})^{3/2}}$ with respect to $x$ .

See Solution

Problem 10405

Find the derivative of $y=5^{-5 x^{6}}$ .

See Solution

Problem 10406

Find the rate of change of sales given by $S(t)=106-90 e^{-0.3 t}$ at $t=1$ and $t=5$ years. What happens as $t$ increases? Does it ever equal zero?

See Solution

Problem 10407

Estimate the population in 2010 using $p(t)=37.47(1.023)^{t}$ and find the instantaneous rate at $t=10$ .

See Solution

Problem 10408

Find the growth function $G(t)$ for online course students since 2002 with $k=0.0474$ and initial enrollment 1.674 million. Then, find enrollment for 2004, 2006, and 2013, and describe the growth rate over time.

See Solution

Problem 10409

A runaway semi at $40 \mathrm{~m/s}$ starts $400 \mathrm{~m}$ high. How high will it go before stopping? Assume no friction.

See Solution

Problem 10410

Calculate the roofing area of the barn using the function $y=31-10\left(e^{x / 20}+e^{-x / 20}\right)$ from $x=-20$ to $x=20$ .

See Solution

Problem 10411

Find the difference quotient for the function $f(x)=4x-2$ : $\frac{f(x+h)-f(x)}{h}$ .

See Solution

Problem 10412

Find $c$ from the Mean Value Theorem for $f(x)=2 \sqrt{x}$ on $[0,25]$ . Options: a. $c=5$ , b. $c=\frac{1}{5}$ , c. $c=0$ , d. $c=\frac{25}{4}$ , e. None.

See Solution

Problem 10413

Evaluate the integral: $\int_{0}^{\sqrt{3}} \frac{1}{(1+x^{2})^{\frac{5}{2}}} \, dx$

See Solution

Problem 10414

Find the smallest value of $M$ such that $f(1) \leq M$ for all differentiable $f$ with $f' \leq 4$ and $f(-1)=3$ .

See Solution

Problem 10415

Find the limits as $x$ approaches $a$ : (a) $f(x)+6g(x)$ , (b) $f(x)g(x)$ , (c) $\sqrt[3]{3+f(x)}$ , (d) $\frac{4g(x)}{9f(x)+g(x)}$ .

See Solution

Problem 10416

Find the minimum value of $f(x)=\frac{1}{3} x^{3}-5 x^{2}+9 x+3$ on $[0,3]$ .

See Solution

Problem 10417

Find the smallest value of $M$ such that $f(1) \leq M$ for all differentiable $f$ with $f' \leq 4$ and $f(-1)=3$ .

See Solution

Problem 10418

Find the linear approximation $L(x)$ of $f(x)=e^{x} \cos x+\sin x$ at $x=0$ . Which is correct?

See Solution

Problem 10419

Find the real zero of $f(x)=x^{3}+4 x^{2}+x-9$ using the intermediate value theorem. Round to two decimal places.

See Solution

Problem 10420

Find all values of $c$ for the function $f(t)=\frac{t}{t-6}$ on the interval [-2, 0] using the Mean Value Theorem.

See Solution

Problem 10421

Find $a_{2}$ in the second degree Taylor polynomial of $y=f(x)g(x)$ at $x=3$ given $f(3)=g(3)=0$ , $f'(3)=2$ , $g'(3)=1$ .

See Solution

Problem 10422

A box with square base $x$ and height $y$ has $\frac{d x}{d t}=3 \mathrm{~cm/s}$ , $\frac{d y}{d t}=7 \mathrm{~cm/s}$ . Find $\frac{d V}{d t}$ when $x=2 \mathrm{~cm}$ , $y=4 \mathrm{~cm}$ .

See Solution

Problem 10423

Find the integrals: $\int x(x+1)^{5/2} \, dx$ and $\int \frac{1}{1+e^{x}} \, dx$ .

See Solution

Problem 10424

Use integration by parts to solve the integral $\int x(x+1)^{5/2} \mathrm{~d} x$ . What is the result?

See Solution

Problem 10425

Find all $x$ where the graph of $f(x)=\frac{1}{3} x^{3}-2 x^{2}+7$ has a point of inflection.

See Solution

Problem 10426

Find the rectangle with the largest area that fits inside a circle of radius $r$ . What are its dimensions?

See Solution

Problem 10427

Determine where the graph of $y=x^{2}-4 \cos x$ changes concavity in the interval $(-\pi / 2, \pi / 2)$ .

See Solution

Problem 10428

Find the second derivative, $f^{\prime \prime}$ , of the function $f(x)=x \cos(x^{2}) + 4x^{2}$ .

