Calculus

Problem 6701

Berechne die Steigung von $f$ an $x_0$ mit der h-Methode für: a) $f(x)=x^{2}, x_{0}=1$ b) $f(x)=2 x^{2}, x_{0}=-1$ c) $f(x)=x^{3}, x_{0}=2$ d) $f(x)=2 x, x_{0}=1$

See Solution

Problem 6702

Find the equations of (a) the tangent line and (b) the normal line to $y=f(x)$ at $x=2$ , given $f(2)=3$ and $f'(2)=5$ .

See Solution

Problem 6703

Given the function $y=f(x)$ with $f(x)=\frac{14}{3+x^{2}}$ , find the slope and equation of the tangent line at point $P(2,2)$ .

See Solution

Problem 6704

Given $y=f(x)=\frac{14}{3+x^{2}}$ , $x \geq 0$ , and point $P(2,2)$ , find the slope $m$ and equation $y=$ of the tangent line to $f^{-1}$ at $P$ .

See Solution

Problem 6705

Calculate the volume of the solid formed by rotating the area between $y = x^3$ , $y = 1$ , and $x = 2$ around $y = -3$ .

See Solution

Problem 6706

Find $\left(f^{-1}\right)^{\prime}(4)$ given $f(-5)=4$ and $f^{\prime}(-5)=-4$ .

See Solution

Problem 6707

Find $\frac{d y}{d x}$ for $14 x^{6}+4 x^{28} y+y^{10}=19$ and the tangent line at $(1,1)$ in $m x+b$ form.

See Solution

Problem 6708

Find $\frac{d y}{d x}$ for the cone volume $V=\frac{1}{3} \pi x^{2} y$ at $x=7$ , $y=21$ , with $V=343 \pi$ . $\frac{d y}{d x} =$

See Solution

Problem 6709

Find the derivative $\frac{d y}{d x}$ for the function $y=\sec^{-1}\left(\frac{1}{x}\right)$ .

See Solution

Problem 6710

Bestimme die Integrale: a) $\int(3-6 x^{2}) dx$ b) $\int_{0}^{2}(2 x-x^{2}) dx$

See Solution

Problem 6711

Berechne die Flächeninhalte unter $f(x) = \sin(x)$ für die Intervalle: $[0 ; 2 \pi]$ , $[0 ; 5 \pi]$ , $[0 ; n \cdot \pi]$ , $[2 \pi ; 8 \pi]$ .

See Solution

Problem 6712

Find the derivative of the function $f(x)=\frac{4}{5 x^{5}}$ in simplest form without negative exponents.

See Solution

Problem 6713

Find the derivative of the function $y=-\sqrt[3]{x^{4}}$ , expressed in radical form and simplified.

See Solution

Problem 6714

Find the derivative of the function $y=-\frac{1}{x}$ and express it without negative exponents.

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Problem 6715

Find the derivative of the function $f(x)=-\frac{1}{2 x^{3}}$ and express it without negative exponents.

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Problem 6716

Bestimme die Ableitung für $x_{0}=1$ mit einer Näherungstabelle für den Differenzenquotienten für: a) $f(x)=x^{2}$ , b) $f(x)=x^{3}$ , c) $f(x)=x^{2}+3$ , d) $f(x)=2 x^{3}-1$ .

See Solution

Problem 6717

Bestimme die Ableitung von $f(x)=x^{2}$ an $x_{0}=4$ durch Näherungstabellen für den Differenzenquotienten.

See Solution

Problem 6718

Bestimmen Sie Grenzwerte für $x \rightarrow+\infty$ und $x \rightarrow-\infty$ , asymptotische Gleichungen und Schnittpunkte der Funktionen a) bis d).

See Solution

Problem 6719

Determine the behavior of $f(x) = \frac{1}{x-7}$ as $x \rightarrow \infty$ . Options: a) 0 from above, b) undefined, c) 0 from below, d) 1/3.

See Solution

Problem 6720

Find the derivative of the function $y=x^{2}(2-\ln x^{2})$ .

See Solution

Problem 6721

Find where the function $f(x)=\frac{4 x}{x^{2}+4}$ has a horizontal tangent.

See Solution

Problem 6722

Find the derivative of $y=\frac{\sqrt{x}}{2}-\frac{1}{6x^3}$ .

See Solution

Problem 6723

Gegeben ist die Funktion $f(x)=x^2-3x$ . Zeichnen Sie den Graphen für $-1 \leq x \leq 4$ und bestimmen Sie Steigung und Winkel bei $x_0=2$ .

See Solution

Problem 6724

Berechne die Steigung von $f$ bei $x_{0}$ für: a) $f(x)=x^{2}, x_{0}=-1$ b) $f(x)=0,5 x^{2}, x_{0}=2$ c) $f(x)=a x^{2}, x_{0}=1$

See Solution

Problem 6725

Berechnen Sie die Steigung von $f(x)=x^{2}$ bei $x_{0}=-1$ .

See Solution

Problem 6726

Berechne die Steigung von $f$ an $x_0$ : a) $f(x)=x^{2}, x_{0}=-1$ b) $f(x)=0,5 x^{2}, x_{0}=2$ c) $f(x)=a x^{2}, x_{0}=1$

See Solution

Problem 6727

Find the derivative of the function $f(x)=\frac{6 x+3}{3 x+5}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 6728

Find the derivative of the function $y=13^{-x} \sin x$ .

