Calculus

Problem 26701

Differentiate the equation $x^{2}+y^{5}=-7$ implicitly to find $\frac{d y}{d x}$ . $\frac{d y}{d x}=$

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Problem 26702

Find the derivative $f'(x)$ of $f(x)=\sqrt{4x^2+5x+5}$ and calculate $f'(3)$ .

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Problem 26703

Find the derivative of the function $f(x)=\frac{8 x+5}{4 x+5}$ . What is $f^{\prime}(x) ?$

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Problem 26704

Die C-14-Methode zur Altersbestimmung hat folgende Aufgaben: a) Halbwertszeit von C-14 berechnen. b) Alter eines Organismus bei 10% C-14. c) Prozent C-14 in einem 1000 Jahre alten Fossil. d) Zerfallsgesetz in der Form $N(t)=N_{0} \cdot a^{t}$ umformen.

See Solution

Problem 26705

Find the derivative $f'(x)$ of the function $f(x)=\sqrt{2 x^{2}+3 x+3}$ and evaluate $f'(2)$ .

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Problem 26706

How much must Ashley invest at a continuous interest rate of $5.6\%$ to have \$2,080 in 16 years?

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Problem 26707

Find the derivative of the function $f(x)=6+\frac{8}{x}+\frac{5}{x^{2}}$ and evaluate $f^{\prime}(5)$ .

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Problem 26708

Find the local minimum and maximum of the function $f(x)=-2 x^{3}+27 x^{2}-108 x+1$ .

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Problem 26709

How much must Alang invest at a $5.7\%$ continuous interest rate to reach $\$113,000$ in 5 years?

See Solution

Problem 26710

Find where the function $f(x) = -4x^3 + 48x^2 + 108x - 6$ is increasing by analyzing its first derivative.

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Problem 26711

Find the growth rate of the bacterial colony $p(t)=4 t^{2}+20 t+200$ after 2 hours. Answer in bacteria/hour.

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Problem 26712

How much must Addison invest at $6\%$ continuous interest to have $\$94,000$ in 5 years? Round to the nearest \$10.

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Problem 26713

Calculate the integral $\int \frac{d x}{\sqrt{64 x^{2}-81}}$ for $x > \frac{9}{8}$ .

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Problem 26714

Evaluate the integral: $\int \frac{d x}{\sqrt{49 x^{2}-81}}$ , where $x > \frac{9}{7}$ .

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Problem 26715

Find $x$ values where the tangent line of $f(x)=2 x^{3}-36 x^{2}+210 x-7$ is horizontal. Smaller $x=$ , larger $x=$ .

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Problem 26716

Evaluate the integral $\int \frac{d x}{x^{2} \sqrt{x^{2}-169}}$ for $x > 13$ .

See Solution

Problem 26717

Find the tangent line equation for the function $f$ at $x=\frac{1}{5}$ , given $f\left(\frac{1}{5}\right)=-1.4$ and $f^{\prime}\left(\frac{1}{5}\right)=3$ .

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Problem 26718

Find the first and second derivatives of $f(x)=x^{3}+3 x^{2}-189 x+8$ and determine intervals for increasing, decreasing, concave up, and concave down.

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Problem 26719

Find the critical value of $f(x)=(2-5x)^{6}$ at $A=$ . Determine the behavior of $f(x)$ for $x<A$ and $x>A$ .

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Problem 26720

Evaluate the integral $\int \frac{d x}{x^{2} \sqrt{x^{2}-121}}$ for $x > 11$ .

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Problem 26721

Evaluate the integral $\int_{0}^{\frac{5 \sqrt{3}}{2}} \frac{d x}{\sqrt{25-x^{2}}}$ .

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Problem 26722

A ball rolls off a $0.84-\mathrm{m}$ table at $0.580 \mathrm{~m/s}$ . Find its speed just before hitting the ground.

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Problem 26723

Compute the limits: (a) $\lim_{x \to 5} (x^3 + x)$ , (b) $\lim_{x \to -4} \frac{x^2 + x - 12}{x + 4}$ , (c) $\lim_{x \to 2} \frac{x^2 + 5x - 14}{x^2 - 4x + 4}$ , (d) $\lim_{x \to \infty} \frac{19x^4 + 30x^2 - 3x + 1}{x^6 + 15x^3}$ , (e) $\lim_{x \to -\infty} \frac{10x^5 - 4x^3 + 5}{2x^5 + 14x^2 - 36x}$ .

See Solution

Problem 26724

Evaluate the expressions: Round to the nearest thousandth. Use $e^{-0.35}$ and $125 e^{0.7}$ .

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Problem 26725

Compute the derivatives of these functions: (a) $f(x)=\sin (x) \cos (x)$ , (b) $g(x)=\frac{1}{\sqrt{x}-1}$ , (c) $h(x)=(x^{3}+\sin (x))^{2}$ , (d) $s(x)=\ln (\sqrt[3]{x^{5}})$ , (e) $r(x)=x^{2} e^{3 x}$ .

