Calculus

Problem 14801

Estimate $\Delta y$ for $y=\sin(2x)$ at $x=0$ with $\Delta x=0.1$ using linear approximation and find error \%.

See Solution

Problem 14802

Find the tangent line equation at the point on the curve where $x=t^{3}+1$ , $y=t^{10}+t$ , for $t=-1$ .

See Solution

Problem 14803

Find the equation of the tangent line to $f(x)=\frac{5}{3}-2x-x^{2}-\frac{1}{3}x^{3}$ with maximum slope.

See Solution

Problem 14804

Find the derivative of the function $\frac{1}{x}$ with respect to $x$ .

See Solution

Problem 14805

Find the maximum area of a rectangle with one corner on the graph of $y=\frac{10}{x^{2}+1}$ and adjacent sides on the axes.

See Solution

Problem 14806

Find the absolute max and min of $f(x)=2-6 x^{2}$ for $-3 \leq x \leq 2$ . What are their values and locations?

See Solution

Problem 14807

Find the derivative of $y$ where $y=\cos^{-1}(6x^3)$ .

See Solution

Problem 14808

Find the derivative of $\ln \left(\frac{x}{x+4}\right)$ with respect to $x$ .

See Solution

Problem 14809

Apply Rolle's Theorem to $f(x)=2 x^{2}-8 x-3$ on $[0,4]$ . How many $c$ satisfy $f^{\prime}(c)=0$ ? What is $c$ ? Also, which condition fails for $[-3,11]$ : $f(a) \neq f(b)$ , continuity, or differentiability?

See Solution

Problem 14810

Determine the local min and max of the function $f(x)=1+3x+27x^{-1}$ .

See Solution

Problem 14811

Find the average slope of $f(x)=5 x^{3}-5 x$ on $[-2,2]$ and determine the two values of $c$ where $f^{\prime}(c)$ equals this slope.

See Solution

Problem 14812

Apply Rolle's Theorem to $f(x)=2 x^{2}-8 x-3$ on $[0,4]$ . How many $c$ values exist where $f'(c)=0$ ? What is $c$ ?

See Solution

Problem 14813

Find the derivative of $h(x)=e^{x^{4}-x+5}$ .

See Solution

Problem 14814

Find the inflection points of $f(x) = 12x^5 + 30x^4 - 160x^3 + 2$ and determine $D$ , $E$ , and $F$ .

See Solution

Problem 14815

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{\ln (x)}{x^{20}}=$

See Solution

Problem 14816

Apply Rolle's Theorem to $f(x)=2x^{2}-8x-7$ on $[0,4]$ . How many $c$ values exist where $f'(c)=0$ ? What is $c$ ?

See Solution

Problem 14817

Evaluate these limits: (a) $\lim_{x \to \infty}(-27x^2 + 26x^3) =$ and (b) $\lim_{x \to -\infty}(-27x^2 + 26x^3) =$ .

See Solution

Problem 14818

Evaluate these limits: (a) $\lim _{x \rightarrow \infty} \frac{\sqrt{5+5 x^{2}}}{6+4 x}=$ ; (b) $\lim _{x \rightarrow-\infty} \frac{\sqrt{5+5 x^{2}}}{6+4 x}=$

See Solution

Problem 14819

Show there are no tangents to $y=6 x^{3}+2 x^{2}$ with slope -5. Verify with a graphing calculator.

See Solution

Problem 14820

Find the first and second derivatives of $g(x)=6 x^{3}-9 x^{2}-108 x$ . Evaluate $g^{\prime \prime}(3)$ .

See Solution

Problem 14821

Find functions $g$ where $g^{\prime}(x)=\frac{5 x^{2}+3 x+5}{\sqrt{x}}$ .

See Solution

Problem 14822

Find the horizontal asymptotes by calculating these limits:
1. $\lim _{x \rightarrow \infty} \frac{-3 x}{2+2 x}$
2. $\lim _{x \rightarrow-\infty} \frac{5 x-9}{x^{3}+11 x-12}$
3. $\lim _{x \rightarrow \infty} \frac{x^{2}-3 x-6}{13-15 x^{2}}$
4. $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+6 x}}{12-9 x}$
5. $\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}+6 x}}{12-9 x}$

See Solution

Problem 14823

Evaluate the limit using L'Hôpital's Rule: $\lim _{x \rightarrow 0}\left(\cot (21 x)-\frac{1}{21 x}\right) =$

See Solution

Problem 14824

Find $f^{\prime}(x)$ from $f^{\prime \prime}(x)=8 x-3$ with $f^{\prime}(-3)=-1$ , and $f(-3)=2$ . Then find $f(2)$ .

See Solution

Problem 14825

Find the antiderivative $F(x)$ of $f(x)=\frac{7}{x^{2}}-\frac{7}{x^{7}}$ with $F(1)=0$ .

See Solution

Problem 14826

Find $f(t)$ given that $f^{\prime \prime}(t)=2(3 t-2)$ and the conditions $f^{\prime}(1)=2$ , $f(1)=4$ .

See Solution

Problem 14827

Find $f(0)$ if $f'(t)=3 \sin(2t)$ and $f\left(\frac{\pi}{2}\right)=6$ .

See Solution

Problem 14828

Identify which functions are anti-derivatives of $f(x)=\sin (x) \cos (x)$ from the given options: (A) $F_{1}(x)$ , (B) $F_{2}(x)$ , (C) $F_{3}(x)$ .

