Calculus

Problem 5901

Determine the limit as $x$ approaches infinity for $\frac{-10 x^{18}}{x^{18}+\log _{4} x}$ . Options: 0, does not exist, or exists and not 0.

See Solution

Problem 5902

Find the horizontal asymptotes of the function $f(x)=\frac{6 x-e^{x}}{3 x-2 x^{2}}$ .

See Solution

Problem 5903

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{12 \sin ^{2}(x)-9 \sin (x)}{5 x}$

See Solution

Problem 5904

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{-10 x}{9 \cos (x)-9}$ .

See Solution

Problem 5905

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{-3-\sin (x)}{4 x+5}$

See Solution

Problem 5906

Find the limit as $x$ approaches 0 for the expression $\frac{3 \sin (x)}{5 x^{2}}$ .

See Solution

Problem 5907

Is the function $f(x)$ continuous at $x=-2$ where $f(x)=\begin{cases} 2+2 x^{2}, & x<-2 \\ 6-2 x, & x \geq-2 \end{cases}$ ?

See Solution

Problem 5908

Is the function $f(x)$ continuous at $x=2$ where $f(x)=\left\{\begin{array}{ll}19-5 x^{2}, & x>-2 \\ -5-x, & x \leq-2\end{array}\right.$ ?

See Solution

Problem 5909

Check if the function $f(x)$ is continuous at $x=-3$ , where $f(x)=11-x^{2}$ for $x>-3$ and $f(x)=11+3x$ for $x \leq -3$ .

See Solution

Problem 5910

Find $\frac{d V}{d v}$ for $V(v)=-B l v$ with $B=8$ , $l=0.8$ . Then find $\frac{d V}{d t}$ if $v=4t+9$ .

See Solution

Problem 5911

A balloon's volume increases at $8 \mathrm{~m}^{3}/\mathrm{min}$ . Find the radius growth rate when the diameter is $2 \mathrm{~m}$ .

See Solution

Problem 5912

Calculate the half-life of cobalt-60 if its mass decreased from $0.800 \mathrm{~g}$ to $0.100 \mathrm{~g}$ in 15.75 years.

See Solution

Problem 5913

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

See Solution

Problem 5914

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin x}{\tan x}$ .

See Solution

Problem 5915

Find when the object changes direction given $v(t)=8(t-5)^{2}(t-3)^{3}(t-2)$ . Options: $t=0, t=2, t=3, t=5$ or none.

See Solution

Problem 5916

Using Ohm's Law $V=IR$ with $V=19$ , find the average rate of change of $I$ from $R=7$ to $R=7.1$ , and the rates at $R=7$ and $I=1.6$ .

See Solution

Problem 5917

Find when the object changes direction given $v(t)=8(t-5)^{2}(t-3)^{3}(t-2)$ . Options: $t=2, t=3, t=5$ .

See Solution

Problem 5918

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos x}{\tan x}$ .

See Solution

Problem 5919

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos ^{2} x}{x^{2}}$ .

See Solution

Problem 5920

Find $\theta^{\prime}(t)$ for the minimum angle between clock hands at 90 minutes. Answer in radians per minute.

See Solution

Problem 5921

Find the slope of the tangent line for $f(x)=x^{2}-11$ at $x=-4$ .

See Solution

Problem 5922

Find the slope of the tangent line for $f(x)=-4 x^{2}+10 x-6$ at $x=8$ .

See Solution

Problem 5923

Calculate the rate of change of BSA $=\frac{\sqrt{h m}}{60}$ for constant height $h=181$ with respect to mass $m$ . What are the units? Find the rate at $m=72$ and $m=85$ . BSA increases less rapidly with higher or lower mass?

See Solution

Problem 5924

Find the slope of the tangent line for the function $f(x)=x^{2}+2$ at the point where $x=8$ .

See Solution

Problem 5925

Find the tangent line equation for the function $f(x)=4 x^{2}+6 x-2$ at $x=6$ .

See Solution

Problem 5926

Evaluate the integral $\int \frac{x}{\sqrt{1-x}} \, dx$ using the substitution $u=\sqrt{1-x}$ .

See Solution

Problem 5927

Find the derivative $\frac{d y}{d x}$ using implicit differentiation for the equation $8 \sqrt{x}+\sqrt{y}=6$ .

See Solution

Problem 5928

Calculate the integral from 0 to 1 of $\frac{x^{2}}{\sqrt[3]{1+7 x^{3}}}$ . Provide your answer as a single fraction.

See Solution

Problem 5929

Find the first, second, and third derivatives of $y(x)=\frac{15}{49+x}$ . Show your work for $y'(x)$ , $y''(x)$ , and $y'''(x)$ .

See Solution

Problem 5930

Ein Körper fällt frei. a. Bestimme die Ableitung von $s(t)=\frac{1}{2} g t^{2}$ und interpretiere sie. b. Berechne die Aufprallgeschwindigkeit aus $100 \mathrm{~m}$ .

See Solution

Problem 5931

Find the third derivative of $y(x)=(3 x^{2}+6)(4 x^{3}+11)$ . What is $y^{\prime \prime \prime}(x)$ ?