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Problem 26726

A carpenter builds an open box with a square base and surface area of $8 \mathrm{~m}^{2}$ . Find dimensions for maximum volume.

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Problem 26727

Find the sum of the series: $\sum_{n=1}^{\infty}(-1)^{n} \frac{\pi^{2 n}}{(2 n) ! 4^{n} 9^{n}}.$

See Solution

Problem 26728

Find the global max and min for these functions on the specified intervals: (a) $f(x)=\frac{x^{2}}{x-2}$ on $[-1, 1]$ (b) $g(x)=(\ln (x)-1)^{2/3}$ on $[1, e^{2}]$

See Solution

Problem 26729

Find global max and min for: (a) $f(x)=\frac{x^{2}}{x-2}$ on $[-1,1]$ ; (b) $g(x)=(\ln (x)-1)^{2/3}$ on $[1,e^{2}]$ .

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Problem 26730

Bestimme die Halbwertszeit der C-14 Zerfallsgleichung $y(t)=100 \cdot e^{-0,00012097 \cdot t}$ . Finde das Alter eines Baumstamms mit 60\% C-14.

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Problem 26731

A bicyclist moving at $20.0 \mathrm{~m/s}$ descends a hill. Find the maximum height he can reach, ignoring friction.

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Problem 26732

Aufgabe: Apfelsaft kühlt nach $T(t)=5+18 \cdot e^{-0,1 t}$ ab. Bestimme a) den Abnahmeprozess, b) die Kühlschranktemperatur, c) die Anfangstemperatur und d) $T(10)$ .

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Problem 26733

What is the maximum value of $\int_{0}^{2} f(x) \, dx$ if $0 \leq f(x) \leq 4$ for a continuous function $f$ ?

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Problem 26734

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{10+h}-\sqrt{10}}{h}$ .

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Problem 26735

Find the average velocity of a particle with $v(t)=e^{t}+t e^{t}$ from $t=0$ to $t=3$ .

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Problem 26736

Find $\int_{0}^{10} f(x) d x$ given $\int_{0}^{5} f(x) d x=-5$ and $\int_{6}^{10} f(x) d x=8$ .

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Problem 26737

Find $F'(x)$ if $F(x)=\int_{1}^{x^{2}} \sqrt{1+t^{3}} dt$ . Choices: (A) $2 x \sqrt{1+x^{6}}$ , (B) $2 x \sqrt{1+x^{3}}$ , (C) $\sqrt{1+x^{6}}$ , (D) $\sqrt{1+x^{3}}$ , (E) $\int_{1}^{x^{2}} \frac{3 t^{2}}{2 \sqrt{1+t^{3}}} dt$ .

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Problem 26738

Find $G'(1)$ and $G(2)$ for $G(x)=\int_{2}^{2x} f(t) dt$ using $f$ from the graph on $[0,7]$ .

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Problem 26739

Find $g(6)$ if $g(x)$ is an antiderivative of $f(x)=\frac{e^{x}}{x^{2}}$ and $g(3)=0$ . Choices: (A) 8.488 (B) 8.975 (C) 10.206 (D) 11.206 (E) 15.513

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Problem 26740

Find the integral of the function $e^{x^3}$ with respect to $x$ : $\int e^{x^3} \, dx$ .

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Problem 26741

Determine if $f(x)=\frac{1}{x-2}$ is continuous at $f(2)$ ; if not, identify the type of discontinuity.

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Problem 26742

Evaluate the series: $\sum_{k=1}^{\infty} 75\left(\frac{1}{6}\right)^{k}$

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Problem 26743

Find $\frac{d y}{d x}$ for $y=\int_{1}^{6}(3 t^{2}-5 t) dt$ . Choose from the options (A) to (E).

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Problem 26744

If $\int_{a}^{b} \xi(x) d x=4 a+b$ , find $\int_{a}^{b}(g(x)+7) d x$ .

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Problem 26745

Find $-\frac{d y}{d x}$ for $y=\int_{1}^{6}(3 t^{2}-5 t) dt$ . Choose from options (A) to (E).

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Problem 26746

Find intervals where $f$ is decreasing given $f^{\prime}(x)=\frac{1}{x}-e^{x} \sin x$ for $0<x \leq 4$ .

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Problem 26747

Find the position of a particle at time $t=1$ if $v(t)=3 t^{2}+6 t$ and $x(0)=2$ . Choices: (A) 4 (B) 6 (C) 9 (D) 11 (E) 12

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Problem 26748

Find the integral: $\int\left(5 \sec ^{2} t+\sqrt[3]{t}\right) d t=$ (A) (B) (C) (D) (E)

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Problem 26749

Find $y^{\prime \prime}(0)$ if $y(x)=\int_{0}^{x^{2}} e^{t^{2}} d t$ .

