Calculus

Problem 30001

Find the stationary points of the function $y=\frac{1}{3} x^{3}-x^{2}-15 x-9$ . Options: $x=5, x=3$ ; $x=5, x=-3$ ; $x=-5, x=3$ ; $x=-5, x=-3$ .

See Solution

Problem 30002

Consider the function $f(x)=\frac{x^{2}}{1+x+x^{2}}$ for $x > -2$ . (a) Does $f$ have a maximum? (YES / NO) (b) Does $f$ have a minimum? (YES / NO) (c) For which $x$ is $f$ decreasing?

See Solution

Problem 30003

Find $f_{y y}$ for the function $z=e^{\ln x}(x^{2}-y^{2})$ .

See Solution

Problem 30004

Find the maximum value of $f(x)=\frac{x^{2}+2 x-3}{x^{2}+1}$ at its stationary point: $-1+\sqrt{5}$ , $1+\sqrt{5}$ , 2, or $2-\sqrt{5}$ .

See Solution

Problem 30005

Find the derivative of $y=e^{2-x} \ln e^{\sqrt{2 x}}$ . Simplify first.

See Solution

Problem 30006

Consider the function $f(x)=\frac{x^{2}}{1+x+x^{2}}$ for $x > -2$ . Does it have a maximum (YES/NO), a minimum (YES/NO), and where is it decreasing?

See Solution

Problem 30007

Evaluate the integral: $\int \frac{18 \tan ^{2} x \sec ^{2} x}{\left(2+\tan ^{3} x\right)^{2}} d x$

See Solution

Problem 30008

Find the values of $x \in \mathbb{R}$ for which $f(x)=\sum_{n=0}^{\infty} e^{-2 n x}$ is defined, and compute $f^{\prime}(x)$ .

See Solution

Problem 30009

Determine the nature of the function $z=-x^{2}+xy-y^{2}-7$ : max $z=-7$ , min $z=0$ , saddle $z=8$ , max $z=0$ .

See Solution

Problem 30010

Find the incorrect statement for extreme values of $z=xy$ with constraint $x^{2}+2y^{2}=1$ : $y= \pm \frac{1}{2}$ , $x= \pm \frac{\sqrt{2}}{2}$ , $z= \pm 2$ , $\lambda= \pm \frac{\sqrt{2}}{4}$ .

See Solution

Problem 30011

Find the derivative of $y=x^{3}-12 x+4$ , find turning points $A$ and $B$ , and the tangent at $C(1,-7)$ in $y=p x+q$ form.

See Solution

Problem 30012

Find the extreme value of $z=49-x^{2}-y^{2}$ subject to $x+3y=10$ . Which statement about the extrema is correct?

See Solution

Problem 30013

Calculate the value of $\int_{a}^{b}\left(e^{x}+\frac{1}{x}\right)$ . Choose the correct expression from the options.

See Solution

Problem 30014

Evaluate the integral $\int_{-1}^{0} \frac{x^{3}}{\sqrt{x^{4}+9}} d x$ .

See Solution

Problem 30015

Find the tangent line to the function $y \ln (2 x-y)+x^{2} y-7 x^{2}+16=0$ at the point (2,3).

See Solution

Problem 30016

Find the slope of the tangent line to $y=F(x)$ at $x=3$ , where $F(x)=\int_{2}^{x} e^{1/t} dt$ .

See Solution

Problem 30017

Berechne die Ableitung der Funktion $f(x)=-0,5 x^{2}+4 x$ an der Stelle $x=3$ mit der $h$ -Methode.

See Solution

Problem 30018

Berechne $f^{\prime}(3)$ für die Funktion $f(x)=-0,5x^2+4x$ mit dem Differenzenquotienten.

See Solution

Problem 30019

Evaluate the integral: $\int_{1}^{4} \frac{d y}{2 \sqrt{y}(1+\sqrt{y})^{2}}$

See Solution

Problem 30020

Find the derivative of the function $y=\frac{3x+9}{2-x}$ .

