Calculus

Problem 3801

Fichte Wachstum: a) Berechne $f(30)$ und interpretiere. b) Bestimme Alter und maximale Wachstumsgeschwindigkeit. c) Zeige, dass Höhe nach 20 Jahren < 20m und berechne Höhe. d) Beweise Wendestelle von $F$ . e) Maximalhöhe der Fichte ermitteln.

See Solution

Problem 3802

Finde den Funktionswert von $f(30)$ , bestimme das Alter mit maximalem Wachstum, und berechne die Höhe nach 20 Jahren.

See Solution

Problem 3803

Ein Skateboardfahrer fährt über einen Wall, modelliert durch $f(x)=e^{-0,1 x^{2}}-0,2$ . Bestimme die maximale Höhe $h$ und die Breite des Walls.

See Solution

Problem 3804

Skizzieren Sie den Graphen der Ableitungsfunktion $f'$ für die gegebene Funktion $f$ .

See Solution

Problem 3805

Ein Skateboardfahrer fährt über einen Wall, modelliert durch $f(x)=e^{-0,1 x^{2}}-0,2$ .
a) Bestimme die maximale Höhe $h$ und die Breite des Walls. b) Finde die Punkte, an denen die Steigung des Walls am größten ist.

See Solution

Problem 3806

Berechnen Sie das Volumen des Walls mit einer Tiefe von $2,5 \mathrm{~m}$ , modelliert durch $f(x)=e^{-0,1 x^{2}}-0,2$ .

See Solution

Problem 3807

Find the integral of $\theta \cos \theta$ with respect to $\theta$ : $\int \theta \cos \theta d \theta$ .

See Solution

Problem 3808

Calculate the integral $\int x e^{-6 x} d x$ .

See Solution

Problem 3809

Calculate the integral $\int x e^{-6 x} d x$ .

See Solution

Problem 3810

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

See Solution

Problem 3811

Berechne die Punkte, wo die Funktion $f(x)=\frac{1}{3} x^{3}-4 x^{2}+18 x+1$ eine Steigung von 3 hat.

See Solution

Problem 3812

Bestimmen Sie das Integral von $f(x)=x^{\frac{2}{3}}$ .

See Solution

Problem 3813

Bestimmen Sie die Stammfunktion von $f(x)=-2 \sqrt[3]{x}$ .

See Solution

Problem 3814

Find the derivative of $g(f(x))$ at $x=7$ with $f(x)=4(x-6)^2$ and $g(y)=3y^4+5y$ using the chain rule.

See Solution

Problem 3815

Find the derivative of $g(f(x))$ at $x=7$ , where $f(x)=4(x-6)^{2}$ and $g(y)=3y^{4}+5y$ . Evaluate $f'(7)$ and $g'(f(7))$ .

See Solution

Problem 3816

Find the derivative of $f(x)=3 x^{6}+8 x^{5}+12 x^{1/4}+5 x^{-5}+7 x+32$ and determine coefficients $a$ through $k$ .

See Solution

Problem 3817

Find the derivative of the function $f(x)=6 e^{3 x}+3 \ln (x^{5}+6 x^{4}(x-2)^{2}+\frac{6}{x}+29)$ and determine coefficients $a$ through $n$ .

See Solution

Problem 3818

Find the derivative of $f(x)=5 x^{6}+5 x^{5}+20 x^{1/5}+6 x^{-2}+5 x+29$ and determine coefficients $a$ through $k$ .

See Solution

Problem 3819

Find the derivative of $g(f(x))$ at $x=12$ where $f(x)=6(x-7)^{2}-144$ and $g(y)=8y^{2}+5y$ . Use the chain rule.

See Solution

Problem 3820

Find the derivative of $f(x)=8 e^{4 x}+7 \ln(x^{4}+3 x^{3}(x-2)^{2}+\frac{6}{x}+30)$ and determine coefficients $a$ to $n$ .

See Solution

Problem 3821

Find the derivative of $q(x)=g(x) f\left(x^{2}+4 x\right)$ and calculate $q^{\prime}(-3)$ . Also, find the derivative of $e^{3x}$ .

See Solution

Problem 3822

Show that there exists a $\delta>0$ so that if $x^{2}+y^{2}<\delta^{2}$ , then $|x^{2}+y^{2}+3xy+180xy^{5}|<\frac{1}{10,000}$ .

See Solution

Problem 3823

Find the derivative of $f(x)=8 e^{4 x}+7 \ln \left(x^{4}+3 x^{3}(x-2)^{2}+\frac{6}{x}+30\right)$ and determine coefficients $a$ to $n$ .

See Solution

Problem 3824

Find the derivative of $f(r)=(r-5)^{r}$ at $r=7$ and round to the nearest integer.

See Solution

Problem 3825

Find the derivative of $f(r)=(r-5)^{r}$ at $r=7$ and round to the nearest integer.

See Solution

Problem 3826

Ein PKW hat ein v-t-Diagramm:
a) Bestimme die Beschleunigung in den Abschnitten I (0-7s) und III (22-26s). b) Berechne den Weg bis $t=26 \mathrm{~s}$ .

See Solution

Problem 3827

Bestimme die Nullstellen und lokalen Extrema der Funktion $f(x)=x^{3}-3 x^{2}$ .