See Solution

Problem 10429

Ein Stein fällt aus $45 \mathrm{~m}$ Höhe. Berechne nach 2 s Höhe und Geschwindigkeit. Wann trifft er den Boden und mit welcher Geschwindigkeit?

See Solution

Problem 10430

A point moves on the curve $3 x^{3}+4 y^{3}=x y$ . At $P=\left(\frac{1}{7}, \frac{1}{7}\right)$ , $y$ increases at 4 units/sec. Find the speed and direction of $x$ .

See Solution

Problem 10431

Find $d y / d x$ at the point $(1,0)$ for the equation $5 e^{6 y}=2 x y+5 x$ .

See Solution

Problem 10432

Skizzieren Sie die Graphen von $f(x)=e^{x}$ und $g(x)=e^{-x}$ . Wie hängen die Graphen zusammen? Zeigen Sie, dass $g^{\prime}(x)=-e^{-x}$ .

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Problem 10433

Find the average rate of change of $f(t)=6t+8$ over $[7,8]$ and compare it to the instantaneous rates at the endpoints.

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Problem 10434

Maximize $z=20xy-(x^2+y^2)$ with the constraint $x+y=4$ .

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Problem 10435

Berechnen Sie den Grenzwert $\lim _{x \rightarrow \frac{2}{3}} \frac{9 x^{2}-6 x}{3 x-2}$ und überprüfen Sie mit einer Wertetabelle. Bestimmen Sie die Art der Definitionslücke bei $x=\frac{2}{3}$ .

See Solution

Problem 10436

Find the derivative of the function $f(x)=\frac{4}{\sqrt[3]{x}}+2 \cos x$ .

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Problem 10437

Find the rate of change of the function $y=3x+3$ .

See Solution

Problem 10438

Find the rate of change of the function $y=-2x-2$ .

See Solution

Problem 10439

Bestimme $a$ , sodass $\lim_{x \to 2} f(x) = -2$ für $f(x) = 3ax^2 - 4a$ .

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Problem 10440

Check if Rolle's Theorem applies for each function. State reasons for conditions a & b, and values for c.
5. $f(x)=x^{2}+2 \quad[-2,2]$ a. continuous on $[-2,2]$ ? Yes b. differentiable on $(a, b)$ ? Yes c. $f(a)=f(b)$ ? Yes
6. $f(x)=\frac{x^{2}-9}{x-3} \quad[0,6]$ a. continuous on $[a,b]$ ? No b. differentiable on $(a,b)$ ? c. $f(a)=f(b)$ ?
7. $f(x)=|x-3| \quad[0,6]$ a. continuous on $[a,b]$ ? b. differentiable on $(a,b)$ ? c. $f(a)=f(b)$ ?

See Solution

Problem 10441

Find the derivative of $f(x)=\frac{x^{8}(x-7)^{3}}{(x^{2}+4)^{5}}$ using logarithmic differentiation.

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Problem 10442

Berechne die Grenzwerte für die folgenden Funktionen: a) $\lim _{x \rightarrow-3} \frac{2 x^{2}-18}{x+3}$ b) $\lim _{x \rightarrow 5} \frac{x^{2}-7 x+10}{x-5}$ c) $\lim _{x \rightarrow 1} \frac{x^{2}-x}{x-1}$ d) $\lim _{x \rightarrow x_{0}} \frac{x^{2}-x_{0}^{2}}{x-x_{0}}$

See Solution

Problem 10443

A hunter accidentally fires a bullet straight up at $200 \mathrm{~m/s}$ . How long until it falls back and how high does it go?

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Problem 10444

Familie Stein wacht um 8.00 Uhr bei $20^{\circ} \mathrm{C}$ auf. Die Temperatur wird durch $T(t)=\frac{52}{t+2}-6$ beschrieben.
a) Wie tief könnte die Temperatur fallen? b) Wann wird der Gefrierpunkt erreicht? c) Wann fiel die Heizung aus?

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Problem 10445

Josue invested \$3,300 at a 7.5% continuous interest rate. Fatoumata invested \$3,300 at a 7.75% annual rate. How much longer to triple?

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Problem 10446

Evaluate the double integral $\int_{0}^{\sqrt{3}} \int_{u^{2}}^{3} \frac{e^{v}}{\sqrt{v}} dv \, du$ .

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Problem 10447

Find the values of $a$ and $b$ that maximize the function $F(a, b)=\int_{0}^{a} \int_{0}^{b} 4-x^{2}-y^{2} d y d x$ . Calculate the product $a b$ .

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Problem 10448

Find the second derivative $f^{\prime \prime}(t)$ of the function $f(t)=\int_{0}^{t} x^{1+x^{3}} d x$ for $t>0$ .

See Solution

Problem 10449

Find the limits of integration for the triple integral $\int_{0}^{2} \int_{2 z}^{4} \int_{1}^{3} f(x, y, z) d x d y d z$ when reordered.

