Calculus

Problem 17001

Find the derivative of $y=\left(x^{2}+3\right)^{2}\left(x^{3}-2 x\right)^{3}$ .

See Solution

Problem 17002

Find the derivative $y^{\prime}$ for $y=(x^{2}+3)^{2}(x^{3}-2x)^{3}$ .

See Solution

Problem 17003

Find the tangent line equation to the curve $y=2 e^{x} \cos (x)$ at the point $(0,2)$ .

See Solution

Problem 17004

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

See Solution

Problem 17005

Find the derivative of $\frac{h(x)}{x}$ at $x=2$ given $h(2)=5$ and $h^{\prime}(2)=-2$ .

See Solution

Problem 17006

Does the Intermediate Value Theorem ensure a real zero for $G(x)=2 x^{4}-6 x^{3}+106 x^{2}-46 x+210$ between 1 and 2?

See Solution

Problem 17007

Sketch the function $y=2 x^{3}-3 x^{2}-x-2$ and label key points. Find: $y$ -intercept, $x$ -intercepts, critical points, inflection points, and intervals of increase/decrease.

See Solution

Problem 17008

Find the derivative $S^{\prime}(100)$ of the function $S(A)=0.882 A^{0.842}$ and interpret the result.

See Solution

Problem 17009

Find the derivative of $y=\sqrt{23 \arctan (x)}$ and simplify if possible.

See Solution

Problem 17010

A kite is $100 \mathrm{ft}$ high and moves at $14 \mathrm{ft/s}$ . Find the rate (in rad/s) at which the angle decreases when $200 \mathrm{ft}$ of string is out.

See Solution

Problem 17011

Find the derivative of the function using logarithmic differentiation: $y=\sqrt{\frac{x-3}{x^{8}+5}}$ .

See Solution

Problem 17012

Compute the integral of $4-x^{2}$ from -2 to 2 using the Riemann sum limit:
$\int_{-2}^{2} 4-x^{2} dx = \lim_{n \rightarrow \infty} \sum_{k=1}^{n}\left(4-\left(\frac{4k}{n}-2\right)^{2}\right)\left(\frac{4}{n}\right)$

See Solution

Problem 17013

Find the tangent line equation for $y=x^{13 x}$ at $x=e$ .

See Solution

Problem 17014

Find the tangent line equation for $y=(x+3)(x+4)^{2}(x+5)^{2}$ at $x=0$ .

See Solution

Problem 17015

Simplify the series $\sum_{i=1}^{n} \frac{4}{n} \cdot\left(4-\left(-2+\frac{i 4}{n}\right)^{2}\right)$ .

See Solution

Problem 17016

Find an antiderivative $F(x)$ of the function $f(x)=\frac{-4}{\sqrt{1-x^{2}}}$ .

See Solution

Problem 17017

Find the derivative of $y$ for $y=(2 x+3)^{x}$ . What is $\frac{dy}{dx}$ ?

See Solution

Problem 17018

Find the function whose derivative is $f^{\prime}(x)=e^{5 x}$ and passes through the point $P=\left(0, \frac{6}{5}\right)$ .

See Solution

Problem 17019

Find the time for \$ 2125 to double at an 8.5% continuous compound interest rate. Round to two decimal places.

See Solution

Problem 17020

Find $y^{\prime}$ using logarithmic differentiation for $y=\sqrt[3]{\frac{6\left(x^{3}+1\right)^{2}}{x^{6} e^{-5 x}}}$ .

See Solution

Problem 17021

Find $y^{\prime}$ using logarithmic differentiation for $y=\sqrt[3]{\frac{6(x^{3}+1)^{2}}{x^{6} e^{-5 x}}}$ .

See Solution

Problem 17022

Differentiate the function using logarithmic differentiation: $f(x)=\frac{(x+1)(6 x+1)(8 x+1)}{\sqrt{4 x+1}}$ .

See Solution

Problem 17023

Use Newton's method to find the root of $x^{3}+x^{2}+10=0$ accurate to five decimal places. Sketch a graph. Root is $x \approx$

See Solution

Problem 17024

Differentiate implicitly: $\ln(3xy) = e^{3xy}$ , find $\frac{dy}{dx}$ , with $y \neq 0$ .

See Solution

Problem 17025

Two soap bubbles contain 6 liters of air. Let $V$ be the left bubble's volume. Show that the surface area is $S=(36 \pi)^{\frac{1}{3}}\left(V^{\frac{2}{3}}+(6-V)^{\frac{2}{3}}\right)$ . Find extreme values of $S$ and discuss stability.

See Solution

Problem 17026

Find the antiderivative of $f(x) = -4x$ .

See Solution

Problem 17027

Calculate the displacement and distance traveled by an object with velocity $\frac{d x}{d t}=4+3 \cos (0.5 t)$ from $0$ to $2\pi$ .

See Solution

Problem 17028

Differentiate implicitly: $(1+e^{4 x})^{2}=4+\ln (x+y)$ , find $\frac{d y}{d x}$ .

See Solution

Problem 17029

Find the half-life of a substance with decay function $y=y_{0} e^{-0.048 t}$ . Round to the nearest tenth.