See Solution

Problem 3828

Find the derivative of $f(t)=(t-1)^{t}$ at $t=5$ and round to the nearest integer.

See Solution

Problem 3829

Find the derivative of $f(x)=4 x^{6}+7 x^{5}+20 x^{1/4}+6 x^{-2}+7 x+22$ and determine coefficients $a$ to $k$ .

See Solution

Problem 3830

Calculate the average rate of change of $f(x) = x^{3} + x$ on the intervals [-1,1] and [2,3].

See Solution

Problem 3831

Bestimme die Stammfunktion von $f(x)=-\frac{12}{49} x^{2}$ .

See Solution

Problem 3832

Find the derivative of $g(f(x))$ where $f(x)=3(x-8)^{2}-21$ and $g(y)=3y^{3}+3y$ , then evaluate at $x=11$ .

See Solution

Problem 3833

Find the derivative of the function $f(x)=x^{4}(8 x+8)$ using the Product or Quotient Rule.

See Solution

Problem 3834

Find the derivative of $f(x)=(x^{3}+1)(-2x^{2}-6)$ using the Product or Quotient Rule.

See Solution

Problem 3835

Find the derivative of $f(x)=\frac{-8 x}{2 x+3}$ using the Product or Quotient Rule.

See Solution

Problem 3836

Find the average rate of change of $h(t)=\sin (2 \pi t)+4$ on intervals $[\frac{1}{4}, \frac{3}{4}]$ and $[4,5]$ .

See Solution

Problem 3837

Find the derivative of the function using the Quotient Rule: $f(x)=\frac{-x^{2}+1}{7 x^{5}+7}$ .

See Solution

Problem 3838

Find the derivative of the function $f(x)=7 e^{6 x}+5 \ln (x^{6}+6 x^{5}(x-2)^{2}+\frac{7}{x}+34)$ and express it as $\frac{d f}{d x}(x)=a \cdot e^{b \cdot x}+c \cdot \ln x+d \cdot x^{7}+e \cdot x^{6}+g \cdot x^{5}+h \cdot x^{4}+j \cdot x+k \cdot x^{-1}+l \cdot x^{-2}+m \cdot x^{-6}+n$ . Find coefficients $a$ through $n$ .

See Solution

Problem 3839

Find the derivative of $f(x)=\frac{3 x^{2}-6}{8 x^{4}+10}$ using the Product or Quotient Rule.

See Solution

Problem 3840

Calculate the average rate of change of $q(x)=\sqrt{4x+1}$ on the intervals [0,1] and [0,2].

See Solution

Problem 3841

Find the population growth rate in thousands 21 years from now, given $P(t)=(0.6 t-9)(0.2 t+10)+20$ . Round to two decimal places.

See Solution

Problem 3842

Die Funktion $f(x)=\frac{1}{9} x^{3}-x^{2}+\frac{8}{3} x$ beschreibt das Höhenprofil einer $6 \mathrm{~km}$ langen Radtour.
a) Bestimme die durchschnittliche Steigung vor dem Abstieg. b) Wie viel Prozent größer ist die größte Steigung als die durchschnittliche? c) Über welche Distanz geht es bergab? d) Welcher Höhenunterschied wird dabei überwunden? e) Wo ist der steilste Abstieg? f) Wo ist die Steigung am größten?

See Solution

Problem 3843

Calculate the average rate of change of $p(x)=\frac{1}{x}$ on $\left[\frac{3}{4}, 1\right]$ and $\left[\frac{1}{4}, \frac{1}{2}\right]$ .

See Solution

Problem 3844

Find the limit: $\lim _{x \rightarrow 2} \frac{x^{2}-4}{x-2}=$

See Solution

Problem 3845

Find the value of $F(x)$ where $F(x)=\int_{0}^{4} \sqrt{x} \, dx$ .

See Solution

Problem 3846

Find $f'(t)$ for $f(t)=\frac{2}{t}$ at $t=a$ and $t=a+h$ . Answers: (a) $\frac{-2 h}{a(a+h)}$ , (b) $\frac{-2}{a(a+h)}$ .

See Solution

Problem 3847

Find the growth rate of the city's population, given by $P(t)=(0.7 t-5)(0.4 t+9)+75$ , in 16 years. Round to two decimal places.

See Solution

Problem 3848

Find the rate of temperature decrease at $h=1050$ ft, with $T(h)=65 e^{-0.000065 h}$ and rising at $807$ ft/hr.

See Solution

Problem 3849

Find the marginal revenue for the demand function $D(x)=\frac{143}{4 x+4}$ at $x=5$ . Round to the nearest cent.

See Solution

Problem 3850

Find the marginal revenue for the demand function $D(x)=\frac{155}{5 x+4}$ at $x=6$ , rounded to the nearest cent.

See Solution

Problem 3851

Find the derivative of $h(x)=\frac{4 \sqrt{x}}{3 x-8}$ using the Product or Quotient Rule.

See Solution

Problem 3852

Find the derivative of $s = t^{23} \ln |t|$ .

See Solution

Problem 3853

Find the derivative of the function $s=t^{17} \ln |t|$ .