See Solution

Problem 14829

Evaluate the limit: $\lim _{x \rightarrow 0}\left(\frac{1}{x^{2}}-\csc ^{2}(x)\right)=$

See Solution

Problem 14830

True or false: For a differentiable function $f$ , is it true that $\frac{d}{d x}[f(\sqrt{x})]=\frac{f^{\prime}(x)}{2 \sqrt{x}}$ ?

See Solution

Problem 14831

Given functions $f$ and $g$ with values in the table, compute:
1. $\left.\frac{d}{d x} f(g(x))\right|_{x=1}$
2. $\left.\frac{d}{d x} g(f(x))\right|_{x=1}$
3. $\left.\frac{d}{d x} f\left(x^{3}-x^{2}\right)\right|_{x=2}$
4. $\left.\frac{d}{d x}\left(g(x)^{3}-g(x)^{2}\right)\right|_{x=2}$

See Solution

Problem 14832

True or false: For a one-to-one function, is the derivative of the inverse given by $\frac{d}{dx}[f^{-1}(x)] = -f^{-2}(x) f'(x)$ ?

See Solution

Problem 14833

Choose the option equal to the derivative of $\sin^{-1}(a x)$ :
A: $\frac{a}{1+(a x)^{2}}$ B: $\frac{a}{\sqrt{1-(a x)^{2}}$ C: $-a \sin^{-2}(a x)$ D: $a \cos(a x)$ E: $-a \sin^{-2}(a x) \cos(a x)$ F: None of the above

See Solution

Problem 14834

Given the functions $f$ and $g$ , calculate the following derivatives at specified points:
1. $\left.\frac{d}{d x} f(g(x))\right|_{x=1}$
2. $\left.\frac{d}{d x} g(f(x))\right|_{x=1}$
3. $\left.\frac{d}{d x} f\left(x^{3}-x^{2}\right)\right|_{x=2}$
4. $\left.\frac{d}{d x}\left(g(x)^{3}-g(x)^{2}\right)\right|_{x=2}$

See Solution

Problem 14835

Find the unique anti-derivative $F$ of $f(x)=\frac{3 e^{3 x}+2 e^{2 x}+4 e^{-x}}{e^{2 x}}$ with $F(0)=0$ .

See Solution

Problem 14836

Given $a(t)=5 \sin(t)$ and $v(0)=-9 \ \mathrm{m/s}$ , find $v(t)$ , displacement from 0 to 3, and total distance traveled.

See Solution

Problem 14837

Find the derivative of $\ln\left(\frac{13}{x}\right)$ with respect to $x$ .

See Solution

Problem 14838

Evaluate the limit: $\lim _{x \rightarrow \infty} x^{13 / x}=$

See Solution

Problem 14839

Find the displacement and total distance of the object with velocity $v(t)=t^{2}-2t-3$ from $t=0$ to $t=7$ .

See Solution

Problem 14840

An object's acceleration is $a(t)=4 \sin (t)$ with initial velocity $v(0)=-2 \mathrm{~m/s}$ . Find $v(t)$ , displacement, and total distance from $t=0$ to $t=3$ .

See Solution

Problem 14841

Object accelerates with $a(t)=4 \sin(t)$ , $v(0)=-2$ . Find $v(t)$ , displacement from 0 to 3, and total distance traveled.

See Solution

Problem 14842

Given values for functions $f$ and $g$ , compute the following derivatives at specified points:
1. $\left.\frac{d}{d x} f(g(x))\right|_{x=1}$
2. $\left.\frac{d}{d x} g(f(x))\right|_{x=1}$
3. $\left.\frac{d}{d x} f\left(x^{3}-x^{2}\right)\right|_{x=2}$
4. $\left.\frac{d}{d x}\left(g(x)^{3}-g(x)^{2}\right)\right|_{x=2}$

See Solution

Problem 14843

Determine the horizontal asymptotes by calculating these limits:
1. $\lim _{x \rightarrow \infty} \frac{-4 x}{2+2 x}$
2. $\lim _{x \rightarrow-\infty} \frac{5 x-6}{x^{3}+9 x-10}$
3. $\lim _{x \rightarrow \infty} \frac{x^{2}-13 x-12}{8-10 x^{2}}$
4. $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+5 x}}{7-6 x}$
5. $\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}+5 x}}{7-6 x}$

See Solution

Problem 14844

Evaluate the limit: $\lim _{x \rightarrow 1} \frac{e^{x}-e}{4 \ln (x)}$ .

See Solution

Problem 14845

Find the limit: $\lim _{x \rightarrow 0} \frac{15 x^{3}}{\sin (x)-x}$ .

See Solution

Problem 14846

Find $H(b)=\lim _{x \rightarrow \infty} \frac{\ln \left(1+b^{x}\right)}{x}$ for $b>1$ and $0<b \leq 1$ .

See Solution

Problem 14847

Evaluate the limit: $\lim _{x \rightarrow \infty} e^{-x}(x^{3}-19 x^{2}+4)=$

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Problem 14848

Approximate $\sqrt{64.4}$ using the tangent line of $f(x)=\sqrt{x}$ at $x=64$ . Find $L(x)=$ . Provide 9 significant figures.

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Problem 14849

Given the speed function $f(t)=\frac{44000}{(t+20)^{2}}$ for $0 \leq t \leq 6$ , explain the distance estimates using sums and averages.

