Calculus

Problem 4301

Berechne die Steigung von $g(x)=x^{3}+x$ im Punkt $P(\alpha / 2)$ mit der $h$ -Methode.

See Solution

Problem 4302

Berechne die Steigung von $g(x)=x^{3}+x$ am Punkt $P(1, 2)$ mit der $h$ -Methode.

See Solution

Problem 4303

Bestimme die Steigung von $g(x)=x^{3}+x$ im Punkt $P(2)$ mit der $h$ -Methode.

See Solution

Problem 4304

Berechne die Steigung von $g(x)=x^{3}+x$ am Punkt $P(2 \mid g(2))$ mit der $h$ -Methode.

See Solution

Problem 4305

Berechne die Steigung von $g(x)=x^{3}+x$ am Punkt $P(1, 2)$ mit der $h$ -Methode.

See Solution

Problem 4306

Find the rate of change of $z$ with respect to $x$ at the point $(1,1,7)$ for the paraboloid $2z - 3x^2 - 4y^2 = 7$ .

See Solution

Problem 4307

Find the difference quotient, $\frac{f(x+h)-f(x)}{h}$ , for the function $f(x)=-7 x^{2}-6 x-5$ .

See Solution

Problem 4308

Choose $f(x)$ and $g(x)$ from $\{\sqrt{x}, x^{3/2}\}$ such that both limits go to $\infty$ and solve: (a) $\lim_{x \to \infty}(f(x)-g(x))=\infty$ . (b) $\lim_{x \to \infty}(f(x)-g(x))$ exists. (c) $\lim_{x \to \infty}(f(x)-g(x))=-\infty$ .

See Solution

Problem 4309

Examine "little o" and "big O" notation. Show $f(x)=\sqrt{1+2x^{2}}$ is $o(x^{2})$ and not $o(x)$ . Then prove $f(x)$ is $\dot{o}(x^{2})$ if $f(x)$ is $\mathcal{O}(x)$ .

See Solution

Problem 4310

Find the first and second-order partial derivatives of $F(x, y)=6 x^{9}+11 y^{14}$ with respect to $y$ and $x$ .

See Solution

Problem 4311

Show that $f(x)=\sqrt{1+2 x^{2}}$ is $o(x^{2})$ and not $o(x)$ . Use the definition involving limits.

See Solution

Problem 4312

A substance has a half-life of 11 years and starts with 130 grams. Find the amount left after 8 years and time until $35\%$ remains.

See Solution

Problem 4313

Given the function
$H(x, y, z, t)=e^{x^{z+1}}+(5y+x)^{z-y}+t \cdot \ln(x \cdot y^{6})$ ,
find the derivatives:
1. $H_{tx}^{\prime\prime}(1,2,3,4)$
2. $H_{ty}^{\prime\prime}(1,2,3,4)$
3. $H_{txy}^{\prime\prime\prime}(1,2,3,4)$
Also, identify the correct statement for swapping partial derivatives:
A. $\frac{\partial^{2} f}{\partial x_{i} \partial x_{j}}=\frac{\partial^{2} f}{\partial x_{j} \partial x_{i}}$

See Solution

Problem 4314

Approximate $g(3.95,-5.02)$ given $g(4,-5)=-5$ , $g_{1}^{\prime}(4,-5)=-5$ , $g_{2}^{\prime}(4,-5)=-1$ :
$g(3.95,-5.02) \approx$

See Solution

Problem 4315

Find the slope $m$ of the tangent line to $y=f(x)$ at $(7, f(7))$ where $f(x) = 9x^2 - 10$ .

See Solution

Problem 4316

Find the critical points of the function $F(t)=\frac{1}{3} t^{3}-7.5 t^{2}+54 t+22$ and characterize them.

See Solution

Problem 4317

Find constants $A$ and $B$ such that $\frac{f(6+h)-f(6)}{h} = Ah + B$ for $f(x)=4x^2-4$ .

See Solution

Problem 4318

Find the extremum of the function $g(z)=-29\left(e^{5 z}+5 e^{-2 z}\right)^{5}$ and round to two digits.

See Solution

Problem 4319

Evaluate $\lim _{n \rightarrow \infty} \int_{0}^{\pi}\left|x^{2} \sin n x\right| d x$ using given steps.

See Solution

Problem 4320

Analyze the piecewise function $f(x)=\left\{\begin{array}{l}5 x, \text { if } x<0 \\ 2 x^{2}, \text { if } 0<x \leq 5 \\ 2 x+5, \text { if } 5<x\end{array}\right.$ . Determine its continuity and discontinuities.

See Solution

Problem 4321

Find the limit: $\lim _{x \rightarrow 2} \frac{x^{2}-25}{x-2}$ . Choose from 0, 7, Does not exist, $-\frac{1}{7}$ , 3.

See Solution

Problem 4322

Find the limit: $\lim _{x \rightarrow 121} \frac{x-121}{\sqrt{x}-11}$ . Options: $\frac{1}{22}$ , 11, 1, 22, $\frac{1}{11}$ , Does not exist.