See Solution

Problem 3854

Find the limit as $x$ approaches 0 from the left: $\lim _{x \rightarrow 0^{-}} e^{\frac{\sin x}{1-\cos x}}$ .

See Solution

Problem 3855

Find the derivative of $s=t^{9} \ln |t|$ .

See Solution

Problem 3856

Find the limit: $\lim _{x \rightarrow 1} \frac{x^{3}-x}{x-1}$ .

See Solution

Problem 3857

Find the limit: $\lim _{x \rightarrow 1} \frac{x^{2}-x}{x-1}$ .

See Solution

Problem 3858

Find the derivative of $h(t)=\left(3-\frac{5}{t^{2}}\right)\left(t^{3}+3 t+7\right)$ using the Product or Quotient Rule.

See Solution

Problem 3859

Find the derivative of $g(t)=\left(6+\frac{7}{t^{3}}\right)\left(4 t^{4}+2 t^{3}+5\right)$ using the Product or Quotient Rule.

See Solution

Problem 3860

Bestimmen Sie die ersten drei Ableitungen von $f$ für die Funktionen: a) $(2x-3)e^{x}$ , b) $(x^{2}+1)e^{x}$ , c) $(5x^{2}+1)e^{-x}$ , d) $5x^{2}e^{-\frac{1}{4}x}$ . Erklären Sie den Vorteil des Ausklammerns.

See Solution

Problem 3861

Find the limit: $\lim _{x \rightarrow -2} \frac{x^{4}-16}{x+2}$ .

See Solution

Problem 3862

Find the derivative of $h(t)=\left(7-\frac{6}{t^{2}}\right)\left(3 t^{2}+2 t-7\right)$ using the Product or Quotient Rule.

See Solution

Problem 3863

Find the derivative of the function: $\ln \left(\frac{2 x}{\sqrt[3]{4-x^{3}}}\right)$ .

See Solution

Problem 3864

Find the derivative of $h(t)=\left(4+\frac{1}{t}\right)\left(t^{4}-\frac{1}{7}\right)$ using the Product or Quotient Rule.

See Solution

Problem 3865

Find the limit: $\lim _{x \rightarrow 0} \frac{2 x \sin x}{\cos ^{2} x-\cos x}$

See Solution

Problem 3866

Find the derivative of $h(t)=\left(9+\frac{1}{t}\right)\left(t^{2}-\frac{1}{7}\right)$ using the Product or Quotient Rule.

See Solution

Problem 3867

Find the limit: $\lim _{x \rightarrow -2} \frac{(x+2)(x-2)(x+2)(x-2)}{x+2}$ .

See Solution

Problem 3868

Find the derivative of $g(u)=\left(5+\frac{1}{u}\right)\left(u^{5}-\frac{1}{6}\right)$ using Product or Quotient Rule.

See Solution

Problem 3869

Evaluate the integral: $\int_{1}^{9} \frac{2 x^{2}+x^{2} \sqrt{x}-1}{x^{2}} d x$

See Solution

Problem 3870

Bestimmen Sie den Grenzwert von $f(x)=\frac{x^{2}-9}{2x-6}$ für $x \to 3$ von links und rechts.

See Solution

Problem 3871

Find the tangent and normal line equations for the curve $y=x^{4}+5 e^{x}$ at the point $(0,5)$ .

See Solution

Problem 3872

Find the derivative of the function using the Product or Quotient Rule: $f(x)=\frac{-2 x^{4}-5}{-9 x+4}$ .

See Solution

Problem 3873

Find the rate of change of rock bass length $L$ at age $A=16$ using the model $L=0.0155 A^{3}-0.372 A^{2}+3.95 A+1.21$ . Round to three decimal places.

See Solution

Problem 3874

Find the derivative of the function $h(t)=(8 t^{2}+5 t)^{-4}$ using positive, negative, and fractional exponents.

See Solution

Problem 3875

Find $S^{\prime}(106)$ for the model $S(A)=0.882 A^{0.842}$ . Round to five decimal places. Interpret the result.

See Solution

Problem 3876

Find the 114th derivative of $\sin x$ .

See Solution

Problem 3877

Find the limit: $\lim _{x \rightarrow-1} \frac{x^{3}-x}{x+1}$ .

See Solution

Problem 3878

Find the derivative of $f(x)=(7 x^{3}-7 x+5)^{\frac{1}{2}}$ using positive, negative, and fractional exponents.

See Solution

Problem 3879

Find the derivative of the function $y=\sqrt[3]{7 x^{3}-3}$ using positive, negative, and fractional exponents.

See Solution

Problem 3880

Find the derivative of the function $f(x)=(2-3x)^{4}$ using positive, negative, and fractional exponents.

See Solution

Problem 3881

Find the slope of the tangent line to the parabola $y=5x-x^{2}$ at the point $(1,4)$ .

See Solution

Problem 3882

Bestimmen Sie die 1. und 2. Ableitung der Funktionen: a) $f(x)=e^{x}+1$ b) $f(x)=e^{x}+x$ c) $f(x)=e^{x}+2 x^{2}$ d) $f(x)=-e^{x}+1$ e) $f(x)=2 e^{x}+3 x^{2}$ f) $f(x)=-5 e^{x}-0,5 x^{3}$ g) $f(x)=-\frac{1}{2}(e^{x}-x^{3})$ h) $f(x)=\frac{1}{4} e^{x}+\sin(x)$

See Solution

Problem 3883

Find the average rate of change of $f$ on the interval $[50,70]$ for the given curve.