See Solution

Problem 6729

Vermuten Sie den Grenzwert $g$ der Folge $a_{n}=\frac{n+4}{2 n}$ und bestimmen Sie $n$ , sodass $\left|a_{n}-g\right|<0,001$ . Beweisen Sie die Konvergenz.

See Solution

Problem 6730

Bestimme die Integrale: a) $\int(3-6 x^{2}) \, dx$ b) $\int_{0}^{2}(2 x-x^{2}) \, dx$ c) $\int_{0}^{2} e^{-x} \, dx$

See Solution

Problem 6731

Find the derivative $f^{\prime}(0)$ for the piecewise function: $f(x)=4x-2x^{2}$ for $x<0$ and $f(x)=8x^{2}+4x$ for $x \geq 0$ .

See Solution

Problem 6732

Find critical numbers for the function $f(x)=\frac{x+9}{x+4}$ .

See Solution

Problem 6733

Determine the interval where the function $f(x)=\frac{x+9}{x+4}$ is increasing.

See Solution

Problem 6734

Find the derivative of $f(x)=\sqrt{x+5}$ using the definition. Simplify the difference quotient and evaluate the limit:
$f^{\prime}(x)=\lim _{h \rightarrow 0}\left(\frac{f(x+h)-f(x)}{h}\right)=\lim _{h \rightarrow 0}(\square)=\square$

See Solution

Problem 6735

Leiten Sie die Funktion $f$ ab für: a) $f(x)=x \cdot \sin (x)$ , b) $f(x)=3 x \cdot \cos (x)$ , d) $f(x)=\sqrt{x} \cdot(2 x-3)$ , e) $f(x)=\sqrt{x} \cdot \cos (x)$ , h) $f(x)=\sin (x) \cdot \cos (x)$ , k) $f(x)=(x^{2}+3 x) \cdot \sin (x)$ , c) $f(x)=(3 x+2) \cdot \sqrt{x}$ , f) $f(x)=(5-3 x) \cdot \sin (x)$ , i) $f(x)=x^{2} \cdot \sin (x)$ , l) $f(x)=\sqrt{x} \cdot(x^{5}-2 x^{3})$ .

See Solution

Problem 6736

2 Ergänzen Sie die Ableitungen: a) $f(x)=x^{2} \cdot \sin (x) ; f^{\prime}(x)=\square \cdot \sin (x)+x^{2}$ . b) $f(x)=(2 x-3) \cdot \cos (4 x) ; f^{\prime}(x)=\square \cdot \cos (4 x)-(2 x-3) \cdot \sin (4 x)$ . c) $f(x)=\sin (2-x) \cdot \sqrt{3 x-1} ; f^{\prime}(x)=-1 \cdot \square \cdot \sqrt{3 x-1}+\sin (2-x) \cdot \frac{\Delta}{2 \cdot \sqrt{3 x-1}}$ . d) $f(x)=\sqrt{x} \cdot(4 x+1) ; f^{\prime}(x)=\square \cdot(4 x+1)+\Delta \cdot 4$ . e) $f(x)=(x+2)^{3} \cdot \cos (-3 x) ; f^{\prime}(x)=3 \cdot(x+2)^{2} \cdot \cos (\square)+(x+2) \cdot(-\sin (-3 x))$ .

See Solution

Problem 6737

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{6 x^{3}-6 x^{2}+3 x}{-x^{3}-2 x+6}$

See Solution

Problem 6738

Find $\delta > 0$ such that for all $x$ , if $0 < |x - 2| < \delta$ , then $|f(x) + 17| < 0.01$ for $f(x) = -9x + 1$ .

See Solution

Problem 6739

Find the limit as $x$ approaches infinity for $\frac{\sqrt[3]{x}-7 x-4}{5 x+x^{2 / 3}+2}$ by simplifying.

See Solution

Problem 6740

Find the slope of the tangent to $f(x) = x^2 - 3$ at $P(-4, 13)$ using $m_{\tan }=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ . Then, find the tangent line's equation.

See Solution

Problem 6741

Find the limit as x approaches infinity: $\lim _{x \rightarrow \infty} \frac{x^{2}-9 x+9}{x^{3}-6 x^{2}+19}$

See Solution

Problem 6742

Berechnen Sie die Fläche unter der Kurve von $f(x) = x^3$ im Intervall [0, 2].

See Solution

Problem 6743

Find the limit of $f(x)=\frac{-2+\frac{6}{x}}{4-\frac{5}{x^{2}}}$ as $x \to \infty$ and $x \to -\infty$ .

See Solution

Problem 6744

Find the derivative of $y=\frac{3 x^{4}+2 x+5}{\sqrt{x}}$ : a) $\frac{d y}{d x}$

See Solution

Problem 6745

Calculate the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{4 x^{2}+3}}{5 x+1}$ .

See Solution

Problem 6746

Is the function $f(x)=\frac{x^{2}-36}{x-6}$ for $x \neq 6$ and $f(6)=10$ continuous at $x=6$ ?

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Problem 6747

How much must lan invest at a continuous interest rate of $6.2\%$ to reach \$182,000 in 10 years? Round to the nearest hundredth.