See Solution

Problem 5932

Leiten Sie die folgenden Funktionen ab: a) $f(x)=3 x^{2} e^{-4 x}$ b) $f(x)=\frac{1}{2} x^{3} e^{2 x}$

See Solution

Problem 5933

Differentiate the function $f(x)=\cos (\ln (4 x))$ .

See Solution

Problem 5934

Find the rate of change of $y$ with respect to $x$ at $x=\frac{\pi}{6}$ for $y=\csc x-4 \cos x$ .

See Solution

Problem 5935

Evaluate the integral: $\int_{0}^{1} \frac{x^{3}-2 x^{2}+x}{\sqrt{4 x+1}} d x$

See Solution

Problem 5936

Find the derivatives of these functions: 1. $y=\sin \frac{6 x}{7}$ , 2. $y=\frac{\sin x}{\cos x}$ , 3. $y=\frac{\tan x}{1-\sec x}$ .

See Solution

Problem 5937

Find the derivatives of these functions: 1. $y=\sin \frac{3 x}{2}$ , 2. $y=\cos \frac{2 x^{3}}{3}$ , 3. $y=\tan \left(x^{5}\right)$ .

See Solution

Problem 5938

Find the function $f(x)$ and the value of $a$ from the derivative expression $\lim_{h \to 0} \frac{\sqrt[3]{8+h}-2}{h}$ .

See Solution

Problem 5939

Find the derivatives of these functions: 1. $y=\sin \frac{6 x}{7}$ , 2. $y=\frac{\sin x}{\cos x}$ , 3. $y=\frac{\tan x}{1-\sec x}$ .

See Solution

Problem 5940

Find the function $f(x)$ and the point $a$ where the derivative is given by $\lim _{x \rightarrow-7} \frac{\sqrt{16+x}-3}{x+7}$ .

See Solution

Problem 5941

Find the derivative of $\Gamma f(x)=1.5 \cdot e^{\frac{1}{30} x-1}+1^{5}+e^{-\frac{1}{30} x+1}-25$ .

See Solution

Problem 5942

Find the derivative of $y=\sec^{2}(3x)$ .

See Solution

Problem 5943

Find the function $f(x)$ and the value of $a$ given the derivative at $x=a$ as $\lim_{h \to 0} \frac{\sqrt[3]{8+h}-2}{h}$ .

See Solution

Problem 5944

Given continuous functions $f(x)$ , $g(x)$ , and $h(x)$ where $g(-1)=3$ , $h(-1)=2$ , and $\lim _{x \rightarrow-1} \frac{(f(x))^{2}-2 g(x)}{3 h(x)}=5$ , find $f(-1)$ if $f(x)$ is odd and $f(1)>0$ .

See Solution

Problem 5945

Find the function $f(x)$ if its derivative at $x=a$ is $\lim_{h \to 0} \frac{\sqrt[3]{27+h}-3}{h}$ .

See Solution

Problem 5946

Find the derivatives of these functions: 1. $y=\frac{1+\sin x}{1+\cos x}$ 2. $y=\sec ^{2} 3 x$

See Solution

Problem 5947

Find $f(-1)$ given that $g(-1)=4$ , $h(-1)=5$ , $\lim_{x \to -1} \frac{(f(x))^2 - 5g(x)}{4h(x)}=19$ , and $f(x)$ is odd with $f(1)>0$ .

See Solution

Problem 5948

Bestimme die Steigung und den Steigungswinkel von $f$ an $x_{0}$ für die Funktionen a) bis f) mit den gegebenen $x_{0}$ .

See Solution

Problem 5949

Find the function $f(x)$ and the point $a$ where the derivative is $\lim _{x \rightarrow -5} \frac{\sqrt{9+x}-2}{x+5}$ .

See Solution

Problem 5950

Finde die Wendetangente der Funktion $f(x) = 5(x+1)e^x$ .

See Solution

Problem 5951

Find the second derivative $y^{\prime \prime}$ of the function $y=\tan 3x$ .

See Solution

Problem 5952

Leiten Sie die Funktionen ab: a) $f(x)=\frac{4}{(2 x+1)^{2}}$ b) $f(x)=\frac{x}{(3 x+2)^{2}}$

See Solution

Problem 5953

Evaluate the integral $\int_{1}^{8} \frac{\sin \sqrt{x+1}}{\sqrt{x+1}} d x$ using the substitution $\sqrt{x+1}=t$ .

See Solution

Problem 5954

Bestimmen Sie die Ableitungsfunktionen für: a) $f(x)=5 x \cdot e^{x^{2}}$ , b) $f(x)=\cos \left(x^{3}-2 x\right)$ , c) $f(x)=\frac{3 x+1}{x^{3}-1}$ .

See Solution

Problem 5955

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt[3]{8+h}-2}{h}$ .

See Solution

Problem 5956

Gegeben ist die Funktion $f(x)=x^{3}-2 x$ . Bestimmen Sie die Tangente bei $P(0 | f(0))$ , Punkte mit Steigung 1 und 7, und ob Tangenten parallel zu $g: y=10 x-3$ sind.