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Problem 10450

Find the limits of integration for the integral of $f(x, y, z)=x y^{2} z^{3}$ over the domain $\frac{x^{2}}{4}+y^{2}+\frac{z^{2}}{9} \leq 1$ , $x \geq 0$ , $y \geq 0$ , $z \geq 0$ .

See Solution

Problem 10451

Find the derivative of $y = \cos(x) - 2\tan(x)$ using the chain rule.

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Problem 10452

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ for the equation $y + y^{2} + x^{3} = 0$ .

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Problem 10453

Find the derivative of $-\sin(-3x)$ with respect to $x$ .

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Problem 10454

Find the first derivative of $f(x)=x^{t-3}$ .

See Solution

Problem 10455

Bestimmen Sie die Ableitung von $f(b)=4ab$ .

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Problem 10456

Find the power series for $f(x)=\frac{3}{(1-3x)^{2}}$ and its radius of convergence. Use $\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$ .

See Solution

Problem 10457

Differentiate the function $G(y)=\ln \left(\frac{(4 y+1)^{3}}{\sqrt{y^{2}+1}}\right)$ . Find $G^{\prime}(y)$ .

See Solution

Problem 10458

Find the first derivative of $f(z)=\frac{4}{z}+\sqrt{z}$ .

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Problem 10459

Use the Intermediate Value Theorem to check if $f(x)=-7 x^{4}+2 x^{2}+10$ has a zero between -2 and -1.

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Problem 10460

Find the derivative of $y = x^{6} \cos(x)$ using logarithmic differentiation. What is $y'$ ?

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Problem 10461

Use Newton's method for $f(x)=x^{2}-5$ with $x_{0}=2$ to find $x_{1}$ and $x_{2}$ . Choose the correct formula.

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Problem 10462

Find $\frac{d y}{d x}$ for the function $y=\sqrt[4]{x}+\frac{x^{2}-8}{x^{2}}$ .

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Problem 10463

Find the derivative of $y=(\cos(9x))^x$ using logarithmic differentiation. Set $y'(x)=1$ .

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Problem 10464

How far does a ball fall in 5 seconds? Use the formula for distance: $d = \frac{1}{2} g t^2$ , where $g \approx 9.8 \, m/s^2$ .

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Problem 10465

Find the formula for Newton's method for $f(x)=4-\tan(3x)$ , starting with $x_{0}=1$ . Compute $x_{1}$ .

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Problem 10466

Find the derivative of $2 \sin x - x e^{-10 x}$ with respect to $x$ .

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Problem 10467

Calculate the integral $\int_{0}^{t} x^{1+x^{3}} dx$ for $t>0$ and determine $f^{\prime \prime}(t)$ .

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Problem 10468

Find Newton's method formula for $f(x)=4-\tan(3x)$ with $x_{0}=1$ . Compute $x_{1}$ and round to six decimal places.

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Problem 10469

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\sqrt[3]{x^{5}}+\pi^{2}-\frac{5}{x}+2^{x}$ .

See Solution

Problem 10470

Find the derivative of the function $f(x)=x^{2} \cos \left(\sqrt{x^{3}-1}+2\right)$ .

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Problem 10471

Find the derivative $f^{\prime}(x)$ for the function $f(x)=e^{-x} \arctan(4 x^{2})$ .

See Solution

Problem 10472

Find the tangent line equation to $7 x^{2}-5 x y+y^{3}=3$ at point $(1,1)$ . (5 points)

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Problem 10473

Find the derivative $\left.\frac{d}{d x}\left(\frac{x f(x)}{g(x)}\right)\right|_{x=1}$ using the given values of $f$ and $g$ .

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Problem 10474

Find the derivative $\left.\frac{d}{d x}(f(x)+2 g(x))\right|_{x=3}$ using the given values of $f(x)$ and $g(x)$ .

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Problem 10475

Find quantities for $f(x)=3 x^{2}$ : (A) slope of secant line at $(2, f(2))$ , (B) slope at $(2, f(2))$ , (C) tangent line equation.

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Problem 10476

Find $h^{\prime}(3)$ where $h(x)=(f \circ g)(x)$ using the given values of $f$ and $g$ .

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Problem 10477

Find the slope of the secant line and tangent line for $f(x)=3x^{2}$ at $x=2$ .

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Problem 10478

Find the derivative $f^{\prime}(x)$ for $f(x)=x^{1/x}$ using logarithmic differentiation.

See Solution

Problem 10479

A cyclist rides 100 km, reaching 72 km in 2 hours at 40 km/h. Find average velocity for the first 2 hours and the slope of $f(x)$ .