See Solution

Problem 30021

Calculate the integral $\int_{2}^{4} \frac{d x}{x \ln x}$ .

See Solution

Problem 30022

Calculate the limit: $\lim _{x \rightarrow-1^{-}} e^{\frac{-1}{1-x^{2}}}$ . If it doesn't exist, state that.

See Solution

Problem 30023

Find the integral of $\frac{1}{x^{2}}$ with respect to $x$ .

See Solution

Problem 30024

Find the integral of $\frac{x^{3}+5}{x^{2}}$ with respect to $x$ .

See Solution

Problem 30025

Calculate the integral $\int_{1}^{2} \frac{x-4}{x^{2}} d x$ .

See Solution

Problem 30026

Find the derivative: $\frac{d}{d x}\left(\int_{0}^{2 x} \ln \left(t^{3}+1\right) d t\right)$ for $x>0$ .

See Solution

Problem 30027

Calculate the integral from -1 to 3 of the function $x^{2}-2x$ . What is $\int_{-1}^{3}\left(x^{2}-2 x\right) d x$ ?

See Solution

Problem 30028

Find $F^{\prime}(x)$ if $F(x)=\int_{4}^{x^{2}} \sqrt{t} dt$ for $x>0$ .

See Solution

Problem 30029

Determine if the series $\sum_{n=1}^{\infty} \frac{\cos (n)}{n^{2}}$ converges or diverges.

See Solution

Problem 30030

Find the limit: $\lim _{n \rightarrow \infty} \ln \left(\frac{4 n^{2}+a}{2 n^{2}+1}\right) \cdot 2 n^{2}-1$ .

See Solution

Problem 30031

A ball is dropped from a height of $3.5 \mathrm{~m}$ . Calculate the time it takes to hit the sidewalk.

See Solution

Problem 30032

A stone is thrown upward at $20.0 \mathrm{~m/s}$ from a $50.0 \mathrm{~m}$ building. Find its velocity after $5.00 \mathrm{~s}$ .

See Solution

Problem 30033

Find the radius of convergence for the series $\sum_{n=1}^{\infty}\left(\frac{3^{n}(n !)^{2}}{(2 n) !}\right) x^{n}$ .

See Solution

Problem 30034

Find points where the tangent line to $y=f(x)=\sqrt{e^{x}-x}$ is horizontal and the tangent line at $(1, \sqrt{e-1})$ .

See Solution

Problem 30035

Find the speed of a roller coaster at points B, C, and D, starting from 120 m high with $g=10 \mathrm{~m/s}^2$ .

See Solution

Problem 30036

Găsiți valoarea marginală și elasticitatea cererii pentru funcțiile: $Q = a + b \cdot p$ , $Q = a \cdot e^{p}$ , $Q = \frac{a p}{b + p}$ , $Q = 6 + 8 p + 3 p^{2}$ , $Q = 3 \cdot \frac{2 + p}{3 + p}$ .

See Solution

Problem 30037

Determine the intervals where the function is increasing, given its max of 6 at $x=3$ and min of -2 at $x=-5$ .

See Solution

Problem 30038

A paint brush falls from a ladder. How do its velocity and acceleration change? Choose two answers for each.

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Problem 30039

Studiați funcția de cerere $C(p)=500-2p$ . Determinați elasticitatea $E_{C}(5)$ și analizați elasticitatea cererii.

See Solution

Problem 30040

Find $T_{1}(x), T_{2}(x), \ldots, T_{5}(x)$ for $f(x)=e^{x}$ at $a=1$ .

See Solution

Problem 30041

Gegeben ist die Funktion $f(t) = 2 \cdot 1,15^{t}$ . Zeichne den Graphen für $0 \leq t \leq 8$ und berechne die mittlere Änderungsrate für die Intervalle $[0; 2]$ und $[5; 8]$ .