See Solution

Problem 5957

Bestimmen Sie die Ableitungsfunktionen für die folgenden Funktionen: a) $f(x)=5 x \cdot e^{x^{2}}$ , b) $f(x)=\cos(x^{3}-2x)$ , c) $f(x)=\frac{3x+1}{x^{3}-1}$ .

See Solution

Problem 5958

Finde die Gleichung der Wendetangente für $f(x)=5 \cdot(x+1) \cdot e^{-\frac{x}{2}}$ .

See Solution

Problem 5959

Gegeben ist die Funktion $f(x)=x^{3}-2 x$ .
a) Finde die Tangentengleichung bei $P(0, f(0))$ . b) Bestimme die Punkte mit Steigung 1 und 7. c) Gibt es Punkte, wo Tangenten parallel zu $y=10 x-3$ sind?

See Solution

Problem 5960

Eine Flüssigkeit kühlt bei $20^{\circ}C$ ab.
a) Welche Art von Abkühlungsprozess ist das?
b) Was war die Anfangstemperatur?
c) Nach 3 Minuten war die Temperatur $73^{\circ}C$ . Wann sinkt die Temperatur erstmals um weniger als $1^{\circ}$ pro Minute?

See Solution

Problem 5961

Ein Ball wird mit $v_{0}=20 \frac{\mathrm{m}}{\mathrm{s}}$ nach oben geworfen. Berechne Wurfzeit $T$ , Steigzeit $t_{s}$ , Steighöhe $s_{H}$ und Zeitpunkte $t_{1}, t_{2}$ bei $s_{1}=5 \mathrm{~m}$ .

See Solution

Problem 5962

Leiten Sie die Funktionen ab: a) $f(x)=\frac{3}{1+e^{x}}$ b) $f(x)=\frac{4}{1-e^{-x}}$

See Solution

Problem 5963

A bucket catches water at $0.5 \mathrm{~cm}^{3}$ per minute. Find $\frac{dh}{dt}$ when $h=10 \mathrm{~cm}$ from $V=\frac{\pi h}{3}(h^{2}+3h+300)$ .

See Solution

Problem 5964

Solve the equation $x y' = y + x e^{2y/x}$ using the substitution $v = y/x$ . Find $y$ in terms of $x$ .

See Solution

Problem 5965

Find the rate of increase of a cube's edge length when its volume grows at $144 \mathrm{~cm}^{3}/\mathrm{s}$ and edge length is $4 \mathrm{~cm}$ .

See Solution

Problem 5966

Zeigen Sie, dass $F$ eine Stammfunktion von $6x^2 + 4$ ist, und berechnen Sie dann $\int^{3}(6x^2 + 4) dx$ .

See Solution

Problem 5967

Sand is dumped at $1.2 \mathrm{~m}^{3} / \mathrm{min}$ forming a cone. Find height growth rate when height is $3 \mathrm{~m}$ .

See Solution

Problem 5968

Zeigen Sie, dass F eine Stammfunktion ist und berechnen Sie die Integrale: a) $\int_{1}^{3}(6 x^{2}+4) dx ; F: x \mapsto 2 x^{3}+4 x-1$ b) $\int_{-4}^{4}(-x^{3}+3 x) dx ; F: x \mapsto -\frac{1}{4} x^{4}+1,5 x^{2}$

See Solution

Problem 5969

Berechne die Integrale und zeichne die Graphen von $\mathrm{f}$ im Intervall. a) $\int_{1}^{2}(x^{2}+1) dx$ b) $\int_{-1}^{2}(x-2) dx$ c) $\int_{0}^{3}(2-\frac{1}{2} x^{2}) dx$ d) $\int_{0}^{4} \sqrt{x} dx$ e) $\int_{-1}^{1} x^{3} dx$ f) $\int_{-1}^{2}(x^{3}-4x) dx$

See Solution

Problem 5970

Sand is dumped at $1.2 \mathrm{~m}^{3}/\mathrm{min}$ forming a cone. Find the height growth rate when the height is $3 \mathrm{~m}$ .

See Solution

Problem 5971

A toy car rolls off a 1.35 m table at 2.5 m/s. How long until it hits the ground?

See Solution

Problem 5972

Find $\frac{\partial f}{\partial s}$ for $f(x,y,z)=xyz$ where $x=\sin s \cos t$ , $y=\sin s \sin t$ , $z=\cos s$ at $s=\frac{\pi}{4}, t=\frac{\pi}{6}$ . Round to three decimal places.

See Solution

Problem 5973

Find the tangent line at $x=3$ using $g(3)=17$ and $g'(3)=-1$ to approximate $g(3.2)$ .

See Solution

Problem 5974

How far from the cliff base will your keys land if thrown horizontally at $7.5 \mathrm{~m/s}$ from a height of $70.8 \mathrm{~m}$ ?

See Solution

Problem 5975

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

See Solution

Problem 5976

Boyle's Law states $PV=C$ . Given $V=600 \mathrm{~cm}^3$ , $P=150 \mathrm{kPa}$ , and $\frac{dP}{dt}=20 \mathrm{kPa/min}$ , find $\frac{dV}{dt}$ .