See Solution

Problem 3884

Bestimmen Sie die 1. und 2. Ableitung der Funktionen: a) $f(x)=e^{x}+1$ , b) $f(x)=e^{x}+x$ , c) $f(x)=e^{x}+2 x^{2}$ , d) $f(x)=-e^{x}+1$ , e) $f(x)=2 e^{x}+3 x^{2}$ , f) $f(x)=-5 e^{x}-0,5 x^{3}$ , g) $f(x)=-\frac{1}{2}(e^{x}-x^{3})$ , h) $f(x)=\frac{1}{4} e^{x}+\sin(x)$ .

See Solution

Problem 3885

Estimate the value of $f’(50)$ for the function $f$ described by its graph points.

See Solution

Problem 3886

Find the net change and average rate of change for $f(t)=\frac{8}{t}$ from $t=a$ to $t=a+h$ in terms of $a$ and $h$ .

See Solution

Problem 3887

Gegeben ist die Kostenfunktion $K(x)=2 x^{3}-18 x^{7}+62 x+32$ .
a) Erläutere dem Abteilungsleiter die Bedeutung des Graphen für Produktionskosten und erkläre den Wendepunkt.
b) Bei 1000 Bleistiften erzielt man 50 €. In welchem Stückzahlbereich wird Gewinn erzielt und wo ist dieser maximal?

See Solution

Problem 3888

Estimate the derivative of the function $f$ at $f(50)$ given its graph points: $(0,600),(10,400),(20,300),(30,250),(40,200),(50,400),(60,700),(70,900),(80,800)$ .

See Solution

Problem 3889

Find the limit as $x$ approaches 1 for the expression $\frac{x^{2}-x}{x-1}$ .

See Solution

Problem 3890

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{3 x}-e^{2 x}}{x^{2}-3 x}$ .

See Solution

Problem 3891

Find the limit as $x$ approaches $x_0$ of $\frac{x^2 - x_0^2}{x - x_0}$ .

See Solution

Problem 3892

Differentiate the function $\frac{1+c e^{-\frac{x^{2}}{2}}}{1-c e^{-\frac{x^{2}}{2}}}$ with respect to $x$ .

See Solution

Problem 3893

Gegeben ist die Funktion $f(x)=x^{2}-2$ .
a) Finde die Tangentengleichung bei $P(0,5 \mid f(0,5))$ . b) Bestimme Punkte mit Steigung 4 und 0. c) Wo ist die Tangente parallel zu $g: y=-2x+3$ ?

See Solution

Problem 3894

Find the limit as $x$ approaches 0 for the expression $\frac{1}{x^{2}}$ .

See Solution

Problem 3895

Find the derivative of $F(x)=(4x+8)^{3}(x^{2}-7x+8)^{4}$ . What is $F'(x)$ ?

See Solution

Problem 3896

Find the limit: $\lim _{x \rightarrow-\infty} 2^{x}$ .

See Solution

Problem 3897

Find these limits: a. $\lim _{x \rightarrow 2} \frac{x^{3}-4 x}{x^{2}-12 x+20}$ , b. $\lim _{x \rightarrow 0} \frac{\sqrt{4+x}-2}{x}$ , c. $\lim _{x \rightarrow 0} \frac{\sin ^{2}(4 x)}{5 x^{2}}$ , d. $\lim _{x \rightarrow 0} \frac{\sin \left(3 x^{2}\right) \cos (5 x)}{2 x^{2}}$ , e. $\lim _{x \rightarrow 0} \frac{\sin (2 x) \cos ^{2}(5 x)}{6 x}$ , f. $\lim _{x \rightarrow-3} \frac{x^{2}+3 x}{x^{2}-9}$ , g. $\lim _{x \rightarrow 0} \frac{\sin (5 x)}{\tan (9 x)}$ , h. $\lim _{x \rightarrow \infty}\left(\frac{10 x^{7}+4 x^{3}-1}{5 x^{8}+2 x^{10}+4}\right)$ , i. $\lim _{x \rightarrow \infty}\left(\frac{10 x^{9}+4 x^{3}-1}{5 x^{9}+2 x^{6}+4}\right)$ .

See Solution

Problem 3898

Find the limit as $x$ approaches 0 from the right for the expression $\frac{1}{\sqrt{x}}$ .

See Solution

Problem 3899

Bestimme die Grenzwerte: $\lim _{x \rightarrow 5} \frac{x^{2}-25}{x-5}$ , $\lim _{x \rightarrow 3} \frac{3 x^{2}-27}{x-3}$ , $\lim _{x \rightarrow 1} \frac{x^{3}-x}{x-1}$ , $\lim _{x \rightarrow-2} \frac{x^{4}-16}{x+2}$ .

See Solution

Problem 3900

Bestimme die Grenzwerte: a) $\lim _{x \rightarrow 5} \frac{x^{2}-25}{x-5}$ , b) $\lim _{x \rightarrow 3} \frac{3 x^{2}-27}{x-3}$ , c) $\lim _{x \rightarrow 1} \frac{x^{3}-x}{x-1}$ , d) $\lim _{x \rightarrow -2} \frac{x^{4}-16}{x+2}$ .