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Problem 14850

Find the local max and min of $f(x)=2+9x+36x^{-1}$ . Determine the $x$ -values and their function values.

See Solution

Problem 14851

Find $\frac{d x}{d t}$ at $x=-4$ given $y=2 x^{2}-4$ and $\frac{d y}{d t}=-4$ .

See Solution

Problem 14852

Evaluate the integral: $\int \frac{5+4 x}{1+x^{2}} d x$

See Solution

Problem 14853

Find $H(b)=\lim _{x \rightarrow \infty} \frac{\ln(1+b^{x})}{x}$ for $b>1$ and $0<b \leq 1$ .

See Solution

Problem 14854

Find the limit as $x$ approaches 0 from the right: $\lim _{x \rightarrow 0^{+}} x^{\sin (2 x)}$ .

See Solution

Problem 14855

Estimate the area between the $\mathrm{x}$ -axis and $y=\sin x$ for $0 \leq x \leq \pi$ .

See Solution

Problem 14856

Estimate $\Delta y$ for $y=\sin(4x)$ with $\Delta x=0.2$ at $x=0$ using linear approximation. Find the percentage error.

See Solution

Problem 14857

Find the inflection points of $f(x) = 12x^{5} + 60x^{4} - 240x^{3} + 6$ . What are the values of $D, E$ , and $F$ ?

See Solution

Problem 14858

Find the total number of cars passing an intersection from $t=0$ to $t=2$ for $r(t)=300+700t-150t^{2}$ .

See Solution

Problem 14859

Evaluate the limit using L'Hôpital's Rule:
$\lim _{x \rightarrow 1}(1+\ln (x))^{4 /(x-1)}=$

See Solution

Problem 14860

Bestimmen Sie die Ableitung der Funktion $g(x) = -\frac{10}{x}$ , also $g'(x)$ .

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Problem 14861

Find the total words memorized in 15 minutes using the rate $M^{\prime}(t)=-0.002 t^{2}+0.8 t$ .

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Problem 14862

Find the displacement formula for a particle with velocity $v(t)=14-5t$ and total distance traveled at $t=0.5$ seconds.

See Solution

Problem 14863

Find the displacement and total distance for the velocity $v(t)=t^{2}-t-20$ from $t=0$ to $t=8$ .

See Solution

Problem 14864

Find the displacement formula for a particle with velocity $v(t)=12-6t$ and total distance at $t=0.5$ seconds.

See Solution

Problem 14865

Find the number of cars passing through an intersection from 6 am to 9 am given $r(t)=500+1000 t-210 t^{2}$ .

See Solution

Problem 14866

Find the average slope of the function $f(x)=3 x^{3}-6 x$ on $[-2,2]$ and identify the two values of $c$ where $f'(c)$ equals this slope.

See Solution

Problem 14867

Evaluate these limits: (a) $\lim _{x \rightarrow \infty} \frac{\sqrt{11+6 x^{2}}}{2+9 x}=$ (b) $\lim _{x \rightarrow-\infty} \frac{\sqrt{11+6 x^{2}}}{2+9 x}=$

See Solution

Problem 14868

Kreuzen Sie an, ob die Ableitungen für die Funktionen $f_{4}(x)=x^{76}+98$ und $f_{5}(x)=\sqrt[4]{x^{2}}$ korrekt sind.

See Solution

Problem 14869

Find the third derivative of $f(x) = x^{4} - 4.5x^{2} + 5.0625$ .

See Solution

Problem 14870

An object moves with velocity $v(t)=t^{2}+4t-5$ . Find displacement and total distance from $t=0$ to $t=7$ .

See Solution

Problem 14871

Find an antiderivative of the function $f(x)=4 x^{8}+5 x^{5}-9 x^{2}-4$ . Do not include $+c$ .

See Solution

Problem 14872

Evaluate these limits: (a) $\lim _{x \rightarrow \infty} \frac{\sqrt{11+6 x^{2}}}{2+9 x}=$ , (b) $\lim _{x \rightarrow-\infty} \frac{\sqrt{11+6 x^{2}}}{2+9 x}=$ .

See Solution

Problem 14873

Find values of $c$ where $f'(c)=0$ for $f(x)=2x^2-20x-6$ on $[3,7]$ . What condition fails for $[0,14]$ ?

See Solution

Problem 14874

Find the average slope of $f(x)=3 x^{3}-6 x$ on $[-2,2]$ and the values of $c$ where $f^{\prime}(c)$ equals this slope.

See Solution

Problem 14875

Determine the extrema and inflection points of the function $f(x)=x^{4}-4.5 x^{2}+5.0625$ .

See Solution

Problem 14876

The object's acceleration is $a(t)=6 \sin (t)$ and initial velocity is $v(0)=-3 \mathrm{~m/s}$ .
a) Find $v(t)$ . b) Calculate displacement from time 0 to 3. c) Determine total distance traveled from time 0 to 3.

See Solution

Problem 14877

Berechnen Sie die Summe: $\frac{\pi}{2 n} \sum_{i=1}^{n} \cos \left(\frac{i \pi}{2 n}\right)$ .

See Solution

Problem 14878

Find the derivative of the function $f(x) = e^{5 x^{2}}$ .

See Solution

Problem 14879

Find an antiderivative of the function $f(x)=4 x^{8}+5 x^{5}-9 x^{2}-4$ . Do not include $+c$ .