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Problem 26750

Determine where the function $f(x)=2 x^{3}-3 x^{2}-12 x$ is increasing or decreasing.

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Problem 26751

Find where the function $f(x)=x-3 x^{\frac{1}{3}}$ is increasing or decreasing.

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Problem 26752

Determine where the function $f(x)=x^{4}-4 x^{2}+3$ is increasing or decreasing.

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Problem 26753

Show that the function $y=c\left(e^{x}\right)$ is its own derivative for any constant $c$ .

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Problem 26754

Find the average rate of change of $f(x) = \cos x$ from 0 to $\frac{4 \pi}{3}$ .

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Problem 26755

Calculate the average rate of change of $f(x)=\tan(2x)$ from 0 to $\frac{11\pi}{12}$ .

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Problem 26756

Find the second derivative of these functions: (a) $f(x): x^{4}-\sin x$ (b) $f(x): \sin(x^{2})$ (c) $f(x)=e^{4x-2}$
Also, find the derivative of: (a) $f(x)=(2x^{3}+3)(8x^{2}+4)$ (b) $f(x)=(4x+2)^{3}$ (c) $f(x)=\frac{8x^{4}+2x-14}{4x^{2}+3}$

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Problem 26757

Find the derivatives of these functions: a. $f(x)=(2 x^{3}+3)(8 x^{2}+4)$ b. $f(x)=(4 x+2)^{3}$ c. $f(x)=\frac{8 x^{4}+2 x-14}{4 x^{2}+3}$

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Problem 26758

Find the second derivative of these functions: (a) $f(x) = x^{4} - \sin x$ , (b) $f(x) = \sin(x^{2})$ , (c) $f(x) = e^{4x - 2}$ .

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Problem 26759

Find the equation of the tangent line to $f(x)=3 x^{2}+2 x-14$ with a slope of 4.

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Problem 26760

Find the average rate of change of $f(x)=\tan(2x)$ from 0 to $\frac{\pi}{12}$ .

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Problem 26761

Calculate the average rate of change of $f(x) = \tan(2x)$ from $0$ to $\frac{11\pi}{12}$ .

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Problem 26762

Evaluate these limits: a. $\lim _{x \rightarrow 4} \frac{x^{2}+12 x+32}{x+4}$ b. $\lim _{x \rightarrow-4} \frac{x^{2}+12 x+32}{x+4}$

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Problem 26763

A car of mass $m$ starts with velocity $V_0$ . Given a constant force $F_0$ , find its velocity at time $t$ .

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Problem 26764

Bestimme die Ableitungsfunktion von $f(x)=x^{16}$ . Was ist $f^{\prime}(x)$ ?

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Problem 26765

Bestimme die Tangentengleichung von $f(x)=x^{6}$ bei $x_{0}=2$ .

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Problem 26766

Estimate $f(1.9)$ using local linear approximation at $x=2$ given $f(2)=-3$ and $f'(2)=4$ . Is it an underestimate or overestimate?

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Problem 26767

Bestimme die Ableitung von $f(x)=(x^{-2})^{-3}$ und fasse sie zuerst mit Potenzgesetzen zusammen. A: $f(x)=\square$ , $f'(x)=\square$

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Problem 26768

Calculate the volume of a cylinder using $V = \pi r^{2} h$ and an oblique cylinder with height $h$ and radius $r$ .

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Problem 26769

Bestimme die Ableitungsfunktion von $f(x)=(x^{-3})^{-5}$ . Fasse zuerst zu einer Potenz zusammen. A $f(x)=\square$ , $f^{\prime}(x)=\square$

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Problem 26770

Gegeben ist die Funktion $f(x)=\frac{1}{4}(x^{2}-2x+1)$ . Berechnen Sie die Steigung der Sekanten in den Intervallen a) $[-1;5]$ , b) $[0;4]$ , c) $[1;3]$ , d) $[1.5;2.5]$ , e) $[1.75;2.25]$ , f) $[1.9;2.1]$ , g) $[1.99;2.01]$ . Bestimmen Sie den Grenzwert des Differenzenquotienten für $h \rightarrow 0$ in $[2-h;2+h]$ .

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Problem 26771

Bestimme die Ableitung von $f(x)=x^{-5}$ . Finde $f^{\prime}(x)$ .

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Problem 26772

Bestimme die Steigung der Funktion $f(x)=x^{-6}$ an der Stelle $x=2$ .
A $m=\square$

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Problem 26773

A yo-yo's height is $h(t)=t^{3}-6 t^{2}+5 t+30$ . Answer these: a. Is height increasing/decreasing at $t=2$ ? b. Average rate of change from $t=0$ to $t=4$ ? c. When is there no acceleration? d. Displacement in first 4 seconds? When is the tangent of $f(x)=2^{x}-6$ parallel to $y=7+3x$ ?