See Solution

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Calculus

Problem 3801

Fichte Wachstum: a) Berechne f(30)f(30)f(30) und interpretiere. b) Bestimme Alter und maximale Wachstumsgeschwindigkeit. c) Zeige, dass Höhe nach 20 Jahren < 20m und berechne Höhe. d) Beweise Wendestelle von FFF. e) Maximalhöhe der Fichte ermitteln.

Problem 3802

Finde den Funktionswert von f(30)f(30)f(30), bestimme das Alter mit maximalem Wachstum, und berechne die Höhe nach 20 Jahren.

Problem 3803

Ein Skateboardfahrer fährt über einen Wall, modelliert durch f(x)=e−0,1x2−0,2f(x)=e^{-0,1 x^{2}}-0,2f(x)=e−0,1x2−0,2. Bestimme die maximale Höhe hhh und die Breite des Walls.

Problem 3804

Skizzieren Sie den Graphen der Ableitungsfunktion f′f'f′ für die gegebene Funktion fff.

Problem 3805

Ein Skateboardfahrer fährt über einen Wall, modelliert durch f(x)=e−0,1x2−0,2f(x)=e^{-0,1 x^{2}}-0,2f(x)=e−0,1x2−0,2. a) Bestimme die maximale Höhe hhh und die Breite des Walls. b) Finde die Punkte, an denen die Steigung des Walls am größten ist.

Problem 3806

Berechnen Sie das Volumen des Walls mit einer Tiefe von 2,5 m2,5 \mathrm{~m}2,5 m, modelliert durch f(x)=e−0,1x2−0,2f(x)=e^{-0,1 x^{2}}-0,2f(x)=e−0,1x2−0,2.

Problem 3807

Find the integral of θcos⁡θ\theta \cos \thetaθcosθ with respect to θ\thetaθ: ∫θcos⁡θdθ\int \theta \cos \theta d \theta∫θcosθdθ.

Problem 3808

Calculate the integral ∫xe−6xdx\int x e^{-6 x} d x∫xe−6xdx.

Problem 3809

Calculate the integral ∫xe−6xdx\int x e^{-6 x} d x∫xe−6xdx.

Problem 3810

Find the first derivative of f(x)=2x+2x2−3xf(x)=\frac{2x+2}{x^{2}-3x}f(x)=x2−3x2x+2​.

Problem 3811

Berechne die Punkte, wo die Funktion f(x)=13x3−4x2+18x+1f(x)=\frac{1}{3} x^{3}-4 x^{2}+18 x+1f(x)=31​x3−4x2+18x+1 eine Steigung von 3 hat.

Problem 3812

Bestimmen Sie das Integral von f(x)=x23f(x)=x^{\frac{2}{3}}f(x)=x32​.

Problem 3813

Bestimmen Sie die Stammfunktion von f(x)=−2x3f(x)=-2 \sqrt[3]{x}f(x)=−23x​.

Problem 3814

Find the derivative of g(f(x))g(f(x))g(f(x)) at x=7x=7x=7 with f(x)=4(x−6)2f(x)=4(x-6)^2f(x)=4(x−6)2 and g(y)=3y4+5yg(y)=3y^4+5yg(y)=3y4+5y using the chain rule.

Problem 3815

Find the derivative of g(f(x))g(f(x))g(f(x)) at x=7x=7x=7, where f(x)=4(x−6)2f(x)=4(x-6)^{2}f(x)=4(x−6)2 and g(y)=3y4+5yg(y)=3y^{4}+5yg(y)=3y4+5y. Evaluate f′(7)f'(7)f′(7) and g′(f(7))g'(f(7))g′(f(7)).

Problem 3816

Find the derivative of f(x)=3x6+8x5+12x1/4+5x−5+7x+32f(x)=3 x^{6}+8 x^{5}+12 x^{1/4}+5 x^{-5}+7 x+32f(x)=3x6+8x5+12x1/4+5x−5+7x+32 and determine coefficients aaa through kkk.

Problem 3817

Find the derivative of the function f(x)=6e3x+3ln⁡(x5+6x4(x−2)2+6x+29)f(x)=6 e^{3 x}+3 \ln (x^{5}+6 x^{4}(x-2)^{2}+\frac{6}{x}+29)f(x)=6e3x+3ln(x5+6x4(x−2)2+x6​+29) and determine coefficients aaa through nnn.

Problem 3818

Find the derivative of f(x)=5x6+5x5+20x1/5+6x−2+5x+29f(x)=5 x^{6}+5 x^{5}+20 x^{1/5}+6 x^{-2}+5 x+29f(x)=5x6+5x5+20x1/5+6x−2+5x+29 and determine coefficients aaa through kkk.

Problem 3819

Find the derivative of g(f(x))g(f(x))g(f(x)) at x=12x=12x=12 where f(x)=6(x−7)2−144f(x)=6(x-7)^{2}-144f(x)=6(x−7)2−144 and g(y)=8y2+5yg(y)=8y^{2}+5yg(y)=8y2+5y. Use the chain rule.

