Calculus

Problem 3301

Find the limit as $x$ approaches 5 for $\frac{x+7}{x+3}$ . A. $\lim _{x \rightarrow 5} \frac{x+7}{x+3}=\square$ B. Limit does not exist.

See Solution

Problem 3302

Find points where the tangent line to $xy + 5y^2 = -9$ is vertical using implicit differentiation.

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Problem 3303

Find the limit as $y$ approaches -3 for $(24-y)^{\frac{5}{3}}$ . Choose A (integer/fraction) or B (limit does not exist).

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Problem 3304

Calculate the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{2 h+1}-1}{h}$ . Is it a number or does it not exist?

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Problem 3305

Find the limit as $x$ approaches 8 for the expression $\frac{x-8}{x^{2}-64}$ . Does it exist?

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Problem 3306

Find the limit: $\lim _{x \rightarrow-6} \frac{x^{2}+2 x-24}{x+6}$ . Simplify if possible.

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Problem 3307

Calculate the limit as $t$ approaches 20 for the expression $\frac{t^{2}+t-420}{t^{2}-400}$ .

See Solution

Problem 3308

Find the limit:
$\lim _{x \rightarrow-6} \frac{x^{2}+2 x-24}{x+6}$
Choose A (simplify) or B (limit does not exist).

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Problem 3309

Find the limit: $\lim _{x \rightarrow 144} \frac{\sqrt{x}-12}{x-144}$ . Choose A for the limit value or B if it doesn't exist.

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Problem 3310

Find the limit as $v$ approaches 9: $\lim _{v \rightarrow 9} \frac{\frac{1}{v}-\frac{1}{9}}{v-9}$ . Options: A. $\square$ , B. Does not exist.

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Problem 3311

Find the limit as $x$ approaches 53 for $\frac{x-53}{\sqrt{x+11}-8}$ . What is the answer? A. or B.

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Problem 3312

Find the limit as $x$ approaches 0: $\lim _{x \rightarrow 0} \sec x$ . Choose A or B.

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Problem 3313

Find the limit as x approaches 0: $\lim _{x \rightarrow 0} \frac{4+7 x+\sin x}{5 \cos x}$ . Simplify if possible.

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Problem 3314

Find the limit: $\lim _{x \rightarrow-4 \pi} \sqrt{x+15} \cos (x+4 \pi)$ . A. $\square$ B. Limit does not exist.

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Problem 3315

Find the limit: $\lim _{x \rightarrow-2 \pi} \sqrt{x+15} \cos (x+2 \pi)$ . Choose A or B.

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Problem 3316

Given $\lim _{x \rightarrow 6} f(x)=9$ and $\lim _{x \rightarrow 6} g(x)=-9$ , find these limits: a. $\lim _{x \rightarrow 6}[f(x) g(x)]$ , b. $\lim _{x \rightarrow 6}[6 f(x) g(x)]$ , c. $\lim _{x \rightarrow 6}[f(x)+2 g(x)]$ , d. $\lim _{x \rightarrow 6}\left[\frac{f(x)}{f(x)-g(x)}\right]$ .

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Problem 3317

Find the limits as $x$ approaches -3: a. $p(x)+r(x)+s(x)$ b. $p(x) \cdot r(x) \cdot s(x)$

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Problem 3318

Analyze the long-term behavior of these functions:
1. $\frac{x^{2}+1}{x^{2}+2}$
2. $\frac{x^{3}+1}{x^{2}+2}$
3. $\frac{x^{2}+1}{x^{3}+2}$

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Problem 3319

Determine the long run behavior of these functions:
1. $\frac{x^{2}+1}{x^{2}+2}$
2. $\frac{x^{3}+1}{x^{2}+2}$
3. $\frac{x^{2}+1}{x^{3}+2}$

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Problem 3320

Given limits: $\lim _{x \rightarrow-3} p(x)=3$ , $\lim _{x \rightarrow-3} r(x)=0$ , $\lim _{x \rightarrow-3} s(x)=-8$ . Find the limits:
a. $\lim _{x \rightarrow-3}(p(x)+r(x)+s(x))$ b. $\lim _{x \rightarrow-3}(p(x) \cdot r(x) \cdot s(x))$ c. $\lim _{x \rightarrow-3} \frac{-5 p(x)+9 r(x)}{s(x)}$

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Problem 3321

Find the limit as $h$ approaches 0 for $\frac{f(2+h)-f(2)}{h}$ where $f(x)=-8x-1$ . What is the result?

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Problem 3322

Find the derivative of $f(x)=\frac{x+\sin x}{|x|-1}$ .

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Problem 3323

Find $\lim _{x \rightarrow 0} g(x)$ given $3-3 x^{2} \leq g(x) \leq 3 \cos x$ . What is the limit?

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Problem 3324

(a) What does $1-\frac{x^{2}}{6}<\frac{x \sin x}{2-2 \cos x}<1$ imply about $\lim _{x \rightarrow 0} \frac{x \sin x}{2-2 \cos x}$ ? Explain. (b) Graph $y=1-\frac{x^{2}}{6}, y=\frac{x \sin x}{2-2 \cos x}, y=1$ for $-2 \leq x \leq 2$ and discuss their behavior as $x \rightarrow 0$ .

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Problem 3325

Evaluate the limit of $f(x)=\frac{x^{2}-1}{|x|-1}$ as $x \rightarrow -1$ using tables and graphs.