See Solution

Problem 17030

Find the original functions for Cameron's derivatives:
a. $2.54 + \frac{dy}{dx}$ b. $\frac{dy}{dx} \sec^2 y$ c. $\sqrt{5y} \left(\frac{dy}{dx}\right)$ d. $\frac{7}{y^2} \left(\frac{dy}{dx}\right)$ e. $(y - \cos x) \left(\frac{dy}{dx} + \sin x\right)$

See Solution

Problem 17031

Kelly thinks $\ln \left(x^{4}\right)$ is the antiderivative of $\frac{1}{x^{4}}$ . Explain why this is incorrect.

See Solution

Problem 17032

Find the tangent line equation for $g(x)=(3x^{2}-4)f(x)$ at $x=-2$ given $f(-2)=-2$ and $f'(-2)=-3$ .

See Solution

Problem 17033

Find the function $g(x)$ such that $g^{\prime}(x)=-512-x^{3}$ and the maximum value of $g(x)$ is 3.

See Solution

Problem 17034

Calculate the area between the curves $y=x^{1/2}$ and $y=x^{1/4}$ from $x=0$ to $x=1$ . Find area =

See Solution

Problem 17035

Find the tangent line to $f(x)=\frac{1}{3} x^{3}+\frac{3 x^{2}}{2}-10 x$ at $x=-1$ .

See Solution

Problem 17036

Aufgabe 1: Finde eine Stammfunktion für $f(x)=6 x^{7}-\frac{7}{x^{8}}$ .
Aufgabe 2: Bestimme $F$ mit $F(-1)=1$ für $f(x)=2 x^{3}-3 x^{2}$ .
Aufgabe 3: Berechne $\int_{1}^{3}(x^{2}-2 x) dx$ und interpretiere das Ergebnis.
Aufgabe 4: Finde $a$ , sodass $\int_{a}^{\infty} \frac{100}{x^{2}} dx=20$ .

See Solution

Problem 17037

Aufgabe 1: Finde die Stammfunktion von $f(x)=6 x^{7}-\frac{7}{x^{8}}$ .
Aufgabe 2: Bestimme $F$ mit $F(-1)=1$ für $f(x)=2 x^{3}-3 x^{2}$ .
Aufgabe 3: Berechne $\int_{1}^{3}(x^{2}-2 x) dx$ und interpretiere das Ergebnis.
Aufgabe 4: Finde $a$ für $\int_{a}^{\infty} \frac{100}{x^{2}} dx=20$ .
Aufgabe 5: a) Schätze $f(4)$ . b) Berechne $\int_{0}^{2} f(x) dx$ und $\int_{0}^{2} F(x) dx$ . c) Zeige, dass $f'$ eine Nullstelle in $[-2; 2]$ hat. d) Zeige, dass $f'$ in $[1; 3]$ negativ ist. e) Skizziere den Graph von $f$ .

See Solution

Problem 17038

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

See Solution

Problem 17039

Berechnen Sie die Integrale: a) $\int_{0}^{2}(2+x)^{3} dx$ b) $\int_{2}^{3}\left(1+\frac{1}{x^{2}}\right) dx$

See Solution

Problem 17040

Evaluate the integral of $e^{u}$ with respect to $u$ .

See Solution

Problem 17041

Calculate the integral $\int_{1}^{2}(x y^{2}+x+2) \, dx$ .

See Solution

Problem 17042

Bestimme die Punkte, an denen der Graph von $f$ die Steigung $m$ hat, und gib die Tangentengleichungen an.

See Solution

Problem 17043

Bestimme die Punkte, an denen der Graph von $f$ die Steigung $m$ hat, und nenne die Tangentengleichungen für: 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 17044

Finde die Stammfunktionen für die folgenden Funktionen: a) $f(x)=x^{2}$ , b) $f(x)=x^{3}$ , c) $f(x)=3x$ , f) $f(x)=x^{4}$ , g) $f(x)=0,1x^{3}$ , h) $f(x)=x$ .

See Solution

Problem 17045

Solve the integral $\int_{k}^{k+1}(k^{2}+2t)dt=9$ for the variable $k$ .

See Solution

Problem 17046

Find the value of $k$ such that $\int_{1}^{2} k x^{3} dx = 75$ .

See Solution

Problem 17047

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 17048

Finde eine Stammfunktion für die folgenden Funktionen: a) $f(x)=x^{2}$ , b) $f(x)=x^{3}$ , c) $f(x)=3 x$ , d) $f(x)=x^{5}$ , e) $f(x)=5 x^{2}$ , f) $f(x)=x^{4}$ , g) $f(x)=0,1 x^{3}$ , h) $f(x)=x$ , i) $f(x)=2$ , j) $f(x)=2 x^{5}$ .

See Solution

Problem 17049

Berechne die verbleibende Menge des Medikaments nach 6 Stunden, wenn die Anfangsmenge $400 \mathrm{mg}$ und die Halbwertszeit 2 Stunden beträgt.

See Solution

Problem 17050

Berechne die Integrale $I=\int_{-2}^{0} f(x) d x$ und $I=\int_{-2}^{3} f(x) d x$ . Finde dann die Stammfunktion $F$ für $f(x)=-5 \cdot x^{4}+8$ mit $F(1)=10$ .