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Problem 6748

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(x-\sqrt{x^{2}-6 x}\right)$ . Hint: Multiply by $\frac{x+\sqrt{x^{2}-6 x}}{x+\sqrt{x^{2}-6 x}}$ .

See Solution

Problem 6749

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \sqrt{\frac{16 x^{2}+x-3}{(x-9)(x+1)}}$

See Solution

Problem 6750

Find the derivative of $f(x)=1-9 x^{2}$ at $x=3$ using the limit definition. Simplify and evaluate the limit.

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Problem 6751

Find $\lim _{x \rightarrow a}\left[\frac{-6 f(x)-4 g(x)}{-7+g(x)}\right]$ given $\lim _{x \rightarrow a} f(x)=-2$ and $\lim _{x \rightarrow a} g(x)=5$ .

See Solution

Problem 6752

Find the derivative of $f(x)=1-9 x^{2}$ using the limit definition: $f^{\prime}(x)=\lim _{h \rightarrow 0}\left(\frac{f(x+h)-f(x)}{h}\right)$ .

See Solution

Problem 6753

Find the derivative of $f(x)=1-9 x^{2}$ using the definition: $f^{\prime}(x)=\lim _{h \rightarrow 0}\left(\frac{f(x+h)-f(x)}{h}\right)$ . Simplify and evaluate the limit.

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Problem 6754

Find the limit as x approaches -7 for the expression $4x - 5$ .

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Problem 6755

Check if the function $f(x)=\frac{5 x^{2}+23 x+12}{x^{2}-8 x}$ is continuous at $a=8$ .

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Problem 6756

Find where the function $f(x)=\frac{x^{2}}{x^{2}-4}$ has a horizontal tangent line.

See Solution

Problem 6757

Calculate the limit: $\lim _{x \rightarrow 2^{+}} \frac{1}{x^{2}-4}$ .

See Solution

Problem 6758

Bestimme die Stammfunktion $F$ von $f(x)=3x^2-2x$ , die durch den Punkt $P(2|-1)$ verläuft. Wähle $C$ passend.

See Solution

Problem 6759

Find the tangent line to $f(x)=1-9 x^{2}$ at $x=4$ with slope $f^{\prime}(4)=-72$ and point $(4,-143)$ . What is $y=$ ?

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Problem 6760

Find the tangent line to $f(x)=x^{2}-6x+7$ that is:
a) Parallel to $y=2x+4$
b) Perpendicular to $y=2x+4$

See Solution

Problem 6761

Find $f^{\prime \prime}(x)$ for the function $f(x)=\frac{x^{3}-3 x^{2}-4 x-1}{2 x}$ .

See Solution

Problem 6762

Find the derivative of $f(x)=\sqrt{x+5}$ using the limit definition. Simplify and evaluate the limit:
$f^{\prime}(x)=\lim _{h \rightarrow 0}\left(\frac{f(x+h)-f(x)}{h}\right)=\lim _{h \rightarrow 0}(\square)=$
Hint: use the conjugate trick.

See Solution

Problem 6763

Find the maximum profit given the cost function $C(q) = 80 + 19q$ and price function $p = 63 - 2q$ .

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Problem 6764

A function is continuous at a point if it satisfies these three conditions: 1) $f(c)$ is defined, 2) $\lim_{x \to c} f(x)$ exists, 3) $\lim_{x \to c} f(x) = f(c)$ .

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Problem 6765

Evaluate the limit: $\lim _{h \rightarrow 0}\left(\frac{f(7+h)-f(7)}{h}\right)$ for $f(x)=\frac{9}{x-2}$ .

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Problem 6766

Elliot's roast cools from $165^{\circ} \mathrm{F}$ to $140^{\circ} \mathrm{F}$ in 12 min. How long to reach $125^{\circ} \mathrm{F}$ ? Use $T(t)=70+\left(165-70\right)e^{-kt}$ . Find $t$ rounded to the nearest minute.

See Solution

Problem 6767

Find the derivative $f^{\prime}(0)$ using the limits $\lim _{x \rightarrow 0^{-}}$ and $\lim _{x \rightarrow 0^{+}}$ for the piecewise function $f(x)$ .

See Solution

Problem 6768

Find $f^{\prime}(0)$ for the piecewise function: $f(x)=4x-2x^{2}$ for $x<0$ and $f(x)=8x^{2}+4x$ for $x \geq 0$ .

See Solution

Problem 6769

Find $h^{\prime}(4)$ for $h(x)=5 f(x)-\frac{2}{3} g(x)$ and $h(x)=3+8 f(x)$ using given values.

See Solution

Problem 6770

Find the points where the function $f(x)=x^{3}-6 x^{2}+9 x+4$ has horizontal tangents.

See Solution

Problem 6771

Find the limit of the series $\sum_{n=1}^{\infty}\left(\frac{3 n^{5}}{15 n^{4}+5}\right)$ . Options: 0, 0.2, 5, $\infty$ .

See Solution

Problem 6772

Find the limit of the series $\sum_{n=1}^{\infty}\left(\frac{3 n^{4}}{15 n^{4}+5}\right)$ . Options: 0, 0.2, 5, $\infty$ .