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Problem 3326

Evaluate the difference quotient $\frac{f(9+h)-f(9)}{h}$ for $f(x)=x^{3}$ and simplify.

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Problem 3327

Find the derivative of $x^7 - x^4 - 3x^2 + 32$ .

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Problem 3328

A person jumps from 5000 ft and opens a parachute after 10 sec. Find speed, height at deployment, and time to ground.
Speed: $=\square \mathrm{ft} / \mathrm{sec}$ , Height: $=\square \mathrm{ft}$ , Time: $\approx \square \text{ sec}$ .

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Problem 3329

A person jumps from 5000 ft, deploys a parachute after 10 sec. Find speed, height at deployment, and time to ground.

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Problem 3330

Find the derivative of $y=\sin (x) \cos (x)$ with respect to $x$ : $\frac{d y}{d x}$ .

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Problem 3331

Find the value of $\frac{d}{d x}\left(\frac{6}{x^{4}}+\frac{1}{x^{2}}\right)$ when $x=-1$ .

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Problem 3332

Find the tangent line equation for the function $g$ at $x=7$ given $g(7)=-3$ and $g^{\prime}(7)=-1$ .

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Problem 3333

Find the derivative of $x^{\frac{5}{4}}$ and evaluate it at $x=16$ .

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Problem 3334

Find $F^{\prime}(0)$ for $F(x)=\frac{g(x)}{h(x)}$ given $g(0)=-3$ , $h(0)=2$ , $g^{\prime}(0)=-5$ , $h^{\prime}(0)=-2$ .

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Problem 3335

A plane drops a package from 625 m at 352 km/h. How long (in seconds) until it hits the sea? Ignore air resistance.

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Problem 3336

A plane drops a package from $625 \mathrm{~m}$ at $352 \mathrm{~km/h}$ . How long to reach sea level? (Ignore air resistance)

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Problem 3337

Find the derivative of $r(x)=3 \sqrt{x+1}$ or solve for $x$ in $r(x)=3 \sqrt{x+1}$ .

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Problem 3338

Find intersections of $f(x)=\sqrt{6}$ and $g(x)=2 \sqrt{3} \sin \left(\frac{x}{2}\right)$ in $[0,2\pi]$ , area between curves, solve $\frac{dy}{dx}=\frac{1}{x}-3$ with $y=0$ at $x=1$ , and find derivative of $h(x)=5x^2$ .

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Problem 3339

Prove that $\overrightarrow{B^{\prime}} \cdot \vec{T}=0$ given $\vec{N}=\frac{\vec{T}^{\prime}}{\left|\vec{T}^{\prime}\right|}$ and $\vec{B}=\vec{T} \times \vec{N}$ .

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Problem 3340

Find the horizontal asymptotes for the function $f(x)=\frac{2 x^{2}-4 x}{x^{5}-3 x^{2}}$ .

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Problem 3341

Calculate the average rate of change of $k(x)=6 \sqrt{x}$ from $x=13$ to $x=20$ . Round your answer to the nearest tenth.

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Problem 3342

Find the horizontal asymptotes of the function $f(x)=\frac{\sqrt{16 x^{4}+16 x^{2}}}{3 x^{2}-4}$ .

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Problem 3343

Calculate the average rate of change of $k(x)=\frac{18}{x}$ from $x=1$ to $x=8$ . Round your answer to the nearest tenth.

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Problem 3344

Find the horizontal asymptotes of the function $f(x)=\frac{4 x^{2}+e^{-x}}{3-x^{2}}$ .

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Problem 3345

Find when the news reaches the highest spread rate, given $N(t)=\frac{10,000}{1+50 e^{-0.4 t}}$ .

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Problem 3346

Find the derivative of the function $f(x)=5 x^{3}+3 x$ using the limit definition, without simplifying.

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Problem 3347

Calculate the average rate of change of $f(x)=\frac{20}{x}$ from $x=4$ to $x=8$ . Round to the nearest tenth.

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Problem 3348

Calculate the average rate of change of $f(x)=x^{3}+12 x^{2}+16 x$ from $x=-10$ to $x=-9$ . Round your answer to the nearest tenth.

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Problem 3349

Calculate the average rate of change of $f(x)=9 \sqrt{x+7}$ on $[0,6]$ . Round your answer to the nearest tenth.

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Problem 3350

Calculate the average rate of change of $g(x)=-11 \sqrt{x}+15$ from $x=4$ to $x=10$ . Round to the nearest tenth.

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Problem 3351

Calculate the average rate of change of $k(x)=\frac{1}{-2 x}$ from $x=1$ to $x=4$ , rounding to the nearest tenth.

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Problem 3352

Calculate the average rate of change of $g(x)=-x^{3}+17 x-17$ from $x=-6$ to $x=-3$ . Round to the nearest tenth.

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Problem 3353

Calculate the average rate of change of $f(x)=-3 \sqrt{x}+13$ on $[10,14]$ . Round your answer to the nearest tenth.

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Problem 3354

Calculate the average rate of change of $k(x)=8 \sqrt{x}-7$ from $x=3$ to $x=10$ . Round to the nearest tenth.

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Problem 3355

Calculate the average rate of change of $f(x)=\frac{8}{x+17}$ from $x=-20$ to $x=-16$ . Round to the nearest tenth.