See Solution

Problem 17051

Berechne die Integrale $I=\int_{-2}^{0} f(x) dx$ und $I=\int_{-2}^{3} f(x) dx$ . Finde die Stammfunktion $F$ von $f(x)=-5 \cdot x^{4}+8$ mit $F(1)=10$ .

See Solution

Problem 17052

Gegeben ist die Funktion $f(x)=e^{-2 x} \cdot\left(x^{2}-2\right)$ .
a) Bestimme den Definitionsbereich und das Grenzverhalten.
b) Überprüfe auf Symmetrien.
c) Zeige, dass $f^{\prime}(x)=e^{-2 x} \cdot\left(-2 x^{2}+2 x+4\right)$ die Ableitung ist.
d) Berechne Nullstellen und Extrempunkte.
e) Finde die Tangentengleichung im Wendepunkt W(-0,58/-5,31).

See Solution

Problem 17053

Gegeben ist ein Wasserbecken mit Zuflussgeschwindigkeit $z$ .
a) (1) Beschreiben Sie die Wassermenge in $I_{1}=[1 ; 6]$ und $I_{2}=[6 ; 9,5]$ . (2) Zeigen Sie, dass die Wassermenge in $I_{3}=[0 ; 9,5]$ um $7 m^{3}$ zugenommen hat.
b) Bestimmen Sie $z$ in $[9,5 ; 15]$ für die Funktion $h$ .
c) (1) Erklären Sie, warum der Füllstand bei $t=11$ maximal ist. (2) Finden Sie die Anfangswassermenge bei $t=0$ . (3) Berechnen Sie die Veränderung der Wassermenge von $t=12,5$ bis $t=14,2$ mit dem Integral von $h$ .

See Solution

Problem 17054

Ein Elektronikunternehmen verkauft Geräte. Die Verkaufsfunktion ist $f(x)=-\frac{1}{2700} \cdot x^{3}+\frac{8}{45} \cdot x^{2}+\frac{20}{3} \cdot x+1000$ für $0 \leq x \leq 360$ .
a) (1) Wie viele Geräte wurden am 40. Tag $(x=40)$ verkauft? (2) Schätzen Sie die Gesamtzahl der verkauften Geräte im Intervall $I_{1}=[0; 40]$ .
b) (1) Warum beschreibt die Funktion $F(x)=-\frac{1}{10800} \cdot x^{4}+\frac{8}{135} \cdot x^{3}+\frac{20}{6} \cdot x^{2}+1000 \cdot x$ die Gesamtzahl der verkauften Geräte bis $x$ ? (2) Berechnen Sie die Anzahl der verkauften Geräte im Intervall $I_{2}=[240,340]$ .

See Solution

Problem 17055

Berechne den Flächeninhalt zwischen dem Graphen $f(x) = x^{2}-5$ und der $x$ -Achse von $x = -4$ bis $x = 4$ .

See Solution

Problem 17056

Berechnen Sie die Integrale: (1) $I=\int_{-2}^{0} f(x) d x$ , (2) $I=\int_{-2}^{3} f(x) d x$ mit $A1=0,3$ , $A2=0,8$ , $A3=2,9$ , $A4=1,1$ .

See Solution

Problem 17057

Berechnen Sie die Schnittpunkte von $G_{a}$ mit den Achsen und das Verhalten von $f_{a}$ für $x \rightarrow+\infty$ .

See Solution

Problem 17058

Bestimme den Wendepunkt und die Wendetangente der Funktion $f(x)=x^{3}+3 x^{2}-4$ . Finde die Länge der Strecke $\overline{\mathrm{PQ}}$ .

See Solution

Problem 17059

Bestimme den Wendepunkt und die Wendetangente der Funktion $f(x)=x^{3}+3 x^{2}-4$ . Berechne die Länge von $\overline{\mathrm{PQ}}$ .

See Solution

Problem 17060

Untersuchen Sie den Extrempunkt und das Krümmungsverhalten des Graphen $G_a$ von $f_a(x) = e^x(x-a)$ . Bestimmen Sie den Wendepunkt.

See Solution

Problem 17061

Berechne die Beschleunigung $a(t)$ mit $a(t)=\frac{v2 - v1}{t2 - t1}$ für die Punkte (0.2s, 1.9591 m/s) und (0.2s, 3.901654 m/s).

See Solution

Problem 17062

Find $s^{\prime}(t)$ for $s(t)=3 t^{\frac{1}{3}}+2 t^{2}$ . Choices: $t^{\frac{2}{3}}+4 t$ , $4 t-\frac{1}{t^{\frac{2}{3}}}$ , $4 t+\frac{1}{t^{\frac{2}{3}}}$ , $4 t+\frac{3}{t^{\frac{2}{3}}}$ .

See Solution

Problem 17063

Find $f^{\prime}(-1)$ for $f(x)=\frac{1-x}{x^{2}}$ . Options: $-1$ , $0$ , $3$ , $1$ .

See Solution

Problem 17064

Bestimmen Sie die Ableitungen $f^{\prime}$ für die Funktionen und berechnen Sie $f^{\prime}(2)$ für jede: a) $f(x)=2 x^{2}$ , b) $f(x)=4 x^{2}$ , c) $f(x)=-3 x^{2}$ , d) $f(x)=\frac{1}{2} x^{2}$ .

