Calculus

Problem 30601

Find $f(5)$ if $f^{\prime \prime}(x)=2 x+8 \sin (x)$ , $f(0)=2$ , and $f^{\prime}(0)=4$ .

See Solution

Problem 30602

Differentiate the function $s(t)=\tan (\sqrt{t})$ .

See Solution

Problem 30603

Find the slope of the tangent line to $y=\frac{3-x}{2 x}$ at $x=1$ . Choices: a) $m=\frac{3}{2}$ b) $m=-\frac{3}{2}$ c) $m=3$ d) $m=-3$

See Solution

Problem 30604

Find the function $f(t)$ given that $f''(t)=2 e^{t}+3 \sin (t)$ , with conditions $f(0)=-2$ and $f(\pi)=6$ .

See Solution

Problem 30605

Find the antiderivative $F(t)$ of $f(t)=8 \sec ^{2}(t)-10 t^{3}$ with $F(0)=0$ . Calculate $F(1.3)$ .

See Solution

Problem 30606

Finde die Nullstellen der Funktionen mit dem Newton-Verfahren in den angegebenen Intervallen: a) $f(x)=x^{3}+2 x-2 ;[0 ; 2]$ b) $f(x)=x^{5}-x+6 ;[-1 ;-2]$ c) $f(x)=-x^{3}+4 x^{2}-1 ;[3 ; 5]$ d) $f(x)=x^{3}-3 x^{2}+3 x-1 ;[0 ; 4]$

See Solution

Problem 30607

Find the derivative of the function $y=\cos \left(x^{3}\right)$ .

See Solution

Problem 30608

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{3} \sqrt{2 x+1}$ .

See Solution

Problem 30609

Differentiate the function: $y=\sec ^{2}(x)+\tan ^{2}(x)$

See Solution

Problem 30610

Find the derivative $f^{\prime}(x)$ for the function $f(x)=(2x^{2}+5)^{7}$ .

See Solution

Problem 30611

Find the derivative of the function $y=\sqrt[3]{x^{2}+x}$ .

See Solution

Problem 30612

Given $f^{\prime \prime}(x)=-6 \cos (x)$ , find $f^{\prime}(x)$ and $f(x)$ using constants $\mathrm{C}$ and $\mathrm{D}$ .

See Solution

Problem 30613

Find the derivative of $y = e^{x^2 + 3x}$ .

See Solution

Problem 30614

Find the derivative of the function $f(x)=(3 \sin x+6 \cos x) \tan ^{-1} x$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 30615

Bestimmen Sie die 1., 2. und 3. Ableitung für die Funktionen a) bis f) und berechnen Sie die Steigung von $f$ an $A(2 \mid f(2))$ .

See Solution

Problem 30616

Berechnen Sie die Steigung von $f$ an $A(2 \mid f(2))$ für die Funktionen: a) $f(x)=\frac{3}{2} x^{2}$ , b) $f(t)=\frac{1}{4} t^{4}-\frac{5}{3} t^{3}$ , c) $f(z)=\frac{3}{4} z^{2}-3 z^{-1}$ , d) $f(x)=-x-\frac{2}{x^{4}}$ , e) $f(x)=2 x^{-1}-4 x^{-2}$ , f) $f(x)=\frac{3}{x}+4 \sqrt{x}-2$ .

See Solution

Problem 30617

Bestimme die Steigung von $f(z) = \frac{3}{4} z^{2} - 3 z^{-1}$ am Punkt $A(2|f(2))$ .

See Solution

Problem 30618

Find the value of the series: $\sum_{n=1}^{+\infty}\left(\sqrt[n]{3}-1\right)$ .

See Solution

Problem 30619

Graph the yield function $V=6.7 e^{-48.1/t}$ , find its horizontal asymptote, and determine time for $V=1.3$ .

See Solution

Problem 30620

Finde die Stellen, an denen $f$ die Steigung $m$ hat: a) $f(x)=\frac{1}{4} x^{4}-6 x, m=2$ b) $f(x)=-\frac{1}{6} x^{3}+x^{2}, m=-2,5$ c) $f(x)=\frac{2}{x}-x, m=-3$ d) $f(x)=3 \sqrt{x}, m=3$

See Solution

Problem 30621

Berechnen Sie die Steigung von $f$ bei $A(2 \mid f(2))$ für: a) $f(x)=\frac{3}{2} x^{2}$ , b) $f(t)=\frac{1}{4} t^{4}-\frac{5}{3} t^{3}$ , c) $f(z)=\frac{3}{4} z^{2}-3 z^{-1}$ .

See Solution

Problem 30622

Begründen Sie, warum die e-Funktion $f(x) = \mathrm{e}^{x}$ keine Nullstellen, Symmetrie, Extrem- oder Wendepunkte hat.

See Solution

Problem 30623

Togets avstand fra stasjonen er gitt ved $s(t)=\left(0,800 \mathrm{~m} / \mathrm{s}^{2}\right) t^{2}+(21,0 \mathrm{~m} / \mathrm{s}) t+(150 \mathrm{~m})$ . Finn koeffisientene, tiden for 35,0 m/s, og avstanden da.