See Solution

Problem 30042

Find the max at $x=3$ , min at $x=-5$ . Identify intervals of increase/decrease, domain, and range of the function.

See Solution

Problem 30043

Fie funcția $R(p)=p \cdot C(p)$ . Arătați că: a) dacă cererea e elastică, $R$ e descrescătoare; b) dacă e inelastică, $R$ e crescătoare. Indiciu: $R^{\prime}(p)=C(p)(1-E_{c}(p))$ .

See Solution

Problem 30044

Calculați derivatele parțiale pentru funcțiile date în punctele (1, 1) și (1, 1, 1).

See Solution

Problem 30045

Calculate the integral $\int_{\pi / 4}^{\pi / 2} \cot t \, dt$ .

See Solution

Problem 30046

Find the average rate of change of $g(x)=-5 x^{3}-2$ from $x=-4$ to $x=3$ .

See Solution

Problem 30047

Calculați derivatele parțiale pentru funcțiile date în punctele $(1,1)$ și $(1,1,1)$ . Răspunsuri incluse.

See Solution

Problem 30048

Gegeben ist die Funktion $f(t)=2 \cdot 1,15^{t}$ . a) Zeichne den Graphen für $0 \leq t \leq 8$ . b) Berechne die mittlere Änderungsrate für die Intervalle $[0 ; 2]$ und $[5 ; 8]$ .

See Solution

Problem 30049

Find the average rate of change of $g(x)=7x^{3}+\frac{5}{x^{2}}$ on the interval $[-2,1]$ .

See Solution

Problem 30050

Find the value of c for which $\lim _{x \rightarrow 2} f(x)$ exists, where $f(x)=\begin{cases} \frac{1}{2} x+c, & x<2 \\ -x+9, & x>2 \end{cases}$ .

See Solution

Problem 30051

Evaluate the integral: $\int \frac{\sin \sqrt{\theta}}{\sqrt{\theta \cos ^{3} \sqrt{\theta}}} d \theta$

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Problem 30052

Bestimmen Sie die Ableitung der Funktion $y=\frac{1}{3}(x^{3}-6 x^{2}+1)$ .

See Solution

Problem 30053

Bestimmen Sie die erste Ableitung von $y=(4 x-2)(x-7)$ .

See Solution

Problem 30054

Find the simplified difference quotient for $f(x)=x^{2}$ and complete the table for $x=5$ and various $h$ values. $\frac{f(x+h)-f(x)}{h}=$

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Problem 30055

Find the difference quotient for $f(x)=x^{2}$ and fill in the table for $x=5$ and various $h$ values.

See Solution

Problem 30056

(a) Simplify the difference quotient for $f(x)=x^{2}$ . (b) Complete the table for $x=5$ and various $h$ values.

See Solution

Problem 30057

(a) Simplify the difference quotient for $f(x)=-5x^{2}$ and (b) complete the table for $x=3$ and various $h$ values.

See Solution

Problem 30058

Calculați productivitățile marginale pentru funcțiile de producție la punctul $(1,1,2)$ : (a) $f(x, y, z)=3 x^{2} y^{4} z^{3}$ , (b) $f(p, q, r)=4 p^{2} \sqrt{q} r$ , (c) $f(x, y, z)=2 x+y-5 z$ .

See Solution

Problem 30059

(a) Simplify the difference quotient for $f(x)=-5x+2$ . (b) Complete the table for $x=4$ and various $h$ values.

See Solution

Problem 30060

Find the tangent line equation for $f(x)=\frac{4}{x}$ at the point $\left(-8,-\frac{1}{2}\right)$ . Use $y=\square$ .

See Solution

Problem 30061

Find the tangent line to $f(x)=x^{3}$ at the point $(2,8)$ . The equation is $y=\square$ .

See Solution

Problem 30062

Finde die Tangentengleichung der Funktion $f(x) = x^2 - 5x + 4 + 2\ln(x)$ bei $x = 1$ .