See Solution

Problem 5977

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation $11x^{2} + y^{2} = 8$ using Implicit Differentiation.

See Solution

Problem 5978

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation $2 x + 4 y = \sin(y)$ .

See Solution

Problem 5979

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation $\cos y + 5 x = 5 y$ .

See Solution

Problem 5980

Find the first three derivatives of $f(x)$ and the general formula for the derivative if $f(x)=\frac{1}{x}$ .

See Solution

Problem 5981

Calculate the limit: $\lim _{x \rightarrow 0} \frac{\tan x-\sin x}{x \cos x}$ .

See Solution

Problem 5982

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos x}{-\tan x}$ .

See Solution

Problem 5983

Find the rate of change of respiration $R(g, t)$ for an amoeba at $(-1,0, \pi^{2})$ using given $g$ and $t$ formulas.

See Solution

Problem 5984

Trouver la vitesse instantanée de l'objet à $t=7$ pour la position $y(t)=\frac{1}{6 t}$ . $V_{t=7}=$

See Solution

Problem 5985

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation $x y+y^{2}=17$ .

See Solution

Problem 5986

Gegeben ist die Funktion $f_{a}(x)=\sqrt{3-a x}$ mit $a>0$ . Bestimmen Sie die Definitionsmenge, die Ableitung $f_{a}^{\prime}(x)$ und den Wert von $a$ , sodass $f_{a}^{\prime}(2)=-0,5$ .

See Solution

Problem 5987

Find the value of $\frac{\partial f}{\partial y}$ for the function $f(x,y,z)$ at the point $(1,2,3)$ .

See Solution

Problem 5988

Calculez la vitesse instantanée de l'objet à $t=a$ pour $y(t)=\frac{1}{6 t}$ , avec $a>0$ . $V_{t=a}=$

See Solution

Problem 5989

Dérivez $g(x)=x^{4}+\frac{1}{x^{4}}$ et trouvez les valeurs de $x$ pour lesquelles $g^{\prime}(x)=0$ . Séparez par des virgules. $x=$

See Solution

Problem 5990

Bestimmen Sie die Intervalle, in denen $f(x)=x^{3}+x$ monoton wachsend oder fallend ist, und untersuchen Sie das Krümmungsverhalten.

See Solution

Problem 5991

Find the Taylor series for $f(x)=3x^{4}-5x^{3}+x^{2}-100$ at $x=-2$ .

See Solution

Problem 5992

Graph a function with: domain $(-\infty, \infty)$ , $f(2)=1$ , limits at 2 and -1, no asymptotes.

See Solution

Problem 5993

Berechne die Integrale: a) $\int_{0}^{4 \pi} \cos x \, dx$ b) $\int_{-2}^{-1} (-2 x^{-3} + 3 x^{-3}) \, dx$

See Solution

Problem 5994

Find $x$ where the derivative $f^{\prime}(x)=0$ for $f(x)=\frac{x}{x^{2}+64}$ . Answers: $x=$

See Solution

Problem 5995

Estimate the cost of making 71 bicycles if $C(70)=5000$ and $C'(70)=65$ .

See Solution

Problem 5996

Estimate the cost of manufacturing 71 bicycles per day if $C(5000)$ and $C'(70) = 65$ .

See Solution

Problem 5997

Bestimmen Sie die Koordinaten von $P$ für maximalen Flächeninhalt des Rechtecks unter den Funktionen:
a) $f(x)=4-\frac{1}{2} x^{3}$ für $x \in[0 ; 2]$
b) $f(x)=\frac{1}{x^{2}}$ für $x \in[1 ; 3]$
c) $f(x)=\sqrt{4-x}$ für $x \in[0 ; 4]$

See Solution

Problem 5998

Berechnen Sie die 1. Ableitung von $f$ an $x_0$ als Grenzwert des Differenzenquotienten für: a) $f(x)=0,25 x^{2}$ , $x_{0}=2$ b) $f(x)=0,5 x^{2}-1$ , $x_{0}=-1$

See Solution

Problem 5999

A rectangle's width increases by 6 in/sec and length by 2 in/sec. Find the area increase rate at width 3 in and length 9 in. Let $W(t)$ and $L(t)$ be width and length.

See Solution

Problem 6000

Differentiate $y=\frac{6}{\pi}+\frac{4}{x^{2}+5}$ with respect to $x$ and simplify your answer.
$y^{\prime}=$

See Solution

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Calculus

Problem 5901

Determine the limit as xxx approaches infinity for −10x18x18+log⁡4x\frac{-10 x^{18}}{x^{18}+\log _{4} x}x18+log4​x−10x18​. Options: 0, does not exist, or exists and not 0.

Problem 5902

Find the horizontal asymptotes of the function f(x)=6x−ex3x−2x2f(x)=\frac{6 x-e^{x}}{3 x-2 x^{2}}f(x)=3x−2x26x−ex​.

Problem 5903

Evaluate the limit: lim⁡x→012sin⁡2(x)−9sin⁡(x)5x\lim _{x \rightarrow 0} \frac{12 \sin ^{2}(x)-9 \sin (x)}{5 x}limx→0​5x12sin2(x)−9sin(x)​

Problem 5904

Evaluate the limit: lim⁡x→0−10x9cos⁡(x)−9\lim _{x \rightarrow 0} \frac{-10 x}{9 \cos (x)-9}limx→0​9cos(x)−9−10x​.