See Solution

Problem 30624

Bestimmen Sie die Ableitungsfunktion von $f(x) = ax + c$ .

See Solution

Problem 30625

Evaluate the series $\sum_{n=1}^{+\infty}(\sqrt[n]{n}-1)^{\alpha}$ .

See Solution

Problem 30626

Calculate the integral: $\int \frac{(x+1) d x}{x^{2}(x-1)}$

See Solution

Problem 30627

Calculate the integral: $\int \frac{2 x d x}{(x-1)(x^{2}+1)}$ .

See Solution

Problem 30628

Bestimmen Sie die Ableitung und die Steigungen bei $x_{1}=1$ und $x_{2}=0$ für $f(x)=-4 e^{x}+2 x$ und $f(x)=2^{x}$ .

See Solution

Problem 30629

Evaluate the integral: $\int \frac{(x+1) d x}{x^{3}+x^{2}-6 x}$ .

See Solution

Problem 30630

Bestimmen Sie die Ableitung und die Steigungen an $x_{1}=1$ und $x_{2}=0$ für: a) $f(x)=2 e^{x}$ , b) $f(x)=-4 e^{x}+2 x$ , c) $f(x)=2^{x}$ .

See Solution

Problem 30631

Evaluate the integral: $\int x \cos^{2} x \, dx$

See Solution

Problem 30632

Find the integral of $x^{2} \ln^{2} x \, dx$ .

See Solution

Problem 30633

Given the function $f(x)=\frac{2x+1}{x-3}$ , find $f'(x)$ , $f'(4)$ , the tangent line at $x=4$ , and tangents from $(6,-5)$ .

See Solution

Problem 30634

Calculate the integral: $\int \frac{d x}{x^{2}+7 x+6}$

See Solution

Problem 30635

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

See Solution

Problem 30636

Bestimme die Fläche A zwischen den Schnittpunkten der Funktionen $f(x)=1-x^{2}$ und $g(x)=x^{2}-2x+1$ .

See Solution

Problem 30637

Evaluate the integral $\int_{\pi / 3}^{\pi / 6} \frac{\cos x-x \sin x}{x \cos x} d x$ and choose the correct answer.

See Solution

Problem 30638

Find the derivative of the function $f(x) = x^{2} + 4x + 1$ .

See Solution

Problem 30639

Find $\frac{d f^{-1}}{d x}$ for $f(x)=x^{2}-4 x+3$ at $x=3$ , where $x \geq 2$ . Choices: $1/4$ , $1/2$ , $1/3$ , 4.

See Solution

Problem 30640

Bestimme die Funktion $f(x)=a \cdot e^{k \cdot x}$ für das Wachstum einer Sonnenblume und die momentane Geschwindigkeit bei $x=20$ .

See Solution

Problem 30641

Haziq's ice cylinder melts at $4 \mathrm{~cm}^{2} \mathrm{~s}^{-1}$ . Find the surface area change rate when radius is $8 \mathrm{~cm}$ .

See Solution

Problem 30642

Berechne den Flächeninhalt $A$ zwischen $f(x)=\frac{1}{2} x^{2}+1$ und $g(x)=-x^{2}+\frac{3}{2} x+4$ von $x=-1$ bis $x=2$ .

See Solution

Problem 30643

What height is needed for an $823 \mathrm{~g}$ falcon to reach $68 \mathrm{~m/s}$ in a dive under gravity?

See Solution

Problem 30644

Berechne die Fläche zwischen den Funktionen $f(x)=-\frac{1}{64} x^{4}+\frac{1}{8} x^{2}+4$ und $g(x)=\frac{1}{8} x^{2}+1$ von $-4$ bis $4$ .

See Solution

Problem 30645

Ein Patient nimmt um 20 Uhr $2 \mathrm{mg}$ eines Medikaments ein. Berechne die verbleibende Menge nach $1, 2, 3$ Stunden und finde die Funktion zur Basis $e$ .

See Solution

Problem 30646

Calculate the integral: $\int_{-4}^{4}\left(-\frac{1}{64} x^{4}-3\right) d x$

See Solution

Problem 30647

Найдите производную функции $y = \frac{\cos 2x}{\ln(x^{2} + 4x)}$ .

See Solution

Problem 30648

If $f'(x)=2x^3$ , find $\frac{d f^{-1}}{d x}$ at $x=f(2)$ . Options: a. 4, b. 16, c. $\frac{1}{24}$ , d. $\frac{1}{8}$ , e. none.

See Solution

Problem 30649

Rewrite the integral using $u$ and $du$ , then evaluate: $\int (u)^{7} du, \quad u=2x-1$

See Solution

Problem 30650

$x^{\frac{2}{3}}+y^{\frac{2}{3}}=4$ için $y^{\prime}$ türevi $x=8$ 'de nedir? A) 0 B) 1 C) 2 D) 4 E) 8

See Solution

Problem 30651

Rewrite the integral using $u$ and $du$ , then solve: $\int t \sqrt{6 t^{2}+1} dt$

See Solution

Problem 30652

Find the limit as $x$ approaches $\frac{3}{2}$ from the right for $\frac{x^{2}+8x-20}{2x^{2}+x-6}$ .