See Solution

Problem 30063

Find the tangent line equation for $f(x)=-3-3 x^{2}$ at the point $(-5,-78)$ . It's $y=\square$ in terms of $\mathrm{x}$ .

See Solution

Problem 30064

Find the tangent line to $f(x)=\frac{2}{x}$ at the point $\left(9, \frac{2}{9}\right)$ . The equation is $y=\square$ .

See Solution

Problem 30065

Find the tangent line equation to $f(x)=4-2 x^{2}$ at the point $(4,-28)$ . The equation is $y=\square$ .

See Solution

Problem 30066

Find the limit: $\lim _{h \rightarrow 0} \frac{\left(1+2(a+h)^{2}\right)^{3}-\left(1+2 a^{2}\right)^{3}}{h}$ . Options: a. $12 a\left(1+2 a^{2}\right)^{2}$ , b. $4 a\left(1+2 a^{2}\right)^{2}$ , c. $\left(1+2 a^{2}\right)^{3}$ , d. None.

See Solution

Problem 30067

Find the tangent line equation for $f(x)=2 x^{3}$ at the point $(-2,-16)$ . The equation is $y=\square$ .

See Solution

Problem 30068

Find the derivative $f^{\prime}(x)$ using $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ for $f(x)=2x^2$ and calculate $f^{\prime}(6)$ .

See Solution

Problem 30069

Calculaţi derivatele parţiale de ordinul 1, 2 şi 3 pentru funcţiile: (a) $f(x, y)=2 x^{2} y^{4}$ la $a=(-2,2)$ ; (b) $f(x, y)=2 \sqrt{x} y$ la $a=(1,1)$ .

See Solution

Problem 30070

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{5}{3-x}$ .

See Solution

Problem 30071

Find the derivative $f^{\prime}(x)$ of the function $f(x)=4 x^{2}$ .

See Solution

Problem 30072

Find the derivative $g^{\prime}(t)$ of the function $g(t)=\frac{9}{t^{4}}$ .

See Solution

Problem 30073

Find the derivative $H^{\prime}(x)$ of the function $H(x)=\frac{3}{\sqrt{x+1}}$ .

See Solution

Problem 30074

Find the derivative $g^{\prime}(t)$ of the function $g(t)=\frac{2}{t^{3}}$ . What is $g^{\prime}(t)=?$

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Problem 30075

Find the derivative $f^{\prime}(x)$ of the function $f(x)=11 x^{4}$ .

See Solution

Problem 30076

Find the derivative $H^{\prime}(x)$ of the function $H(x)=\frac{9}{\sqrt{x+2}}$ .

See Solution

Problem 30077

Find $f^{\prime}(5)$ using $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ for $f(x)=6 x^{2}$ .

See Solution

Problem 30078

Graph $f(x)=|2x-6|$ and find $\lim_{x \to 3} f(x)$ and $\lim_{x \to 1} f(x)$ . State if any limit does not exist.

See Solution

Problem 30079

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{2}{6-x}$ .

See Solution

Problem 30080

Gegeben sind die Funktionen $f_{1}(t)=100 e^{-0,2 t}$ und $f_{2}(t)=100-50 e^{-0,4 t}$ . Bestimme die Ableitungen und deren Graphen.

See Solution

Problem 30081

Znajdź promień zbieżności szeregu potęgowego $\sum_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^{n^{2}} x^{n}.$

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Problem 30082

Find $f(g'(x))$ for $f(x)=1-2x$ and $g(x)=3x^2-x$ . What is the result? A. $1-6x^2-x$ B. $-(6x^2-2x-1)$ C. $12x^2-10x+2$ D. $-(6x^2+2x+1)$ E. $-6x^3+5x^2-x$

See Solution

Problem 30083

Find the derivative of $(7x^2 - 5)^{10}$ with respect to $x$ .

See Solution

Problem 30084

Graph $g(x)=x^{2}-3$ and find $\lim _{x \rightarrow-1} g(x)$ and $\lim _{x \rightarrow-2} g(x)$ .