See Solution

Problem 17065

Berechnen Sie die Integrale: a) $\int_{0}^{4} x^{2} d x$ , b) $\int_{2}^{4} x^{2} d x$ , c) $\int_{-1}^{5} 2 x d x$ , d) $\int_{10}^{11} 0,5 x d x$ , e) $\int_{10}^{20} 5 d x$ .

See Solution

Problem 17066

Bestimme die momentane Änderungsrate der Wirkstoffkonzentration $f(t)=0,25 t^{3}-3 t^{2}+9 t$ bei $t=4$ h.

See Solution

Problem 17067

Untersuchen Sie die Funktion $f(t)=0,25 t^{3}-3 t^{2}+9 t$ für $0 \leq t \leq 6$ :
a) Wann steigt oder fällt die Konzentration? b) Bestimmen Sie das Intervall, in dem $f(t) \geq 3,7$ . c) Finden Sie das Hochpunkt-Koordinaten und zeigen Sie, dass $f(6)$ ein Minimum ist. d) Berechnen Sie die momentane Änderungsrate bei $t=4$ .

See Solution

Problem 17068

Calculate the integral of $f(x)=\frac{x^{2}+2 x}{x^{5}}$ .

See Solution

Problem 17069

Bestimmen Sie die Funktion $f$ mit Basis e und die ersten beiden Ableitungen für $f(x)=4^{x}$ , $f(x)=2 \cdot 6^{x}$ , $f(x)=2^{x+2}$ , $f(x)=2,5^{x} \cdot 3^{x}$ .

See Solution

Problem 17070

Evaluate the line integral $\int_{C} \boldsymbol{F} \cdot \boldsymbol{d r}$ for $\boldsymbol{F}=\left\langle x^{2} y, x y^{2}\right\rangle$ from $(1,6)$ to $(6,1)$ .

See Solution

Problem 17071

Calculate the integral $\int_{2}^{4} \frac{-1}{x^{3}} d x$ .

See Solution

Problem 17072

Calculate the integral from 2 to 4 of the function $-\frac{1}{x^{3}}$ .

See Solution

Problem 17073

Find the Jacobian for the transformation: $x=u^{2}+uv-v^{2}$ , $y=u^{2}-v+4w$ , $z=3u+w^{2}$ .

See Solution

Problem 17074

Evaluate the line integral $\int_{C} \boldsymbol{G} \bullet \boldsymbol{d r}$ from $(1,6)$ to $(6,1)$ for $\boldsymbol{G}=\left\langle x y^{2}-1, x^{2} y+2\right\rangle$ . Is $\boldsymbol{G}$ a gradient? Is $C$ parameterized?

See Solution

Problem 17075

Evaluate the integral: $\int \frac{3 e^{3x}+2 e^{2x}+e^{x}}{(e^{3x}+e^{2x}+e^{x}+1)^{\frac{1}{3}}} \, dx$

See Solution

Problem 17076

Evaluate the line integral $\int_{C} \boldsymbol{G} \bullet \boldsymbol{d r}$ for $\boldsymbol{G}=\left\langle x y^{2}-1, x^{2} y+2\right\rangle$ from $(1,6)$ to $(6,1)$ .

See Solution

Problem 17077

Find a scalar function $h(x, y, z)$ where $\nabla h=3 x^{2} z \mathrm{i}-2 y z^{4} \mathbf{j}+\left(x^{3}-4 y^{2} z^{3}\right) \mathbf{k}$ , and evaluate $\int_{C} \nabla h \bullet d \boldsymbol{r}$ for $C: \boldsymbol{r}(t)=(4 \cos t+\sin t) \mathrm{i}+(3 \cos 2 t-7 \sin t) \mathbf{j}+(2 \sin t-\cos t) \mathbf{k}$ , $0 \leq t \leq \frac{\pi}{2}$ .

See Solution

Problem 17078

Halla el valor de $c$ tal que el área de la región $R$ bajo $y=\frac{x}{x^{2}+2}$ y $x=c$ sea $\ln 3$ .

See Solution

Problem 17079

Finde die steilste Stelle der Rutsche beschrieben durch $f(x)=(x+3)(x-3)^{2}$ und bestimme den zugehörigen Winkel.

See Solution

Problem 17080

Calculate the indefinite integral: $\int x^{6} \, dx$ (use $C$ for the constant).

See Solution

Problem 17081

Calculate the indefinite integral and include the constant $C$ : $\int x^{3/4} \, dx$ .

See Solution

Problem 17082

Find the first derivatives of $f(x, y) = x^{2}y - 3y^{4}$ and $f(x, t) = \sqrt{3x + 4t}$ .

See Solution

Problem 17083

Calculate the indefinite integral and include the constant $C$ : $\int \frac{d w}{w^{6}}$ .

See Solution

Problem 17084

Find the indefinite integral: $\int \frac{d z}{\sqrt{z}}$ (Use $C$ for the constant of integration).

See Solution

Problem 17085

Calculate the indefinite integral: $\int 4 x^{3} dx$ and use $C$ for the constant of integration.

See Solution

Problem 17086

Bestimme den Wasserstand $f(x)=0,05 x^{3}-0,8 x^{2}+3 x$ eines Stausees nach 10 Tagen. Beantworte die Fragen A-E.