See Solution

Problem 14880

Untersuchen Sie die Funktion $f_{2}(t)=-0,01 a t^{3}+0,3 a t^{2}+50$ für $a=2$ . Bestimmen Sie den maximalen Kurs und die Trendwende.

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Problem 14881

Eine Aktie kostet 50 \ $. Untersuche die Funktion$ f_{a}(t)=-0,01 a t^{3}+0,3 a t^{2}+50 $für$ a=2 $und$ a=-1$. Bestimme Maximalwerte und Trendwenden.

See Solution

Problem 14882

Find a value for $a$ if $F$ is an antiderivative of $f$ :
a) $f(x)=3 x^{2} ; F(x)=x^{a}$
b) $f(x)=2 x ; F(x)=x^{2}-a$
c) $f(x)=2 x ; F(x)=x^{2}+1+a$
d) $f(x)=(a+1) \cdot x ; F(x)=x^{a+1}$
Use a graphing calculator.

See Solution

Problem 14883

Gegeben ist die Funktion $h(t)=-\frac{1}{3} t^{3}+2 t^{2}+21 t+10$ . Bestimmen Sie Extrempunkte, Höhe zu $t=0$ , Wachstumsraten und vergleichen Sie mit $27 \mathrm{~cm}$ /Woche. Ermitteln Sie die Höhe der "Goldene Gloria" und den größten Höhenunterschied zu "Sonnenstern".

See Solution

Problem 14884

Bestimmen Sie den Berührpunkt und die Gleichung der Tangente an $f(x) = \frac{1}{3}x^3 - \frac{3}{2}x^2$ , die durch $P(0, \frac{9}{8})$ und $N(\frac{1}{2}, 0)$ verläuft.

See Solution

Problem 14885

Ein Heißluftballon hat die Geschwindigkeit $v(t)=-0,12 t^{2}+1,2 t$ und startet bei $h(0)=520 \mathrm{~m}$ .
a) Finde die Höhenfunktion $h(t)$ . b) Bestimme die maximale Höhe. c) Wann erreicht der Ballon die Start-Höhe?

See Solution

Problem 14886

Find the limit as $x$ approaches 3 from the left of $\frac{2}{|3-x|}$ .

See Solution

Problem 14887

Find the limit as $x$ approaches -1 for the expression $\frac{x^{2}+6 x+5}{x^{2}-3 x-4}$ .

See Solution

Problem 14888

Berechnen Sie die Ableitungen der Funktionen:
a) $f(x)=(5-x)(3+x)$
b) $g(x)=3x(0,5x+1)^{2}$
c) $h(x)=\frac{1}{x}(x^{2}+5x)$ mit verschiedenen Ableitungsregeln.

See Solution

Problem 14889

Finde die Extrem- und Wendepunkte für die Funktionen: a) $f(x)=x \cdot(x+3)^{2}$ , b) $f(t)=(3 t-1)^{2}$ , c) $g(x)=(2 x-5)^{3}-150 x$ , d) $g(t)=t^{2} \cdot(2 t+5)$ , e) $h(x)=(3-4 x)^{2}+32 x$ , f) $h(t)=\left(\frac{1}{3} t+2\right)^{2} \cdot t$ .

See Solution

Problem 14890

Ein Bogenschütze schießt einen Pfeil. Höhe: $h(t)=60 t-5 t^{2}$ . Fragen: a) Abschussgeschwindigkeit? b) Gipfelhöhe und Flugzeit? c) Geschwindigkeit $100 \mathrm{~km/h}$ ? d) Abschusswinkel $\alpha$ und Maximalhöhe bei $f(x)=1,70+0,035 x-0,0005 x^{2}$ ? Trifft er das Ziel (Durchmesser $10 \mathrm{~cm}$ , M(80|1,34)?

See Solution

Problem 14891

Gegeben sind die Funktionen $f(x)=-x^{2}+4$ und $g(x)=x^{2}-5x+6$ . Bestimme Ableitungen, Steigungen, Extrempunkte, Tangente und Schnittwinkel.

See Solution

Problem 14892

Find the derivative of $\cos(y^2)$ with respect to $x$ , using the chain rule, where $y$ is a function of $x$ .

See Solution

Problem 14893

Find $\int_{2}^{6} f(4-x) d x$ given $\int_{-2}^{6} f(x) d x=10$ and $\int_{2}^{6} f(x) d x=3$ .

See Solution

Problem 14894

Berechnen Sie die Fläche unter der Kurve $f(x) = x^4 - 5x^2 + 4$ im Intervall von -2 bis 0.

See Solution

Problem 14895

Berechnen Sie den Inhalt der gefärbten Fläche mit dem Integral $A_{1}=\left|\int_{-2}^{0}\left(-x^{3}+4 x\right) d x\right|$ .

See Solution

Problem 14896

Berechne die Fläche unter der Funktion $f(x) = -x^4 + 2x^3 + x^2$ im Intervall von 0 bis 2: $A=\int_{0}^{2} f(x) d x$ .

See Solution

Problem 14897

Zeigen Sie, dass $F(x)=\left(-2 x^{2}-8 x-16\right) \cdot e^{-\frac{1}{2} x}$ eine Stammfunktion zu $f(x)=x^{2} \cdot e^{-\frac{1}{2} x}$ ist. Berechnen Sie die Flächeninhalte für $x=2$ und $x=10, 20, 50, \ldots$ .