See Solution

Problem 30653

Rewrite the integral as $u$ and $du$ , then evaluate $\int \sin(u) \, du$ , where $u=9\theta-5$ .

See Solution

Problem 30654

Evaluate the integral from 0 to $\frac{\pi}{6}$ of $\tan(2x)$ . What is the result?

See Solution

Problem 30655

Rewrite the integral using $u$ and $du$ , then evaluate: $\int \frac{(\ln (x))^{6}}{x} dx, \quad u=\ln (x)$

See Solution

Problem 30656

Determine where the function $x^{4}-4 x^{2}+3$ is increasing and decreasing.

See Solution

Problem 30657

Evaluate the integral $\int_{0}^{\pi / 3} \frac{4 \sin \theta}{1-4 \cos \theta} d \theta$ .

See Solution

Problem 30658

Find when the particle changes direction for the position $s(t) = -2t^3 + 6t^2 - 3$ , where $t \geq 0$ .

See Solution

Problem 30659

Find the derivative of $y$ with respect to $x$ , $t$ , or $\theta$ for $y=\ln (\sec \theta+\tan \theta)$ .

See Solution

Problem 30660

Find the first and second derivatives of these functions: a. $f(x)=x^{2}+1$ (first principles), b. $f(x)=\cos(x)e^{x}$ , c. $f(x)=(x^{2}-3x+2)^{3}\sin(2x^{2})$ (first derivative only).

See Solution

Problem 30661

How long for 75% of iodine-131 (half-life 8.02 days) to decay?

See Solution

Problem 30662

Approximate Bolt's velocity at $t=2$ seconds using the position equation $s(t)=0.567 t^{4}-10.2 t^{3}+57.2 t^{2}-60.4 t$ .

See Solution

Problem 30663

Evaluate the limits: a) $\frac{x^{2}-4 x+3}{x-3}$ b) $\frac{x^{2}-4 x+3}{x-3}$

See Solution

Problem 30664

Calcula el límite de $\frac{x^{2}-4 x+3}{x-3}$ cuando $x$ tiende a 3.

See Solution

Problem 30665

Differentiate the function $f(x) = x \sqrt{x^{2}+1}$ .

See Solution

Problem 30666

Find the first and second derivatives of these functions: a) $f(x)=x^{2}+1$ , b) $f(x)=\cos(x)e^{x}$ , c) $f(x)=2\sin(x^{2})$ .

See Solution

Problem 30667

Find the slope of the tangent to $y=x^{2}+9 x$ where it intersects the line $y=3 x$ .

See Solution

Problem 30668

Find the first derivative using first principles for: a) $f(x)=7 x-2$ b) $f(x)=3 x^{2}+7 x+14$

See Solution

Problem 30669

Find the derivative of the function $e^{3x}$ .

See Solution

Problem 30670

Differentiate the function: $f(x) = (1-3x) \sqrt{2x+5}$ .

See Solution

Problem 30671

The position of an object is $s(t)=3 t^{3}-40.5 t^{2}+162 t$ . Find velocity, acceleration, and when it's stationary, advancing, or retreating.

See Solution

Problem 30672

Find velocity and acceleration of $s(t)=3 t^{3}-40.5 t^{2}+162 t$ for $0 \leq t \leq 8$ . When is it stationary, advancing, or retreating? When is velocity constant, decreasing, or increasing?

See Solution

Problem 30673

Find the derivative of $y = \frac{3}{e^{x^{4}}}$ .

See Solution

Problem 30674

Find the velocity and acceleration of the object described by $s(t)=3 t^{3}-40.5 t^{2}+162 t$ for $0 \leq t \leq 8$ .

See Solution

Problem 30675

Find the velocity and acceleration of an object with position $s(t)=3 t^{3}-40.5 t^{2}+162 t$ for $0 \leq t \leq 8$ .

See Solution

Problem 30676

Find the asymptotes and local extrema for the function $y=\frac{8}{x^{2}-9}$ .

See Solution

Problem 30677

Find the derivatives of these functions: a. $y=2 \sin x-3 \cos 5 x$ , b. $y=\sin(\cos x^{2})$ , c. $y=(\sin 2x+1)^{4}$ .

See Solution

Problem 30678

Amira wants to know how much to invest at a $2.4\%$ continuous interest rate to reach \$320 in 17 years.

See Solution

Problem 30679

Calculate the integral $\int(2 x+1) \, dx$ .

See Solution

Problem 30680

Find $y$ in terms of $x$ given $\frac{\mathrm{d} y}{\mathrm{~d} x}=2 x+1$ and $y=5$ when $x=1$ .