Problem 5905

Evaluate the limit: lim⁡x→0−3−sin⁡(x)4x+5\lim _{x \rightarrow 0} \frac{-3-\sin (x)}{4 x+5}limx→0​4x+5−3−sin(x)​

Problem 5906

Find the limit as xxx approaches 0 for the expression 3sin⁡(x)5x2\frac{3 \sin (x)}{5 x^{2}}5x23sin(x)​.

Problem 5907

Is the function f(x)f(x)f(x) continuous at x=−2x=-2x=−2 where f(x)={2+2x2,x<−26−2x,x≥−2f(x)=\begin{cases} 2+2 x^{2}, & x<-2 \\ 6-2 x, & x \geq-2 \end{cases}f(x)={2+2x2,6−2x,​x<−2x≥−2​?

Problem 5908

Is the function f(x)f(x)f(x) continuous at x=2x=2x=2 where f(x)={19−5x2,x>−2−5−x,x≤−2f(x)=\left\{\begin{array}{ll}19-5 x^{2}, & x>-2 \\ -5-x, & x \leq-2\end{array}\right.f(x)={19−5x2,−5−x,​x>−2x≤−2​?

Problem 5909

Check if the function f(x)f(x)f(x) is continuous at x=−3x=-3x=−3, where f(x)=11−x2f(x)=11-x^{2}f(x)=11−x2 for x>−3x>-3x>−3 and f(x)=11+3xf(x)=11+3xf(x)=11+3x for x≤−3x \leq -3x≤−3.

Problem 5910

Find dVdv\frac{d V}{d v}dvdV​ for V(v)=−BlvV(v)=-B l vV(v)=−Blv with B=8B=8B=8, l=0.8l=0.8l=0.8. Then find dVdt\frac{d V}{d t}dtdV​ if v=4t+9v=4t+9v=4t+9.

Problem 5911

A balloon's volume increases at 8 m3/min8 \mathrm{~m}^{3}/\mathrm{min}8 m3/min. Find the radius growth rate when the diameter is 2 m2 \mathrm{~m}2 m.

Problem 5912

Calculate the half-life of cobalt-60 if its mass decreased from 0.800 g0.800 \mathrm{~g}0.800 g to 0.100 g0.100 \mathrm{~g}0.100 g in 15.75 years.

Problem 5913

Find the limit: lim⁡x→02sin⁡x−sin⁡2xxcos⁡x\lim _{x \rightarrow 0} \frac{2 \sin x - \sin 2x}{x \cos x}limx→0​xcosx2sinx−sin2x​.

Problem 5914

Find the limit: lim⁡x→0sin⁡xtan⁡x\lim _{x \rightarrow 0} \frac{\sin x}{\tan x}limx→0​tanxsinx​.

Problem 5915

Find when the object changes direction given v(t)=8(t−5)2(t−3)3(t−2)v(t)=8(t-5)^{2}(t-3)^{3}(t-2)v(t)=8(t−5)2(t−3)3(t−2). Options: t=0,t=2,t=3,t=5t=0, t=2, t=3, t=5t=0,t=2,t=3,t=5 or none.

Problem 5916

Using Ohm's Law V=IRV=IRV=IR with V=19V=19V=19, find the average rate of change of III from R=7R=7R=7 to R=7.1R=7.1R=7.1, and the rates at R=7R=7R=7 and I=1.6I=1.6I=1.6.

Problem 5917

Find when the object changes direction given v(t)=8(t−5)2(t−3)3(t−2)v(t)=8(t-5)^{2}(t-3)^{3}(t-2)v(t)=8(t−5)2(t−3)3(t−2). Options: t=2,t=3,t=5t=2, t=3, t=5t=2,t=3,t=5.

Problem 5918

Find the limit: lim⁡x→01−cos⁡xtan⁡x\lim _{x \rightarrow 0} \frac{1-\cos x}{\tan x}limx→0​tanx1−cosx​.

Problem 5919

Find the limit: lim⁡x→01−cos⁡2xx2\lim _{x \rightarrow 0} \frac{1-\cos ^{2} x}{x^{2}}limx→0​x21−cos2x​.

Problem 5920

Find θ′(t)\theta^{\prime}(t)θ′(t) for the minimum angle between clock hands at 90 minutes. Answer in radians per minute.

Problem 5921

Find the slope of the tangent line for f(x)=x2−11f(x)=x^{2}-11f(x)=x2−11 at x=−4x=-4x=−4.

Problem 5922

Find the slope of the tangent line for f(x)=−4x2+10x−6f(x)=-4 x^{2}+10 x-6f(x)=−4x2+10x−6 at x=8x=8x=8.

Problem 5923

Calculate the rate of change of BSA =hm60=\frac{\sqrt{h m}}{60}=60hm​​ for constant height h=181h=181h=181 with respect to mass mmm. What are the units? Find the rate at m=72m=72m=72 and m=85m=85m=85. BSA increases less rapidly with higher or lower mass?