See Solution

Problem 6773

Maximize or minimize $f(x, y, z)=8xy+6xz+6yz$ with constraint $xyz-504=0$ using Lagrange multipliers. Check points $(1,1,504)$ and $(1,\sqrt{504},\sqrt{504})$ . Verify $f\left(\sqrt[3]{378}, \sqrt[3]{378}, \frac{4}{3}\sqrt[3]{378}\right) \approx 1045.58$ .

See Solution

Problem 6774

Find the limit of the series $\sum_{n=1}^{\infty}\left(\frac{3 n^{3}}{15 n^{4}+5}\right)$ . Options: 0, 0.2, 5, $\infty$ .

See Solution

Problem 6775

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

See Solution

Problem 6776

Analyze the series $\frac{3}{10}+\frac{3}{2}+\frac{15}{2}+\frac{75}{2}+\frac{325}{2}+\ldots$ for convergence or divergence.

See Solution

Problem 6777

Find the derivative of $y=(3x^2-1)^3 \tan(3x)$ .

See Solution

Problem 6778

Determine if the series with $a_{n}=\frac{4^{n}}{5^{n+1} \cdot n}$ converges or diverges using the ratio test.

See Solution

Problem 6779

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

See Solution

Problem 6780

Find the net change and average rate of change of a function between points (-1,2) and (5,3).

See Solution

Problem 6781

Calculate the integral $\int \frac{x^{4} d x}{\sqrt{x^{2}-4}}$ .

See Solution

Problem 6782

Find the derivative of $y=\sqrt{\frac{x^{2}+5}{x^{2}-5}}$ .

See Solution

Problem 6783

For the function $y=4 x^{2}$ , find the average rate of change over $[5,8]$ and the instantaneous rate at $x=5$ .

See Solution

Problem 6784

A smoothie machine starts with 300g of protein in 100L water. After 9 mins, find protein left with 10L/min water added, 20L/min dispensed.
(a) Find smoothie volume function over time. (b) Define protein density $D(t) = \frac{P(t)}{V(t)}$ . (c) Set up a differential equation for protein loss. (d) Solve it and find protein amount after 9 mins.

See Solution

Problem 6785

A particle moves with $s(t)=t^{3}-6t^{2}+4t$ . Find velocity $v(t)$ , $v(3)$ , when at rest, and when moving positively.

See Solution

Problem 6786

Find the derivative of $y=5^{x^{2}} \csc ^{3}(6 x^{2})$ .

See Solution

Problem 6787

Find the derivative of $f(x)=4x$ .

See Solution

Problem 6788

Find the derivative of $y=8^{x^{3}}+\sqrt[3]{x^{5}}+\log _{8}(x^{3}+5)-\ln(8)$ .

See Solution

Problem 6789

Find the derivative of $y=\ln(\cos x)^{3}+\ln^{4}(\cos x)$ .

See Solution

Problem 6790

Find the derivative of $y=\ln \left(\frac{(3 x+1)^{2} \sqrt{6-x^{2}}}{(5-x^{3})^{4}}\right)$ .

See Solution

Problem 6791

Calculate the integral: $\int 2^{4} \sec ^{5} \theta \, d\theta$

See Solution

Problem 6792

Find the tangent line equation to the hyperbola $y=\frac{3}{x}$ at the point $(3,1)$ .

See Solution

Problem 6793

Find the derivative of $y=\frac{x^{3}+\ln x}{e^{x^{2}}(3 x+1)^{4}}$ .

See Solution

Problem 6794

Find the tangent and normal line equations to $y=x^{2} \ln x$ at the point $(1,0)$ .

See Solution

Problem 6795

A ball is thrown with an initial velocity of $44 \mathrm{ft/s}$ . Its height after $t$ seconds is $y=44t-16t^2$ .
(a) Find the average velocity for $t=2$ over: (i) 0.5 seconds (ii) 0.1 seconds (iii) 0.05 seconds (iv) 0.01 seconds
(b) Estimate the instantaneous velocity at $t=2$ .

See Solution

Problem 6796

Explain why the function $M(x)=\sqrt{1+\frac{1}{x}}$ is continuous in its domain and state the domain in interval notation.

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Problem 6797

Find the derivative of the function $f(x)=-4x-2$ .

See Solution

Problem 6798

Evaluate the limit using continuity: $\lim _{x \rightarrow 5} x \sqrt{41-x^{2}}$ .

See Solution

Problem 6799

Find where the functions have horizontal tangents: a) $f(x)=x^{3} e^{2 x}$ , c) $f(x)=\frac{x^{2}}{\sqrt{x^{2}-1}}$ .

See Solution

Problem 6800

Find the derivatives of these functions: 1) $y=3x^{2}+3x+3$ 2) $f(x)=\sqrt{4x-5}$

See Solution

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Calculus

Problem 6701

Berechne die Steigung von fff an x0x_0x0​ mit der h-Methode für: a) f(x)=x2,x0=1f(x)=x^{2}, x_{0}=1f(x)=x2,x0​=1 b) f(x)=2x2,x0=−1f(x)=2 x^{2}, x_{0}=-1f(x)=2x2,x0​=−1 c) f(x)=x3,x0=2f(x)=x^{3}, x_{0}=2f(x)=x3,x0​=2 d) f(x)=2x,x0=1f(x)=2 x, x_{0}=1f(x)=2x,x0​=1

Problem 6702

Find the equations of (a) the tangent line and (b) the normal line to y=f(x)y=f(x)y=f(x) at x=2x=2x=2, given f(2)=3f(2)=3f(2)=3 and f′(2)=5f'(2)=5f′(2)=5.