See Solution

Problem 30681

Soit la fonction $g(x)=\frac{1-x}{x+2}$ .
1. Montrez que $D_{g} = ]-\infty ; -2[ \cup ]-2 ; +\infty[$ .
2. Montrez que $g'(x) = -\frac{3}{(x+2)^2}$ .
3. Étudiez le sens de variation de $g$ sur $(-1 ; 5)$ .
4. Dressez le tableau de variation de $g$ sur $[-1 ; 5]$ .
5. Complétez le tableau: $x$ : -1, 0, 1, 3, 5; $g(x)$ : ?
6. Représentez graphiquement $g$ sur $[-1 ; 5]$ .

See Solution

Problem 30682

Find $y$ for each case given the conditions: a) $\frac{dy}{dx}=3-6x$ , $y=1$ at $x=2$ b) $\frac{dy}{dx}=3x^2-x$ , $y=41$ at $x=4$ c) $\frac{dy}{dx}=x^2+4x^x+1$ , $y=4$ at $x=-3$ d) $\frac{dy}{dx}=7-5x-x^3$ , $y=0$ at $x=2$ e) $\frac{dy}{dx}=8x-\frac{2}{x^2}$ , $y=-1$ at $x=\frac{1}{2}$ f) $\frac{dy}{dx}=3-\sqrt{x}$ , $y=8$ at $x=4$

See Solution

Problem 30683

Find the integral of $\cos ^{3} x$ with respect to $x$ .

See Solution

Problem 30684

Evaluate the series: $\sum_{n=1}^{\infty}(-1)^{n} \sin \frac{1}{n}$ .

See Solution

Problem 30685

Find the function $f(x)$ given that $f'(x)=3+2x-x^{2}$ and it passes through the point $(3,5)$ .

See Solution

Problem 30686

Das Wachstum einer Bakterienkultur wird durch $f(t)=-\frac{1}{2} t^{3}+6 t^{2}$ für $t$ in Stunden beschrieben.
a) Beschreiben Sie das Wachstum. b) Bestimmen Sie die Fläche. c) Berechnen Sie die Kulturgröße nach 2, 4 und 6 Stunden. d) Verwenden Sie $g(t)=15 t$ für die Berechnung und vergleichen Sie diese Ergebnisse mit c).

See Solution

Problem 30687

Find $y$ when $x=4$ given $\frac{\mathrm{d} y}{\mathrm{~d} x}=10 x^{\frac{3}{2}}-2 x^{-\frac{1}{2}}$ and $y=7$ at $x=0$ .

See Solution

Problem 30688

1. Given $f'(-1)=4$ and $f'(x)=2x^3-x-8$ , find $f(x)$ and the tangent at $x=2$ .
2. With $f'(x)=3x^2-8x-5$ and passing through origin, find other $x$ -axis crossing points.
3. Given $\frac{dy}{dx}=3x+\frac{2}{x^2}$ , find $y$ and its value when $x=\frac{1}{2}$ .
4. For $C$ with $\frac{dy}{dx}=3x^2+kx$ , find $k$ and the curve's equation, passing through $(1,6)$ and $(2,1)$ .

See Solution

Problem 30689

Find $y$ given $\frac{d y}{d x}=3-\sqrt{x}$ and $y=8$ at $x=4$ .

See Solution

Problem 30690

Find the volume of the solid formed by rotating the triangle with vertices $(0,0),(6,6)$ , and $(9,3)$ around $y=x$ .

See Solution

Problem 30691

Find the derivative of the curve $y=x^{3}-4 x^{2}+5 x+4$ and the $x$ values where the gradient is 1.

See Solution

Problem 30692

Bestimme das unbestimmte Integral von $f(x)=x^{\frac{2}{3}}-x$ .

See Solution

Problem 30693

How many years does it take for an investment to double at a continuous rate of $3\%$ ? Round to one decimal place.

See Solution

Problem 30694

A particle $P$ moves along a line with displacement $s=t+\frac{36}{t}+4$ for $t>1$ .
(a) Find $v$ by differentiating $s$ .
(b) Determine $t$ when $v=0$ .
(c) Calculate $a$ when $t=2$ .

See Solution

Problem 30695

Find the limit: $\lim _{x \rightarrow 3}(6-x)$ . If it doesn't exist, write DNE.

See Solution

Problem 30696

Find the limit: $\lim_{x \rightarrow 1} f(x)$ for $f(x)=\begin{cases} x^{2}-2 & \text{if } x \neq 1 \\ -3 & \text{if } x=1 \end{cases}$ .

See Solution

Problem 30697

A particle $P$ moves along a line with displacement $s=t+\frac{36}{t}+4$ for $t>1$ .
(a) Find the velocity $v$ by differentiating $s$ .
(b) Determine when $v=0$ .
(c) Calculate acceleration $a$ when $t=2$ .

See Solution

Problem 30698

Find the limit as $x$ approaches -3 for the expression $\frac{|x+3|}{x+3}$ .

See Solution

Problem 30699

Find the limit: $\lim _{x \rightarrow 7} f(x)$ for the piecewise function $f(x)=\left\{\begin{array}{ll} 9-x, & x \neq 7 \\ 0, & x=7 \end{array}\right.$ .