Problem 5924

Find the slope of the tangent line for the function f(x)=x2+2f(x)=x^{2}+2f(x)=x2+2 at the point where x=8x=8x=8.

Problem 5925

Find the tangent line equation for the function f(x)=4x2+6x−2f(x)=4 x^{2}+6 x-2f(x)=4x2+6x−2 at x=6x=6x=6.

Problem 5926

Evaluate the integral ∫x1−x dx\int \frac{x}{\sqrt{1-x}} \, dx∫1−x​x​dx using the substitution u=1−xu=\sqrt{1-x}u=1−x​.

Problem 5927

Find the derivative dydx\frac{d y}{d x}dxdy​ using implicit differentiation for the equation 8x+y=68 \sqrt{x}+\sqrt{y}=68x​+y​=6.

Problem 5928

Calculate the integral from 0 to 1 of x21+7x33\frac{x^{2}}{\sqrt[3]{1+7 x^{3}}}31+7x3​x2​. Provide your answer as a single fraction.

Problem 5929

Find the first, second, and third derivatives of y(x)=1549+xy(x)=\frac{15}{49+x}y(x)=49+x15​. Show your work for y′(x)y'(x)y′(x), y′′(x)y''(x)y′′(x), and y′′′(x)y'''(x)y′′′(x).

Problem 5930

Ein Körper fällt frei. a. Bestimme die Ableitung von s(t)=12gt2s(t)=\frac{1}{2} g t^{2}s(t)=21​gt2 und interpretiere sie. b. Berechne die Aufprallgeschwindigkeit aus 100 m100 \mathrm{~m}100 m.

Problem 5931

Find the third derivative of y(x)=(3x2+6)(4x3+11)y(x)=(3 x^{2}+6)(4 x^{3}+11)y(x)=(3x2+6)(4x3+11). What is y′′′(x)y^{\prime \prime \prime}(x)y′′′(x)?

Problem 5932

Leiten Sie die folgenden Funktionen ab: a) f(x)=3x2e−4xf(x)=3 x^{2} e^{-4 x}f(x)=3x2e−4x b) f(x)=12x3e2xf(x)=\frac{1}{2} x^{3} e^{2 x}f(x)=21​x3e2x

Problem 5933

Differentiate the function f(x)=cos⁡(ln⁡(4x))f(x)=\cos (\ln (4 x))f(x)=cos(ln(4x)).

Problem 5934

Find the rate of change of yyy with respect to xxx at x=π6x=\frac{\pi}{6}x=6π​ for y=csc⁡x−4cos⁡xy=\csc x-4 \cos xy=cscx−4cosx.

Problem 5935

Evaluate the integral: ∫01x3−2x2+x4x+1dx\int_{0}^{1} \frac{x^{3}-2 x^{2}+x}{\sqrt{4 x+1}} d x∫01​4x+1​x3−2x2+x​dx

Problem 5936

Find the derivatives of these functions: 1. y=sin⁡6x7y=\sin \frac{6 x}{7}y=sin76x​, 2. y=sin⁡xcos⁡xy=\frac{\sin x}{\cos x}y=cosxsinx​, 3. y=tan⁡x1−sec⁡xy=\frac{\tan x}{1-\sec x}y=1−secxtanx​.

Problem 5937

Find the derivatives of these functions: 1. y=sin⁡3x2y=\sin \frac{3 x}{2}y=sin23x​, 2. y=cos⁡2x33y=\cos \frac{2 x^{3}}{3}y=cos32x3​, 3. y=tan⁡(x5)y=\tan \left(x^{5}\right)y=tan(x5).

Problem 5938

Find the function f(x)f(x)f(x) and the value of aaa from the derivative expression lim⁡h→08+h3−2h\lim_{h \to 0} \frac{\sqrt[3]{8+h}-2}{h}limh→0​h38+h​−2​.

Problem 5939

Find the derivatives of these functions: 1. y=sin⁡6x7y=\sin \frac{6 x}{7}y=sin76x​, 2. y=sin⁡xcos⁡xy=\frac{\sin x}{\cos x}y=cosxsinx​, 3. y=tan⁡x1−sec⁡xy=\frac{\tan x}{1-\sec x}y=1−secxtanx​.

Problem 5940

Determine the limit as $x$ approaches infinity for $\frac{-10 x^{18}}{x^{18}+\log _{4} x}$ . Options: 0, does not exist, or exists and not 0.

Find the horizontal asymptotes of the function $f(x)=\frac{6 x-e^{x}}{3 x-2 x^{2}}$ .

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{12 \sin ^{2}(x)-9 \sin (x)}{5 x}$

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{-10 x}{9 \cos (x)-9}$ .

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{-3-\sin (x)}{4 x+5}$

Find the limit as $x$ approaches 0 for the expression $\frac{3 \sin (x)}{5 x^{2}}$ .

Is the function $f(x)$ continuous at $x=-2$ where $f(x)=\begin{cases} 2+2 x^{2}, & x<-2 \\ 6-2 x, & x \geq-2 \end{cases}$ ?