See Solution

Problem 4323

Find the Taylor series for $h(x)=x^{3} \ln (1-3 x)$ , discuss critical points, and classify them using derivatives.

See Solution

Problem 4324

Determine which functions allow the use of the Intermediate Value Theorem to find roots in the interval $[-2,2]$ : I. $f(x)=\frac{x-4}{x}$ II. $f(x)=4x^{3}-3x$ III. $f(x)=x^{2}+3$ Options: I and II, I and III, I only, II and III, I, II and III, II only.

See Solution

Problem 4325

Find the limit $\lim _{x \rightarrow 0} \frac{2 \sin ^{2}(3 x)}{5 x^{2}}$ . Options: $\frac{6}{5}$ , $\frac{18}{5}$ , 0, $\frac{18}{25}$ , None of the above.

See Solution

Problem 4326

Find the velocity function $v(t)$ and position function $s(t)$ for $a(t)=2+\cos(t)$ , with $s(0)=0$ and $s(2\pi)=0$ .

See Solution

Problem 4327

Find the expression for the derivative $f^{\prime}(x)$ of $f(x)=\frac{x}{5 x+2}$ from the options given.

See Solution

Problem 4328

Find $\frac{d y}{d x}$ for the equation $x^{2}+y^{2}=1$ at the point $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ .

See Solution

Problem 4329

Find the expression for $f^{\prime}(x)$ given $f(x)=\frac{x}{5 x+2}$ . Which limit defines the derivative?

See Solution

Problem 4330

Find the derivatives: (a) of $\ln (\sec (x)+\tan (x))$ , (b) of $\ln (-\sec (x)-\tan (x))$ , (c) deduce the derivative of $\ln |\sec (x)+\tan (x)|$ .

See Solution

Problem 4331

Find the slope of the tangent line to $f(x)=x^{4}+5 x^{3}-x^{2}+8$ at $x=1$ .

See Solution

Problem 4332

For $f(x)=x+x^{3}$ on $[0,4]$ , find: a) $R_{4}$ , b) $R_{n}$ , c) $\lim _{n \rightarrow \infty} R_{n}$ .

See Solution

Problem 4333

Find $\frac{d y}{d x}$ for $y=\frac{5 x^{2}}{x+6}$ .

See Solution

Problem 4334

Find the derivative of $f(x)=3 \tan (2 x)+5 \sec (6 x)$ : $f^{\prime}(x)=?$

See Solution

Problem 4335

Find the second derivative of $f(x)=5 \sqrt{x}+5 x^{\frac{1}{3}}+\frac{4}{x^{2}}$ .

See Solution

Problem 4336

Find the second derivative $f^{\prime \prime}(x)$ of the function $f(x)=5 \sqrt{x}+5 x^{\frac{1}{3}}+\frac{4}{x^{2}}$ .

See Solution

Problem 4337

Find the derivative of the function $f(x)=3 x^{5} \sin(4 x^{3})$ .

See Solution

Problem 4338

Find the bacteria population change rate on $[0,6]$ and $[6,12]$ for $P(t)=-40 t^{2}+480 t+7560$ . How many after 6 days?

See Solution

Problem 4339

Check if the function $f(x) = \tan(x) + 2$ is continuous.

See Solution

Problem 4340

Find the derivative of the integral $\int_{x^{2}}^{1} \sqrt{1-t^{2}} d t$ with respect to $x$ .

See Solution

Problem 4341

A substance decreases by $22\%$ in 27 hours. Find its half-life. Answer: 2.79 hours.

See Solution

Problem 4342

Find the tangent line equation at $(-2,-2)$ for $x^{2}+x y+2 y^{2}=16$ .

See Solution

Problem 4343

Find first and second-order approximations of $f(x)=9 x^{4}$ around $x_{0}=1$ and compare coefficients.

See Solution

Problem 4344

Invest \$6800 at 5% interest compounded continuously. Find the account balance after 10 years and the rate for \$8100.

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Problem 4345

Find $\frac{d y}{d x}$ for the equation $4 x^{3}+4 y^{3}-2 x y=0$ .

See Solution

Problem 4346

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

See Solution

Problem 4347

Find first and second-order approximations of $f(x) = 9x^4$ around $x_0=1$ . Use both to estimate $f(1.1)$ .

See Solution

Problem 4348

Find the linear approximation of $f(x)=x^{3}+2 \cdot x$ at $x_{0}=3$ and $x_{0}=3.1$ . Calculate $L(3.1)$ and $I(3)$ .

See Solution

Problem 4349

Find points $\hat{x}_{0}$ and $\tilde{x}_{0}$ where $L(x) = 10x + 16$ approximates $f(x) = 6x^3 - 152x - 308$ and $g(x) = 8 \ln(5x) - 30x + 24$ . Round to two digits.

See Solution

Problem 4350

Calculate the area between the curve $y(x)=x^{2}+2x-15$ and the $x$ -axis from $x=1$ to $x=4$ .