See Solution

Problem 10480

Find the derivative of $f(x)=x^{1/x}$ using logarithmic differentiation.

See Solution

Problem 10481

Find the tangent line equation for $f(x)=\frac{1}{2+x^{2}}$ at $x=1$ where the slope is $-\frac{2}{9}$ .

See Solution

Problem 10482

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x^{2}+2 x y-y^{2}+x=2$ .

See Solution

Problem 10483

Find the average rate of change of $f(x)=3 x^{2}+2 x+1$ over the interval $[2, 4]$ .

See Solution

Problem 10484

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x^{2}+2 x y-y^{2}+x=2$ .

See Solution

Problem 10485

Find the tangent line equation for $f(x)=\frac{1}{2+x^{2}}$ at $x=3$ with slope $-\frac{6}{121}$ .

See Solution

Problem 10486

Find the tangent line equation for $f(x)=\frac{1}{7+x^{2}}$ at $x=2$ with slope $-\frac{4}{121}$ .

See Solution

Problem 10487

Find the average rate of change of $f(x)=3 x^{2}+2 x+1$ from $x=2$ to $x=4$ .

See Solution

Problem 10488

Find $s^{\prime}(x)$ for $s(x)=7x-8$ and calculate $s^{\prime}(1)$ , $s^{\prime}(2)$ , and $s^{\prime}(3)$ .

See Solution

Problem 10489

Find where the function $f(x)=11 x^{2}+x^{4}$ is concave up and concave down.

See Solution

Problem 10490

Calculate the limit: $\lim _{h \rightarrow 0} \frac{(-2+h)^{2}-4}{h}$ .

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Problem 10491

True or False: a) The $n$ th derivative of $5x^3 + 2x + 5$ is 0 for $n \geq 3$ . d) Use the Chain Rule to differentiate $f(x)=e^{\sqrt{x+1}}$ . e) Is $\frac{d}{dx}(\sqrt{2}^x) = x \sqrt{2}^{x-1}$ ?

See Solution

Problem 10492

Find $f^{\prime}(x)$ for $f(x)=2 x^{2}-9 x+10$ and then calculate $f^{\prime}(3)$ , $f^{\prime}(5)$ , and $f^{\prime}(9)$ .

See Solution

Problem 10493

Find the length of the curve $\mathbf{r}(t)=\langle 2 t, \sin t, \cos t\rangle$ from $(0,0,1)$ to $(2\pi,0,-1)$ .

See Solution

Problem 10494

Find the slopes of secant lines between points and the tangent line at $(2, f(2))$ for $f(x)=x^{2}+x$ .

See Solution

Problem 10495

Find the extreme values of $y=f(x)=x^{3}-3x$ for $0 \leq x \leq 2$ .

See Solution

Problem 10496

Differentiate $x^{3}-4 x^{2}+3$ .

See Solution

Problem 10497

Calculate the integral from 2 to 3 of the function $\frac{1}{x}$ .

See Solution

Problem 10498

Find the derivative of $\sin y = x^3 + y^4$ using implicit differentiation.

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Problem 10499

Find the limit statements for the end behavior of $y=\frac{(4 x+3)^{2}}{(3 x-1)(2 x+5)}$ .

See Solution

Problem 10500

Determine the end behavior limits for the function $y=\frac{2 x^{3}-5 x+6}{6 x^{3}+10 x^{2}-4 x-12}$ .

See Solution

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Calculus

Problem 10401

Use the substitution x=48sec⁡θx=\sqrt{48} \sec \thetax=48​secθ to solve the integral ∫24x2x2−48dx\int \frac{24}{x^{2} \sqrt{x^{2}-48}} dx∫x2x2−48​24​dx.

Problem 10402

Find the critical number AAA for the function f(x)=−4x2+4x−4f(x)=-4x^{2}+4x-4f(x)=−4x2+4x−4. Is it a local min (LMIN), local max (LMAX), or neither (NEITHER)?

Problem 10403

Find the local minimum and maximum of the function f(x)=−2x3+36x2−192x+9f(x)=-2 x^{3}+36 x^{2}-192 x+9f(x)=−2x3+36x2−192x+9.

Problem 10404

Compute the integral of 3(3+x2)3/2\frac{3}{(3+x^{2})^{3/2}}(3+x2)3/23​ with respect to xxx.

Problem 10405

Find the derivative of y=5−5x6y=5^{-5 x^{6}}y=5−5x6.

Problem 10406

Find the rate of change of sales given by S(t)=106−90e−0.3tS(t)=106-90 e^{-0.3 t}S(t)=106−90e−0.3t at t=1t=1t=1 and t=5t=5t=5 years. What happens as ttt increases? Does it ever equal zero?