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Problem 3356

Calculate the average rate of change of $k(x)=15\sqrt{x}+11$ from $x=10$ to $x=17$ . Round to the nearest tenth.

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Problem 3357

1. (a) For the hyperbola $y=\frac{2 x-2}{x-3}$ : (i) Write in general form. (ii) Find $y$ at $x=2$ . (iii) Find $x$ when $y=3$ . (iv) Show $\frac{d y}{d x}=\frac{-4}{(x-3)^{2}}$ . (v) Find the tangent line at $x=2$ . (b) Derive the quadratic formula from $a x^{2}+b x+c=0$ .

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Problem 3358

Ordnen Sie die Funktionen A, B, D, G, E ihren ersten Ableitungen 4, F, 1 und der zweiten Ableitung C zu.

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Problem 3359

Gegeben ist der Graph einer Funktion $f$ .
a) Finde die Nullstellen von $f'$ .
b) Bestimme, wo $f'$ positiv oder negativ ist.
c) Skizziere den Graphen von $f'$ .
d) Nenne die Tangentengleichung bei $P(2,5 \mid-1)$ .

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Problem 3360

Find the first and second derivatives of these functions:
c) $f(x) = 4 \sin (x)-1$
d) $f(x) = \frac{1}{2} x^{2} (x-8)$
e) $f(x) = \frac{7 x^{4}-5 x^{2}}{3 x}$

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Problem 3361

Berechnen Sie $f^{\prime}(x)$ und $f^{\prime \prime}(x)$ für die Funktionen: a) $3 x^{2}-5 x+2$ , b) $\frac{2}{x}+3 \sqrt{x}$ , c) $4 \sin (x)-1$ , d) $\frac{1}{2} x^{2} \cdot(x-8)$ , e) $\frac{7 x^{4}-5 x^{3}}{3 x}$ .

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Problem 3362

Find the antiderivative of $i(x)=\frac{1}{3} e^{3 x+1}$ and explain how you solved it.

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Problem 3363

Bestätigen Sie die Ableitung von $f(a)=\left(-\frac{1}{2} a^{2}+a \sqrt{2}\right)^{4}$ und prüfen Sie den Fehler in $f^{\prime}(a)=4 \cdot(-a+\sqrt{2}) \cdot\left(-\frac{1}{2} a^{2}+a \sqrt{2}\right)$ .

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Problem 3364

Was sagt das über das Schaubild von $f$ aus, wenn $f(3)=2$ und $f^{\prime}(3)=-1$ ?

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Problem 3365

Find local minima/maxima and intervals of increase/decrease for $f$ using $f^{\prime}(x)=\frac{x+6}{x^{2}(x-18)}$ . Sketch a possible graph.

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Problem 3366

Nennen Sie drei Funktionen, die für $x \rightarrow+\infty$ keinen Grenzwert haben.

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Problem 3367

Berechne den Grenzwert von $f(x)=0,25 x^{2}-x+2$ für $x_{0}=2$ .

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Problem 3368

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

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Problem 3369

In einer Stadt gibt es ein Bobbycar-Rennen. Berechne $f(0), f(10), f(20), f(30)$ und $f(40)$ für $f(t)=0,0003 t^{4}-0,024 t^{3}+0,605 t^{2}$ . Wann ist die Höchstgeschwindigkeit?

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Problem 3370

Find the antiderivative and zeros of $k(t)=t \cdot(t-4)^{2}$ .

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Problem 3371

Berechnen Sie die Fläche zwischen dem Graphen von $k(t)=t \cdot(t-4)^{2}$ und der x-Achse.

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Problem 3372

Gegeben ist die Funktion $f(x)=e^{x}$ . Bestimme die Steigungen an $x_{1}=-1$ , $x_{2}=0$ , $x_{3}=1$ , $x_{4}=2$ und die Tangentengleichung bei $P(a, e^{a})$ .

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Problem 3373

Jährliches Bobbycar-Rennen:
a) Berechne $f(0), f(10), f(20), f(30), f(40)$ . b) Finde den Zeitpunkt der Höchstgeschwindigkeit. c) Berechne $\frac{f(40)-f(20)}{20}$ und deute. d) Wo hat die Strecke die schärfste Kurve?

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Problem 3374

Calculate the area between the curve $f(x)=x^{3}-4x$ and the x-axis. The answer is 8FE.

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Problem 3375

Bestimmen Sie die Tangentengleichung der Funktion $f(x)=\frac{1}{2} x^{2}-3 x+1$ an den Punkten: a) $P(4 \mid-3)$ , b) $P(1 \mid-1,5)$ , c) $P(-4 \mid 21)$ , d) $P(0 \mid 1)$ .

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Problem 3376

Bestimmen Sie das unbestimmte Integral für die Funktionen: a) $f(x)=x^{5}$ , b) $f(x)=2 x^{3}$ , c) $f(x)=3 x^{4}-6 x+8$ .

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Problem 3377

Finde die Stammfunktionen für die folgenden Funktionen: a) $f(x)=2 x^{3}$ , b) $f(x)=3 x^{4}-6 x+8$ , c) $f(x)=\frac{1}{x^{3}}$ .

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Problem 3378

Calculate the area between the curve $A(x) = 6 + x - x^{2}$ and the x-axis.