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Problem 26774

Gegeben ist die Funktion $f(x)=x^{-6}$ . Finde die Steigung des Graphen bei $x=-3$ . A: $m=1$

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Problem 26775

Evaluate the series: $\sum_{n=1}^{\infty} \frac{5 \sqrt{n}-1}{n^{2}+3 n+1}$ .

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Problem 26776

Bestimme die Ableitung von $f(x)=(x^{-4})^{3}$ nach Vereinfachung der Potenz. A: $f(x)=\square$ , $f'(x)=\square$

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Problem 26777

Bestimme die Tangentengleichung von $f(x)=x^{-4}$ bei $x_{0}=1$ .

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Problem 26778

Bei welcher $x$ -Stelle hat die Funktion $f(x)=x^{-6}$ eine Steigung von 6? A: Bei $x=\square$ .

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Problem 26779

Bestimme die Ableitungsfunktion von $f(x)=-\frac{4}{x}+3 x^{6}$ mit Faktor- und Summenregel.

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Problem 26780

Brownville's population grows at $2.2\%$ per year. If it's 720,000 now, how long until it reaches 1,000,000?

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Problem 26781

Bestimme die Ableitungsfunktion von $f(x)=(x^{-5})^{3}$ . Fasse zuerst zu einer Potenz zusammen und leite ab.

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Problem 26782

Find the first 5 non-zero terms of the Taylor series for $\frac{1}{\sqrt{1+x}}$ and $\frac{1}{\sqrt{1-x^{2}}$, then use it to approximate $\arcsin x$.

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Problem 26783

Bestimme die Ableitungsfunktion von $f(x)=-\frac{4}{x}-4 \cdot \sin (x)$ mit Faktor- und Summenregel.

See Solution

Problem 26784

Bestimme die Ableitungsfunktion von $f(x)=-6 \cdot \cos (x)-4 \sqrt{x}$ mithilfe der Regeln.

See Solution

Problem 26785

What does the slope of a position vs. time graph represent: Mass, Force, Velocity, or Acceleration?

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Problem 26786

Find the local maximum and minimum points of the function $f(x)=2 x^{3}-54 x$ using the first derivative test.

See Solution

Problem 26787

Find the average rate of change of the function $h(x)=-x^{2}+6 x+11$ from $x=-3$ to $x=6$ .

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Problem 26788

Find $\frac{d y}{d t}$ given $z=\frac{x y}{2}$ , $\frac{d z}{d t}=-12$ , $\frac{d x}{d t}=3$ , $z=4$ , $y=6$ .

See Solution

Problem 26789

Find the limit: $\lim _{x \rightarrow 2} \frac{3 f(x)-15}{x-2}$ given that $f(2)=5$ .

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Problem 26790

Find the local maximum and minimum points of the function $f(x)=2 x^{3}-24 x$ using the first derivative test.

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Problem 26791

Find the relative maximum and minimum points of the function $f(x)=-x^{3}+6x^{2}+1$ using the first derivative.

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Problem 26792

Find the relative maximum and minimum points of the function $f(x)=-x^{3}+12 x^{2}-45 x+1$ using the first derivative test.

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Problem 26793

Find the relative maximum and minimum points of $f(x)=-x^{3}-12 x^{2}+2$ using the first-derivative test.

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Problem 26794

Find the local linear approximation to $f(x)=\int_{1/2}^{\frac{3}{2} x^{2}-1} \frac{\sin \pi \theta}{\theta} d\theta$ at $x=1$ .

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Problem 26795

Find the relative maximum and minimum points of the function $f(x)=-x^{3}-6 x^{2}+3$ using the first-derivative test.

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Problem 26796

Find the limit as $x$ approaches 0 from the right of $(\sin(x))^x$ .

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Problem 26797

Find the average rate of change of $f(x)=3 x^{2}+4 x+1$ on the interval $[1,3]$ .

See Solution

Problem 26798

Find the interval(s) where the function $f(x)=x^{2}-5x-6$ has a negative rate of change.

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Problem 26799

Evaluate the integral $\int_{-1}^{1}\left|x^{3}-x\right| d x$ . What is the result?

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Problem 26800

Which definite integrals equal $\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(-1+\frac{4 k}{n}\right)^{2} \frac{4}{n}$ ? Options: I. $\int_{-1}^{3} x^{2} dx$ II. $\int_{0}^{4}(-1+x)^{2} dx$ III. $\int_{0}^{1} 4(-1+4 x)^{2} dx$ (A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III only

See Solution

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Calculus

Problem 26701

Differentiate the equation x2+y5=−7x^{2}+y^{5}=-7x2+y5=−7 implicitly to find dydx\frac{d y}{d x}dxdy​. dydx=\frac{d y}{d x}=dxdy​=

Problem 26702

Find the derivative f′(x)f'(x)f′(x) of f(x)=4x2+5x+5f(x)=\sqrt{4x^2+5x+5}f(x)=4x2+5x+5​ and calculate f′(3)f'(3)f′(3).