Problem 10407

Estimate the population in 2010 using p(t)=37.47(1.023)tp(t)=37.47(1.023)^{t}p(t)=37.47(1.023)t and find the instantaneous rate at t=10t=10t=10.

Problem 10408

Find the growth function G(t)G(t)G(t) for online course students since 2002 with k=0.0474k=0.0474k=0.0474 and initial enrollment 1.674 million. Then, find enrollment for 2004, 2006, and 2013, and describe the growth rate over time.

Problem 10409

A runaway semi at 40 m/s40 \mathrm{~m/s}40 m/s starts 400 m400 \mathrm{~m}400 m high. How high will it go before stopping? Assume no friction.

Problem 10410

Calculate the roofing area of the barn using the function y=31−10(ex/20+e−x/20)y=31-10\left(e^{x / 20}+e^{-x / 20}\right)y=31−10(ex/20+e−x/20) from x=−20x=-20x=−20 to x=20x=20x=20.

Problem 10411

Find the difference quotient for the function f(x)=4x−2f(x)=4x-2f(x)=4x−2: f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​.

Problem 10412

Find ccc from the Mean Value Theorem for f(x)=2xf(x)=2 \sqrt{x}f(x)=2x​ on [0,25][0,25][0,25]. Options: a. c=5c=5c=5, b. c=15c=\frac{1}{5}c=51​, c. c=0c=0c=0, d. c=254c=\frac{25}{4}c=425​, e. None.

Problem 10413

Evaluate the integral: ∫031(1+x2)52 dx\int_{0}^{\sqrt{3}} \frac{1}{(1+x^{2})^{\frac{5}{2}}} \, dx∫03​​(1+x2)25​1​dx

Problem 10414

Find the smallest value of MMM such that f(1)≤Mf(1) \leq Mf(1)≤M for all differentiable fff with f′≤4f' \leq 4f′≤4 and f(−1)=3f(-1)=3f(−1)=3.

Problem 10415

Find the limits as xxx approaches aaa: (a) f(x)+6g(x)f(x)+6g(x)f(x)+6g(x), (b) f(x)g(x)f(x)g(x)f(x)g(x), (c) 3+f(x)3\sqrt[3]{3+f(x)}33+f(x)​, (d) 4g(x)9f(x)+g(x)\frac{4g(x)}{9f(x)+g(x)}9f(x)+g(x)4g(x)​.

Problem 10416

Find the minimum value of f(x)=13x3−5x2+9x+3f(x)=\frac{1}{3} x^{3}-5 x^{2}+9 x+3f(x)=31​x3−5x2+9x+3 on [0,3][0,3][0,3].

Problem 10417

Find the smallest value of MMM such that f(1)≤Mf(1) \leq Mf(1)≤M for all differentiable fff with f′≤4f' \leq 4f′≤4 and f(−1)=3f(-1)=3f(−1)=3.

Problem 10418

Find the linear approximation L(x)L(x)L(x) of f(x)=excos⁡x+sin⁡xf(x)=e^{x} \cos x+\sin xf(x)=excosx+sinx at x=0x=0x=0. Which is correct?

Problem 10419

Find the real zero of f(x)=x3+4x2+x−9f(x)=x^{3}+4 x^{2}+x-9f(x)=x3+4x2+x−9 using the intermediate value theorem. Round to two decimal places.

Problem 10420

Find all values of ccc for the function f(t)=tt−6f(t)=\frac{t}{t-6}f(t)=t−6t​ on the interval [-2, 0] using the Mean Value Theorem.

Problem 10421

Find a2a_{2}a2​ in the second degree Taylor polynomial of y=f(x)g(x)y=f(x)g(x)y=f(x)g(x) at x=3x=3x=3 given f(3)=g(3)=0f(3)=g(3)=0f(3)=g(3)=0, f′(3)=2f'(3)=2f′(3)=2, g′(3)=1g'(3)=1g′(3)=1.

Problem 10422

A box with square base xxx and height yyy has dxdt=3 cm/s\frac{d x}{d t}=3 \mathrm{~cm/s}dtdx​=3 cm/s, dydt=7 cm/s\frac{d y}{d t}=7 \mathrm{~cm/s}dtdy​=7 cm/s. Find dVdt\frac{d V}{d t}dtdV​ when x=2 cmx=2 \mathrm{~cm}x=2 cm, y=4 cmy=4 \mathrm{~cm}y=4 cm.

Problem 10423

Find the integrals: ∫x(x+1)5/2 dx \int x(x+1)^{5/2} \, dx ∫x(x+1)5/2dx and ∫11+ex dx \int \frac{1}{1+e^{x}} \, dx ∫1+ex1​dx.