See Solution

Problem 14898

Evaluate $\int_{0}^{1} \cos x^{2} dx$ and $\int_{0}^{1} \sin x^{2} dx$ using Maclaurin series. Approximate $\int_{0}^{1} \tan x^{2} dx$ .

See Solution

Problem 14899

Find the trapezoidal approximation of $\int_{1}^{12} f(x) dx$ using intervals $[1,3],[3,5],[5,9],[9,12]$ . Choices: (A) 97 (B) 115 (C) 128 (D) 136.

See Solution

Problem 14900

Differentiate the function: $5 x^{2} - e^{(4 y^{2})}$ . What is $\frac{d}{d x}$ of this expression?

See Solution

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Calculus

Problem 14801

Estimate Δy\Delta yΔy for y=sin⁡(2x)y=\sin(2x)y=sin(2x) at x=0x=0x=0 with Δx=0.1\Delta x=0.1Δx=0.1 using linear approximation and find error \%.

Problem 14802

Find the tangent line equation at the point on the curve where x=t3+1x=t^{3}+1x=t3+1, y=t10+ty=t^{10}+ty=t10+t, for t=−1t=-1t=−1.

Problem 14803

Find the equation of the tangent line to f(x)=53−2x−x2−13x3f(x)=\frac{5}{3}-2x-x^{2}-\frac{1}{3}x^{3}f(x)=35​−2x−x2−31​x3 with maximum slope.

Problem 14804

Find the derivative of the function 1x\frac{1}{x}x1​ with respect to xxx.

Problem 14805

Find the maximum area of a rectangle with one corner on the graph of y=10x2+1y=\frac{10}{x^{2}+1}y=x2+110​ and adjacent sides on the axes.

Problem 14806

Find the absolute max and min of f(x)=2−6x2f(x)=2-6 x^{2}f(x)=2−6x2 for −3≤x≤2-3 \leq x \leq 2−3≤x≤2. What are their values and locations?

Problem 14807

Find the derivative of yyy where y=cos⁡−1(6x3)y=\cos^{-1}(6x^3)y=cos−1(6x3).

Problem 14808

Find the derivative of ln⁡(xx+4)\ln \left(\frac{x}{x+4}\right)ln(x+4x​) with respect to xxx.

Problem 14809

Apply Rolle's Theorem to f(x)=2x2−8x−3f(x)=2 x^{2}-8 x-3f(x)=2x2−8x−3 on [0,4][0,4][0,4]. How many ccc satisfy f′(c)=0f^{\prime}(c)=0f′(c)=0? What is ccc? Also, which condition fails for [−3,11][-3,11][−3,11]: f(a)≠f(b)f(a) \neq f(b)f(a)=f(b), continuity, or differentiability?

Problem 14810

Determine the local min and max of the function f(x)=1+3x+27x−1f(x)=1+3x+27x^{-1}f(x)=1+3x+27x−1.

Problem 14811

Find the average slope of f(x)=5x3−5xf(x)=5 x^{3}-5 xf(x)=5x3−5x on [−2,2][-2,2][−2,2] and determine the two values of ccc where f′(c)f^{\prime}(c)f′(c) equals this slope.

Problem 14812

Apply Rolle's Theorem to f(x)=2x2−8x−3f(x)=2 x^{2}-8 x-3f(x)=2x2−8x−3 on [0,4][0,4][0,4]. How many ccc values exist where f′(c)=0f'(c)=0f′(c)=0? What is ccc?

Problem 14813

Find the derivative of h(x)=ex4−x+5h(x)=e^{x^{4}-x+5}h(x)=ex4−x+5.

Problem 14814

Find the inflection points of f(x)=12x5+30x4−160x3+2f(x) = 12x^5 + 30x^4 - 160x^3 + 2f(x)=12x5+30x4−160x3+2 and determine DDD, EEE, and FFF.

Problem 14815

Evaluate the limit: lim⁡x→∞ln⁡(x)x20=\lim _{x \rightarrow \infty} \frac{\ln (x)}{x^{20}}=x→∞lim​x20ln(x)​=

Problem 14816

Apply Rolle's Theorem to f(x)=2x2−8x−7f(x)=2x^{2}-8x-7f(x)=2x2−8x−7 on [0,4][0,4][0,4]. How many ccc values exist where f′(c)=0f'(c)=0f′(c)=0? What is ccc?

Problem 14817

Evaluate these limits: (a) lim⁡x→∞(−27x2+26x3)=\lim_{x \to \infty}(-27x^2 + 26x^3) = limx→∞​(−27x2+26x3)= and (b) lim⁡x→−∞(−27x2+26x3)=\lim_{x \to -\infty}(-27x^2 + 26x^3) = limx→−∞​(−27x2+26x3)=.

Problem 14818

Evaluate these limits: (a) lim⁡x→∞5+5x26+4x=\lim _{x \rightarrow \infty} \frac{\sqrt{5+5 x^{2}}}{6+4 x}=limx→∞​6+4x5+5x2​​=; (b) lim⁡x→−∞5+5x26+4x=\lim _{x \rightarrow-\infty} \frac{\sqrt{5+5 x^{2}}}{6+4 x}=limx→−∞​6+4x5+5x2​​=

Problem 14819

Show there are no tangents to y=6x3+2x2y=6 x^{3}+2 x^{2}y=6x3+2x2 with slope -5. Verify with a graphing calculator.