Problem 3820

Find the derivative of f(x)=8e4x+7ln⁡(x4+3x3(x−2)2+6x+30)f(x)=8 e^{4 x}+7 \ln(x^{4}+3 x^{3}(x-2)^{2}+\frac{6}{x}+30)f(x)=8e4x+7ln(x4+3x3(x−2)2+x6​+30) and determine coefficients aaa to nnn.

Problem 3821

Find the derivative of q(x)=g(x)f(x2+4x)q(x)=g(x) f\left(x^{2}+4 x\right)q(x)=g(x)f(x2+4x) and calculate q′(−3)q^{\prime}(-3)q′(−3). Also, find the derivative of e3xe^{3x}e3x.

Problem 3822

Show that there exists a δ>0\delta>0δ>0 so that if x2+y2<δ2x^{2}+y^{2}<\delta^{2}x2+y2<δ2, then ∣x2+y2+3xy+180xy5∣<110,000|x^{2}+y^{2}+3xy+180xy^{5}|<\frac{1}{10,000}∣x2+y2+3xy+180xy5∣<10,0001​.

Problem 3823

Find the derivative of f(x)=8e4x+7ln⁡(x4+3x3(x−2)2+6x+30)f(x)=8 e^{4 x}+7 \ln \left(x^{4}+3 x^{3}(x-2)^{2}+\frac{6}{x}+30\right)f(x)=8e4x+7ln(x4+3x3(x−2)2+x6​+30) and determine coefficients aaa to nnn.

Problem 3824

Find the derivative of f(r)=(r−5)rf(r)=(r-5)^{r}f(r)=(r−5)r at r=7r=7r=7 and round to the nearest integer.

Problem 3825

Find the derivative of f(r)=(r−5)rf(r)=(r-5)^{r}f(r)=(r−5)r at r=7r=7r=7 and round to the nearest integer.

Problem 3826

Ein PKW hat ein v-t-Diagramm: a) Bestimme die Beschleunigung in den Abschnitten I (0-7s) und III (22-26s). b) Berechne den Weg bis t=26 st=26 \mathrm{~s}t=26 s.

Problem 3827

Bestimme die Nullstellen und lokalen Extrema der Funktion f(x)=x3−3x2f(x)=x^{3}-3 x^{2}f(x)=x3−3x2.

Problem 3828

Find the derivative of f(t)=(t−1)tf(t)=(t-1)^{t}f(t)=(t−1)t at t=5t=5t=5 and round to the nearest integer.

Problem 3829

Find the derivative of f(x)=4x6+7x5+20x1/4+6x−2+7x+22f(x)=4 x^{6}+7 x^{5}+20 x^{1/4}+6 x^{-2}+7 x+22f(x)=4x6+7x5+20x1/4+6x−2+7x+22 and determine coefficients aaa to kkk.

Problem 3830

Calculate the average rate of change of f(x)=x3+xf(x) = x^{3} + xf(x)=x3+x on the intervals [-1,1] and [2,3].

Problem 3831

Bestimme die Stammfunktion von f(x)=−1249x2f(x)=-\frac{12}{49} x^{2}f(x)=−4912​x2.

Problem 3832

Find the derivative of g(f(x))g(f(x))g(f(x)) where f(x)=3(x−8)2−21f(x)=3(x-8)^{2}-21f(x)=3(x−8)2−21 and g(y)=3y3+3yg(y)=3y^{3}+3yg(y)=3y3+3y, then evaluate at x=11x=11x=11.

Problem 3833

Find the derivative of the function f(x)=x4(8x+8)f(x)=x^{4}(8 x+8)f(x)=x4(8x+8) using the Product or Quotient Rule.

Problem 3834

Find the derivative of f(x)=(x3+1)(−2x2−6)f(x)=(x^{3}+1)(-2x^{2}-6)f(x)=(x3+1)(−2x2−6) using the Product or Quotient Rule.

Problem 3835

Find the derivative of f(x)=−8x2x+3f(x)=\frac{-8 x}{2 x+3}f(x)=2x+3−8x​ using the Product or Quotient Rule.

Problem 3836

Find the average rate of change of h(t)=sin⁡(2πt)+4h(t)=\sin (2 \pi t)+4h(t)=sin(2πt)+4 on intervals [14,34][\frac{1}{4}, \frac{3}{4}][41​,43​] and [4,5][4,5][4,5].

Problem 3837

Find the derivative of the function using the Quotient Rule: f(x)=−x2+17x5+7f(x)=\frac{-x^{2}+1}{7 x^{5}+7}f(x)=7x5+7−x2+1​.

Problem 3838

Problem 3839

Find the derivative of f(x)=3x2−68x4+10f(x)=\frac{3 x^{2}-6}{8 x^{4}+10}f(x)=8x4+103x2−6​ using the Product or Quotient Rule.

Problem 3840

Calculate the average rate of change of q(x)=4x+1q(x)=\sqrt{4x+1}q(x)=4x+1​ on the intervals [0,1] and [0,2].

Fichte Wachstum: a) Berechne $f(30)$ und interpretiere. b) Bestimme Alter und maximale Wachstumsgeschwindigkeit. c) Zeige, dass Höhe nach 20 Jahren < 20m und berechne Höhe. d) Beweise Wendestelle von $F$ . e) Maximalhöhe der Fichte ermitteln.