See Solution

Problem 30700

Bestimmen Sie die Stammfunktion $F(x)$ mit $F(1)=5$ für $f(x)=2 x^{3}-x^{-2}$ .

See Solution

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Calculus

Problem 30601

Find f(5)f(5)f(5) if f′′(x)=2x+8sin⁡(x)f^{\prime \prime}(x)=2 x+8 \sin (x)f′′(x)=2x+8sin(x), f(0)=2f(0)=2f(0)=2, and f′(0)=4f^{\prime}(0)=4f′(0)=4.

Problem 30602

Differentiate the function s(t)=tan⁡(t)s(t)=\tan (\sqrt{t})s(t)=tan(t​).

Problem 30603

Find the slope of the tangent line to y=3−x2xy=\frac{3-x}{2 x}y=2x3−x​ at x=1x=1x=1. Choices: a) m=32m=\frac{3}{2}m=23​ b) m=−32m=-\frac{3}{2}m=−23​ c) m=3m=3m=3 d) m=−3m=-3m=−3

Problem 30604

Find the function f(t)f(t)f(t) given that f′′(t)=2et+3sin⁡(t)f''(t)=2 e^{t}+3 \sin (t)f′′(t)=2et+3sin(t), with conditions f(0)=−2f(0)=-2f(0)=−2 and f(π)=6f(\pi)=6f(π)=6.

Problem 30605

Find the antiderivative F(t)F(t)F(t) of f(t)=8sec⁡2(t)−10t3f(t)=8 \sec ^{2}(t)-10 t^{3}f(t)=8sec2(t)−10t3 with F(0)=0F(0)=0F(0)=0. Calculate F(1.3)F(1.3)F(1.3).

Problem 30606

Problem 30607

Find the derivative of the function y=cos⁡(x3)y=\cos \left(x^{3}\right)y=cos(x3).

Problem 30608

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=x32x+1y=x^{3} \sqrt{2 x+1}y=x32x+1​.

Problem 30609

Differentiate the function: y=sec⁡2(x)+tan⁡2(x)y=\sec ^{2}(x)+\tan ^{2}(x)y=sec2(x)+tan2(x)

Problem 30610

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=(2x2+5)7f(x)=(2x^{2}+5)^{7}f(x)=(2x2+5)7.

Problem 30611

Find the derivative of the function y=x2+x3y=\sqrt[3]{x^{2}+x}y=3x2+x​.

Problem 30612

Given f′′(x)=−6cos⁡(x)f^{\prime \prime}(x)=-6 \cos (x)f′′(x)=−6cos(x), find f′(x)f^{\prime}(x)f′(x) and f(x)f(x)f(x) using constants C\mathrm{C}C and D\mathrm{D}D.

Problem 30613

Find the derivative of y=ex2+3xy = e^{x^2 + 3x}y=ex2+3x.

Problem 30614

Find the derivative of the function f(x)=(3sin⁡x+6cos⁡x)tan⁡−1xf(x)=(3 \sin x+6 \cos x) \tan ^{-1} xf(x)=(3sinx+6cosx)tan−1x. What is f′(x)f^{\prime}(x)f′(x)?

Problem 30615

Bestimmen Sie die 1., 2. und 3. Ableitung für die Funktionen a) bis f) und berechnen Sie die Steigung von fff an A(2∣f(2))A(2 \mid f(2))A(2∣f(2)).

Problem 30616

Problem 30617

Bestimme die Steigung von f(z)=34z2−3z−1f(z) = \frac{3}{4} z^{2} - 3 z^{-1}f(z)=43​z2−3z−1 am Punkt A(2∣f(2))A(2|f(2))A(2∣f(2)).

Problem 30618

Find the value of the series: ∑n=1+∞(3n−1)\sum_{n=1}^{+\infty}\left(\sqrt[n]{3}-1\right)n=1∑+∞​(n3​−1).

Problem 30619

Graph the yield function V=6.7e−48.1/tV=6.7 e^{-48.1/t}V=6.7e−48.1/t, find its horizontal asymptote, and determine time for V=1.3V=1.3V=1.3.

Problem 30620

Problem 30621

Berechnen Sie die Steigung von fff bei A(2∣f(2))A(2 \mid f(2))A(2∣f(2)) für: a) f(x)=32x2f(x)=\frac{3}{2} x^{2}f(x)=23​x2, b) f(t)=14t4−53t3f(t)=\frac{1}{4} t^{4}-\frac{5}{3} t^{3}f(t)=41​t4−35​t3, c) f(z)=34z2−3z−1f(z)=\frac{3}{4} z^{2}-3 z^{-1}f(z)=43​z2−3z−1.

Problem 30622

Begründen Sie, warum die e-Funktion f(x)=exf(x) = \mathrm{e}^{x}f(x)=ex keine Nullstellen, Symmetrie, Extrem- oder Wendepunkte hat.