See Solution

Problem 17087

A mass $m$ on a table is connected by springs to points $A$ and $B$ . Derive the motion equation and find equilibrium, frequency, and displacement.

See Solution

Problem 17088

Calculate the indefinite integral of $(x-7)^{2}$ with respect to $x$ .

See Solution

Problem 17089

Untersuche das Verhalten von $f(x)$ für $x \rightarrow \infty$ für die Funktionen: a) $-x^{4}-x^{3}+2$ , b) $\frac{1}{3} x^{3}-x^{2}$ , c) $-x^{5}+4 x^{3}-x$ .

See Solution

Problem 17090

Calculate the integral: $\int \frac{18 x^{6}-18 x^{3}+x}{x} d x$ (Include constant $C$ )

See Solution

Problem 17091

Calculate the indefinite integral: $\int(1+20 w) \sqrt{w} \, d w$ (use $C$ for the constant).

See Solution

Problem 17092

Calculate the indefinite integral: $\int (x-6)(x+8) \, dx$ . Include the constant of integration C.

See Solution

Problem 17093

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

See Solution

Problem 17094

Identify which statements about series convergence/divergence are implied by the $n$ -th term test.

See Solution

Problem 17095

Calculate the indefinite integral: $\int \frac{x^{2}-9}{x+3} \, dx$ (include constant $C$ ).

See Solution

Problem 17096

Die Funktion $g(x)=0,15 x^{2}-1,6 x+3$ beschreibt den Wasserfluss eines Stausees. Beantworte die Fragen A-D mathematisch.

See Solution

Problem 17097

Find the infinite sum $S=\sum_{n=1}^{\infty} a_{n}$ given $S_{T}=\frac{2 T-3}{T+5}$ . Options: 0, 1, 2, 3, 4, 5.

See Solution

Problem 17098

Sketch the graph of $y=f(x)$ where $f(x)=11 x(x-1)^{3}$ . Find local max/min and inflection points.

See Solution

Problem 17099

Calculate the indefinite integral: $\int 30 x^{4}(x-1) \, dx$

See Solution

Problem 17100

Calculate the integral: $\int \frac{\operatorname{sen}^{3} x}{\sqrt{\cos x}} d x$

See Solution

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Calculus

Problem 17001

Find the derivative of y=(x2+3)2(x3−2x)3y=\left(x^{2}+3\right)^{2}\left(x^{3}-2 x\right)^{3}y=(x2+3)2(x3−2x)3.

Problem 17002

Find the derivative y′y^{\prime}y′ for y=(x2+3)2(x3−2x)3y=(x^{2}+3)^{2}(x^{3}-2x)^{3}y=(x2+3)2(x3−2x)3.

Problem 17003

Find the tangent line equation to the curve y=2excos⁡(x)y=2 e^{x} \cos (x)y=2excos(x) at the point (0,2)(0,2)(0,2).

Problem 17004

Find the derivative of y=2x2+1(3x2−5)4y=\sqrt{2x^{2}+1}(3x^{2}-5)^{4}y=2x2+1​(3x2−5)4.

Problem 17005

Find the derivative of h(x)x\frac{h(x)}{x}xh(x)​ at x=2x=2x=2 given h(2)=5h(2)=5h(2)=5 and h′(2)=−2h^{\prime}(2)=-2h′(2)=−2.

Problem 17006

Does the Intermediate Value Theorem ensure a real zero for G(x)=2x4−6x3+106x2−46x+210G(x)=2 x^{4}-6 x^{3}+106 x^{2}-46 x+210G(x)=2x4−6x3+106x2−46x+210 between 1 and 2?

Problem 17007

Sketch the function y=2x3−3x2−x−2y=2 x^{3}-3 x^{2}-x-2y=2x3−3x2−x−2 and label key points. Find: yyy-intercept, xxx-intercepts, critical points, inflection points, and intervals of increase/decrease.

Problem 17008

Find the derivative S′(100)S^{\prime}(100)S′(100) of the function S(A)=0.882A0.842S(A)=0.882 A^{0.842}S(A)=0.882A0.842 and interpret the result.

Problem 17009

Find the derivative of y=23arctan⁡(x)y=\sqrt{23 \arctan (x)}y=23arctan(x)​ and simplify if possible.

Problem 17010

A kite is 100ft100 \mathrm{ft}100ft high and moves at 14ft/s14 \mathrm{ft/s}14ft/s. Find the rate (in rad/s) at which the angle decreases when 200ft200 \mathrm{ft}200ft of string is out.

Problem 17011

Find the derivative of the function using logarithmic differentiation: y=x−3x8+5y=\sqrt{\frac{x-3}{x^{8}+5}}y=x8+5x−3​​.

Problem 17012

Problem 17013

Find the tangent line equation for y=x13xy=x^{13 x}y=x13x at x=ex=ex=e.

Problem 17014

Find the tangent line equation for y=(x+3)(x+4)2(x+5)2y=(x+3)(x+4)^{2}(x+5)^{2}y=(x+3)(x+4)2(x+5)2 at x=0x=0x=0.