See Solution

Problem 30085

Graph the function $F(x)=\{3x+11 \text{ for } x<-4; x \text{ for } x \geq-4\}$ and find the limits at $x=-4$ .

See Solution

Problem 30086

Berechnen Sie die Ableitung der Funktion $C(x) = (7x^2 - 5)^{10}$ .

See Solution

Problem 30087

Berechne die Grenzwerte von $k(x)$ und $h(x)$ für große $x$ . Ab wann ist das Kunststoffleitwerk günstiger?

See Solution

Problem 30088

Calculate the limit: $\lim _{h \rightarrow 0}\left(\frac{1}{h^{\frac{1}{3}}}\right)^{2}=$

See Solution

Problem 30089

Graph $f(x)=|x-2|$ and find $\lim _{x \rightarrow 2} f(x)$ and $\lim _{x \rightarrow-2} f(x)$ . Limit may not exist.

See Solution

Problem 30090

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{4}{9 x^{2}}$ .

See Solution

Problem 30091

Graph the function $g(x)=x^{2}-8$ and find the limits: $\lim _{x \rightarrow 2} g(x)$ and $\lim _{x \rightarrow-3} g(x)$ .

See Solution

Problem 30092

Calculați derivatele parțiale de ordinul 1, 2 și 3 pentru funcțiile date și valorile lor în punctele specificate.

See Solution

Problem 30093

Find the derivative of the function $y=5 x^{-\frac{3}{2}}+8 x^{-\frac{1}{2}}+x^{6}-7$ . What is $y^{\prime}=?$

See Solution

Problem 30094

Graph the function $F(x)=\begin{cases} 3x-8, & x<2 \\ x, & x \geq 2 \end{cases}$ and find $\lim_{x \to 2^{-}} F(x)$ , $\lim_{x \to 2^{+}} F(x)$ , $\lim_{x \to 2} F(x)$ .

See Solution

Problem 30095

Calculați derivatele parțiale de ordin 1, 2 și 3 pentru funcțiile date la punctele specificate.

See Solution

Problem 30096

Find the average rate of change of $f(x)=\sqrt{2 x}$ from $x=2$ to $x=8$ . Options: a. 16 b. 12 c. 60 d. $\frac{1}{3}$

See Solution

Problem 30097

Find the derivative of $\sqrt[5]{x} - \frac{8}{x}$ . What is $\frac{d}{d x}\left(\sqrt[5]{x}-\frac{8}{x}\right)$ ?

See Solution

Problem 30098

Differentiate the function: $y=\frac{x}{4}-\frac{4}{x}$ . Find $\frac{d}{dx}\left(\frac{x}{4}-\frac{4}{x}\right)$ .

See Solution

Problem 30099

Calculaţi derivatele parţiale 1, 2 şi 3 pentru funcţiile: (c) $f(x, y)=3 x^{5}-2 x y^{2}+y^{3}$ la $a=(2,1)$ ; (d) $f(x, y)=x^{3} y+x y^{3}$ la $a=(-1,1)$ ; (e) $f(x, y)=\frac{2 x}{y}-\frac{y}{\sqrt{x}}$ la $a=(1,1)$ .

See Solution

Problem 30100

Find the derivative of $y = x^{-11}$ with respect to $x$ : $\frac{d y}{d x} =$

See Solution

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Calculus

Problem 30001

Find the stationary points of the function y=13x3−x2−15x−9y=\frac{1}{3} x^{3}-x^{2}-15 x-9y=31​x3−x2−15x−9. Options: x=5,x=3x=5, x=3x=5,x=3; x=5,x=−3x=5, x=-3x=5,x=−3; x=−5,x=3x=-5, x=3x=−5,x=3; x=−5,x=−3x=-5, x=-3x=−5,x=−3.