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Problem 3379

Berechnen Sie die Fläche zwischen der Funktion $f$ und der x-Achse für die Intervalle: a) $f(x)=x+3, I=[0; 4]$ b) $f(x)=2x^{2}+1, I=[1; 2]$ c) $f(x)=(2-x)^{2}, I=[1; 3]$ Zeichnen Sie eine Skizze.

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Problem 3380

Berechne die Steigung und den Winkel der Funktion $f$ an $x_{0}$ für die folgenden Fälle: a) $f(x)=2 x^{3}$ , $x_{0}=-1$ ; b) $f(x)=-2 \sqrt{x}$ , $x_{0}=3$ ; c) $f(x)=-x^{2}-5 x$ , $x_{0}=0$ ; d) $f(x)=\frac{4}{x^{\prime}}$ , $x_{0}=-2$ ; e) $f(x)=(2 x+1)^{2}$ , $x_{0}=-\frac{1}{2}$ ; f) $f(x)=\frac{1}{2 x^{2}}$ , $x_{0}=-3$ .

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Problem 3381

8 a) Was bedeutet $f(5)=82,0$ und $\frac{f(6)-f(5,5)}{6-5,5} \approx-0,1$ ? Einheiten angeben. b) Was bedeutet $v(5)=25$ und $v^{\prime}(8)=16$ ? Einheiten angeben. Was ist $v^{\prime}(t)$ ? 9 Nennen Sie eine Funktion $f$ mit: a) immer positiver Steigung. b) Ableitung $f'$ hat genau eine Nullstelle.

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Problem 3382

Bestimme die Steigung und den Steigungswinkel von $f$ an den gegebenen Stellen: a) $f(x)=2 x^{3}$ bei $x_{0}=-1$ ; b) $f(x)=-2 \sqrt{x}$ bei $x_{0}=3$ ; c) $f(x)=-x^{2}-5 x$ ; d) $f(x)=-\frac{4}{x^{\prime}}$ bei $x_{0}=-2$ ; e) $f(x)=(2 x+1)^{2}$ bei $x_{0}=-\frac{1}{2}$ ; f) $f(x)=\frac{1}{2 x^{2}}$ bei $x_{0}=0$ und $x_{0}=-3$ .

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Problem 3383

Finde die Punkte, wo der Graph von $f$ die Steigung $m$ hat, und gib die Tangentengleichungen an. a) $f(x)=\frac{1}{3} x^{3}-8 x ; m=1$ b) $f(x)=(2 x+1)^{2} ; m=8$ c) $f(x)=x^{3}-3 x^{2}+6 ; m=0$ d) $f(x)=-\frac{4}{x} ; m=1$ e) $f(x)=\frac{1}{x^{2}}+2 x ; m=\frac{9}{4}$ f) $f(x)=x^{5}+5 x^{3}+3 ; m=4$

See Solution

Problem 3384

Eine Kugel fällt von einem Garagendach. Bestimme die Ableitung der Funktion $H(t)$ für $t=0,5$ , $t=1,5$ , $t=2,5$ und untersuche die Differenzierbarkeit bei $t=1$ und $t=1,8$ .

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Problem 3385

Berechnen Sie das Integral von $f(x) = -12(2x - 3)$ .

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Problem 3386

Find the derivative of $f(x)=2\left(\frac{1}{5} x^{2}+\frac{3}{5} x^{6}\right)$ .

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Problem 3387

Gegeben ist die Funktion $f(x)$ .
a) Finde die Gleichung der Geraden $f$ .
b) Bestimme die Flächeninhaltsfunktion $A_{0}$ von $f$ mit unterer Grenze 0.

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Problem 3388

Gegeben ist die Funktion $f(x)=4 x^{3}-48 x^{2}+84 x$ . Finde Extremstellen und Krümmungsverhalten mit Ableitungen.

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Problem 3389

A stone is thrown up at $9.8 \mathrm{~ms^{-1}}$ . After 1s, it falls and lands in the sea at $29.4 \mathrm{~ms^{-1}}$ after 4s.
a) Draw a velocity-time graph. b) Find: (i) distance to highest point, (ii) distance to sea, (iii) height of the cliff.

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Problem 3390

Berechne die Ableitung der Funktion $f$ für die folgenden Fälle: a) $f(x)=(8 x^{4}+2)^{3}$ , b) $f(x)=(\frac{1}{2}-5 x)^{3}$ , c) $f(x)=(x+2)^{4}$ , d) $f(x)=\frac{1}{4}(x^{2}-5)^{2}$ , e) $f(x)=(8 x-7)^{-1}$ , f) $f(x)=(5-x)^{-4}$ , g) $f(x)=(15 x^{3}-3)^{-2}$ , h) $f(x)=(15 x-3 x^{2})^{-2}$ .

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Problem 3391

Zeigen Sie, dass $A_{0}$ eine Flächeninhaltsfunktion von $f$ ist für: a) $f(x)=3$ , b) $f(x)=3x$ , c) $f(x)=2x+2$ , d) $f(x)=4x^3+x$ .

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Problem 3392

Berechnen Sie die Fläche zwischen der Kurve $f$ und der X-Achse über den Intervallen I. Skizzieren Sie die Graphen. a) $f(x)=x+3, I=[0 ; 4]$ b) $f(x)=2 x^{2}+1, I=[1 ; 2]$ c) $f(x)=(2-x)^{2}, I=[1 ; 3]$

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Problem 3393

Use the epsilon-delta definition of limits to find $\delta$ for $\lim_{x \to 3}(2x-1)=5$ and the interval for $\lim_{x \to -2}(2x-1)=-5$ with $\varepsilon=0.2$ .