Problem 6703

Given the function y=f(x)y=f(x)y=f(x) with f(x)=143+x2f(x)=\frac{14}{3+x^{2}}f(x)=3+x214​, find the slope and equation of the tangent line at point P(2,2)P(2,2)P(2,2).

Problem 6704

Given y=f(x)=143+x2y=f(x)=\frac{14}{3+x^{2}}y=f(x)=3+x214​, x≥0x \geq 0x≥0, and point P(2,2)P(2,2)P(2,2), find the slope mmm and equation y=y=y= of the tangent line to f−1f^{-1}f−1 at PPP.

Problem 6705

Calculate the volume of the solid formed by rotating the area between y=x3y = x^3y=x3, y=1y = 1y=1, and x=2x = 2x=2 around y=−3y = -3y=−3.

Problem 6706

Find (f−1)′(4)\left(f^{-1}\right)^{\prime}(4)(f−1)′(4) given f(−5)=4f(-5)=4f(−5)=4 and f′(−5)=−4f^{\prime}(-5)=-4f′(−5)=−4.

Problem 6707

Find dydx\frac{d y}{d x}dxdy​ for 14x6+4x28y+y10=1914 x^{6}+4 x^{28} y+y^{10}=1914x6+4x28y+y10=19 and the tangent line at (1,1)(1,1)(1,1) in mx+bm x+bmx+b form.

Problem 6708

Find dydx\frac{d y}{d x}dxdy​ for the cone volume V=13πx2yV=\frac{1}{3} \pi x^{2} yV=31​πx2y at x=7x=7x=7, y=21y=21y=21, with V=343πV=343 \piV=343π. dydx= \frac{d y}{d x} = dxdy​=

Problem 6709

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=sec⁡−1(1x)y=\sec^{-1}\left(\frac{1}{x}\right)y=sec−1(x1​).

Problem 6710

Bestimme die Integrale: a) ∫(3−6x2)dx\int(3-6 x^{2}) dx∫(3−6x2)dx b) ∫02(2x−x2)dx\int_{0}^{2}(2 x-x^{2}) dx∫02​(2x−x2)dx

Problem 6711

Berechne die Flächeninhalte unter f(x)=sin⁡(x)f(x) = \sin(x)f(x)=sin(x) für die Intervalle: [0;2π][0 ; 2 \pi][0;2π], [0;5π][0 ; 5 \pi][0;5π], [0;n⋅π][0 ; n \cdot \pi][0;n⋅π], [2π;8π][2 \pi ; 8 \pi][2π;8π].

Problem 6712

Find the derivative of the function f(x)=45x5f(x)=\frac{4}{5 x^{5}}f(x)=5x54​ in simplest form without negative exponents.

Problem 6713

Find the derivative of the function y=−x43y=-\sqrt[3]{x^{4}}y=−3x4​, expressed in radical form and simplified.

Problem 6714

Find the derivative of the function y=−1xy=-\frac{1}{x}y=−x1​ and express it without negative exponents.

Problem 6715

Find the derivative of the function f(x)=−12x3f(x)=-\frac{1}{2 x^{3}}f(x)=−2x31​ and express it without negative exponents.

Problem 6716

Bestimme die Ableitung für x0=1x_{0}=1x0​=1 mit einer Näherungstabelle für den Differenzenquotienten für: a) f(x)=x2f(x)=x^{2}f(x)=x2, b) f(x)=x3f(x)=x^{3}f(x)=x3, c) f(x)=x2+3f(x)=x^{2}+3f(x)=x2+3, d) f(x)=2x3−1f(x)=2 x^{3}-1f(x)=2x3−1.

Problem 6717

Bestimme die Ableitung von f(x)=x2f(x)=x^{2}f(x)=x2 an x0=4x_{0}=4x0​=4 durch Näherungstabellen für den Differenzenquotienten.

Problem 6718

Bestimmen Sie Grenzwerte für x→+∞x \rightarrow+\inftyx→+∞ und x→−∞x \rightarrow-\inftyx→−∞, asymptotische Gleichungen und Schnittpunkte der Funktionen a) bis d).

Problem 6719

Determine the behavior of f(x)=1x−7f(x) = \frac{1}{x-7}f(x)=x−71​ as x→∞x \rightarrow \inftyx→∞. Options: a) 0 from above, b) undefined, c) 0 from below, d) 1/3.

Problem 6720

Find the derivative of the function y=x2(2−ln⁡x2)y=x^{2}(2-\ln x^{2})y=x2(2−lnx2).

Problem 6721

Find where the function f(x)=4xx2+4f(x)=\frac{4 x}{x^{2}+4}f(x)=x2+44x​ has a horizontal tangent.

Problem 6722

Find the derivative of y=x2−16x3y=\frac{\sqrt{x}}{2}-\frac{1}{6x^3}y=2x​​−6x31​.

Problem 6723

Gegeben ist die Funktion f(x)=x2−3xf(x)=x^2-3xf(x)=x2−3x. Zeichnen Sie den Graphen für −1≤x≤4-1 \leq x \leq 4−1≤x≤4 und bestimmen Sie Steigung und Winkel bei x0=2x_0=2x0​=2.