Problem 30002

Consider the function f(x)=x21+x+x2f(x)=\frac{x^{2}}{1+x+x^{2}}f(x)=1+x+x2x2​ for x>−2x > -2x>−2. (a) Does fff have a maximum? (YES / NO) (b) Does fff have a minimum? (YES / NO) (c) For which xxx is fff decreasing?

Problem 30003

Find fyyf_{y y}fyy​ for the function z=eln⁡x(x2−y2)z=e^{\ln x}(x^{2}-y^{2})z=elnx(x2−y2).

Problem 30004

Find the maximum value of f(x)=x2+2x−3x2+1f(x)=\frac{x^{2}+2 x-3}{x^{2}+1}f(x)=x2+1x2+2x−3​ at its stationary point: −1+5-1+\sqrt{5}−1+5​, 1+51+\sqrt{5}1+5​, 2, or 2−52-\sqrt{5}2−5​.

Problem 30005

Find the derivative of y=e2−xln⁡e2xy=e^{2-x} \ln e^{\sqrt{2 x}}y=e2−xlne2x​. Simplify first.

Problem 30006

Consider the function f(x)=x21+x+x2f(x)=\frac{x^{2}}{1+x+x^{2}}f(x)=1+x+x2x2​ for x>−2x > -2x>−2. Does it have a maximum (YES/NO), a minimum (YES/NO), and where is it decreasing?

Problem 30007

Evaluate the integral: ∫18tan⁡2xsec⁡2x(2+tan⁡3x)2dx\int \frac{18 \tan ^{2} x \sec ^{2} x}{\left(2+\tan ^{3} x\right)^{2}} d x∫(2+tan3x)218tan2xsec2x​dx

Problem 30008

Find the values of x∈Rx \in \mathbb{R}x∈R for which f(x)=∑n=0∞e−2nxf(x)=\sum_{n=0}^{\infty} e^{-2 n x}f(x)=∑n=0∞​e−2nx is defined, and compute f′(x)f^{\prime}(x)f′(x).

Problem 30009

Determine the nature of the function z=−x2+xy−y2−7z=-x^{2}+xy-y^{2}-7z=−x2+xy−y2−7: max z=−7z=-7z=−7, min z=0z=0z=0, saddle z=8z=8z=8, max z=0z=0z=0.

Problem 30010

Find the incorrect statement for extreme values of z=xyz=xyz=xy with constraint x2+2y2=1x^{2}+2y^{2}=1x2+2y2=1: y=±12y= \pm \frac{1}{2}y=±21​, x=±22x= \pm \frac{\sqrt{2}}{2}x=±22​​, z=±2z= \pm 2z=±2, λ=±24\lambda= \pm \frac{\sqrt{2}}{4}λ=±42​​.

Problem 30011

Find the derivative of y=x3−12x+4y=x^{3}-12 x+4y=x3−12x+4, find turning points AAA and BBB, and the tangent at C(1,−7)C(1,-7)C(1,−7) in y=px+qy=p x+qy=px+q form.

Problem 30012

Find the extreme value of z=49−x2−y2z=49-x^{2}-y^{2}z=49−x2−y2 subject to x+3y=10x+3y=10x+3y=10. Which statement about the extrema is correct?

Problem 30013

Calculate the value of ∫ab(ex+1x)\int_{a}^{b}\left(e^{x}+\frac{1}{x}\right)∫ab​(ex+x1​). Choose the correct expression from the options.

Problem 30014

Evaluate the integral ∫−10x3x4+9dx\int_{-1}^{0} \frac{x^{3}}{\sqrt{x^{4}+9}} d x∫−10​x4+9​x3​dx.

Problem 30015

Find the tangent line to the function yln⁡(2x−y)+x2y−7x2+16=0y \ln (2 x-y)+x^{2} y-7 x^{2}+16=0yln(2x−y)+x2y−7x2+16=0 at the point (2,3).

Problem 30016

Find the slope of the tangent line to y=F(x)y=F(x)y=F(x) at x=3x=3x=3, where F(x)=∫2xe1/tdtF(x)=\int_{2}^{x} e^{1/t} dtF(x)=∫2x​e1/tdt.