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Problem 4351

Find the linear approximation of $f(x) = x^m$ near $x_0=1$ and estimate:
(a) $0.99^{26}$
(b) $\sqrt[3]{1.03}$

See Solution

Problem 4352

Find $x$ where the slope of the tangent to $f(x)=2 x^{3}-27 x^{2}+5$ is $m=-120$ . Choices: $x=4,0$ ; $x=5,-5$ ; $x=4,5$ ; $x=-4,-5$ ; $x=4,-4$ ; None.

See Solution

Problem 4353

Find $\frac{d y}{d z}$ for the equation $4 x^{3}+4 y^{3}-2 x y=0$ .

See Solution

Problem 4354

Evaluate the integral: $\int_{0}^{1}\left(\frac{1}{1+\sqrt{x}}\right)^{4} d x$

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Problem 4355

Find first and second order approximations of $f(x)=8x^5$ around $x_0=1$ and compare values for $x=1.1$ .

See Solution

Problem 4356

Find the derivative of the function $f(x)=5 x \sqrt{2 x^{2}+10 x+5}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 4357

Find first and second-order approximations for $f(x)=8x^5$ around $x_0=1$ . Round coefficients to two digits.

See Solution

Problem 4358

Find the derivative of the function $f(x)=5 x \sqrt{2 x^{2}+10 x+5}$ .

See Solution

Problem 4359

Find the linear approximation $L(x)=A+B \cdot (x-3)$ for $f(x)=e^{2x}+4x$ at $x_0=3$ . Calculate $A$ , $B$ , and $f(3.2) \approx L(3.2)$ .

See Solution

Problem 4360

Find the linear approximation of $f(x)=e^{2x} +4x$ at $x_{0}=3.2$ in the form $I(x)=a+b \cdot\left(x-x_{0}\right)$ . What are $a$ and $b$ ?

See Solution

Problem 4361

Find the first derivative of $f(x)=3 x^{2}+5 x-4$ .

See Solution

Problem 4362

Find the first derivative of $f(x)=\frac{5}{x}+\frac{7}{x^{2}}-\sqrt[3]{x}$ .

See Solution

Problem 4363

Calculate the first derivative of $f(x)=\frac{7 x^{1/3}+5 x^{2}-6}{2 x^{1/2}}$ .

See Solution

Problem 4364

Find the first derivative of $f(x)=(16x^{2}-3x+2)^{5}$ .

See Solution

Problem 4365

Find the first derivative of $f(x) = e^{6x+2}$ .

See Solution

Problem 4366

Find the linear approximations of $f(x)=e^{4x}+12x$ at $x_0=1$ and $x_0=1.4$ , then approximate $f(1.4)$ and $f(1)$ .

See Solution

Problem 4367

Find the linear approximation of $f(x) = x^{m}$ near $x_{0} = 1$ . Use it to estimate (a) $0.98^{30}$ and (b) $\sqrt[4]{1.15}$ .

See Solution

Problem 4368

Estimate $g^{\prime}(3)$ for $g(x)=\ln(2x)$ using average rate of change over $[2.9,3.1]$ . Round to 4 decimals.

See Solution

Problem 4369

Find the instantaneous rate of change of $f(x)=x^{3}$ at a given point. Choose from: a. -15.75621 b. -13.17107 c. 7.23165 d. 4.25 e. -7.23165

See Solution

Problem 4370

Find the instantaneous rate of change of $f(x)=x^{3}-2 e^{x}$ at $x=3$ . Choose from: a. -15.75621 b. -13.17107 c. 7.23165 d. 4.25 e. -7.23165

See Solution

Problem 4371

Find the slope of the secant line for $f(x)=1.3 x^{3}-33 x^{2}+261 x+558$ from $x_{1}=0$ to $x_{2}=4$ . How does it compare to the actual increase of 140?

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Problem 4372

A ball is thrown up with initial speed $60 \mathrm{ft/s}$ . Find average velocity from $t=1$ for (i) 0.1s, (ii) 0.01s, (iii) 0.001s. Guess instantaneous velocity at $t=1$ .

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Problem 4373

Find the curve where $\frac{d y}{d x}=k x^{2}-\frac{6}{x^{3}}$ , passing through $(1,6)$ with gradient 9.

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Problem 4374

Find the equation of the curve given $\frac{d y}{d x}=k x^{2}-12 x+5$ and points (1,-3) and (3,11).

See Solution

Problem 4375

Find the curve with $\frac{d y}{d x}=5 x \sqrt{x}+2$ passing through $(1,3)$ and the tangent at $x=4$ .

See Solution

Problem 4376

A girl on a swing is 3.2 m high at her highest point. How high is she at her lowest point if her velocity is $4.2 \mathrm{~m/s}$ ? $(2.3 \mathrm{~m})$

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Problem 4377

Find the slope of the secant line $PQ$ for $x=5.1$ , $5.01$ , $4.9$ , and $4.99$ on the curve $y=x^{2}+x+6$ . Guess the tangent slope at $P(5,36)$ .

See Solution

Problem 4378

Für welches $x$ wird das Volumen einer Schachtel aus einem $16 \mathrm{~cm} \times 10 \mathrm{~cm}$ Rechteck maximal?