Problem 6724

Berechne die Steigung von fff bei x0x_{0}x0​ für: a) f(x)=x2,x0=−1f(x)=x^{2}, x_{0}=-1f(x)=x2,x0​=−1 b) f(x)=0,5x2,x0=2f(x)=0,5 x^{2}, x_{0}=2f(x)=0,5x2,x0​=2 c) f(x)=ax2,x0=1f(x)=a x^{2}, x_{0}=1f(x)=ax2,x0​=1

Problem 6725

Berechnen Sie die Steigung von f(x)=x2f(x)=x^{2}f(x)=x2 bei x0=−1x_{0}=-1x0​=−1.

Problem 6726

Berechne die Steigung von fff an x0x_0x0​: a) f(x)=x2,x0=−1f(x)=x^{2}, x_{0}=-1f(x)=x2,x0​=−1 b) f(x)=0,5x2,x0=2f(x)=0,5 x^{2}, x_{0}=2f(x)=0,5x2,x0​=2 c) f(x)=ax2,x0=1f(x)=a x^{2}, x_{0}=1f(x)=ax2,x0​=1

Problem 6727

Find the derivative of the function f(x)=6x+33x+5f(x)=\frac{6 x+3}{3 x+5}f(x)=3x+56x+3​. What is f′(x)f^{\prime}(x)f′(x)?

Problem 6728

Find the derivative of the function y=13−xsin⁡xy=13^{-x} \sin xy=13−xsinx.

Problem 6729

Vermuten Sie den Grenzwert ggg der Folge an=n+42na_{n}=\frac{n+4}{2 n}an​=2nn+4​ und bestimmen Sie nnn, sodass ∣an−g∣<0,001\left|a_{n}-g\right|<0,001∣an​−g∣<0,001. Beweisen Sie die Konvergenz.

Problem 6730

Bestimme die Integrale: a) ∫(3−6x2) dx\int(3-6 x^{2}) \, dx∫(3−6x2)dx b) ∫02(2x−x2) dx\int_{0}^{2}(2 x-x^{2}) \, dx∫02​(2x−x2)dx c) ∫02e−x dx\int_{0}^{2} e^{-x} \, dx∫02​e−xdx

Problem 6731

Find the derivative f′(0)f^{\prime}(0)f′(0) for the piecewise function: f(x)=4x−2x2f(x)=4x-2x^{2}f(x)=4x−2x2 for x<0x<0x<0 and f(x)=8x2+4xf(x)=8x^{2}+4xf(x)=8x2+4x for x≥0x \geq 0x≥0.

Problem 6732

Find critical numbers for the function f(x)=x+9x+4f(x)=\frac{x+9}{x+4}f(x)=x+4x+9​.

Problem 6733

Determine the interval where the function f(x)=x+9x+4f(x)=\frac{x+9}{x+4}f(x)=x+4x+9​ is increasing.

Problem 6734

Problem 6735

Problem 6736

Problem 6737

Find the limit as xxx approaches infinity: lim⁡x→∞6x3−6x2+3x−x3−2x+6\lim _{x \rightarrow \infty} \frac{6 x^{3}-6 x^{2}+3 x}{-x^{3}-2 x+6}x→∞lim​−x3−2x+66x3−6x2+3x​

Problem 6738

Find δ>0\delta > 0δ>0 such that for all xxx, if 0<∣x−2∣<δ0 < |x - 2| < \delta0<∣x−2∣<δ, then ∣f(x)+17∣<0.01|f(x) + 17| < 0.01∣f(x)+17∣<0.01 for f(x)=−9x+1f(x) = -9x + 1f(x)=−9x+1.

Problem 6739

Find the limit as xxx approaches infinity for x3−7x−45x+x2/3+2\frac{\sqrt[3]{x}-7 x-4}{5 x+x^{2 / 3}+2}5x+x2/3+23x​−7x−4​ by simplifying.

Problem 6740

Find the slope of the tangent to f(x)=x2−3f(x) = x^2 - 3f(x)=x2−3 at P(−4,13)P(-4, 13)P(−4,13) using mtan⁡=lim⁡h→0f(a+h)−f(a)hm_{\tan }=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}mtan​=limh→0​hf(a+h)−f(a)​. Then, find the tangent line's equation.

Problem 6741

Find the limit as x approaches infinity: lim⁡x→∞x2−9x+9x3−6x2+19\lim _{x \rightarrow \infty} \frac{x^{2}-9 x+9}{x^{3}-6 x^{2}+19}x→∞lim​x3−6x2+19x2−9x+9​

Berechne die Steigung von $f$ an $x_0$ mit der h-Methode für: a) $f(x)=x^{2}, x_{0}=1$ b) $f(x)=2 x^{2}, x_{0}=-1$ c) $f(x)=x^{3}, x_{0}=2$ d) $f(x)=2 x, x_{0}=1$

Find the equations of (a) the tangent line and (b) the normal line to $y=f(x)$ at $x=2$ , given $f(2)=3$ and $f'(2)=5$ .

Given the function $y=f(x)$ with $f(x)=\frac{14}{3+x^{2}}$ , find the slope and equation of the tangent line at point $P(2,2)$ .

Given $y=f(x)=\frac{14}{3+x^{2}}$ , $x \geq 0$ , and point $P(2,2)$ , find the slope $m$ and equation $y=$ of the tangent line to $f^{-1}$ at $P$ .