Finde den Funktionswert von $f(30)$ , bestimme das Alter mit maximalem Wachstum, und berechne die Höhe nach 20 Jahren.

Ein Skateboardfahrer fährt über einen Wall, modelliert durch $f(x)=e^{-0,1 x^{2}}-0,2$ . Bestimme die maximale Höhe $h$ und die Breite des Walls.

Skizzieren Sie den Graphen der Ableitungsfunktion $f'$ für die gegebene Funktion $f$ .

Ein Skateboardfahrer fährt über einen Wall, modelliert durch $f(x)=e^{-0,1 x^{2}}-0,2$ .
a) Bestimme die maximale Höhe $h$ und die Breite des Walls. b) Finde die Punkte, an denen die Steigung des Walls am größten ist.

Berechnen Sie das Volumen des Walls mit einer Tiefe von $2,5 \mathrm{~m}$ , modelliert durch $f(x)=e^{-0,1 x^{2}}-0,2$ .

Find the integral of $\theta \cos \theta$ with respect to $\theta$ : $\int \theta \cos \theta d \theta$ .

Calculate the integral $\int x e^{-6 x} d x$ .

Calculate the integral $\int x e^{-6 x} d x$ .

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

Berechne die Punkte, wo die Funktion $f(x)=\frac{1}{3} x^{3}-4 x^{2}+18 x+1$ eine Steigung von 3 hat.

Bestimmen Sie das Integral von $f(x)=x^{\frac{2}{3}}$ .

Bestimmen Sie die Stammfunktion von $f(x)=-2 \sqrt[3]{x}$ .

Find the derivative of $g(f(x))$ at $x=7$ with $f(x)=4(x-6)^2$ and $g(y)=3y^4+5y$ using the chain rule.

Find the derivative of $g(f(x))$ at $x=7$ , where $f(x)=4(x-6)^{2}$ and $g(y)=3y^{4}+5y$ . Evaluate $f'(7)$ and $g'(f(7))$ .

Find the derivative of $f(x)=3 x^{6}+8 x^{5}+12 x^{1/4}+5 x^{-5}+7 x+32$ and determine coefficients $a$ through $k$ .

Find the derivative of the function $f(x)=6 e^{3 x}+3 \ln (x^{5}+6 x^{4}(x-2)^{2}+\frac{6}{x}+29)$ and determine coefficients $a$ through $n$ .

Find the derivative of $f(x)=5 x^{6}+5 x^{5}+20 x^{1/5}+6 x^{-2}+5 x+29$ and determine coefficients $a$ through $k$ .

Find the derivative of $g(f(x))$ at $x=12$ where $f(x)=6(x-7)^{2}-144$ and $g(y)=8y^{2}+5y$ . Use the chain rule.

Find the derivative of $f(x)=8 e^{4 x}+7 \ln(x^{4}+3 x^{3}(x-2)^{2}+\frac{6}{x}+30)$ and determine coefficients $a$ to $n$ .

Find the derivative of $q(x)=g(x) f\left(x^{2}+4 x\right)$ and calculate $q^{\prime}(-3)$ . Also, find the derivative of $e^{3x}$ .

Show that there exists a $\delta>0$ so that if $x^{2}+y^{2}<\delta^{2}$ , then $|x^{2}+y^{2}+3xy+180xy^{5}|<\frac{1}{10,000}$ .

Find the derivative of $f(x)=8 e^{4 x}+7 \ln \left(x^{4}+3 x^{3}(x-2)^{2}+\frac{6}{x}+30\right)$ and determine coefficients $a$ to $n$ .

Find the derivative of $f(r)=(r-5)^{r}$ at $r=7$ and round to the nearest integer.

Find the derivative of $f(r)=(r-5)^{r}$ at $r=7$ and round to the nearest integer.

Ein PKW hat ein v-t-Diagramm:
a) Bestimme die Beschleunigung in den Abschnitten I (0-7s) und III (22-26s). b) Berechne den Weg bis $t=26 \mathrm{~s}$ .

Bestimme die Nullstellen und lokalen Extrema der Funktion $f(x)=x^{3}-3 x^{2}$ .

Find the derivative of $f(t)=(t-1)^{t}$ at $t=5$ and round to the nearest integer.

Find the derivative of $f(x)=4 x^{6}+7 x^{5}+20 x^{1/4}+6 x^{-2}+7 x+22$ and determine coefficients $a$ to $k$ .

Calculate the average rate of change of $f(x) = x^{3} + x$ on the intervals [-1,1] and [2,3].

Bestimme die Stammfunktion von $f(x)=-\frac{12}{49} x^{2}$ .

Find the derivative of $g(f(x))$ where $f(x)=3(x-8)^{2}-21$ and $g(y)=3y^{3}+3y$ , then evaluate at $x=11$ .

Find the derivative of the function $f(x)=x^{4}(8 x+8)$ using the Product or Quotient Rule.

Find the derivative of $f(x)=(x^{3}+1)(-2x^{2}-6)$ using the Product or Quotient Rule.

Find the derivative of $f(x)=\frac{-8 x}{2 x+3}$ using the Product or Quotient Rule.