Problem 26703

Find the derivative of the function f(x)=8x+54x+5f(x)=\frac{8 x+5}{4 x+5}f(x)=4x+58x+5​. What is f′(x)?f^{\prime}(x) ?f′(x)?

Problem 26704

Die C-14-Methode zur Altersbestimmung hat folgende Aufgaben: a) Halbwertszeit von C-14 berechnen. b) Alter eines Organismus bei 10% C-14. c) Prozent C-14 in einem 1000 Jahre alten Fossil. d) Zerfallsgesetz in der Form N(t)=N0⋅atN(t)=N_{0} \cdot a^{t}N(t)=N0​⋅at umformen.

Problem 26705

Find the derivative f′(x)f'(x)f′(x) of the function f(x)=2x2+3x+3f(x)=\sqrt{2 x^{2}+3 x+3}f(x)=2x2+3x+3​ and evaluate f′(2)f'(2)f′(2).

Problem 26706

How much must Ashley invest at a continuous interest rate of 5.6%5.6\%5.6% to have \$2,080 in 16 years?

Problem 26707

Find the derivative of the function f(x)=6+8x+5x2f(x)=6+\frac{8}{x}+\frac{5}{x^{2}}f(x)=6+x8​+x25​ and evaluate f′(5)f^{\prime}(5)f′(5).

Problem 26708

Find the local minimum and maximum of the function f(x)=−2x3+27x2−108x+1f(x)=-2 x^{3}+27 x^{2}-108 x+1f(x)=−2x3+27x2−108x+1.

Problem 26709

How much must Alang invest at a 5.7%5.7\%5.7% continuous interest rate to reach $113,000\$113,000$113,000 in 5 years?

Problem 26710

Find where the function f(x)=−4x3+48x2+108x−6f(x) = -4x^3 + 48x^2 + 108x - 6f(x)=−4x3+48x2+108x−6 is increasing by analyzing its first derivative.

Problem 26711

Find the growth rate of the bacterial colony p(t)=4t2+20t+200p(t)=4 t^{2}+20 t+200p(t)=4t2+20t+200 after 2 hours. Answer in bacteria/hour.

Problem 26712

How much must Addison invest at 6%6\%6% continuous interest to have $94,000\$94,000$94,000 in 5 years? Round to the nearest \$10.

Problem 26713

Calculate the integral ∫dx64x2−81\int \frac{d x}{\sqrt{64 x^{2}-81}}∫64x2−81​dx​ for x>98x > \frac{9}{8}x>89​.

Problem 26714

Evaluate the integral: ∫dx49x2−81\int \frac{d x}{\sqrt{49 x^{2}-81}}∫49x2−81​dx​, where x>97x > \frac{9}{7}x>79​.

Problem 26715

Find xxx values where the tangent line of f(x)=2x3−36x2+210x−7f(x)=2 x^{3}-36 x^{2}+210 x-7f(x)=2x3−36x2+210x−7 is horizontal. Smaller x=x=x=, larger x=x=x=.

Problem 26716

Evaluate the integral ∫dxx2x2−169\int \frac{d x}{x^{2} \sqrt{x^{2}-169}}∫x2x2−169​dx​ for x>13x > 13x>13.

Problem 26717

Find the tangent line equation for the function fff at x=15x=\frac{1}{5}x=51​, given f(15)=−1.4f\left(\frac{1}{5}\right)=-1.4f(51​)=−1.4 and f′(15)=3f^{\prime}\left(\frac{1}{5}\right)=3f′(51​)=3.

Problem 26718

Find the first and second derivatives of f(x)=x3+3x2−189x+8f(x)=x^{3}+3 x^{2}-189 x+8f(x)=x3+3x2−189x+8 and determine intervals for increasing, decreasing, concave up, and concave down.

Problem 26719

Find the critical value of f(x)=(2−5x)6f(x)=(2-5x)^{6}f(x)=(2−5x)6 at A=A=A=. Determine the behavior of f(x)f(x)f(x) for x<Ax<Ax<A and x>Ax>Ax>A.

Problem 26720

Evaluate the integral ∫dxx2x2−121\int \frac{d x}{x^{2} \sqrt{x^{2}-121}}∫x2x2−121​dx​ for x>11x > 11x>11.

Problem 26721

Evaluate the integral ∫0532dx25−x2\int_{0}^{\frac{5 \sqrt{3}}{2}} \frac{d x}{\sqrt{25-x^{2}}}∫0253​​​25−x2​dx​.

Problem 26722

A ball rolls off a 0.84−m0.84-\mathrm{m}0.84−m table at 0.580 m/s0.580 \mathrm{~m/s}0.580 m/s. Find its speed just before hitting the ground.

Problem 26723

Problem 26724

Evaluate the expressions: Round to the nearest thousandth. Use e−0.35e^{-0.35}e−0.35 and 125e0.7125 e^{0.7}125e0.7.