Problem 14820

Find the first and second derivatives of g(x)=6x3−9x2−108xg(x)=6 x^{3}-9 x^{2}-108 xg(x)=6x3−9x2−108x. Evaluate g′′(3)g^{\prime \prime}(3)g′′(3).

Problem 14821

Find functions ggg where g′(x)=5x2+3x+5xg^{\prime}(x)=\frac{5 x^{2}+3 x+5}{\sqrt{x}}g′(x)=x​5x2+3x+5​.

Problem 14822

Problem 14823

Evaluate the limit using L'Hôpital's Rule: lim⁡x→0(cot⁡(21x)−121x)=\lim _{x \rightarrow 0}\left(\cot (21 x)-\frac{1}{21 x}\right) =x→0lim​(cot(21x)−21x1​)=

Problem 14824

Find f′(x)f^{\prime}(x)f′(x) from f′′(x)=8x−3f^{\prime \prime}(x)=8 x-3f′′(x)=8x−3 with f′(−3)=−1f^{\prime}(-3)=-1f′(−3)=−1, and f(−3)=2f(-3)=2f(−3)=2. Then find f(2)f(2)f(2).

Problem 14825

Find the antiderivative F(x)F(x)F(x) of f(x)=7x2−7x7f(x)=\frac{7}{x^{2}}-\frac{7}{x^{7}}f(x)=x27​−x77​ with F(1)=0F(1)=0F(1)=0.

Problem 14826

Find f(t)f(t)f(t) given that f′′(t)=2(3t−2)f^{\prime \prime}(t)=2(3 t-2)f′′(t)=2(3t−2) and the conditions f′(1)=2f^{\prime}(1)=2f′(1)=2, f(1)=4f(1)=4f(1)=4.

Problem 14827

Find f(0)f(0)f(0) if f′(t)=3sin⁡(2t)f'(t)=3 \sin(2t)f′(t)=3sin(2t) and f(π2)=6f\left(\frac{\pi}{2}\right)=6f(2π​)=6.

Problem 14828

Identify which functions are anti-derivatives of f(x)=sin⁡(x)cos⁡(x)f(x)=\sin (x) \cos (x)f(x)=sin(x)cos(x) from the given options: (A) F1(x)F_{1}(x)F1​(x), (B) F2(x)F_{2}(x)F2​(x), (C) F3(x)F_{3}(x)F3​(x).

Problem 14829

Evaluate the limit: lim⁡x→0(1x2−csc⁡2(x))=\lim _{x \rightarrow 0}\left(\frac{1}{x^{2}}-\csc ^{2}(x)\right)=limx→0​(x21​−csc2(x))=

Problem 14830

True or false: For a differentiable function fff, is it true that ddx[f(x)]=f′(x)2x\frac{d}{d x}[f(\sqrt{x})]=\frac{f^{\prime}(x)}{2 \sqrt{x}}dxd​[f(x​)]=2x​f′(x)​?

Problem 14831

Problem 14832

True or false: For a one-to-one function, is the derivative of the inverse given by ddx[f−1(x)]=−f−2(x)f′(x)\frac{d}{dx}[f^{-1}(x)] = -f^{-2}(x) f'(x)dxd​[f−1(x)]=−f−2(x)f′(x)?

Problem 14833

Choose the option equal to the derivative of sin⁡−1(ax)\sin^{-1}(a x)sin−1(ax): A: a1+(ax)2\frac{a}{1+(a x)^{2}}1+(ax)2a​ B: $\frac{a}{\sqrt{1-(a x)^{2}}$ C: $-a \sin^{-2}(a x)$ D: $a \cos(a x)$ E: $-a \sin^{-2}(a x) \cos(a x)$ F: None of the above

Problem 14834

Problem 14835

Find the unique anti-derivative FFF of f(x)=3e3x+2e2x+4e−xe2xf(x)=\frac{3 e^{3 x}+2 e^{2 x}+4 e^{-x}}{e^{2 x}}f(x)=e2x3e3x+2e2x+4e−x​ with F(0)=0F(0)=0F(0)=0.

Problem 14836

Given a(t)=5sin⁡(t)a(t)=5 \sin(t)a(t)=5sin(t) and v(0)=−9 m/sv(0)=-9 \ \mathrm{m/s}v(0)=−9 m/s, find v(t)v(t)v(t), displacement from 0 to 3, and total distance traveled.

Problem 14837

Find the derivative of ln⁡(13x)\ln\left(\frac{13}{x}\right)ln(x13​) with respect to xxx.

Problem 14838

Evaluate the limit: lim⁡x→∞x13/x=\lim _{x \rightarrow \infty} x^{13 / x}=x→∞lim​x13/x=

Problem 14839

Find the displacement and total distance of the object with velocity v(t)=t2−2t−3v(t)=t^{2}-2t-3v(t)=t2−2t−3 from t=0t=0t=0 to t=7t=7t=7.

Problem 14840

An object's acceleration is a(t)=4sin⁡(t)a(t)=4 \sin (t)a(t)=4sin(t) with initial velocity v(0)=−2 m/sv(0)=-2 \mathrm{~m/s}v(0)=−2 m/s. Find v(t)v(t)v(t), displacement, and total distance from t=0t=0t=0 to t=3t=3t=3.