Problem 17015

Simplify the series ∑i=1n4n⋅(4−(−2+i4n)2)\sum_{i=1}^{n} \frac{4}{n} \cdot\left(4-\left(-2+\frac{i 4}{n}\right)^{2}\right)∑i=1n​n4​⋅(4−(−2+ni4​)2).

Problem 17016

Find an antiderivative F(x)F(x)F(x) of the function f(x)=−41−x2f(x)=\frac{-4}{\sqrt{1-x^{2}}}f(x)=1−x2​−4​.

Problem 17017

Find the derivative of yyy for y=(2x+3)xy=(2 x+3)^{x}y=(2x+3)x. What is dydx\frac{dy}{dx}dxdy​?

Problem 17018

Find the function whose derivative is f′(x)=e5xf^{\prime}(x)=e^{5 x}f′(x)=e5x and passes through the point P=(0,65)P=\left(0, \frac{6}{5}\right)P=(0,56​).

Problem 17019

Find the time for \$ 2125 to double at an 8.5% continuous compound interest rate. Round to two decimal places.

Problem 17020

Find y′y^{\prime}y′ using logarithmic differentiation for y=6(x3+1)2x6e−5x3y=\sqrt[3]{\frac{6\left(x^{3}+1\right)^{2}}{x^{6} e^{-5 x}}}y=3x6e−5x6(x3+1)2​​.

Problem 17021

Find y′y^{\prime}y′ using logarithmic differentiation for y=6(x3+1)2x6e−5x3y=\sqrt[3]{\frac{6(x^{3}+1)^{2}}{x^{6} e^{-5 x}}}y=3x6e−5x6(x3+1)2​​.

Problem 17022

Differentiate the function using logarithmic differentiation: f(x)=(x+1)(6x+1)(8x+1)4x+1f(x)=\frac{(x+1)(6 x+1)(8 x+1)}{\sqrt{4 x+1}}f(x)=4x+1​(x+1)(6x+1)(8x+1)​.

Problem 17023

Use Newton's method to find the root of x3+x2+10=0x^{3}+x^{2}+10=0x3+x2+10=0 accurate to five decimal places. Sketch a graph. Root is x≈x \approxx≈

Problem 17024

Differentiate implicitly: ln⁡(3xy)=e3xy\ln(3xy) = e^{3xy}ln(3xy)=e3xy, find dydx\frac{dy}{dx}dxdy​, with y≠0y \neq 0y=0.

Problem 17025

Two soap bubbles contain 6 liters of air. Let VVV be the left bubble's volume. Show that the surface area is S=(36π)13(V23+(6−V)23)S=(36 \pi)^{\frac{1}{3}}\left(V^{\frac{2}{3}}+(6-V)^{\frac{2}{3}}\right)S=(36π)31​(V32​+(6−V)32​). Find extreme values of SSS and discuss stability.

Problem 17026

Find the antiderivative of f(x)=−4xf(x) = -4xf(x)=−4x.

Problem 17027

Calculate the displacement and distance traveled by an object with velocity dxdt=4+3cos⁡(0.5t)\frac{d x}{d t}=4+3 \cos (0.5 t)dtdx​=4+3cos(0.5t) from 000 to 2π2\pi2π.

Problem 17028

Differentiate implicitly: (1+e4x)2=4+ln⁡(x+y)(1+e^{4 x})^{2}=4+\ln (x+y)(1+e4x)2=4+ln(x+y), find dydx\frac{d y}{d x}dxdy​.

Problem 17029

Find the half-life of a substance with decay function y=y0e−0.048ty=y_{0} e^{-0.048 t}y=y0​e−0.048t. Round to the nearest tenth.

Problem 17030

Problem 17031

Kelly thinks ln⁡(x4)\ln \left(x^{4}\right)ln(x4) is the antiderivative of 1x4\frac{1}{x^{4}}x41​. Explain why this is incorrect.

Problem 17032

Find the tangent line equation for g(x)=(3x2−4)f(x)g(x)=(3x^{2}-4)f(x)g(x)=(3x2−4)f(x) at x=−2x=-2x=−2 given f(−2)=−2f(-2)=-2f(−2)=−2 and f′(−2)=−3f'(-2)=-3f′(−2)=−3.

Problem 17033

Find the function g(x)g(x)g(x) such that g′(x)=−512−x3g^{\prime}(x)=-512-x^{3}g′(x)=−512−x3 and the maximum value of g(x)g(x)g(x) is 3.

Problem 17034

Calculate the area between the curves y=x1/2y=x^{1/2}y=x1/2 and y=x1/4y=x^{1/4}y=x1/4 from x=0x=0x=0 to x=1x=1x=1. Find area =

Problem 17035

Find the tangent line to f(x)=13x3+3x22−10xf(x)=\frac{1}{3} x^{3}+\frac{3 x^{2}}{2}-10 xf(x)=31​x3+23x2​−10x at x=−1x=-1x=−1.

Problem 17036

Problem 17037

Problem 17038

Find the antiderivative of f(x)=x−2+x−3f(x) = x^{-2} + x^{-3}f(x)=x−2+x−3.

Problem 17039

Berechnen Sie die Integrale: a) ∫02(2+x)3dx\int_{0}^{2}(2+x)^{3} dx∫02​(2+x)3dx b) ∫23(1+1x2)dx\int_{2}^{3}\left(1+\frac{1}{x^{2}}\right) dx∫23​(1+x21​)dx

Problem 17040

Evaluate the integral of eue^{u}eu with respect to uuu.