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Problem 4379

Leiten Sie die Funktionen ab: a) $a(x)=e^{5x}$ , b) $b(x)=5e^{\frac{1}{7}x}$ , c) $c(x)=4^{x}+7x$ , d) $d(x)=2x \cdot e^{3x+5}$ , e) $f(x)=\sin(x) \cdot e^{2x}$ . Lösungen ohne Rechenweg für: $e^{x}=10$ , $e^{x} \cdot \frac{1}{5}=12$ , $0=(x^{2}-4)(x-\pi)(e^{2x}-1)$ , $0.8^{x}+2=10$ . Ordnen Sie Graphen den Funktionen $a(x)=\frac{1}{2}e^{x}-2$ , $b(x)=\ln(x)$ , $c(x)=x^{2}$ , $d(x)=x^{4}-x^{2}$ , $e(x)=e^{x}$ zu. Erklären Sie die Herleitung der Ableitung $f^{\prime}(x)=\ln(a) a^{x}$ . Leiten Sie $g(x)=3^{x}$ und $h(x)=2x \cdot \left(\frac{1}{2}\right)^{x}$ ab.

See Solution

Problem 4380

A 4.6 g golf ball hits at 50 m/s. Initial velocity $v(m)=\frac{83 m}{m+0.046}$ . Analyze how $v$ changes as club mass $m$ increases.

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Problem 4381

Berechnen Sie die ersten drei Ableitungen von $h(t)=2 t^{2} \cdot e^{-\frac{1}{2} t}$ und den Wasserstand nach 2 Tagen. Bestimmen Sie den Zeitraum, in dem der Pegel $4 \mathrm{~m}$ über Normalmarke steht.

See Solution

Problem 4382

Berechne die ersten drei Ableitungen von $h(t)=2 t^{2} \cdot e^{-\frac{1}{2} t}$ .

See Solution

Problem 4383

Find a recursive relation for $I_{n}=\int \frac{t^{n}}{1+t^{2}} d t$ using $I_{n+2}$ and $I_{n+1}$ .

See Solution

Problem 4384

Find the derivative of the function $f(x)=5 x^{3}+x^{2}-2 x-1$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 4385

Find $\frac{d y}{d x}$ for the cone volume $V=\frac{1}{3} \pi x^{2} y$ at $x=6$ , $y=9$ , given $V=108 \pi$ .

See Solution

Problem 4386

Find $y^{\prime}(4)$ using implicit differentiation for $\sqrt{x}+\sqrt{y}=9$ and $y(4)=49$ .

See Solution

Problem 4387

Find $y^{\prime}(3)$ using implicit differentiation for $4 x^{2}+5 x+x y=3$ with $y(3)=-16$ .

See Solution

Problem 4388

Find $y^{\prime}(1)$ using implicit differentiation for $\sqrt{x}+\sqrt{y}=9$ and $y(1)=64$ .

See Solution

Problem 4389

Berechne die ersten drei Ableitungen von $h(t)=2 t^{2} e^{-\frac{1}{2} t}$ . Bestimme $h(2)$ und den Maximalstand.

See Solution

Problem 4390

Find the first and second derivatives, $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ , for $4 \sqrt{y}-2 x=-3 y$ .

See Solution

Problem 4391

Berechnen Sie die ersten drei Ableitungen von $h(t)=2 t^{2} \cdot e^{-\frac{1}{2} t}$ und den Wasserstand nach 2 Tagen.

See Solution

Problem 4392

Berechne die ersten drei Ableitungen von $h(t)=2 t^{2} \cdot e^{-\frac{1}{2} t}$ und den Wasserstand nach 2 Tagen.

See Solution

Problem 4393

Find $h^{\prime}(5)$ for $h(x)=f(x) \times g(x)$ given $f(5)=4, f^{\prime}(5)=2, g(5)=6, g^{\prime}(5)=-7$ .

See Solution

Problem 4394

Find the tangent line equation for $f(x)=\sqrt{x+1}$ at $x=3$ . Choices: A. $x-4y=-5$ , B. $x+4y=8$ , C. $x-4y=5$ , D. $4x-y=-8$ .

See Solution

Problem 4395

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

See Solution

Problem 4396

$\lim _{n \rightarrow \infty} \int_{0}^{\pi}\left|x^{2} \sin n x\right| d x$ を求める手順を示せ。

See Solution

Problem 4397

Untersuchen Sie die Funktion $f(x)=2 x \cdot e^{-x}$ auf Nullstellen, Extrema, Wendepunkte und Verhalten für $x \rightarrow \infty$ und $x \rightarrow -\infty$ . Graph für $-0,5 \leq x \leq 3$ zeichnen.

See Solution

Problem 4398

Untersuchen Sie die Funktion $f(x)=2 x \cdot e^{-x}$ auf Nullstellen, Extrema, Wendepunkte und das Verhalten für $x \rightarrow \pm\infty$ . Zeichnen Sie den Graphen für $-0,5 \leq x \leq 3$ .

See Solution

Problem 4399

Untersuchen Sie die Funktion $f(x)=2x \cdot e^{-x}$ auf Nullstellen, Extrema, Wendepunkte und Verhalten für $x \to \infty$ und $x \to -\infty$ . Zeichnen Sie den Graphen für $-0,5 \leq x \leq 3$ .