Problem 26725

Problem 26726

A carpenter builds an open box with a square base and surface area of 8 m28 \mathrm{~m}^{2}8 m2. Find dimensions for maximum volume.

Problem 26727

Find the sum of the series: ∑n=1∞(−1)nπ2n(2n)!4n9n.\sum_{n=1}^{\infty}(-1)^{n} \frac{\pi^{2 n}}{(2 n) ! 4^{n} 9^{n}}.n=1∑∞​(−1)n(2n)!4n9nπ2n​.

Problem 26728

Find the global max and min for these functions on the specified intervals: (a) f(x)=x2x−2f(x)=\frac{x^{2}}{x-2}f(x)=x−2x2​ on [−1,1][-1, 1][−1,1] (b) g(x)=(ln⁡(x)−1)2/3g(x)=(\ln (x)-1)^{2/3}g(x)=(ln(x)−1)2/3 on [1,e2][1, e^{2}][1,e2]

Problem 26729

Find global max and min for: (a) f(x)=x2x−2f(x)=\frac{x^{2}}{x-2}f(x)=x−2x2​ on [−1,1][-1,1][−1,1]; (b) g(x)=(ln⁡(x)−1)2/3g(x)=(\ln (x)-1)^{2/3}g(x)=(ln(x)−1)2/3 on [1,e2][1,e^{2}][1,e2].

Problem 26730

Bestimme die Halbwertszeit der C-14 Zerfallsgleichung y(t)=100⋅e−0,00012097⋅ty(t)=100 \cdot e^{-0,00012097 \cdot t}y(t)=100⋅e−0,00012097⋅t. Finde das Alter eines Baumstamms mit 60\% C-14.

Problem 26731

A bicyclist moving at 20.0 m/s20.0 \mathrm{~m/s}20.0 m/s descends a hill. Find the maximum height he can reach, ignoring friction.

Problem 26732

Aufgabe: Apfelsaft kühlt nach T(t)=5+18⋅e−0,1tT(t)=5+18 \cdot e^{-0,1 t}T(t)=5+18⋅e−0,1t ab. Bestimme a) den Abnahmeprozess, b) die Kühlschranktemperatur, c) die Anfangstemperatur und d) T(10)T(10)T(10).

Problem 26733

What is the maximum value of ∫02f(x) dx\int_{0}^{2} f(x) \, dx∫02​f(x)dx if 0≤f(x)≤40 \leq f(x) \leq 40≤f(x)≤4 for a continuous function fff?

Problem 26734

Find the limit: lim⁡h→010+h−10h\lim _{h \rightarrow 0} \frac{\sqrt{10+h}-\sqrt{10}}{h}limh→0​h10+h​−10​​.

Problem 26735

Find the average velocity of a particle with v(t)=et+tetv(t)=e^{t}+t e^{t}v(t)=et+tet from t=0t=0t=0 to t=3t=3t=3.

Problem 26736

Find ∫010f(x)dx\int_{0}^{10} f(x) d x∫010​f(x)dx given ∫05f(x)dx=−5\int_{0}^{5} f(x) d x=-5∫05​f(x)dx=−5 and ∫610f(x)dx=8\int_{6}^{10} f(x) d x=8∫610​f(x)dx=8.

Problem 26737

Problem 26738

Find G′(1)G'(1)G′(1) and G(2)G(2)G(2) for G(x)=∫22xf(t)dtG(x)=\int_{2}^{2x} f(t) dtG(x)=∫22x​f(t)dt using fff from the graph on [0,7][0,7][0,7].

Problem 26739

Find g(6)g(6)g(6) if g(x)g(x)g(x) is an antiderivative of f(x)=exx2f(x)=\frac{e^{x}}{x^{2}}f(x)=x2ex​ and g(3)=0g(3)=0g(3)=0. Choices: (A) 8.488 (B) 8.975 (C) 10.206 (D) 11.206 (E) 15.513

Problem 26740

Find the integral of the function ex3e^{x^3}ex3 with respect to xxx: ∫ex3 dx\int e^{x^3} \, dx∫ex3dx.

Problem 26741

Determine if f(x)=1x−2f(x)=\frac{1}{x-2}f(x)=x−21​ is continuous at f(2)f(2)f(2); if not, identify the type of discontinuity.

Differentiate the equation $x^{2}+y^{5}=-7$ implicitly to find $\frac{d y}{d x}$ . $\frac{d y}{d x}=$

Find the derivative $f'(x)$ of $f(x)=\sqrt{4x^2+5x+5}$ and calculate $f'(3)$ .

Find the derivative of the function $f(x)=\frac{8 x+5}{4 x+5}$ . What is $f^{\prime}(x) ?$

Die C-14-Methode zur Altersbestimmung hat folgende Aufgaben: a) Halbwertszeit von C-14 berechnen. b) Alter eines Organismus bei 10% C-14. c) Prozent C-14 in einem 1000 Jahre alten Fossil. d) Zerfallsgesetz in der Form $N(t)=N_{0} \cdot a^{t}$ umformen.