Is the function $f(x)$ continuous at $x=2$ where $f(x)=\left\{\begin{array}{ll}19-5 x^{2}, & x>-2 \\ -5-x, & x \leq-2\end{array}\right.$ ?

Check if the function $f(x)$ is continuous at $x=-3$ , where $f(x)=11-x^{2}$ for $x>-3$ and $f(x)=11+3x$ for $x \leq -3$ .

Find $\frac{d V}{d v}$ for $V(v)=-B l v$ with $B=8$ , $l=0.8$ . Then find $\frac{d V}{d t}$ if $v=4t+9$ .

A balloon's volume increases at $8 \mathrm{~m}^{3}/\mathrm{min}$ . Find the radius growth rate when the diameter is $2 \mathrm{~m}$ .

Calculate the half-life of cobalt-60 if its mass decreased from $0.800 \mathrm{~g}$ to $0.100 \mathrm{~g}$ in 15.75 years.

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

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin x}{\tan x}$ .

Find when the object changes direction given $v(t)=8(t-5)^{2}(t-3)^{3}(t-2)$ . Options: $t=0, t=2, t=3, t=5$ or none.

Using Ohm's Law $V=IR$ with $V=19$ , find the average rate of change of $I$ from $R=7$ to $R=7.1$ , and the rates at $R=7$ and $I=1.6$ .

Find when the object changes direction given $v(t)=8(t-5)^{2}(t-3)^{3}(t-2)$ . Options: $t=2, t=3, t=5$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos x}{\tan x}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos ^{2} x}{x^{2}}$ .

Find $\theta^{\prime}(t)$ for the minimum angle between clock hands at 90 minutes. Answer in radians per minute.

Find the slope of the tangent line for $f(x)=x^{2}-11$ at $x=-4$ .

Find the slope of the tangent line for $f(x)=-4 x^{2}+10 x-6$ at $x=8$ .

Calculate the rate of change of BSA $=\frac{\sqrt{h m}}{60}$ for constant height $h=181$ with respect to mass $m$ . What are the units? Find the rate at $m=72$ and $m=85$ . BSA increases less rapidly with higher or lower mass?

Find the slope of the tangent line for the function $f(x)=x^{2}+2$ at the point where $x=8$ .

Find the tangent line equation for the function $f(x)=4 x^{2}+6 x-2$ at $x=6$ .

Evaluate the integral $\int \frac{x}{\sqrt{1-x}} \, dx$ using the substitution $u=\sqrt{1-x}$ .

Find the derivative $\frac{d y}{d x}$ using implicit differentiation for the equation $8 \sqrt{x}+\sqrt{y}=6$ .

Calculate the integral from 0 to 1 of $\frac{x^{2}}{\sqrt[3]{1+7 x^{3}}}$ . Provide your answer as a single fraction.

Find the first, second, and third derivatives of $y(x)=\frac{15}{49+x}$ . Show your work for $y'(x)$ , $y''(x)$ , and $y'''(x)$ .

Ein Körper fällt frei. a. Bestimme die Ableitung von $s(t)=\frac{1}{2} g t^{2}$ und interpretiere sie. b. Berechne die Aufprallgeschwindigkeit aus $100 \mathrm{~m}$ .

Find the third derivative of $y(x)=(3 x^{2}+6)(4 x^{3}+11)$ . What is $y^{\prime \prime \prime}(x)$ ?

Leiten Sie die folgenden Funktionen ab: a) $f(x)=3 x^{2} e^{-4 x}$ b) $f(x)=\frac{1}{2} x^{3} e^{2 x}$

Differentiate the function $f(x)=\cos (\ln (4 x))$ .

Find the rate of change of $y$ with respect to $x$ at $x=\frac{\pi}{6}$ for $y=\csc x-4 \cos x$ .

Evaluate the integral: $\int_{0}^{1} \frac{x^{3}-2 x^{2}+x}{\sqrt{4 x+1}} d x$

Find the derivatives of these functions: 1. $y=\sin \frac{6 x}{7}$ , 2. $y=\frac{\sin x}{\cos x}$ , 3. $y=\frac{\tan x}{1-\sec x}$ .

Find the derivatives of these functions: 1. $y=\sin \frac{3 x}{2}$ , 2. $y=\cos \frac{2 x^{3}}{3}$ , 3. $y=\tan \left(x^{5}\right)$ .

Find the function $f(x)$ and the value of $a$ from the derivative expression $\lim_{h \to 0} \frac{\sqrt[3]{8+h}-2}{h}$ .

Find the derivatives of these functions: 1. $y=\sin \frac{6 x}{7}$ , 2. $y=\frac{\sin x}{\cos x}$ , 3. $y=\frac{\tan x}{1-\sec x}$ .

Find the function $f(x)$ and the point $a$ where the derivative is given by $\lim _{x \rightarrow-7} \frac{\sqrt{16+x}-3}{x+7}$ .

Find the derivative of $\Gamma f(x)=1.5 \cdot e^{\frac{1}{30} x-1}+1^{5}+e^{-\frac{1}{30} x+1}-25$ .

Find the derivative of $y=\sec^{2}(3x)$ .