See Solution

Problem 4400

Untersuchen Sie die Funktion $f(x)=2 x \cdot e^{-x}$ auf Nullstellen, Extrema und Wendepunkte. Verhalten für $x \rightarrow \infty$ und $x \rightarrow -\infty$ ? Graph für $-0,5 \leq x \leq 3$ zeichnen.

See Solution

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Calculus

Problem 4301

Berechne die Steigung von g(x)=x3+xg(x)=x^{3}+xg(x)=x3+x im Punkt P(α/2)P(\alpha / 2)P(α/2) mit der hhh-Methode.

Problem 4302

Berechne die Steigung von g(x)=x3+xg(x)=x^{3}+xg(x)=x3+x am Punkt P(1,2)P(1, 2)P(1,2) mit der hhh-Methode.

Problem 4303

Bestimme die Steigung von g(x)=x3+xg(x)=x^{3}+xg(x)=x3+x im Punkt P(2)P(2)P(2) mit der hhh-Methode.

Problem 4304

Berechne die Steigung von g(x)=x3+xg(x)=x^{3}+xg(x)=x3+x am Punkt P(2∣g(2))P(2 \mid g(2))P(2∣g(2)) mit der hhh-Methode.

Problem 4305

Berechne die Steigung von g(x)=x3+xg(x)=x^{3}+xg(x)=x3+x am Punkt P(1,2)P(1, 2)P(1,2) mit der hhh-Methode.

Problem 4306

Find the rate of change of zzz with respect to xxx at the point (1,1,7)(1,1,7)(1,1,7) for the paraboloid 2z−3x2−4y2=72z - 3x^2 - 4y^2 = 72z−3x2−4y2=7.

Problem 4307

Find the difference quotient, f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​, for the function f(x)=−7x2−6x−5f(x)=-7 x^{2}-6 x-5f(x)=−7x2−6x−5.

Problem 4308

Problem 4309

Examine "little o" and "big O" notation. Show f(x)=1+2x2f(x)=\sqrt{1+2x^{2}}f(x)=1+2x2​ is o(x2)o(x^{2})o(x2) and not o(x)o(x)o(x). Then prove f(x)f(x)f(x) is o˙(x2)\dot{o}(x^{2})o˙(x2) if f(x)f(x)f(x) is O(x)\mathcal{O}(x)O(x).

Problem 4310

Find the first and second-order partial derivatives of F(x,y)=6x9+11y14F(x, y)=6 x^{9}+11 y^{14}F(x,y)=6x9+11y14 with respect to yyy and xxx.

Problem 4311

Show that f(x)=1+2x2f(x)=\sqrt{1+2 x^{2}}f(x)=1+2x2​ is o(x2)o(x^{2})o(x2) and not o(x)o(x)o(x). Use the definition involving limits.

Problem 4312

A substance has a half-life of 11 years and starts with 130 grams. Find the amount left after 8 years and time until 35%35\%35% remains.

Problem 4313

Problem 4314

Approximate g(3.95,−5.02)g(3.95,-5.02)g(3.95,−5.02) given g(4,−5)=−5g(4,-5)=-5g(4,−5)=−5, g1′(4,−5)=−5g_{1}^{\prime}(4,-5)=-5g1′​(4,−5)=−5, g2′(4,−5)=−1g_{2}^{\prime}(4,-5)=-1g2′​(4,−5)=−1:g(3.95,−5.02)≈ g(3.95,-5.02) \approx g(3.95,−5.02)≈

Problem 4315

Find the slope mmm of the tangent line to y=f(x)y=f(x)y=f(x) at (7,f(7))(7, f(7))(7,f(7)) where f(x)=9x2−10f(x) = 9x^2 - 10f(x)=9x2−10.

Problem 4316

Find the critical points of the function F(t)=13t3−7.5t2+54t+22F(t)=\frac{1}{3} t^{3}-7.5 t^{2}+54 t+22F(t)=31​t3−7.5t2+54t+22 and characterize them.

Problem 4317

Find constants AAA and BBB such that f(6+h)−f(6)h=Ah+B\frac{f(6+h)-f(6)}{h} = Ah + Bhf(6+h)−f(6)​=Ah+B for f(x)=4x2−4f(x)=4x^2-4f(x)=4x2−4.

Problem 4318

Find the extremum of the function g(z)=−29(e5z+5e−2z)5g(z)=-29\left(e^{5 z}+5 e^{-2 z}\right)^{5}g(z)=−29(e5z+5e−2z)5 and round to two digits.

Problem 4319

Evaluate lim⁡n→∞∫0π∣x2sin⁡nx∣dx\lim _{n \rightarrow \infty} \int_{0}^{\pi}\left|x^{2} \sin n x\right| d xlimn→∞​∫0π​∣∣​x2sinnx∣∣​dx using given steps.

Problem 4320

Problem 4321

Find the limit: lim⁡x→2x2−25x−2\lim _{x \rightarrow 2} \frac{x^{2}-25}{x-2}limx→2​x−2x2−25​. Choose from 0, 7, Does not exist, −17-\frac{1}{7}−71​, 3.