Problem 10424

Use integration by parts to solve the integral ∫x(x+1)5/2 dx\int x(x+1)^{5/2} \mathrm{~d} x∫x(x+1)5/2 dx. What is the result?

Problem 10425

Find all xxx where the graph of f(x)=13x3−2x2+7f(x)=\frac{1}{3} x^{3}-2 x^{2}+7f(x)=31​x3−2x2+7 has a point of inflection.

Problem 10426

Find the rectangle with the largest area that fits inside a circle of radius rrr. What are its dimensions?

Problem 10427

Determine where the graph of y=x2−4cos⁡xy=x^{2}-4 \cos xy=x2−4cosx changes concavity in the interval (−π/2,π/2)(-\pi / 2, \pi / 2)(−π/2,π/2).

Problem 10428

Find the second derivative, f′′f^{\prime \prime}f′′, of the function f(x)=xcos⁡(x2)+4x2f(x)=x \cos(x^{2}) + 4x^{2}f(x)=xcos(x2)+4x2.

Problem 10429

Ein Stein fällt aus 45 m45 \mathrm{~m}45 m Höhe. Berechne nach 2 s Höhe und Geschwindigkeit. Wann trifft er den Boden und mit welcher Geschwindigkeit?

Problem 10430

A point moves on the curve 3x3+4y3=xy3 x^{3}+4 y^{3}=x y3x3+4y3=xy. At P=(17,17)P=\left(\frac{1}{7}, \frac{1}{7}\right)P=(71​,71​), yyy increases at 4 units/sec. Find the speed and direction of xxx.

Problem 10431

Find dy/dxd y / d xdy/dx at the point (1,0)(1,0)(1,0) for the equation 5e6y=2xy+5x5 e^{6 y}=2 x y+5 x5e6y=2xy+5x.

Problem 10432

Skizzieren Sie die Graphen von f(x)=exf(x)=e^{x}f(x)=ex und g(x)=e−xg(x)=e^{-x}g(x)=e−x. Wie hängen die Graphen zusammen? Zeigen Sie, dass g′(x)=−e−xg^{\prime}(x)=-e^{-x}g′(x)=−e−x.

Problem 10433

Find the average rate of change of f(t)=6t+8f(t)=6t+8f(t)=6t+8 over [7,8][7,8][7,8] and compare it to the instantaneous rates at the endpoints.

Problem 10434

Maximize z=20xy−(x2+y2)z=20xy-(x^2+y^2)z=20xy−(x2+y2) with the constraint x+y=4x+y=4x+y=4.

Problem 10435

Berechnen Sie den Grenzwert lim⁡x→239x2−6x3x−2\lim _{x \rightarrow \frac{2}{3}} \frac{9 x^{2}-6 x}{3 x-2}x→32​lim​3x−29x2−6x​ und überprüfen Sie mit einer Wertetabelle. Bestimmen Sie die Art der Definitionslücke bei x=23x=\frac{2}{3}x=32​.

Problem 10436

Find the derivative of the function f(x)=4x3+2cos⁡xf(x)=\frac{4}{\sqrt[3]{x}}+2 \cos xf(x)=3x​4​+2cosx.

Problem 10437

Find the rate of change of the function y=3x+3y=3x+3y=3x+3.

Problem 10438

Find the rate of change of the function y=−2x−2y=-2x-2y=−2x−2.

Problem 10439

Bestimme aaa, sodass lim⁡x→2f(x)=−2\lim_{x \to 2} f(x) = -2limx→2​f(x)=−2 für f(x)=3ax2−4af(x) = 3ax^2 - 4af(x)=3ax2−4a.

Problem 10440

Use the substitution $x=\sqrt{48} \sec \theta$ to solve the integral $\int \frac{24}{x^{2} \sqrt{x^{2}-48}} dx$ .

Find the critical number $A$ for the function $f(x)=-4x^{2}+4x-4$ . Is it a local min (LMIN), local max (LMAX), or neither (NEITHER)?

Find the local minimum and maximum of the function $f(x)=-2 x^{3}+36 x^{2}-192 x+9$ .

Compute the integral of $\frac{3}{(3+x^{2})^{3/2}}$ with respect to $x$ .

Find the derivative of $y=5^{-5 x^{6}}$ .

Find the rate of change of sales given by $S(t)=106-90 e^{-0.3 t}$ at $t=1$ and $t=5$ years. What happens as $t$ increases? Does it ever equal zero?

Estimate the population in 2010 using $p(t)=37.47(1.023)^{t}$ and find the instantaneous rate at $t=10$ .

Find the growth function $G(t)$ for online course students since 2002 with $k=0.0474$ and initial enrollment 1.674 million. Then, find enrollment for 2004, 2006, and 2013, and describe the growth rate over time.