Find the function $f(x)$ and the value of $a$ given the derivative at $x=a$ as $\lim_{h \to 0} \frac{\sqrt[3]{8+h}-2}{h}$ .

Find the function $f(x)$ if its derivative at $x=a$ is $\lim_{h \to 0} \frac{\sqrt[3]{27+h}-3}{h}$ .

Find the derivatives of these functions: 1. $y=\frac{1+\sin x}{1+\cos x}$ 2. $y=\sec ^{2} 3 x$

Find $f(-1)$ given that $g(-1)=4$ , $h(-1)=5$ , $\lim_{x \to -1} \frac{(f(x))^2 - 5g(x)}{4h(x)}=19$ , and $f(x)$ is odd with $f(1)>0$ .

Bestimme die Steigung und den Steigungswinkel von $f$ an $x_{0}$ für die Funktionen a) bis f) mit den gegebenen $x_{0}$ .

Find the function $f(x)$ and the point $a$ where the derivative is $\lim _{x \rightarrow -5} \frac{\sqrt{9+x}-2}{x+5}$ .

Finde die Wendetangente der Funktion $f(x) = 5(x+1)e^x$ .

Find the second derivative $y^{\prime \prime}$ of the function $y=\tan 3x$ .

Leiten Sie die Funktionen ab: a) $f(x)=\frac{4}{(2 x+1)^{2}}$ b) $f(x)=\frac{x}{(3 x+2)^{2}}$

Evaluate the integral $\int_{1}^{8} \frac{\sin \sqrt{x+1}}{\sqrt{x+1}} d x$ using the substitution $\sqrt{x+1}=t$ .

Bestimmen Sie die Ableitungsfunktionen für: a) $f(x)=5 x \cdot e^{x^{2}}$ , b) $f(x)=\cos \left(x^{3}-2 x\right)$ , c) $f(x)=\frac{3 x+1}{x^{3}-1}$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt[3]{8+h}-2}{h}$ .

Gegeben ist die Funktion $f(x)=x^{3}-2 x$ . Bestimmen Sie die Tangente bei $P(0 | f(0))$ , Punkte mit Steigung 1 und 7, und ob Tangenten parallel zu $g: y=10 x-3$ sind.

Bestimmen Sie die Ableitungsfunktionen für die folgenden Funktionen: a) $f(x)=5 x \cdot e^{x^{2}}$ , b) $f(x)=\cos(x^{3}-2x)$ , c) $f(x)=\frac{3x+1}{x^{3}-1}$ .

Finde die Gleichung der Wendetangente für $f(x)=5 \cdot(x+1) \cdot e^{-\frac{x}{2}}$ .

Gegeben ist die Funktion $f(x)=x^{3}-2 x$ .
a) Finde die Tangentengleichung bei $P(0, f(0))$ . b) Bestimme die Punkte mit Steigung 1 und 7. c) Gibt es Punkte, wo Tangenten parallel zu $y=10 x-3$ sind?

Eine Flüssigkeit kühlt bei $20^{\circ}C$ ab.
a) Welche Art von Abkühlungsprozess ist das?
b) Was war die Anfangstemperatur?
c) Nach 3 Minuten war die Temperatur $73^{\circ}C$ . Wann sinkt die Temperatur erstmals um weniger als $1^{\circ}$ pro Minute?

Ein Ball wird mit $v_{0}=20 \frac{\mathrm{m}}{\mathrm{s}}$ nach oben geworfen. Berechne Wurfzeit $T$ , Steigzeit $t_{s}$ , Steighöhe $s_{H}$ und Zeitpunkte $t_{1}, t_{2}$ bei $s_{1}=5 \mathrm{~m}$ .

Leiten Sie die Funktionen ab: a) $f(x)=\frac{3}{1+e^{x}}$ b) $f(x)=\frac{4}{1-e^{-x}}$

A bucket catches water at $0.5 \mathrm{~cm}^{3}$ per minute. Find $\frac{dh}{dt}$ when $h=10 \mathrm{~cm}$ from $V=\frac{\pi h}{3}(h^{2}+3h+300)$ .

Solve the equation $x y' = y + x e^{2y/x}$ using the substitution $v = y/x$ . Find $y$ in terms of $x$ .

Find the rate of increase of a cube's edge length when its volume grows at $144 \mathrm{~cm}^{3}/\mathrm{s}$ and edge length is $4 \mathrm{~cm}$ .

Zeigen Sie, dass $F$ eine Stammfunktion von $6x^2 + 4$ ist, und berechnen Sie dann $\int^{3}(6x^2 + 4) dx$ .

Sand is dumped at $1.2 \mathrm{~m}^{3} / \mathrm{min}$ forming a cone. Find height growth rate when height is $3 \mathrm{~m}$ .

Sand is dumped at $1.2 \mathrm{~m}^{3}/\mathrm{min}$ forming a cone. Find the height growth rate when the height is $3 \mathrm{~m}$ .

Find the tangent line at $x=3$ using $g(3)=17$ and $g'(3)=-1$ to approximate $g(3.2)$ .

How far from the cliff base will your keys land if thrown horizontally at $7.5 \mathrm{~m/s}$ from a height of $70.8 \mathrm{~m}$ ?