Problem 17041

Calculate the integral ∫12(xy2+x+2) dx\int_{1}^{2}(x y^{2}+x+2) \, dx∫12​(xy2+x+2)dx.

Problem 17042

Find the derivative of $y=\left(x^{2}+3\right)^{2}\left(x^{3}-2 x\right)^{3}$ .

Find the derivative $y^{\prime}$ for $y=(x^{2}+3)^{2}(x^{3}-2x)^{3}$ .

Find the tangent line equation to the curve $y=2 e^{x} \cos (x)$ at the point $(0,2)$ .

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

Find the derivative of $\frac{h(x)}{x}$ at $x=2$ given $h(2)=5$ and $h^{\prime}(2)=-2$ .

Does the Intermediate Value Theorem ensure a real zero for $G(x)=2 x^{4}-6 x^{3}+106 x^{2}-46 x+210$ between 1 and 2?

Sketch the function $y=2 x^{3}-3 x^{2}-x-2$ and label key points. Find: $y$ -intercept, $x$ -intercepts, critical points, inflection points, and intervals of increase/decrease.

Find the derivative $S^{\prime}(100)$ of the function $S(A)=0.882 A^{0.842}$ and interpret the result.

Find the derivative of $y=\sqrt{23 \arctan (x)}$ and simplify if possible.

A kite is $100 \mathrm{ft}$ high and moves at $14 \mathrm{ft/s}$ . Find the rate (in rad/s) at which the angle decreases when $200 \mathrm{ft}$ of string is out.

Find the derivative of the function using logarithmic differentiation: $y=\sqrt{\frac{x-3}{x^{8}+5}}$ .

Find the tangent line equation for $y=x^{13 x}$ at $x=e$ .

Find the tangent line equation for $y=(x+3)(x+4)^{2}(x+5)^{2}$ at $x=0$ .

Simplify the series $\sum_{i=1}^{n} \frac{4}{n} \cdot\left(4-\left(-2+\frac{i 4}{n}\right)^{2}\right)$ .

Find an antiderivative $F(x)$ of the function $f(x)=\frac{-4}{\sqrt{1-x^{2}}}$ .

Find the derivative of $y$ for $y=(2 x+3)^{x}$ . What is $\frac{dy}{dx}$ ?

Find the function whose derivative is $f^{\prime}(x)=e^{5 x}$ and passes through the point $P=\left(0, \frac{6}{5}\right)$ .

Find $y^{\prime}$ using logarithmic differentiation for $y=\sqrt[3]{\frac{6\left(x^{3}+1\right)^{2}}{x^{6} e^{-5 x}}}$ .

Find $y^{\prime}$ using logarithmic differentiation for $y=\sqrt[3]{\frac{6(x^{3}+1)^{2}}{x^{6} e^{-5 x}}}$ .

Differentiate the function using logarithmic differentiation: $f(x)=\frac{(x+1)(6 x+1)(8 x+1)}{\sqrt{4 x+1}}$ .

Use Newton's method to find the root of $x^{3}+x^{2}+10=0$ accurate to five decimal places. Sketch a graph. Root is $x \approx$

Differentiate implicitly: $\ln(3xy) = e^{3xy}$ , find $\frac{dy}{dx}$ , with $y \neq 0$ .

Two soap bubbles contain 6 liters of air. Let $V$ be the left bubble's volume. Show that the surface area is $S=(36 \pi)^{\frac{1}{3}}\left(V^{\frac{2}{3}}+(6-V)^{\frac{2}{3}}\right)$ . Find extreme values of $S$ and discuss stability.

Find the antiderivative of $f(x) = -4x$ .

Calculate the displacement and distance traveled by an object with velocity $\frac{d x}{d t}=4+3 \cos (0.5 t)$ from $0$ to $2\pi$ .

Differentiate implicitly: $(1+e^{4 x})^{2}=4+\ln (x+y)$ , find $\frac{d y}{d x}$ .

Find the half-life of a substance with decay function $y=y_{0} e^{-0.048 t}$ . Round to the nearest tenth.

Kelly thinks $\ln \left(x^{4}\right)$ is the antiderivative of $\frac{1}{x^{4}}$ . Explain why this is incorrect.

Find the tangent line equation for $g(x)=(3x^{2}-4)f(x)$ at $x=-2$ given $f(-2)=-2$ and $f'(-2)=-3$ .

Find the function $g(x)$ such that $g^{\prime}(x)=-512-x^{3}$ and the maximum value of $g(x)$ is 3.

Calculate the area between the curves $y=x^{1/2}$ and $y=x^{1/4}$ from $x=0$ to $x=1$ . Find area =

Find the tangent line to $f(x)=\frac{1}{3} x^{3}+\frac{3 x^{2}}{2}-10 x$ at $x=-1$ .

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

Berechnen Sie die Integrale: a) $\int_{0}^{2}(2+x)^{3} dx$ b) $\int_{2}^{3}\left(1+\frac{1}{x^{2}}\right) dx$

Evaluate the integral of $e^{u}$ with respect to $u$ .

Calculate the integral $\int_{1}^{2}(x y^{2}+x+2) \, dx$ .