Problem 4322

Find the limit: lim⁡x→121x−121x−11\lim _{x \rightarrow 121} \frac{x-121}{\sqrt{x}-11}limx→121​x​−11x−121​. Options: 122\frac{1}{22}221​, 11, 1, 22, 111\frac{1}{11}111​, Does not exist.

Problem 4323

Find the Taylor series for h(x)=x3ln⁡(1−3x)h(x)=x^{3} \ln (1-3 x)h(x)=x3ln(1−3x), discuss critical points, and classify them using derivatives.

Problem 4324

Problem 4325

Find the limit lim⁡x→02sin⁡2(3x)5x2\lim _{x \rightarrow 0} \frac{2 \sin ^{2}(3 x)}{5 x^{2}}limx→0​5x22sin2(3x)​. Options: 65\frac{6}{5}56​, 185\frac{18}{5}518​, 0, 1825\frac{18}{25}2518​, None of the above.

Problem 4326

Find the velocity function v(t)v(t)v(t) and position function s(t)s(t)s(t) for a(t)=2+cos⁡(t)a(t)=2+\cos(t)a(t)=2+cos(t), with s(0)=0s(0)=0s(0)=0 and s(2π)=0s(2\pi)=0s(2π)=0.

Problem 4327

Find the expression for the derivative f′(x)f^{\prime}(x)f′(x) of f(x)=x5x+2f(x)=\frac{x}{5 x+2}f(x)=5x+2x​ from the options given.

Problem 4328

Find dydx\frac{d y}{d x}dxdy​ for the equation x2+y2=1x^{2}+y^{2}=1x2+y2=1 at the point (12,12)\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)(2​1​,2​1​).

Problem 4329

Find the expression for f′(x)f^{\prime}(x)f′(x) given f(x)=x5x+2f(x)=\frac{x}{5 x+2}f(x)=5x+2x​. Which limit defines the derivative?

Problem 4330

Find the derivatives: (a) of ln⁡(sec⁡(x)+tan⁡(x))\ln (\sec (x)+\tan (x))ln(sec(x)+tan(x)), (b) of ln⁡(−sec⁡(x)−tan⁡(x))\ln (-\sec (x)-\tan (x))ln(−sec(x)−tan(x)), (c) deduce the derivative of ln⁡∣sec⁡(x)+tan⁡(x)∣\ln |\sec (x)+\tan (x)|ln∣sec(x)+tan(x)∣.

Problem 4331

Find the slope of the tangent line to f(x)=x4+5x3−x2+8f(x)=x^{4}+5 x^{3}-x^{2}+8f(x)=x4+5x3−x2+8 at x=1x=1x=1.

Problem 4332

For f(x)=x+x3f(x)=x+x^{3}f(x)=x+x3 on [0,4][0,4][0,4], find: a) R4R_{4}R4​, b) RnR_{n}Rn​, c) lim⁡n→∞Rn\lim _{n \rightarrow \infty} R_{n}limn→∞​Rn​.

Problem 4333

Find dydx\frac{d y}{d x}dxdy​ for y=5x2x+6y=\frac{5 x^{2}}{x+6}y=x+65x2​.

Problem 4334

Find the derivative of f(x)=3tan⁡(2x)+5sec⁡(6x)f(x)=3 \tan (2 x)+5 \sec (6 x)f(x)=3tan(2x)+5sec(6x): f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 4335

Find the second derivative of f(x)=5x+5x13+4x2f(x)=5 \sqrt{x}+5 x^{\frac{1}{3}}+\frac{4}{x^{2}}f(x)=5x​+5x31​+x24​.

Problem 4336

Find the second derivative f′′(x)f^{\prime \prime}(x)f′′(x) of the function f(x)=5x+5x13+4x2f(x)=5 \sqrt{x}+5 x^{\frac{1}{3}}+\frac{4}{x^{2}}f(x)=5x​+5x31​+x24​.

Problem 4337

Find the derivative of the function f(x)=3x5sin⁡(4x3)f(x)=3 x^{5} \sin(4 x^{3})f(x)=3x5sin(4x3).

Problem 4338

Find the bacteria population change rate on [0,6][0,6][0,6] and [6,12][6,12][6,12] for P(t)=−40t2+480t+7560P(t)=-40 t^{2}+480 t+7560P(t)=−40t2+480t+7560. How many after 6 days?

Problem 4339

Check if the function f(x)=tan⁡(x)+2f(x) = \tan(x) + 2f(x)=tan(x)+2 is continuous.

Problem 4340

Find the derivative of the integral ∫x211−t2dt\int_{x^{2}}^{1} \sqrt{1-t^{2}} d t∫x21​1−t2​dt with respect to xxx.

Problem 4341

A substance decreases by 22%22\%22% in 27 hours. Find its half-life. Answer: 2.79 hours.

Problem 4342

Berechne die Steigung von $g(x)=x^{3}+x$ im Punkt $P(\alpha / 2)$ mit der $h$ -Methode.

Berechne die Steigung von $g(x)=x^{3}+x$ am Punkt $P(1, 2)$ mit der $h$ -Methode.