Calculate the volume of the solid formed by rotating the area between $y = x^3$ , $y = 1$ , and $x = 2$ around $y = -3$ .

Find $\left(f^{-1}\right)^{\prime}(4)$ given $f(-5)=4$ and $f^{\prime}(-5)=-4$ .

Find $\frac{d y}{d x}$ for $14 x^{6}+4 x^{28} y+y^{10}=19$ and the tangent line at $(1,1)$ in $m x+b$ form.

Find $\frac{d y}{d x}$ for the cone volume $V=\frac{1}{3} \pi x^{2} y$ at $x=7$ , $y=21$ , with $V=343 \pi$ . $\frac{d y}{d x} =$

Find the derivative $\frac{d y}{d x}$ for the function $y=\sec^{-1}\left(\frac{1}{x}\right)$ .

Bestimme die Integrale: a) $\int(3-6 x^{2}) dx$ b) $\int_{0}^{2}(2 x-x^{2}) dx$

Berechne die Flächeninhalte unter $f(x) = \sin(x)$ für die Intervalle: $[0 ; 2 \pi]$ , $[0 ; 5 \pi]$ , $[0 ; n \cdot \pi]$ , $[2 \pi ; 8 \pi]$ .

Find the derivative of the function $f(x)=\frac{4}{5 x^{5}}$ in simplest form without negative exponents.

Find the derivative of the function $y=-\sqrt[3]{x^{4}}$ , expressed in radical form and simplified.

Find the derivative of the function $y=-\frac{1}{x}$ and express it without negative exponents.

Find the derivative of the function $f(x)=-\frac{1}{2 x^{3}}$ and express it without negative exponents.

Bestimme die Ableitung für $x_{0}=1$ mit einer Näherungstabelle für den Differenzenquotienten für: a) $f(x)=x^{2}$ , b) $f(x)=x^{3}$ , c) $f(x)=x^{2}+3$ , d) $f(x)=2 x^{3}-1$ .

Bestimme die Ableitung von $f(x)=x^{2}$ an $x_{0}=4$ durch Näherungstabellen für den Differenzenquotienten.

Bestimmen Sie Grenzwerte für $x \rightarrow+\infty$ und $x \rightarrow-\infty$ , asymptotische Gleichungen und Schnittpunkte der Funktionen a) bis d).

Determine the behavior of $f(x) = \frac{1}{x-7}$ as $x \rightarrow \infty$ . Options: a) 0 from above, b) undefined, c) 0 from below, d) 1/3.

Find the derivative of the function $y=x^{2}(2-\ln x^{2})$ .

Find where the function $f(x)=\frac{4 x}{x^{2}+4}$ has a horizontal tangent.

Find the derivative of $y=\frac{\sqrt{x}}{2}-\frac{1}{6x^3}$ .

Gegeben ist die Funktion $f(x)=x^2-3x$ . Zeichnen Sie den Graphen für $-1 \leq x \leq 4$ und bestimmen Sie Steigung und Winkel bei $x_0=2$ .

Berechne die Steigung von $f$ bei $x_{0}$ für: a) $f(x)=x^{2}, x_{0}=-1$ b) $f(x)=0,5 x^{2}, x_{0}=2$ c) $f(x)=a x^{2}, x_{0}=1$

Berechnen Sie die Steigung von $f(x)=x^{2}$ bei $x_{0}=-1$ .

Berechne die Steigung von $f$ an $x_0$ : a) $f(x)=x^{2}, x_{0}=-1$ b) $f(x)=0,5 x^{2}, x_{0}=2$ c) $f(x)=a x^{2}, x_{0}=1$

Find the derivative of the function $f(x)=\frac{6 x+3}{3 x+5}$ . What is $f^{\prime}(x)$ ?

Find the derivative of the function $y=13^{-x} \sin x$ .

Vermuten Sie den Grenzwert $g$ der Folge $a_{n}=\frac{n+4}{2 n}$ und bestimmen Sie $n$ , sodass $\left|a_{n}-g\right|<0,001$ . Beweisen Sie die Konvergenz.

Bestimme die Integrale: a) $\int(3-6 x^{2}) \, dx$ b) $\int_{0}^{2}(2 x-x^{2}) \, dx$ c) $\int_{0}^{2} e^{-x} \, dx$

Find the derivative $f^{\prime}(0)$ for the piecewise function: $f(x)=4x-2x^{2}$ for $x<0$ and $f(x)=8x^{2}+4x$ for $x \geq 0$ .

Find critical numbers for the function $f(x)=\frac{x+9}{x+4}$ .

Determine the interval where the function $f(x)=\frac{x+9}{x+4}$ is increasing.

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{6 x^{3}-6 x^{2}+3 x}{-x^{3}-2 x+6}$

Find $\delta > 0$ such that for all $x$ , if $0 < |x - 2| < \delta$ , then $|f(x) + 17| < 0.01$ for $f(x) = -9x + 1$ .

Find the limit as $x$ approaches infinity for $\frac{\sqrt[3]{x}-7 x-4}{5 x+x^{2 / 3}+2}$ by simplifying.

Find the slope of the tangent to $f(x) = x^2 - 3$ at $P(-4, 13)$ using $m_{\tan }=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ . Then, find the tangent line's equation.