Problem 14841

Object accelerates with a(t)=4sin⁡(t)a(t)=4 \sin(t)a(t)=4sin(t), v(0)=−2v(0)=-2v(0)=−2. Find v(t)v(t)v(t), displacement from 0 to 3, and total distance traveled.

Estimate $\Delta y$ for $y=\sin(2x)$ at $x=0$ with $\Delta x=0.1$ using linear approximation and find error \%.

Find the tangent line equation at the point on the curve where $x=t^{3}+1$ , $y=t^{10}+t$ , for $t=-1$ .

Find the equation of the tangent line to $f(x)=\frac{5}{3}-2x-x^{2}-\frac{1}{3}x^{3}$ with maximum slope.

Find the derivative of the function $\frac{1}{x}$ with respect to $x$ .

Find the maximum area of a rectangle with one corner on the graph of $y=\frac{10}{x^{2}+1}$ and adjacent sides on the axes.

Find the absolute max and min of $f(x)=2-6 x^{2}$ for $-3 \leq x \leq 2$ . What are their values and locations?

Find the derivative of $y$ where $y=\cos^{-1}(6x^3)$ .

Find the derivative of $\ln \left(\frac{x}{x+4}\right)$ with respect to $x$ .

Apply Rolle's Theorem to $f(x)=2 x^{2}-8 x-3$ on $[0,4]$ . How many $c$ satisfy $f^{\prime}(c)=0$ ? What is $c$ ? Also, which condition fails for $[-3,11]$ : $f(a) \neq f(b)$ , continuity, or differentiability?

Determine the local min and max of the function $f(x)=1+3x+27x^{-1}$ .

Find the average slope of $f(x)=5 x^{3}-5 x$ on $[-2,2]$ and determine the two values of $c$ where $f^{\prime}(c)$ equals this slope.

Apply Rolle's Theorem to $f(x)=2 x^{2}-8 x-3$ on $[0,4]$ . How many $c$ values exist where $f'(c)=0$ ? What is $c$ ?

Find the derivative of $h(x)=e^{x^{4}-x+5}$ .

Find the inflection points of $f(x) = 12x^5 + 30x^4 - 160x^3 + 2$ and determine $D$ , $E$ , and $F$ .

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{\ln (x)}{x^{20}}=$

Apply Rolle's Theorem to $f(x)=2x^{2}-8x-7$ on $[0,4]$ . How many $c$ values exist where $f'(c)=0$ ? What is $c$ ?

Evaluate these limits: (a) $\lim_{x \to \infty}(-27x^2 + 26x^3) =$ and (b) $\lim_{x \to -\infty}(-27x^2 + 26x^3) =$ .

Evaluate these limits: (a) $\lim _{x \rightarrow \infty} \frac{\sqrt{5+5 x^{2}}}{6+4 x}=$ ; (b) $\lim _{x \rightarrow-\infty} \frac{\sqrt{5+5 x^{2}}}{6+4 x}=$

Show there are no tangents to $y=6 x^{3}+2 x^{2}$ with slope -5. Verify with a graphing calculator.

Find the first and second derivatives of $g(x)=6 x^{3}-9 x^{2}-108 x$ . Evaluate $g^{\prime \prime}(3)$ .

Find functions $g$ where $g^{\prime}(x)=\frac{5 x^{2}+3 x+5}{\sqrt{x}}$ .

Evaluate the limit using L'Hôpital's Rule: $\lim _{x \rightarrow 0}\left(\cot (21 x)-\frac{1}{21 x}\right) =$

Find $f^{\prime}(x)$ from $f^{\prime \prime}(x)=8 x-3$ with $f^{\prime}(-3)=-1$ , and $f(-3)=2$ . Then find $f(2)$ .

Find the antiderivative $F(x)$ of $f(x)=\frac{7}{x^{2}}-\frac{7}{x^{7}}$ with $F(1)=0$ .

Find $f(t)$ given that $f^{\prime \prime}(t)=2(3 t-2)$ and the conditions $f^{\prime}(1)=2$ , $f(1)=4$ .

Find $f(0)$ if $f'(t)=3 \sin(2t)$ and $f\left(\frac{\pi}{2}\right)=6$ .

Identify which functions are anti-derivatives of $f(x)=\sin (x) \cos (x)$ from the given options: (A) $F_{1}(x)$ , (B) $F_{2}(x)$ , (C) $F_{3}(x)$ .

Evaluate the limit: $\lim _{x \rightarrow 0}\left(\frac{1}{x^{2}}-\csc ^{2}(x)\right)=$

True or false: For a differentiable function $f$ , is it true that $\frac{d}{d x}[f(\sqrt{x})]=\frac{f^{\prime}(x)}{2 \sqrt{x}}$ ?

True or false: For a one-to-one function, is the derivative of the inverse given by $\frac{d}{dx}[f^{-1}(x)] = -f^{-2}(x) f'(x)$ ?

Choose the option equal to the derivative of $\sin^{-1}(a x)$ :
A: $\frac{a}{1+(a x)^{2}}$ B: $\frac{a}{\sqrt{1-(a x)^{2}}$ C: $-a \sin^{-2}(a x)$ D: $a \cos(a x)$ E: $-a \sin^{-2}(a x) \cos(a x)$ F: None of the above

Find the unique anti-derivative $F$ of $f(x)=\frac{3 e^{3 x}+2 e^{2 x}+4 e^{-x}}{e^{2 x}}$ with $F(0)=0$ .