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Problem 3394

Find the limits:
(a) $\lim _{x \rightarrow -2^{+}} f(x)$
(b) $\lim _{x \rightarrow -2^{-}} f(x)$
(c) $\lim _{x \rightarrow -2} f(x)$
Is $f(x)$ continuous at $x=-2$ ? Answer 'Yes' or 'No'.

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Problem 3395

Bestimme die Ableitung der Funktionen: a) $f(x)=x \sin(x)$ , b) $f(x)=3x \cos(x)$ , c) $f(x)=(3x+2) \sqrt{x}$ , d) $f(x)=\sqrt{x}(2x-3)$ , e) $f(x)=\sqrt{x} \cos(x)$ , f) $f(x)=(5-3x) \sin(x)$ , g) $f(x)=\frac{2}{x} \cos(x)$ , h) $f(x)=\sin(x) \cos(x)$ , i) $f(x)=x^2 \sin(x)$ , j) $f(x)=\frac{1}{\sqrt{x}} \cos(x)$ , k) $f(x)=(x^2+3x) \sin(x)$ , l) $f(x)=\sqrt{x}(x^5-2x^3)$ .

See Solution

Problem 3396

Find the limit as x approaches 11 for the expression $\frac{11-x}{x^{2}-121}$ .

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Problem 3397

Find the derivative of $f(x)=\frac{1}{x^{4}}-\frac{2}{x^{2}}+3 x^{4}+7 \sqrt{x}$ .

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Problem 3398

Graph the function and estimate the limit: $\lim _{x \rightarrow 0} \frac{6 \sin (x)}{\sqrt[3]{x}}$ . Use a table and analytic methods.

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Problem 3399

Find the one-sided limit:
$\lim _{x \rightarrow-3^{-}}\left(x^{2}+\frac{9}{x+3}\right)$

See Solution

Problem 3400

Find the limit: $\lim _{x \rightarrow 8} \frac{x-8}{x^{2}-9 x+8}$ and round to four decimal places.

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Calculus

Problem 3301

Find the limit as xxx approaches 5 for x+7x+3\frac{x+7}{x+3}x+3x+7​. A. lim⁡x→5x+7x+3=□\lim _{x \rightarrow 5} \frac{x+7}{x+3}=\squarelimx→5​x+3x+7​=□ B. Limit does not exist.

Problem 3302

Find points where the tangent line to xy+5y2=−9xy + 5y^2 = -9xy+5y2=−9 is vertical using implicit differentiation.

Problem 3303

Find the limit as yyy approaches -3 for (24−y)53(24-y)^{\frac{5}{3}}(24−y)35​. Choose A (integer/fraction) or B (limit does not exist).

Problem 3304

Calculate the limit: lim⁡h→02h+1−1h\lim _{h \rightarrow 0} \frac{\sqrt{2 h+1}-1}{h}limh→0​h2h+1​−1​. Is it a number or does it not exist?

Problem 3305

Find the limit as xxx approaches 8 for the expression x−8x2−64\frac{x-8}{x^{2}-64}x2−64x−8​. Does it exist?

Problem 3306

Find the limit: lim⁡x→−6x2+2x−24x+6\lim _{x \rightarrow-6} \frac{x^{2}+2 x-24}{x+6}limx→−6​x+6x2+2x−24​. Simplify if possible.

Problem 3307

Calculate the limit as ttt approaches 20 for the expression t2+t−420t2−400\frac{t^{2}+t-420}{t^{2}-400}t2−400t2+t−420​.

Problem 3308

Find the limit: lim⁡x→−6x2+2x−24x+6 \lim _{x \rightarrow-6} \frac{x^{2}+2 x-24}{x+6} x→−6lim​x+6x2+2x−24​ Choose A (simplify) or B (limit does not exist).

Problem 3309

Find the limit: lim⁡x→144x−12x−144\lim _{x \rightarrow 144} \frac{\sqrt{x}-12}{x-144}limx→144​x−144x​−12​. Choose A for the limit value or B if it doesn't exist.

Problem 3310

Find the limit as vvv approaches 9: lim⁡v→91v−19v−9\lim _{v \rightarrow 9} \frac{\frac{1}{v}-\frac{1}{9}}{v-9}limv→9​v−9v1​−91​​. Options: A. □\square□, B. Does not exist.

Problem 3311

Find the limit as xxx approaches 53 for x−53x+11−8\frac{x-53}{\sqrt{x+11}-8}x+11​−8x−53​. What is the answer? A. or B.

Problem 3312

Find the limit as xxx approaches 0: lim⁡x→0sec⁡x\lim _{x \rightarrow 0} \sec xlimx→0​secx. Choose A or B.

Problem 3313

Find the limit as x approaches 0: lim⁡x→04+7x+sin⁡x5cos⁡x\lim _{x \rightarrow 0} \frac{4+7 x+\sin x}{5 \cos x}x→0lim​5cosx4+7x+sinx​. Simplify if possible.