Bestimme die Steigung von $g(x)=x^{3}+x$ im Punkt $P(2)$ mit der $h$ -Methode.

Berechne die Steigung von $g(x)=x^{3}+x$ am Punkt $P(2 \mid g(2))$ mit der $h$ -Methode.

Berechne die Steigung von $g(x)=x^{3}+x$ am Punkt $P(1, 2)$ mit der $h$ -Methode.

Find the rate of change of $z$ with respect to $x$ at the point $(1,1,7)$ for the paraboloid $2z - 3x^2 - 4y^2 = 7$ .

Find the difference quotient, $\frac{f(x+h)-f(x)}{h}$ , for the function $f(x)=-7 x^{2}-6 x-5$ .

Examine "little o" and "big O" notation. Show $f(x)=\sqrt{1+2x^{2}}$ is $o(x^{2})$ and not $o(x)$ . Then prove $f(x)$ is $\dot{o}(x^{2})$ if $f(x)$ is $\mathcal{O}(x)$ .

Find the first and second-order partial derivatives of $F(x, y)=6 x^{9}+11 y^{14}$ with respect to $y$ and $x$ .

Show that $f(x)=\sqrt{1+2 x^{2}}$ is $o(x^{2})$ and not $o(x)$ . Use the definition involving limits.

A substance has a half-life of 11 years and starts with 130 grams. Find the amount left after 8 years and time until $35\%$ remains.

Approximate $g(3.95,-5.02)$ given $g(4,-5)=-5$ , $g_{1}^{\prime}(4,-5)=-5$ , $g_{2}^{\prime}(4,-5)=-1$ :
$g(3.95,-5.02) \approx$

Find the slope $m$ of the tangent line to $y=f(x)$ at $(7, f(7))$ where $f(x) = 9x^2 - 10$ .

Find the critical points of the function $F(t)=\frac{1}{3} t^{3}-7.5 t^{2}+54 t+22$ and characterize them.

Find constants $A$ and $B$ such that $\frac{f(6+h)-f(6)}{h} = Ah + B$ for $f(x)=4x^2-4$ .

Find the extremum of the function $g(z)=-29\left(e^{5 z}+5 e^{-2 z}\right)^{5}$ and round to two digits.

Evaluate $\lim _{n \rightarrow \infty} \int_{0}^{\pi}\left|x^{2} \sin n x\right| d x$ using given steps.

Find the limit: $\lim _{x \rightarrow 2} \frac{x^{2}-25}{x-2}$ . Choose from 0, 7, Does not exist, $-\frac{1}{7}$ , 3.

Find the limit: $\lim _{x \rightarrow 121} \frac{x-121}{\sqrt{x}-11}$ . Options: $\frac{1}{22}$ , 11, 1, 22, $\frac{1}{11}$ , Does not exist.

Find the Taylor series for $h(x)=x^{3} \ln (1-3 x)$ , discuss critical points, and classify them using derivatives.

Find the limit $\lim _{x \rightarrow 0} \frac{2 \sin ^{2}(3 x)}{5 x^{2}}$ . Options: $\frac{6}{5}$ , $\frac{18}{5}$ , 0, $\frac{18}{25}$ , None of the above.

Find the velocity function $v(t)$ and position function $s(t)$ for $a(t)=2+\cos(t)$ , with $s(0)=0$ and $s(2\pi)=0$ .

Find the expression for the derivative $f^{\prime}(x)$ of $f(x)=\frac{x}{5 x+2}$ from the options given.

Find $\frac{d y}{d x}$ for the equation $x^{2}+y^{2}=1$ at the point $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ .

Find the expression for $f^{\prime}(x)$ given $f(x)=\frac{x}{5 x+2}$ . Which limit defines the derivative?

Find the derivatives: (a) of $\ln (\sec (x)+\tan (x))$ , (b) of $\ln (-\sec (x)-\tan (x))$ , (c) deduce the derivative of $\ln |\sec (x)+\tan (x)|$ .

Find the slope of the tangent line to $f(x)=x^{4}+5 x^{3}-x^{2}+8$ at $x=1$ .

For $f(x)=x+x^{3}$ on $[0,4]$ , find: a) $R_{4}$ , b) $R_{n}$ , c) $\lim _{n \rightarrow \infty} R_{n}$ .

Find $\frac{d y}{d x}$ for $y=\frac{5 x^{2}}{x+6}$ .

Find the derivative of $f(x)=3 \tan (2 x)+5 \sec (6 x)$ : $f^{\prime}(x)=?$

Find the second derivative of $f(x)=5 \sqrt{x}+5 x^{\frac{1}{3}}+\frac{4}{x^{2}}$ .

Find the second derivative $f^{\prime \prime}(x)$ of the function $f(x)=5 \sqrt{x}+5 x^{\frac{1}{3}}+\frac{4}{x^{2}}$ .

Find the derivative of the function $f(x)=3 x^{5} \sin(4 x^{3})$ .

Find the bacteria population change rate on $[0,6]$ and $[6,12]$ for $P(t)=-40 t^{2}+480 t+7560$ . How many after 6 days?

Check if the function $f(x) = \tan(x) + 2$ is continuous.