A runaway semi at $40 \mathrm{~m/s}$ starts $400 \mathrm{~m}$ high. How high will it go before stopping? Assume no friction.

Calculate the roofing area of the barn using the function $y=31-10\left(e^{x / 20}+e^{-x / 20}\right)$ from $x=-20$ to $x=20$ .

Find the difference quotient for the function $f(x)=4x-2$ : $\frac{f(x+h)-f(x)}{h}$ .

Find $c$ from the Mean Value Theorem for $f(x)=2 \sqrt{x}$ on $[0,25]$ . Options: a. $c=5$ , b. $c=\frac{1}{5}$ , c. $c=0$ , d. $c=\frac{25}{4}$ , e. None.

Evaluate the integral: $\int_{0}^{\sqrt{3}} \frac{1}{(1+x^{2})^{\frac{5}{2}}} \, dx$

Find the smallest value of $M$ such that $f(1) \leq M$ for all differentiable $f$ with $f' \leq 4$ and $f(-1)=3$ .

Find the limits as $x$ approaches $a$ : (a) $f(x)+6g(x)$ , (b) $f(x)g(x)$ , (c) $\sqrt[3]{3+f(x)}$ , (d) $\frac{4g(x)}{9f(x)+g(x)}$ .

Find the minimum value of $f(x)=\frac{1}{3} x^{3}-5 x^{2}+9 x+3$ on $[0,3]$ .

Find the smallest value of $M$ such that $f(1) \leq M$ for all differentiable $f$ with $f' \leq 4$ and $f(-1)=3$ .

Find the linear approximation $L(x)$ of $f(x)=e^{x} \cos x+\sin x$ at $x=0$ . Which is correct?

Find the real zero of $f(x)=x^{3}+4 x^{2}+x-9$ using the intermediate value theorem. Round to two decimal places.

Find all values of $c$ for the function $f(t)=\frac{t}{t-6}$ on the interval [-2, 0] using the Mean Value Theorem.

Find $a_{2}$ in the second degree Taylor polynomial of $y=f(x)g(x)$ at $x=3$ given $f(3)=g(3)=0$ , $f'(3)=2$ , $g'(3)=1$ .

A box with square base $x$ and height $y$ has $\frac{d x}{d t}=3 \mathrm{~cm/s}$ , $\frac{d y}{d t}=7 \mathrm{~cm/s}$ . Find $\frac{d V}{d t}$ when $x=2 \mathrm{~cm}$ , $y=4 \mathrm{~cm}$ .

Find the integrals: $\int x(x+1)^{5/2} \, dx$ and $\int \frac{1}{1+e^{x}} \, dx$ .

Use integration by parts to solve the integral $\int x(x+1)^{5/2} \mathrm{~d} x$ . What is the result?

Find all $x$ where the graph of $f(x)=\frac{1}{3} x^{3}-2 x^{2}+7$ has a point of inflection.

Find the rectangle with the largest area that fits inside a circle of radius $r$ . What are its dimensions?

Determine where the graph of $y=x^{2}-4 \cos x$ changes concavity in the interval $(-\pi / 2, \pi / 2)$ .

Find the second derivative, $f^{\prime \prime}$ , of the function $f(x)=x \cos(x^{2}) + 4x^{2}$ .

Ein Stein fällt aus $45 \mathrm{~m}$ Höhe. Berechne nach 2 s Höhe und Geschwindigkeit. Wann trifft er den Boden und mit welcher Geschwindigkeit?

A point moves on the curve $3 x^{3}+4 y^{3}=x y$ . At $P=\left(\frac{1}{7}, \frac{1}{7}\right)$ , $y$ increases at 4 units/sec. Find the speed and direction of $x$ .

Find $d y / d x$ at the point $(1,0)$ for the equation $5 e^{6 y}=2 x y+5 x$ .

Skizzieren Sie die Graphen von $f(x)=e^{x}$ und $g(x)=e^{-x}$ . Wie hängen die Graphen zusammen? Zeigen Sie, dass $g^{\prime}(x)=-e^{-x}$ .

Find the average rate of change of $f(t)=6t+8$ over $[7,8]$ and compare it to the instantaneous rates at the endpoints.

Maximize $z=20xy-(x^2+y^2)$ with the constraint $x+y=4$ .

Berechnen Sie den Grenzwert $\lim _{x \rightarrow \frac{2}{3}} \frac{9 x^{2}-6 x}{3 x-2}$ und überprüfen Sie mit einer Wertetabelle. Bestimmen Sie die Art der Definitionslücke bei $x=\frac{2}{3}$ .

Find the derivative of the function $f(x)=\frac{4}{\sqrt[3]{x}}+2 \cos x$ .

Find the rate of change of the function $y=3x+3$ .