Problem 30623

Togets avstand fra stasjonen er gitt ved s(t)=(0,800 m/s2)t2+(21,0 m/s)t+(150 m)s(t)=\left(0,800 \mathrm{~m} / \mathrm{s}^{2}\right) t^{2}+(21,0 \mathrm{~m} / \mathrm{s}) t+(150 \mathrm{~m})s(t)=(0,800 m/s2)t2+(21,0 m/s)t+(150 m). Finn koeffisientene, tiden for 35,0 m/s, og avstanden da.

Problem 30624

Bestimmen Sie die Ableitungsfunktion von f(x)=ax+cf(x) = ax + cf(x)=ax+c.

Problem 30625

Evaluate the series ∑n=1+∞(nn−1)α\sum_{n=1}^{+\infty}(\sqrt[n]{n}-1)^{\alpha}∑n=1+∞​(nn​−1)α.

Problem 30626

Calculate the integral: ∫(x+1)dxx2(x−1)\int \frac{(x+1) d x}{x^{2}(x-1)}∫x2(x−1)(x+1)dx​

Problem 30627

Calculate the integral: ∫2xdx(x−1)(x2+1)\int \frac{2 x d x}{(x-1)(x^{2}+1)}∫(x−1)(x2+1)2xdx​.

Problem 30628

Bestimmen Sie die Ableitung und die Steigungen bei x1=1x_{1}=1x1​=1 und x2=0x_{2}=0x2​=0 für f(x)=−4ex+2xf(x)=-4 e^{x}+2 xf(x)=−4ex+2x und f(x)=2xf(x)=2^{x}f(x)=2x.

Problem 30629

Evaluate the integral: ∫(x+1)dxx3+x2−6x\int \frac{(x+1) d x}{x^{3}+x^{2}-6 x}∫x3+x2−6x(x+1)dx​.

Problem 30630

Bestimmen Sie die Ableitung und die Steigungen an x1=1x_{1}=1x1​=1 und x2=0x_{2}=0x2​=0 für: a) f(x)=2exf(x)=2 e^{x}f(x)=2ex, b) f(x)=−4ex+2xf(x)=-4 e^{x}+2 xf(x)=−4ex+2x, c) f(x)=2xf(x)=2^{x}f(x)=2x.

Problem 30631

Evaluate the integral: ∫xcos⁡2x dx\int x \cos^{2} x \, dx∫xcos2xdx

Problem 30632

Find the integral of x2ln⁡2x dxx^{2} \ln^{2} x \, dxx2ln2xdx.

Problem 30633

Given the function f(x)=2x+1x−3f(x)=\frac{2x+1}{x-3}f(x)=x−32x+1​, find f′(x)f'(x)f′(x), f′(4)f'(4)f′(4), the tangent line at x=4x=4x=4, and tangents from (6,−5)(6,-5)(6,−5).

Problem 30634

Calculate the integral: ∫dxx2+7x+6\int \frac{d x}{x^{2}+7 x+6}∫x2+7x+6dx​

Problem 30635

Find the derivative of the function f(x)=x2+εx+2f(x) = x^{2} + \varepsilon x + 2f(x)=x2+εx+2.

Problem 30636

Bestimme die Fläche A zwischen den Schnittpunkten der Funktionen f(x)=1−x2f(x)=1-x^{2}f(x)=1−x2 und g(x)=x2−2x+1g(x)=x^{2}-2x+1g(x)=x2−2x+1.

Problem 30637

Evaluate the integral ∫π/3π/6cos⁡x−xsin⁡xxcos⁡xdx\int_{\pi / 3}^{\pi / 6} \frac{\cos x-x \sin x}{x \cos x} d x∫π/3π/6​xcosxcosx−xsinx​dx and choose the correct answer.

Problem 30638

Find the derivative of the function f(x)=x2+4x+1f(x) = x^{2} + 4x + 1f(x)=x2+4x+1.

Problem 30639

Find df−1dx\frac{d f^{-1}}{d x}dxdf−1​ for f(x)=x2−4x+3f(x)=x^{2}-4 x+3f(x)=x2−4x+3 at x=3x=3x=3, where x≥2x \geq 2x≥2. Choices: 1/41/41/4, 1/21/21/2, 1/31/31/3, 4.

Problem 30640

Bestimme die Funktion f(x)=a⋅ek⋅xf(x)=a \cdot e^{k \cdot x}f(x)=a⋅ek⋅x für das Wachstum einer Sonnenblume und die momentane Geschwindigkeit bei x=20x=20x=20.

Problem 30641

Haziq's ice cylinder melts at 4 cm2 s−14 \mathrm{~cm}^{2} \mathrm{~s}^{-1}4 cm2 s−1. Find the surface area change rate when radius is 8 cm8 \mathrm{~cm}8 cm.

Find $f(5)$ if $f^{\prime \prime}(x)=2 x+8 \sin (x)$ , $f(0)=2$ , and $f^{\prime}(0)=4$ .

Differentiate the function $s(t)=\tan (\sqrt{t})$ .