Find the derivative of the integral $\int_{x^{2}}^{1} \sqrt{1-t^{2}} d t$ with respect to $x$ .

A substance decreases by $22\%$ in 27 hours. Find its half-life. Answer: 2.79 hours.

Find the tangent line equation at $(-2,-2)$ for $x^{2}+x y+2 y^{2}=16$ .

Find first and second-order approximations of $f(x)=9 x^{4}$ around $x_{0}=1$ and compare coefficients.

Find $\frac{d y}{d x}$ for the equation $4 x^{3}+4 y^{3}-2 x y=0$ .

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

Find first and second-order approximations of $f(x) = 9x^4$ around $x_0=1$ . Use both to estimate $f(1.1)$ .

Find the linear approximation of $f(x)=x^{3}+2 \cdot x$ at $x_{0}=3$ and $x_{0}=3.1$ . Calculate $L(3.1)$ and $I(3)$ .

Find points $\hat{x}_{0}$ and $\tilde{x}_{0}$ where $L(x) = 10x + 16$ approximates $f(x) = 6x^3 - 152x - 308$ and $g(x) = 8 \ln(5x) - 30x + 24$ . Round to two digits.

Calculate the area between the curve $y(x)=x^{2}+2x-15$ and the $x$ -axis from $x=1$ to $x=4$ .

Find the linear approximation of $f(x) = x^m$ near $x_0=1$ and estimate:
(a) $0.99^{26}$
(b) $\sqrt[3]{1.03}$

Find $x$ where the slope of the tangent to $f(x)=2 x^{3}-27 x^{2}+5$ is $m=-120$ . Choices: $x=4,0$ ; $x=5,-5$ ; $x=4,5$ ; $x=-4,-5$ ; $x=4,-4$ ; None.

Find $\frac{d y}{d z}$ for the equation $4 x^{3}+4 y^{3}-2 x y=0$ .

Evaluate the integral: $\int_{0}^{1}\left(\frac{1}{1+\sqrt{x}}\right)^{4} d x$

Find first and second order approximations of $f(x)=8x^5$ around $x_0=1$ and compare values for $x=1.1$ .

Find the derivative of the function $f(x)=5 x \sqrt{2 x^{2}+10 x+5}$ . What is $f^{\prime}(x)$ ?

Find first and second-order approximations for $f(x)=8x^5$ around $x_0=1$ . Round coefficients to two digits.

Find the derivative of the function $f(x)=5 x \sqrt{2 x^{2}+10 x+5}$ .

Find the linear approximation $L(x)=A+B \cdot (x-3)$ for $f(x)=e^{2x}+4x$ at $x_0=3$ . Calculate $A$ , $B$ , and $f(3.2) \approx L(3.2)$ .

Find the linear approximation of $f(x)=e^{2x} +4x$ at $x_{0}=3.2$ in the form $I(x)=a+b \cdot\left(x-x_{0}\right)$ . What are $a$ and $b$ ?

Find the first derivative of $f(x)=3 x^{2}+5 x-4$ .

Find the first derivative of $f(x)=\frac{5}{x}+\frac{7}{x^{2}}-\sqrt[3]{x}$ .

Calculate the first derivative of $f(x)=\frac{7 x^{1/3}+5 x^{2}-6}{2 x^{1/2}}$ .

Find the first derivative of $f(x)=(16x^{2}-3x+2)^{5}$ .

Find the first derivative of $f(x) = e^{6x+2}$ .

Find the linear approximations of $f(x)=e^{4x}+12x$ at $x_0=1$ and $x_0=1.4$ , then approximate $f(1.4)$ and $f(1)$ .

Find the linear approximation of $f(x) = x^{m}$ near $x_{0} = 1$ . Use it to estimate (a) $0.98^{30}$ and (b) $\sqrt[4]{1.15}$ .

Estimate $g^{\prime}(3)$ for $g(x)=\ln(2x)$ using average rate of change over $[2.9,3.1]$ . Round to 4 decimals.

Find the instantaneous rate of change of $f(x)=x^{3}$ at a given point. Choose from: a. -15.75621 b. -13.17107 c. 7.23165 d. 4.25 e. -7.23165

Find the instantaneous rate of change of $f(x)=x^{3}-2 e^{x}$ at $x=3$ . Choose from: a. -15.75621 b. -13.17107 c. 7.23165 d. 4.25 e. -7.23165

Find the slope of the secant line for $f(x)=1.3 x^{3}-33 x^{2}+261 x+558$ from $x_{1}=0$ to $x_{2}=4$ . How does it compare to the actual increase of 140?

A ball is thrown up with initial speed $60 \mathrm{ft/s}$ . Find average velocity from $t=1$ for (i) 0.1s, (ii) 0.01s, (iii) 0.001s. Guess instantaneous velocity at $t=1$ .

Find the curve where $\frac{d y}{d x}=k x^{2}-\frac{6}{x^{3}}$ , passing through $(1,6)$ with gradient 9.

Find the equation of the curve given $\frac{d y}{d x}=k x^{2}-12 x+5$ and points (1,-3) and (3,11).