Problem 3314

Find the limit: lim⁡x→−4πx+15cos⁡(x+4π)\lim _{x \rightarrow-4 \pi} \sqrt{x+15} \cos (x+4 \pi)limx→−4π​x+15​cos(x+4π). A. □\square□ B. Limit does not exist.

Problem 3315

Find the limit: lim⁡x→−2πx+15cos⁡(x+2π)\lim _{x \rightarrow-2 \pi} \sqrt{x+15} \cos (x+2 \pi)limx→−2π​x+15​cos(x+2π). Choose A or B.

Problem 3316

Problem 3317

Find the limits as xxx approaches -3: a. p(x)+r(x)+s(x)p(x)+r(x)+s(x)p(x)+r(x)+s(x) b. p(x)⋅r(x)⋅s(x)p(x) \cdot r(x) \cdot s(x)p(x)⋅r(x)⋅s(x)

Problem 3318

Analyze the long-term behavior of these functions: 1. x2+1x2+2\frac{x^{2}+1}{x^{2}+2}x2+2x2+1​ 2. x3+1x2+2\frac{x^{3}+1}{x^{2}+2}x2+2x3+1​ 3. x2+1x3+2\frac{x^{2}+1}{x^{3}+2}x3+2x2+1​

Problem 3319

Determine the long run behavior of these functions: 1. x2+1x2+2\frac{x^{2}+1}{x^{2}+2}x2+2x2+1​ 2. x3+1x2+2\frac{x^{3}+1}{x^{2}+2}x2+2x3+1​ 3. x2+1x3+2\frac{x^{2}+1}{x^{3}+2}x3+2x2+1​

Problem 3320

Problem 3321

Find the limit as hhh approaches 0 for f(2+h)−f(2)h\frac{f(2+h)-f(2)}{h}hf(2+h)−f(2)​ where f(x)=−8x−1f(x)=-8x-1f(x)=−8x−1. What is the result?

Problem 3322

Find the derivative of f(x)=x+sin⁡x∣x∣−1f(x)=\frac{x+\sin x}{|x|-1}f(x)=∣x∣−1x+sinx​.

Problem 3323

Find lim⁡x→0g(x)\lim _{x \rightarrow 0} g(x)limx→0​g(x) given 3−3x2≤g(x)≤3cos⁡x3-3 x^{2} \leq g(x) \leq 3 \cos x3−3x2≤g(x)≤3cosx. What is the limit?

Problem 3324

Problem 3325

Evaluate the limit of f(x)=x2−1∣x∣−1f(x)=\frac{x^{2}-1}{|x|-1}f(x)=∣x∣−1x2−1​ as x→−1x \rightarrow -1x→−1 using tables and graphs.

Problem 3326

Evaluate the difference quotient f(9+h)−f(9)h\frac{f(9+h)-f(9)}{h}hf(9+h)−f(9)​ for f(x)=x3f(x)=x^{3}f(x)=x3 and simplify.

Problem 3327

Find the derivative of x7−x4−3x2+32x^7 - x^4 - 3x^2 + 32x7−x4−3x2+32.

Problem 3328

A person jumps from 5000 ft and opens a parachute after 10 sec. Find speed, height at deployment, and time to ground. Speed: =□ft/sec=\square \mathrm{ft} / \mathrm{sec}=□ft/sec, Height: =□ft=\square \mathrm{ft}=□ft, Time: ≈□ sec\approx \square \text{ sec}≈□ sec.

Problem 3329

A person jumps from 5000 ft, deploys a parachute after 10 sec. Find speed, height at deployment, and time to ground.

Problem 3330

Find the derivative of y=sin⁡(x)cos⁡(x)y=\sin (x) \cos (x)y=sin(x)cos(x) with respect to xxx: dydx\frac{d y}{d x}dxdy​.

Problem 3331

Find the value of ddx(6x4+1x2)\frac{d}{d x}\left(\frac{6}{x^{4}}+\frac{1}{x^{2}}\right)dxd​(x46​+x21​) when x=−1x=-1x=−1.

Problem 3332

Find the tangent line equation for the function ggg at x=7x=7x=7 given g(7)=−3g(7)=-3g(7)=−3 and g′(7)=−1g^{\prime}(7)=-1g′(7)=−1.

Problem 3333

Find the derivative of x54x^{\frac{5}{4}}x45​ and evaluate it at x=16x=16x=16.

Problem 3334

Find F′(0)F^{\prime}(0)F′(0) for F(x)=g(x)h(x)F(x)=\frac{g(x)}{h(x)}F(x)=h(x)g(x)​ given g(0)=−3g(0)=-3g(0)=−3, h(0)=2h(0)=2h(0)=2, g′(0)=−5g^{\prime}(0)=-5g′(0)=−5, h′(0)=−2h^{\prime}(0)=-2h′(0)=−2.

Problem 3335

A plane drops a package from 625 m at 352 km/h. How long (in seconds) until it hits the sea? Ignore air resistance.

Problem 3336

A plane drops a package from 625 m625 \mathrm{~m}625 m at 352 km/h352 \mathrm{~km/h}352 km/h. How long to reach sea level? (Ignore air resistance)

Problem 3337

Find the derivative of r(x)=3x+1r(x)=3 \sqrt{x+1}r(x)=3x+1​ or solve for xxx in r(x)=3x+1r(x)=3 \sqrt{x+1}r(x)=3x+1​.