Find the slope of the tangent line to $y=\frac{3-x}{2 x}$ at $x=1$ . Choices: a) $m=\frac{3}{2}$ b) $m=-\frac{3}{2}$ c) $m=3$ d) $m=-3$

Find the function $f(t)$ given that $f''(t)=2 e^{t}+3 \sin (t)$ , with conditions $f(0)=-2$ and $f(\pi)=6$ .

Find the antiderivative $F(t)$ of $f(t)=8 \sec ^{2}(t)-10 t^{3}$ with $F(0)=0$ . Calculate $F(1.3)$ .

Find the derivative of the function $y=\cos \left(x^{3}\right)$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{3} \sqrt{2 x+1}$ .

Differentiate the function: $y=\sec ^{2}(x)+\tan ^{2}(x)$

Find the derivative $f^{\prime}(x)$ for the function $f(x)=(2x^{2}+5)^{7}$ .

Find the derivative of the function $y=\sqrt[3]{x^{2}+x}$ .

Given $f^{\prime \prime}(x)=-6 \cos (x)$ , find $f^{\prime}(x)$ and $f(x)$ using constants $\mathrm{C}$ and $\mathrm{D}$ .

Find the derivative of $y = e^{x^2 + 3x}$ .

Find the derivative of the function $f(x)=(3 \sin x+6 \cos x) \tan ^{-1} x$ . What is $f^{\prime}(x)$ ?

Bestimmen Sie die 1., 2. und 3. Ableitung für die Funktionen a) bis f) und berechnen Sie die Steigung von $f$ an $A(2 \mid f(2))$ .

Bestimme die Steigung von $f(z) = \frac{3}{4} z^{2} - 3 z^{-1}$ am Punkt $A(2|f(2))$ .

Find the value of the series: $\sum_{n=1}^{+\infty}\left(\sqrt[n]{3}-1\right)$ .

Graph the yield function $V=6.7 e^{-48.1/t}$ , find its horizontal asymptote, and determine time for $V=1.3$ .

Berechnen Sie die Steigung von $f$ bei $A(2 \mid f(2))$ für: a) $f(x)=\frac{3}{2} x^{2}$ , b) $f(t)=\frac{1}{4} t^{4}-\frac{5}{3} t^{3}$ , c) $f(z)=\frac{3}{4} z^{2}-3 z^{-1}$ .

Begründen Sie, warum die e-Funktion $f(x) = \mathrm{e}^{x}$ keine Nullstellen, Symmetrie, Extrem- oder Wendepunkte hat.

Togets avstand fra stasjonen er gitt ved $s(t)=\left(0,800 \mathrm{~m} / \mathrm{s}^{2}\right) t^{2}+(21,0 \mathrm{~m} / \mathrm{s}) t+(150 \mathrm{~m})$ . Finn koeffisientene, tiden for 35,0 m/s, og avstanden da.

Bestimmen Sie die Ableitungsfunktion von $f(x) = ax + c$ .

Evaluate the series $\sum_{n=1}^{+\infty}(\sqrt[n]{n}-1)^{\alpha}$ .

Calculate the integral: $\int \frac{(x+1) d x}{x^{2}(x-1)}$

Calculate the integral: $\int \frac{2 x d x}{(x-1)(x^{2}+1)}$ .

Bestimmen Sie die Ableitung und die Steigungen bei $x_{1}=1$ und $x_{2}=0$ für $f(x)=-4 e^{x}+2 x$ und $f(x)=2^{x}$ .

Evaluate the integral: $\int \frac{(x+1) d x}{x^{3}+x^{2}-6 x}$ .

Bestimmen Sie die Ableitung und die Steigungen an $x_{1}=1$ und $x_{2}=0$ für: a) $f(x)=2 e^{x}$ , b) $f(x)=-4 e^{x}+2 x$ , c) $f(x)=2^{x}$ .

Evaluate the integral: $\int x \cos^{2} x \, dx$

Find the integral of $x^{2} \ln^{2} x \, dx$ .

Given the function $f(x)=\frac{2x+1}{x-3}$ , find $f'(x)$ , $f'(4)$ , the tangent line at $x=4$ , and tangents from $(6,-5)$ .

Calculate the integral: $\int \frac{d x}{x^{2}+7 x+6}$

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

Bestimme die Fläche A zwischen den Schnittpunkten der Funktionen $f(x)=1-x^{2}$ und $g(x)=x^{2}-2x+1$ .

Evaluate the integral $\int_{\pi / 3}^{\pi / 6} \frac{\cos x-x \sin x}{x \cos x} d x$ and choose the correct answer.

Find the derivative of the function $f(x) = x^{2} + 4x + 1$ .

Find $\frac{d f^{-1}}{d x}$ for $f(x)=x^{2}-4 x+3$ at $x=3$ , where $x \geq 2$ . Choices: $1/4$ , $1/2$ , $1/3$ , 4.

Bestimme die Funktion $f(x)=a \cdot e^{k \cdot x}$ für das Wachstum einer Sonnenblume und die momentane Geschwindigkeit bei $x=20$ .