Find the rate of change of the function $y=-2x-2$ .

Bestimme $a$ , sodass $\lim_{x \to 2} f(x) = -2$ für $f(x) = 3ax^2 - 4a$ .

Find the derivative of $f(x)=\frac{x^{8}(x-7)^{3}}{(x^{2}+4)^{5}}$ using logarithmic differentiation.

A hunter accidentally fires a bullet straight up at $200 \mathrm{~m/s}$ . How long until it falls back and how high does it go?

Familie Stein wacht um 8.00 Uhr bei $20^{\circ} \mathrm{C}$ auf. Die Temperatur wird durch $T(t)=\frac{52}{t+2}-6$ beschrieben.
a) Wie tief könnte die Temperatur fallen? b) Wann wird der Gefrierpunkt erreicht? c) Wann fiel die Heizung aus?

Evaluate the double integral $\int_{0}^{\sqrt{3}} \int_{u^{2}}^{3} \frac{e^{v}}{\sqrt{v}} dv \, du$ .

Find the values of $a$ and $b$ that maximize the function $F(a, b)=\int_{0}^{a} \int_{0}^{b} 4-x^{2}-y^{2} d y d x$ . Calculate the product $a b$ .

Find the second derivative $f^{\prime \prime}(t)$ of the function $f(t)=\int_{0}^{t} x^{1+x^{3}} d x$ for $t>0$ .

Find the limits of integration for the triple integral $\int_{0}^{2} \int_{2 z}^{4} \int_{1}^{3} f(x, y, z) d x d y d z$ when reordered.

Find the limits of integration for the integral of $f(x, y, z)=x y^{2} z^{3}$ over the domain $\frac{x^{2}}{4}+y^{2}+\frac{z^{2}}{9} \leq 1$ , $x \geq 0$ , $y \geq 0$ , $z \geq 0$ .

Find the derivative of $y = \cos(x) - 2\tan(x)$ using the chain rule.

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ for the equation $y + y^{2} + x^{3} = 0$ .

Find the derivative of $-\sin(-3x)$ with respect to $x$ .

Find the first derivative of $f(x)=x^{t-3}$ .

Bestimmen Sie die Ableitung von $f(b)=4ab$ .

Find the power series for $f(x)=\frac{3}{(1-3x)^{2}}$ and its radius of convergence. Use $\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$ .

Differentiate the function $G(y)=\ln \left(\frac{(4 y+1)^{3}}{\sqrt{y^{2}+1}}\right)$ . Find $G^{\prime}(y)$ .

Find the first derivative of $f(z)=\frac{4}{z}+\sqrt{z}$ .

Use the Intermediate Value Theorem to check if $f(x)=-7 x^{4}+2 x^{2}+10$ has a zero between -2 and -1.

Find the derivative of $y = x^{6} \cos(x)$ using logarithmic differentiation. What is $y'$ ?

Use Newton's method for $f(x)=x^{2}-5$ with $x_{0}=2$ to find $x_{1}$ and $x_{2}$ . Choose the correct formula.

Find $\frac{d y}{d x}$ for the function $y=\sqrt[4]{x}+\frac{x^{2}-8}{x^{2}}$ .

Find the derivative of $y=(\cos(9x))^x$ using logarithmic differentiation. Set $y'(x)=1$ .

How far does a ball fall in 5 seconds? Use the formula for distance: $d = \frac{1}{2} g t^2$ , where $g \approx 9.8 \, m/s^2$ .

Find the formula for Newton's method for $f(x)=4-\tan(3x)$ , starting with $x_{0}=1$ . Compute $x_{1}$ .

Find the derivative of $2 \sin x - x e^{-10 x}$ with respect to $x$ .

Calculate the integral $\int_{0}^{t} x^{1+x^{3}} dx$ for $t>0$ and determine $f^{\prime \prime}(t)$ .

Find Newton's method formula for $f(x)=4-\tan(3x)$ with $x_{0}=1$ . Compute $x_{1}$ and round to six decimal places.

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\sqrt[3]{x^{5}}+\pi^{2}-\frac{5}{x}+2^{x}$ .

Find the derivative of the function $f(x)=x^{2} \cos \left(\sqrt{x^{3}-1}+2\right)$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=e^{-x} \arctan(4 x^{2})$ .

Find the tangent line equation to $7 x^{2}-5 x y+y^{3}=3$ at point $(1,1)$ . (5 points)

Find the derivative $\left.\frac{d}{d x}\left(\frac{x f(x)}{g(x)}\right)\right|_{x=1}$ using the given values of $f$ and $g$ .

Find the derivative $\left.\frac{d}{d x}(f(x)+2 g(x))\right|_{x=3}$ using the given values of $f(x)$ and $g(x)$ .