Given $a(t)=5 \sin(t)$ and $v(0)=-9 \ \mathrm{m/s}$ , find $v(t)$ , displacement from 0 to 3, and total distance traveled.

Find the derivative of $\ln\left(\frac{13}{x}\right)$ with respect to $x$ .

Evaluate the limit: $\lim _{x \rightarrow \infty} x^{13 / x}=$

Find the displacement and total distance of the object with velocity $v(t)=t^{2}-2t-3$ from $t=0$ to $t=7$ .

An object's acceleration is $a(t)=4 \sin (t)$ with initial velocity $v(0)=-2 \mathrm{~m/s}$ . Find $v(t)$ , displacement, and total distance from $t=0$ to $t=3$ .

Object accelerates with $a(t)=4 \sin(t)$ , $v(0)=-2$ . Find $v(t)$ , displacement from 0 to 3, and total distance traveled.

Evaluate the limit: $\lim _{x \rightarrow 1} \frac{e^{x}-e}{4 \ln (x)}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{15 x^{3}}{\sin (x)-x}$ .

Find $H(b)=\lim _{x \rightarrow \infty} \frac{\ln \left(1+b^{x}\right)}{x}$ for $b>1$ and $0<b \leq 1$ .

Evaluate the limit: $\lim _{x \rightarrow \infty} e^{-x}(x^{3}-19 x^{2}+4)=$

Approximate $\sqrt{64.4}$ using the tangent line of $f(x)=\sqrt{x}$ at $x=64$ . Find $L(x)=$ . Provide 9 significant figures.

Given the speed function $f(t)=\frac{44000}{(t+20)^{2}}$ for $0 \leq t \leq 6$ , explain the distance estimates using sums and averages.

Find the local max and min of $f(x)=2+9x+36x^{-1}$ . Determine the $x$ -values and their function values.

Find $\frac{d x}{d t}$ at $x=-4$ given $y=2 x^{2}-4$ and $\frac{d y}{d t}=-4$ .

Evaluate the integral: $\int \frac{5+4 x}{1+x^{2}} d x$

Find $H(b)=\lim _{x \rightarrow \infty} \frac{\ln(1+b^{x})}{x}$ for $b>1$ and $0<b \leq 1$ .

Find the limit as $x$ approaches 0 from the right: $\lim _{x \rightarrow 0^{+}} x^{\sin (2 x)}$ .

Estimate the area between the $\mathrm{x}$ -axis and $y=\sin x$ for $0 \leq x \leq \pi$ .

Estimate $\Delta y$ for $y=\sin(4x)$ with $\Delta x=0.2$ at $x=0$ using linear approximation. Find the percentage error.

Find the inflection points of $f(x) = 12x^{5} + 60x^{4} - 240x^{3} + 6$ . What are the values of $D, E$ , and $F$ ?

Find the total number of cars passing an intersection from $t=0$ to $t=2$ for $r(t)=300+700t-150t^{2}$ .

Evaluate the limit using L'Hôpital's Rule:
$\lim _{x \rightarrow 1}(1+\ln (x))^{4 /(x-1)}=$

Bestimmen Sie die Ableitung der Funktion $g(x) = -\frac{10}{x}$ , also $g'(x)$ .

Find the total words memorized in 15 minutes using the rate $M^{\prime}(t)=-0.002 t^{2}+0.8 t$ .

Find the displacement formula for a particle with velocity $v(t)=14-5t$ and total distance traveled at $t=0.5$ seconds.

Find the displacement and total distance for the velocity $v(t)=t^{2}-t-20$ from $t=0$ to $t=8$ .

Find the displacement formula for a particle with velocity $v(t)=12-6t$ and total distance at $t=0.5$ seconds.

Find the number of cars passing through an intersection from 6 am to 9 am given $r(t)=500+1000 t-210 t^{2}$ .

Find the average slope of the function $f(x)=3 x^{3}-6 x$ on $[-2,2]$ and identify the two values of $c$ where $f'(c)$ equals this slope.

Evaluate these limits: (a) $\lim _{x \rightarrow \infty} \frac{\sqrt{11+6 x^{2}}}{2+9 x}=$ (b) $\lim _{x \rightarrow-\infty} \frac{\sqrt{11+6 x^{2}}}{2+9 x}=$

Kreuzen Sie an, ob die Ableitungen für die Funktionen $f_{4}(x)=x^{76}+98$ und $f_{5}(x)=\sqrt[4]{x^{2}}$ korrekt sind.

Find the third derivative of $f(x) = x^{4} - 4.5x^{2} + 5.0625$ .

An object moves with velocity $v(t)=t^{2}+4t-5$ . Find displacement and total distance from $t=0$ to $t=7$ .

Find an antiderivative of the function $f(x)=4 x^{8}+5 x^{5}-9 x^{2}-4$ . Do not include $+c$ .

Evaluate these limits: (a) $\lim _{x \rightarrow \infty} \frac{\sqrt{11+6 x^{2}}}{2+9 x}=$ , (b) $\lim _{x \rightarrow-\infty} \frac{\sqrt{11+6 x^{2}}}{2+9 x}=$ .

Find values of $c$ where $f'(c)=0$ for $f(x)=2x^2-20x-6$ on $[3,7]$ . What condition fails for $[0,14]$ ?

Find the average slope of $f(x)=3 x^{3}-6 x$ on $[-2,2]$ and the values of $c$ where $f^{\prime}(c)$ equals this slope.