Problem 3338

Problem 3339

Prove that B′undefined⋅T⃗=0\overrightarrow{B^{\prime}} \cdot \vec{T}=0B′⋅T=0 given N⃗=T⃗′∣T⃗′∣\vec{N}=\frac{\vec{T}^{\prime}}{\left|\vec{T}^{\prime}\right|}N=∣T′∣T′​ and B⃗=T⃗×N⃗\vec{B}=\vec{T} \times \vec{N}B=T×N.

Problem 3340

Find the horizontal asymptotes for the function f(x)=2x2−4xx5−3x2f(x)=\frac{2 x^{2}-4 x}{x^{5}-3 x^{2}}f(x)=x5−3x22x2−4x​.

Problem 3341

Calculate the average rate of change of k(x)=6xk(x)=6 \sqrt{x}k(x)=6x​ from x=13x=13x=13 to x=20x=20x=20. Round your answer to the nearest tenth.

Problem 3342

Find the limit as $x$ approaches 5 for $\frac{x+7}{x+3}$ . A. $\lim _{x \rightarrow 5} \frac{x+7}{x+3}=\square$ B. Limit does not exist.

Find points where the tangent line to $xy + 5y^2 = -9$ is vertical using implicit differentiation.

Find the limit as $y$ approaches -3 for $(24-y)^{\frac{5}{3}}$ . Choose A (integer/fraction) or B (limit does not exist).

Calculate the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{2 h+1}-1}{h}$ . Is it a number or does it not exist?

Find the limit as $x$ approaches 8 for the expression $\frac{x-8}{x^{2}-64}$ . Does it exist?

Find the limit: $\lim _{x \rightarrow-6} \frac{x^{2}+2 x-24}{x+6}$ . Simplify if possible.

Calculate the limit as $t$ approaches 20 for the expression $\frac{t^{2}+t-420}{t^{2}-400}$ .

Find the limit:
$\lim _{x \rightarrow-6} \frac{x^{2}+2 x-24}{x+6}$
Choose A (simplify) or B (limit does not exist).

Find the limit: $\lim _{x \rightarrow 144} \frac{\sqrt{x}-12}{x-144}$ . Choose A for the limit value or B if it doesn't exist.

Find the limit as $v$ approaches 9: $\lim _{v \rightarrow 9} \frac{\frac{1}{v}-\frac{1}{9}}{v-9}$ . Options: A. $\square$ , B. Does not exist.

Find the limit as $x$ approaches 53 for $\frac{x-53}{\sqrt{x+11}-8}$ . What is the answer? A. or B.

Find the limit as $x$ approaches 0: $\lim _{x \rightarrow 0} \sec x$ . Choose A or B.

Find the limit as x approaches 0: $\lim _{x \rightarrow 0} \frac{4+7 x+\sin x}{5 \cos x}$ . Simplify if possible.

Find the limit: $\lim _{x \rightarrow-4 \pi} \sqrt{x+15} \cos (x+4 \pi)$ . A. $\square$ B. Limit does not exist.

Find the limit: $\lim _{x \rightarrow-2 \pi} \sqrt{x+15} \cos (x+2 \pi)$ . Choose A or B.

Find the limits as $x$ approaches -3: a. $p(x)+r(x)+s(x)$ b. $p(x) \cdot r(x) \cdot s(x)$

Analyze the long-term behavior of these functions:
1. $\frac{x^{2}+1}{x^{2}+2}$
2. $\frac{x^{3}+1}{x^{2}+2}$
3. $\frac{x^{2}+1}{x^{3}+2}$

Determine the long run behavior of these functions:
1. $\frac{x^{2}+1}{x^{2}+2}$
2. $\frac{x^{3}+1}{x^{2}+2}$
3. $\frac{x^{2}+1}{x^{3}+2}$

Find the limit as $h$ approaches 0 for $\frac{f(2+h)-f(2)}{h}$ where $f(x)=-8x-1$ . What is the result?

Find the derivative of $f(x)=\frac{x+\sin x}{|x|-1}$ .

Find $\lim _{x \rightarrow 0} g(x)$ given $3-3 x^{2} \leq g(x) \leq 3 \cos x$ . What is the limit?

Evaluate the limit of $f(x)=\frac{x^{2}-1}{|x|-1}$ as $x \rightarrow -1$ using tables and graphs.

Evaluate the difference quotient $\frac{f(9+h)-f(9)}{h}$ for $f(x)=x^{3}$ and simplify.

Find the derivative of $x^7 - x^4 - 3x^2 + 32$ .

A person jumps from 5000 ft and opens a parachute after 10 sec. Find speed, height at deployment, and time to ground.
Speed: $=\square \mathrm{ft} / \mathrm{sec}$ , Height: $=\square \mathrm{ft}$ , Time: $\approx \square \text{ sec}$ .

Find the derivative of $y=\sin (x) \cos (x)$ with respect to $x$ : $\frac{d y}{d x}$ .

Find the value of $\frac{d}{d x}\left(\frac{6}{x^{4}}+\frac{1}{x^{2}}\right)$ when $x=-1$ .

Find the tangent line equation for the function $g$ at $x=7$ given $g(7)=-3$ and $g^{\prime}(7)=-1$ .

Find the derivative of $x^{\frac{5}{4}}$ and evaluate it at $x=16$ .