Find the limit as x approaches infinity: $\lim _{x \rightarrow \infty} \frac{x^{2}-9 x+9}{x^{3}-6 x^{2}+19}$

Berechnen Sie die Fläche unter der Kurve von $f(x) = x^3$ im Intervall [0, 2].

Find the limit of $f(x)=\frac{-2+\frac{6}{x}}{4-\frac{5}{x^{2}}}$ as $x \to \infty$ and $x \to -\infty$ .

Find the derivative of $y=\frac{3 x^{4}+2 x+5}{\sqrt{x}}$ : a) $\frac{d y}{d x}$

Calculate the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{4 x^{2}+3}}{5 x+1}$ .

Is the function $f(x)=\frac{x^{2}-36}{x-6}$ for $x \neq 6$ and $f(6)=10$ continuous at $x=6$ ?

How much must lan invest at a continuous interest rate of $6.2\%$ to reach \$182,000 in 10 years? Round to the nearest hundredth.

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(x-\sqrt{x^{2}-6 x}\right)$ . Hint: Multiply by $\frac{x+\sqrt{x^{2}-6 x}}{x+\sqrt{x^{2}-6 x}}$ .

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \sqrt{\frac{16 x^{2}+x-3}{(x-9)(x+1)}}$

Find the derivative of $f(x)=1-9 x^{2}$ at $x=3$ using the limit definition. Simplify and evaluate the limit.

Find $\lim _{x \rightarrow a}\left[\frac{-6 f(x)-4 g(x)}{-7+g(x)}\right]$ given $\lim _{x \rightarrow a} f(x)=-2$ and $\lim _{x \rightarrow a} g(x)=5$ .

Find the derivative of $f(x)=1-9 x^{2}$ using the limit definition: $f^{\prime}(x)=\lim _{h \rightarrow 0}\left(\frac{f(x+h)-f(x)}{h}\right)$ .

Find the derivative of $f(x)=1-9 x^{2}$ using the definition: $f^{\prime}(x)=\lim _{h \rightarrow 0}\left(\frac{f(x+h)-f(x)}{h}\right)$ . Simplify and evaluate the limit.

Find the limit as x approaches -7 for the expression $4x - 5$ .

Check if the function $f(x)=\frac{5 x^{2}+23 x+12}{x^{2}-8 x}$ is continuous at $a=8$ .

Find where the function $f(x)=\frac{x^{2}}{x^{2}-4}$ has a horizontal tangent line.

Calculate the limit: $\lim _{x \rightarrow 2^{+}} \frac{1}{x^{2}-4}$ .

Bestimme die Stammfunktion $F$ von $f(x)=3x^2-2x$ , die durch den Punkt $P(2|-1)$ verläuft. Wähle $C$ passend.

Find the tangent line to $f(x)=1-9 x^{2}$ at $x=4$ with slope $f^{\prime}(4)=-72$ and point $(4,-143)$ . What is $y=$ ?

Find the tangent line to $f(x)=x^{2}-6x+7$ that is:
a) Parallel to $y=2x+4$
b) Perpendicular to $y=2x+4$

Find $f^{\prime \prime}(x)$ for the function $f(x)=\frac{x^{3}-3 x^{2}-4 x-1}{2 x}$ .

Find the maximum profit given the cost function $C(q) = 80 + 19q$ and price function $p = 63 - 2q$ .

A function is continuous at a point if it satisfies these three conditions: 1) $f(c)$ is defined, 2) $\lim_{x \to c} f(x)$ exists, 3) $\lim_{x \to c} f(x) = f(c)$ .

Evaluate the limit: $\lim _{h \rightarrow 0}\left(\frac{f(7+h)-f(7)}{h}\right)$ for $f(x)=\frac{9}{x-2}$ .

Elliot's roast cools from $165^{\circ} \mathrm{F}$ to $140^{\circ} \mathrm{F}$ in 12 min. How long to reach $125^{\circ} \mathrm{F}$ ? Use $T(t)=70+\left(165-70\right)e^{-kt}$ . Find $t$ rounded to the nearest minute.

Find the derivative $f^{\prime}(0)$ using the limits $\lim _{x \rightarrow 0^{-}}$ and $\lim _{x \rightarrow 0^{+}}$ for the piecewise function $f(x)$ .

Find $f^{\prime}(0)$ for the piecewise function: $f(x)=4x-2x^{2}$ for $x<0$ and $f(x)=8x^{2}+4x$ for $x \geq 0$ .

Find $h^{\prime}(4)$ for $h(x)=5 f(x)-\frac{2}{3} g(x)$ and $h(x)=3+8 f(x)$ using given values.

Find the points where the function $f(x)=x^{3}-6 x^{2}+9 x+4$ has horizontal tangents.

Find the limit of the series $\sum_{n=1}^{\infty}\left(\frac{3 n^{5}}{15 n^{4}+5}\right)$ . Options: 0, 0.2, 5, $\infty$ .

Find the limit of the series $\sum_{n=1}^{\infty}\left(\frac{3 n^{4}}{15 n^{4}+5}\right)$ . Options: 0, 0.2, 5, $\infty$ .

Find the limit of the series $\sum_{n=1}^{\infty}\left(\frac{3 n^{3}}{15 n^{4}+5}\right)$ . Options: 0, 0.2, 5, $\infty$ .