Find the average rate of change of $h(t)=\sin (2 \pi t)+4$ on intervals $[\frac{1}{4}, \frac{3}{4}]$ and $[4,5]$ .

Find the derivative of the function using the Quotient Rule: $f(x)=\frac{-x^{2}+1}{7 x^{5}+7}$ .

Find the derivative of $f(x)=\frac{3 x^{2}-6}{8 x^{4}+10}$ using the Product or Quotient Rule.

Calculate the average rate of change of $q(x)=\sqrt{4x+1}$ on the intervals [0,1] and [0,2].

Find the population growth rate in thousands 21 years from now, given $P(t)=(0.6 t-9)(0.2 t+10)+20$ . Round to two decimal places.

Calculate the average rate of change of $p(x)=\frac{1}{x}$ on $\left[\frac{3}{4}, 1\right]$ and $\left[\frac{1}{4}, \frac{1}{2}\right]$ .

Find the limit: $\lim _{x \rightarrow 2} \frac{x^{2}-4}{x-2}=$

Find the value of $F(x)$ where $F(x)=\int_{0}^{4} \sqrt{x} \, dx$ .

Find $f'(t)$ for $f(t)=\frac{2}{t}$ at $t=a$ and $t=a+h$ . Answers: (a) $\frac{-2 h}{a(a+h)}$ , (b) $\frac{-2}{a(a+h)}$ .

Find the growth rate of the city's population, given by $P(t)=(0.7 t-5)(0.4 t+9)+75$ , in 16 years. Round to two decimal places.

Find the rate of temperature decrease at $h=1050$ ft, with $T(h)=65 e^{-0.000065 h}$ and rising at $807$ ft/hr.

Find the marginal revenue for the demand function $D(x)=\frac{143}{4 x+4}$ at $x=5$ . Round to the nearest cent.

Find the marginal revenue for the demand function $D(x)=\frac{155}{5 x+4}$ at $x=6$ , rounded to the nearest cent.

Find the derivative of $h(x)=\frac{4 \sqrt{x}}{3 x-8}$ using the Product or Quotient Rule.

Find the derivative of $s = t^{23} \ln |t|$ .

Find the derivative of the function $s=t^{17} \ln |t|$ .

Find the limit as $x$ approaches 0 from the left: $\lim _{x \rightarrow 0^{-}} e^{\frac{\sin x}{1-\cos x}}$ .

Find the derivative of $s=t^{9} \ln |t|$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{x^{3}-x}{x-1}$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{x^{2}-x}{x-1}$ .

Find the derivative of $h(t)=\left(3-\frac{5}{t^{2}}\right)\left(t^{3}+3 t+7\right)$ using the Product or Quotient Rule.

Find the derivative of $g(t)=\left(6+\frac{7}{t^{3}}\right)\left(4 t^{4}+2 t^{3}+5\right)$ using the Product or Quotient Rule.

Bestimmen Sie die ersten drei Ableitungen von $f$ für die Funktionen: a) $(2x-3)e^{x}$ , b) $(x^{2}+1)e^{x}$ , c) $(5x^{2}+1)e^{-x}$ , d) $5x^{2}e^{-\frac{1}{4}x}$ . Erklären Sie den Vorteil des Ausklammerns.

Find the limit: $\lim _{x \rightarrow -2} \frac{x^{4}-16}{x+2}$ .

Find the derivative of $h(t)=\left(7-\frac{6}{t^{2}}\right)\left(3 t^{2}+2 t-7\right)$ using the Product or Quotient Rule.

Find the derivative of the function: $\ln \left(\frac{2 x}{\sqrt[3]{4-x^{3}}}\right)$ .

Find the derivative of $h(t)=\left(4+\frac{1}{t}\right)\left(t^{4}-\frac{1}{7}\right)$ using the Product or Quotient Rule.

Find the limit: $\lim _{x \rightarrow 0} \frac{2 x \sin x}{\cos ^{2} x-\cos x}$

Find the derivative of $h(t)=\left(9+\frac{1}{t}\right)\left(t^{2}-\frac{1}{7}\right)$ using the Product or Quotient Rule.

Find the limit: $\lim _{x \rightarrow -2} \frac{(x+2)(x-2)(x+2)(x-2)}{x+2}$ .

Find the derivative of $g(u)=\left(5+\frac{1}{u}\right)\left(u^{5}-\frac{1}{6}\right)$ using Product or Quotient Rule.

Evaluate the integral: $\int_{1}^{9} \frac{2 x^{2}+x^{2} \sqrt{x}-1}{x^{2}} d x$

Bestimmen Sie den Grenzwert von $f(x)=\frac{x^{2}-9}{2x-6}$ für $x \to 3$ von links und rechts.

Find the tangent and normal line equations for the curve $y=x^{4}+5 e^{x}$ at the point $(0,5)$ .

Find the derivative of the function using the Product or Quotient Rule: $f(x)=\frac{-2 x^{4}-5}{-9 x+4}$ .

Find the rate of change of rock bass length $L$ at age $A=16$ using the model $L=0.0155 A^{3}-0.372 A^{2}+3.95 A+1.21$ . Round to three decimal places.

Find the derivative of the function $h(t)=(8 t^{2}+5 t)^{-4}$ using positive, negative, and fractional exponents.