Bestimme die Punkte, an denen der Graph von $f$ die Steigung $m$ hat, und gib die Tangentengleichungen an.

Finde die Stammfunktionen für die folgenden Funktionen: a) $f(x)=x^{2}$ , b) $f(x)=x^{3}$ , c) $f(x)=3x$ , f) $f(x)=x^{4}$ , g) $f(x)=0,1x^{3}$ , h) $f(x)=x$ .

Solve the integral $\int_{k}^{k+1}(k^{2}+2t)dt=9$ for the variable $k$ .

Find the value of $k$ such that $\int_{1}^{2} k x^{3} dx = 75$ .

Berechne die verbleibende Menge des Medikaments nach 6 Stunden, wenn die Anfangsmenge $400 \mathrm{mg}$ und die Halbwertszeit 2 Stunden beträgt.

Berechne die Integrale $I=\int_{-2}^{0} f(x) d x$ und $I=\int_{-2}^{3} f(x) d x$ . Finde dann die Stammfunktion $F$ für $f(x)=-5 \cdot x^{4}+8$ mit $F(1)=10$ .

Berechne die Integrale $I=\int_{-2}^{0} f(x) dx$ und $I=\int_{-2}^{3} f(x) dx$ . Finde die Stammfunktion $F$ von $f(x)=-5 \cdot x^{4}+8$ mit $F(1)=10$ .

Berechne den Flächeninhalt zwischen dem Graphen $f(x) = x^{2}-5$ und der $x$ -Achse von $x = -4$ bis $x = 4$ .

Berechnen Sie die Integrale: (1) $I=\int_{-2}^{0} f(x) d x$ , (2) $I=\int_{-2}^{3} f(x) d x$ mit $A1=0,3$ , $A2=0,8$ , $A3=2,9$ , $A4=1,1$ .

Berechnen Sie die Schnittpunkte von $G_{a}$ mit den Achsen und das Verhalten von $f_{a}$ für $x \rightarrow+\infty$ .

Bestimme den Wendepunkt und die Wendetangente der Funktion $f(x)=x^{3}+3 x^{2}-4$ . Finde die Länge der Strecke $\overline{\mathrm{PQ}}$ .

Bestimme den Wendepunkt und die Wendetangente der Funktion $f(x)=x^{3}+3 x^{2}-4$ . Berechne die Länge von $\overline{\mathrm{PQ}}$ .

Untersuchen Sie den Extrempunkt und das Krümmungsverhalten des Graphen $G_a$ von $f_a(x) = e^x(x-a)$ . Bestimmen Sie den Wendepunkt.

Berechne die Beschleunigung $a(t)$ mit $a(t)=\frac{v2 - v1}{t2 - t1}$ für die Punkte (0.2s, 1.9591 m/s) und (0.2s, 3.901654 m/s).

Find $s^{\prime}(t)$ for $s(t)=3 t^{\frac{1}{3}}+2 t^{2}$ . Choices: $t^{\frac{2}{3}}+4 t$ , $4 t-\frac{1}{t^{\frac{2}{3}}}$ , $4 t+\frac{1}{t^{\frac{2}{3}}}$ , $4 t+\frac{3}{t^{\frac{2}{3}}}$ .

Find $f^{\prime}(-1)$ for $f(x)=\frac{1-x}{x^{2}}$ . Options: $-1$ , $0$ , $3$ , $1$ .

Bestimmen Sie die Ableitungen $f^{\prime}$ für die Funktionen und berechnen Sie $f^{\prime}(2)$ für jede: a) $f(x)=2 x^{2}$ , b) $f(x)=4 x^{2}$ , c) $f(x)=-3 x^{2}$ , d) $f(x)=\frac{1}{2} x^{2}$ .

Berechnen Sie die Integrale: a) $\int_{0}^{4} x^{2} d x$ , b) $\int_{2}^{4} x^{2} d x$ , c) $\int_{-1}^{5} 2 x d x$ , d) $\int_{10}^{11} 0,5 x d x$ , e) $\int_{10}^{20} 5 d x$ .

Bestimme die momentane Änderungsrate der Wirkstoffkonzentration $f(t)=0,25 t^{3}-3 t^{2}+9 t$ bei $t=4$ h.

Calculate the integral of $f(x)=\frac{x^{2}+2 x}{x^{5}}$ .

Bestimmen Sie die Funktion $f$ mit Basis e und die ersten beiden Ableitungen für $f(x)=4^{x}$ , $f(x)=2 \cdot 6^{x}$ , $f(x)=2^{x+2}$ , $f(x)=2,5^{x} \cdot 3^{x}$ .

Evaluate the line integral $\int_{C} \boldsymbol{F} \cdot \boldsymbol{d r}$ for $\boldsymbol{F}=\left\langle x^{2} y, x y^{2}\right\rangle$ from $(1,6)$ to $(6,1)$ .

Calculate the integral $\int_{2}^{4} \frac{-1}{x^{3}} d x$ .

Calculate the integral from 2 to 4 of the function $-\frac{1}{x^{3}}$ .

Find the Jacobian for the transformation: $x=u^{2}+uv-v^{2}$ , $y=u^{2}-v+4w$ , $z=3u+w^{2}$ .