Find $F^{\prime}(0)$ for $F(x)=\frac{g(x)}{h(x)}$ given $g(0)=-3$ , $h(0)=2$ , $g^{\prime}(0)=-5$ , $h^{\prime}(0)=-2$ .

A plane drops a package from $625 \mathrm{~m}$ at $352 \mathrm{~km/h}$ . How long to reach sea level? (Ignore air resistance)

Find the derivative of $r(x)=3 \sqrt{x+1}$ or solve for $x$ in $r(x)=3 \sqrt{x+1}$ .

Prove that $\overrightarrow{B^{\prime}} \cdot \vec{T}=0$ given $\vec{N}=\frac{\vec{T}^{\prime}}{\left|\vec{T}^{\prime}\right|}$ and $\vec{B}=\vec{T} \times \vec{N}$ .

Find the horizontal asymptotes for the function $f(x)=\frac{2 x^{2}-4 x}{x^{5}-3 x^{2}}$ .

Calculate the average rate of change of $k(x)=6 \sqrt{x}$ from $x=13$ to $x=20$ . Round your answer to the nearest tenth.

Find the horizontal asymptotes of the function $f(x)=\frac{\sqrt{16 x^{4}+16 x^{2}}}{3 x^{2}-4}$ .

Calculate the average rate of change of $k(x)=\frac{18}{x}$ from $x=1$ to $x=8$ . Round your answer to the nearest tenth.

Find the horizontal asymptotes of the function $f(x)=\frac{4 x^{2}+e^{-x}}{3-x^{2}}$ .

Find when the news reaches the highest spread rate, given $N(t)=\frac{10,000}{1+50 e^{-0.4 t}}$ .

Find the derivative of the function $f(x)=5 x^{3}+3 x$ using the limit definition, without simplifying.

Calculate the average rate of change of $f(x)=\frac{20}{x}$ from $x=4$ to $x=8$ . Round to the nearest tenth.

Calculate the average rate of change of $f(x)=x^{3}+12 x^{2}+16 x$ from $x=-10$ to $x=-9$ . Round your answer to the nearest tenth.

Calculate the average rate of change of $f(x)=9 \sqrt{x+7}$ on $[0,6]$ . Round your answer to the nearest tenth.

Calculate the average rate of change of $g(x)=-11 \sqrt{x}+15$ from $x=4$ to $x=10$ . Round to the nearest tenth.

Calculate the average rate of change of $k(x)=\frac{1}{-2 x}$ from $x=1$ to $x=4$ , rounding to the nearest tenth.

Calculate the average rate of change of $g(x)=-x^{3}+17 x-17$ from $x=-6$ to $x=-3$ . Round to the nearest tenth.

Calculate the average rate of change of $f(x)=-3 \sqrt{x}+13$ on $[10,14]$ . Round your answer to the nearest tenth.

Calculate the average rate of change of $k(x)=8 \sqrt{x}-7$ from $x=3$ to $x=10$ . Round to the nearest tenth.

Calculate the average rate of change of $f(x)=\frac{8}{x+17}$ from $x=-20$ to $x=-16$ . Round to the nearest tenth.

Calculate the average rate of change of $k(x)=15\sqrt{x}+11$ from $x=10$ to $x=17$ . Round to the nearest tenth.

Gegeben ist der Graph einer Funktion $f$ .
a) Finde die Nullstellen von $f'$ .
b) Bestimme, wo $f'$ positiv oder negativ ist.
c) Skizziere den Graphen von $f'$ .
d) Nenne die Tangentengleichung bei $P(2,5 \mid-1)$ .

Find the first and second derivatives of these functions:
c) $f(x) = 4 \sin (x)-1$
d) $f(x) = \frac{1}{2} x^{2} (x-8)$
e) $f(x) = \frac{7 x^{4}-5 x^{2}}{3 x}$

Find the antiderivative of $i(x)=\frac{1}{3} e^{3 x+1}$ and explain how you solved it.

Was sagt das über das Schaubild von $f$ aus, wenn $f(3)=2$ und $f^{\prime}(3)=-1$ ?

Find local minima/maxima and intervals of increase/decrease for $f$ using $f^{\prime}(x)=\frac{x+6}{x^{2}(x-18)}$ . Sketch a possible graph.

Nennen Sie drei Funktionen, die für $x \rightarrow+\infty$ keinen Grenzwert haben.

Berechne den Grenzwert von $f(x)=0,25 x^{2}-x+2$ für $x_{0}=2$ .

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

In einer Stadt gibt es ein Bobbycar-Rennen. Berechne $f(0), f(10), f(20), f(30)$ und $f(40)$ für $f(t)=0,0003 t^{4}-0,024 t^{3}+0,605 t^{2}$ . Wann ist die Höchstgeschwindigkeit?

Find the antiderivative and zeros of $k(t)=t \cdot(t-4)^{2}$ .

Berechnen Sie die Fläche zwischen dem Graphen von $k(t)=t \cdot(t-4)^{2}$ und der x-Achse.

Gegeben ist die Funktion $f(x)=e^{x}$ . Bestimme die Steigungen an $x_{1}=-1$ , $x_{2}=0$ , $x_{3}=1$ , $x_{4}=2$ und die Tangentengleichung bei $P(a, e^{a})$ .

Calculate the area between the curve $f(x)=x^{3}-4x$ and the x-axis. The answer is 8FE.