Find the derivative $f'(x)$ of the function $f(x)=\sqrt{2 x^{2}+3 x+3}$ and evaluate $f'(2)$ .

How much must Ashley invest at a continuous interest rate of $5.6\%$ to have \$2,080 in 16 years?

Find the derivative of the function $f(x)=6+\frac{8}{x}+\frac{5}{x^{2}}$ and evaluate $f^{\prime}(5)$ .

Find the local minimum and maximum of the function $f(x)=-2 x^{3}+27 x^{2}-108 x+1$ .

How much must Alang invest at a $5.7\%$ continuous interest rate to reach $\$113,000$ in 5 years?

Find where the function $f(x) = -4x^3 + 48x^2 + 108x - 6$ is increasing by analyzing its first derivative.

Find the growth rate of the bacterial colony $p(t)=4 t^{2}+20 t+200$ after 2 hours. Answer in bacteria/hour.

How much must Addison invest at $6\%$ continuous interest to have $\$94,000$ in 5 years? Round to the nearest \$10.

Calculate the integral $\int \frac{d x}{\sqrt{64 x^{2}-81}}$ for $x > \frac{9}{8}$ .

Evaluate the integral: $\int \frac{d x}{\sqrt{49 x^{2}-81}}$ , where $x > \frac{9}{7}$ .

Find $x$ values where the tangent line of $f(x)=2 x^{3}-36 x^{2}+210 x-7$ is horizontal. Smaller $x=$ , larger $x=$ .

Evaluate the integral $\int \frac{d x}{x^{2} \sqrt{x^{2}-169}}$ for $x > 13$ .

Find the tangent line equation for the function $f$ at $x=\frac{1}{5}$ , given $f\left(\frac{1}{5}\right)=-1.4$ and $f^{\prime}\left(\frac{1}{5}\right)=3$ .

Find the first and second derivatives of $f(x)=x^{3}+3 x^{2}-189 x+8$ and determine intervals for increasing, decreasing, concave up, and concave down.

Find the critical value of $f(x)=(2-5x)^{6}$ at $A=$ . Determine the behavior of $f(x)$ for $x<A$ and $x>A$ .

Evaluate the integral $\int \frac{d x}{x^{2} \sqrt{x^{2}-121}}$ for $x > 11$ .

Evaluate the integral $\int_{0}^{\frac{5 \sqrt{3}}{2}} \frac{d x}{\sqrt{25-x^{2}}}$ .

A ball rolls off a $0.84-\mathrm{m}$ table at $0.580 \mathrm{~m/s}$ . Find its speed just before hitting the ground.

Evaluate the expressions: Round to the nearest thousandth. Use $e^{-0.35}$ and $125 e^{0.7}$ .

A carpenter builds an open box with a square base and surface area of $8 \mathrm{~m}^{2}$ . Find dimensions for maximum volume.

Find the sum of the series: $\sum_{n=1}^{\infty}(-1)^{n} \frac{\pi^{2 n}}{(2 n) ! 4^{n} 9^{n}}.$

Find the global max and min for these functions on the specified intervals: (a) $f(x)=\frac{x^{2}}{x-2}$ on $[-1, 1]$ (b) $g(x)=(\ln (x)-1)^{2/3}$ on $[1, e^{2}]$

Find global max and min for: (a) $f(x)=\frac{x^{2}}{x-2}$ on $[-1,1]$ ; (b) $g(x)=(\ln (x)-1)^{2/3}$ on $[1,e^{2}]$ .

Bestimme die Halbwertszeit der C-14 Zerfallsgleichung $y(t)=100 \cdot e^{-0,00012097 \cdot t}$ . Finde das Alter eines Baumstamms mit 60\% C-14.

A bicyclist moving at $20.0 \mathrm{~m/s}$ descends a hill. Find the maximum height he can reach, ignoring friction.

Aufgabe: Apfelsaft kühlt nach $T(t)=5+18 \cdot e^{-0,1 t}$ ab. Bestimme a) den Abnahmeprozess, b) die Kühlschranktemperatur, c) die Anfangstemperatur und d) $T(10)$ .

What is the maximum value of $\int_{0}^{2} f(x) \, dx$ if $0 \leq f(x) \leq 4$ for a continuous function $f$ ?

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{10+h}-\sqrt{10}}{h}$ .

Find the average velocity of a particle with $v(t)=e^{t}+t e^{t}$ from $t=0$ to $t=3$ .

Find $\int_{0}^{10} f(x) d x$ given $\int_{0}^{5} f(x) d x=-5$ and $\int_{6}^{10} f(x) d x=8$ .

Find $G'(1)$ and $G(2)$ for $G(x)=\int_{2}^{2x} f(t) dt$ using $f$ from the graph on $[0,7]$ .