Haziq's ice cylinder melts at $4 \mathrm{~cm}^{2} \mathrm{~s}^{-1}$ . Find the surface area change rate when radius is $8 \mathrm{~cm}$ .

Berechne den Flächeninhalt $A$ zwischen $f(x)=\frac{1}{2} x^{2}+1$ und $g(x)=-x^{2}+\frac{3}{2} x+4$ von $x=-1$ bis $x=2$ .

What height is needed for an $823 \mathrm{~g}$ falcon to reach $68 \mathrm{~m/s}$ in a dive under gravity?

Berechne die Fläche zwischen den Funktionen $f(x)=-\frac{1}{64} x^{4}+\frac{1}{8} x^{2}+4$ und $g(x)=\frac{1}{8} x^{2}+1$ von $-4$ bis $4$ .

Ein Patient nimmt um 20 Uhr $2 \mathrm{mg}$ eines Medikaments ein. Berechne die verbleibende Menge nach $1, 2, 3$ Stunden und finde die Funktion zur Basis $e$ .

Calculate the integral: $\int_{-4}^{4}\left(-\frac{1}{64} x^{4}-3\right) d x$

Найдите производную функции $y = \frac{\cos 2x}{\ln(x^{2} + 4x)}$ .

If $f'(x)=2x^3$ , find $\frac{d f^{-1}}{d x}$ at $x=f(2)$ . Options: a. 4, b. 16, c. $\frac{1}{24}$ , d. $\frac{1}{8}$ , e. none.

Rewrite the integral using $u$ and $du$ , then evaluate: $\int (u)^{7} du, \quad u=2x-1$

$x^{\frac{2}{3}}+y^{\frac{2}{3}}=4$ için $y^{\prime}$ türevi $x=8$ 'de nedir? A) 0 B) 1 C) 2 D) 4 E) 8

Rewrite the integral using $u$ and $du$ , then solve: $\int t \sqrt{6 t^{2}+1} dt$

Find the limit as $x$ approaches $\frac{3}{2}$ from the right for $\frac{x^{2}+8x-20}{2x^{2}+x-6}$ .

Rewrite the integral as $u$ and $du$ , then evaluate $\int \sin(u) \, du$ , where $u=9\theta-5$ .

Evaluate the integral from 0 to $\frac{\pi}{6}$ of $\tan(2x)$ . What is the result?

Rewrite the integral using $u$ and $du$ , then evaluate: $\int \frac{(\ln (x))^{6}}{x} dx, \quad u=\ln (x)$

Determine where the function $x^{4}-4 x^{2}+3$ is increasing and decreasing.

Evaluate the integral $\int_{0}^{\pi / 3} \frac{4 \sin \theta}{1-4 \cos \theta} d \theta$ .

Find when the particle changes direction for the position $s(t) = -2t^3 + 6t^2 - 3$ , where $t \geq 0$ .

Find the derivative of $y$ with respect to $x$ , $t$ , or $\theta$ for $y=\ln (\sec \theta+\tan \theta)$ .

Find the first and second derivatives of these functions: a. $f(x)=x^{2}+1$ (first principles), b. $f(x)=\cos(x)e^{x}$ , c. $f(x)=(x^{2}-3x+2)^{3}\sin(2x^{2})$ (first derivative only).

Approximate Bolt's velocity at $t=2$ seconds using the position equation $s(t)=0.567 t^{4}-10.2 t^{3}+57.2 t^{2}-60.4 t$ .

Evaluate the limits: a) $\frac{x^{2}-4 x+3}{x-3}$ b) $\frac{x^{2}-4 x+3}{x-3}$

Calcula el límite de $\frac{x^{2}-4 x+3}{x-3}$ cuando $x$ tiende a 3.

Differentiate the function $f(x) = x \sqrt{x^{2}+1}$ .

Find the first and second derivatives of these functions: a) $f(x)=x^{2}+1$ , b) $f(x)=\cos(x)e^{x}$ , c) $f(x)=2\sin(x^{2})$ .

Find the slope of the tangent to $y=x^{2}+9 x$ where it intersects the line $y=3 x$ .

Find the first derivative using first principles for: a) $f(x)=7 x-2$ b) $f(x)=3 x^{2}+7 x+14$

Find the derivative of the function $e^{3x}$ .

Differentiate the function: $f(x) = (1-3x) \sqrt{2x+5}$ .

The position of an object is $s(t)=3 t^{3}-40.5 t^{2}+162 t$ . Find velocity, acceleration, and when it's stationary, advancing, or retreating.

Find velocity and acceleration of $s(t)=3 t^{3}-40.5 t^{2}+162 t$ for $0 \leq t \leq 8$ . When is it stationary, advancing, or retreating? When is velocity constant, decreasing, or increasing?

Find the derivative of $y = \frac{3}{e^{x^{4}}}$ .

Find the velocity and acceleration of the object described by $s(t)=3 t^{3}-40.5 t^{2}+162 t$ for $0 \leq t \leq 8$ .