Find $g(6)$ if $g(x)$ is an antiderivative of $f(x)=\frac{e^{x}}{x^{2}}$ and $g(3)=0$ . Choices: (A) 8.488 (B) 8.975 (C) 10.206 (D) 11.206 (E) 15.513

Find the integral of the function $e^{x^3}$ with respect to $x$ : $\int e^{x^3} \, dx$ .

Determine if $f(x)=\frac{1}{x-2}$ is continuous at $f(2)$ ; if not, identify the type of discontinuity.

Evaluate the series: $\sum_{k=1}^{\infty} 75\left(\frac{1}{6}\right)^{k}$

Find $\frac{d y}{d x}$ for $y=\int_{1}^{6}(3 t^{2}-5 t) dt$ . Choose from the options (A) to (E).

If $\int_{a}^{b} \xi(x) d x=4 a+b$ , find $\int_{a}^{b}(g(x)+7) d x$ .

Find $-\frac{d y}{d x}$ for $y=\int_{1}^{6}(3 t^{2}-5 t) dt$ . Choose from options (A) to (E).

Find intervals where $f$ is decreasing given $f^{\prime}(x)=\frac{1}{x}-e^{x} \sin x$ for $0<x \leq 4$ .

Find the position of a particle at time $t=1$ if $v(t)=3 t^{2}+6 t$ and $x(0)=2$ . Choices: (A) 4 (B) 6 (C) 9 (D) 11 (E) 12

Find the integral: $\int\left(5 \sec ^{2} t+\sqrt[3]{t}\right) d t=$ (A) (B) (C) (D) (E)

Find $y^{\prime \prime}(0)$ if $y(x)=\int_{0}^{x^{2}} e^{t^{2}} d t$ .

Determine where the function $f(x)=2 x^{3}-3 x^{2}-12 x$ is increasing or decreasing.

Find where the function $f(x)=x-3 x^{\frac{1}{3}}$ is increasing or decreasing.

Determine where the function $f(x)=x^{4}-4 x^{2}+3$ is increasing or decreasing.

Show that the function $y=c\left(e^{x}\right)$ is its own derivative for any constant $c$ .

Find the average rate of change of $f(x) = \cos x$ from 0 to $\frac{4 \pi}{3}$ .

Calculate the average rate of change of $f(x)=\tan(2x)$ from 0 to $\frac{11\pi}{12}$ .

Find the derivatives of these functions: a. $f(x)=(2 x^{3}+3)(8 x^{2}+4)$ b. $f(x)=(4 x+2)^{3}$ c. $f(x)=\frac{8 x^{4}+2 x-14}{4 x^{2}+3}$

Find the second derivative of these functions: (a) $f(x) = x^{4} - \sin x$ , (b) $f(x) = \sin(x^{2})$ , (c) $f(x) = e^{4x - 2}$ .

Find the equation of the tangent line to $f(x)=3 x^{2}+2 x-14$ with a slope of 4.

Find the average rate of change of $f(x)=\tan(2x)$ from 0 to $\frac{\pi}{12}$ .

Calculate the average rate of change of $f(x) = \tan(2x)$ from $0$ to $\frac{11\pi}{12}$ .

Evaluate these limits: a. $\lim _{x \rightarrow 4} \frac{x^{2}+12 x+32}{x+4}$ b. $\lim _{x \rightarrow-4} \frac{x^{2}+12 x+32}{x+4}$

A car of mass $m$ starts with velocity $V_0$ . Given a constant force $F_0$ , find its velocity at time $t$ .

Bestimme die Ableitungsfunktion von $f(x)=x^{16}$ . Was ist $f^{\prime}(x)$ ?

Bestimme die Tangentengleichung von $f(x)=x^{6}$ bei $x_{0}=2$ .

Estimate $f(1.9)$ using local linear approximation at $x=2$ given $f(2)=-3$ and $f'(2)=4$ . Is it an underestimate or overestimate?

Bestimme die Ableitung von $f(x)=(x^{-2})^{-3}$ und fasse sie zuerst mit Potenzgesetzen zusammen. A: $f(x)=\square$ , $f'(x)=\square$

Calculate the volume of a cylinder using $V = \pi r^{2} h$ and an oblique cylinder with height $h$ and radius $r$ .

Bestimme die Ableitungsfunktion von $f(x)=(x^{-3})^{-5}$ . Fasse zuerst zu einer Potenz zusammen. A $f(x)=\square$ , $f^{\prime}(x)=\square$

Bestimme die Ableitung von $f(x)=x^{-5}$ . Finde $f^{\prime}(x)$ .

Bestimme die Steigung der Funktion $f(x)=x^{-6}$ an der Stelle $x=2$ .
A $m=\square$

Gegeben ist die Funktion $f(x)=x^{-6}$ . Finde die Steigung des Graphen bei $x=-3$ . A: $m=1$