Problem 30017

Berechne die Ableitung der Funktion f(x)=−0,5x2+4xf(x)=-0,5 x^{2}+4 xf(x)=−0,5x2+4x an der Stelle x=3x=3x=3 mit der hhh-Methode.

Problem 30018

Berechne f′(3) f^{\prime}(3) f′(3) für die Funktion f(x)=−0,5x2+4x f(x)=-0,5x^2+4x f(x)=−0,5x2+4x mit dem Differenzenquotienten.

Problem 30019

Evaluate the integral: ∫14dy2y(1+y)2\int_{1}^{4} \frac{d y}{2 \sqrt{y}(1+\sqrt{y})^{2}}∫14​2y​(1+y​)2dy​

Problem 30020

Find the derivative of the function y=3x+92−xy=\frac{3x+9}{2-x}y=2−x3x+9​.

Problem 30021

Calculate the integral ∫24dxxln⁡x\int_{2}^{4} \frac{d x}{x \ln x}∫24​xlnxdx​.

Problem 30022

Calculate the limit: lim⁡x→−1−e−11−x2\lim _{x \rightarrow-1^{-}} e^{\frac{-1}{1-x^{2}}}limx→−1−​e1−x2−1​. If it doesn't exist, state that.

Problem 30023

Find the integral of 1x2\frac{1}{x^{2}}x21​ with respect to xxx.

Problem 30024

Find the integral of x3+5x2\frac{x^{3}+5}{x^{2}}x2x3+5​ with respect to xxx.

Problem 30025

Calculate the integral ∫12x−4x2dx\int_{1}^{2} \frac{x-4}{x^{2}} d x∫12​x2x−4​dx.

Problem 30026

Find the derivative: ddx(∫02xln⁡(t3+1)dt)\frac{d}{d x}\left(\int_{0}^{2 x} \ln \left(t^{3}+1\right) d t\right)dxd​(∫02x​ln(t3+1)dt) for x>0x>0x>0.

Problem 30027

Calculate the integral from -1 to 3 of the function x2−2xx^{2}-2xx2−2x. What is ∫−13(x2−2x)dx\int_{-1}^{3}\left(x^{2}-2 x\right) d x∫−13​(x2−2x)dx?

Problem 30028

Find F′(x)F^{\prime}(x)F′(x) if F(x)=∫4x2tdtF(x)=\int_{4}^{x^{2}} \sqrt{t} dtF(x)=∫4x2​t​dt for x>0x>0x>0.

Problem 30029

Determine if the series ∑n=1∞cos⁡(n)n2\sum_{n=1}^{\infty} \frac{\cos (n)}{n^{2}}n=1∑∞​n2cos(n)​ converges or diverges.

Problem 30030

Find the limit: lim⁡n→∞ln⁡(4n2+a2n2+1)⋅2n2−1\lim _{n \rightarrow \infty} \ln \left(\frac{4 n^{2}+a}{2 n^{2}+1}\right) \cdot 2 n^{2}-1limn→∞​ln(2n2+14n2+a​)⋅2n2−1.

Problem 30031

A ball is dropped from a height of 3.5 m3.5 \mathrm{~m}3.5 m. Calculate the time it takes to hit the sidewalk.

Problem 30032

A stone is thrown upward at 20.0 m/s20.0 \mathrm{~m/s}20.0 m/s from a 50.0 m50.0 \mathrm{~m}50.0 m building. Find its velocity after 5.00 s5.00 \mathrm{~s}5.00 s.

Problem 30033

Find the radius of convergence for the series ∑n=1∞(3n(n!)2(2n)!)xn\sum_{n=1}^{\infty}\left(\frac{3^{n}(n !)^{2}}{(2 n) !}\right) x^{n}∑n=1∞​((2n)!3n(n!)2​)xn.

Problem 30034

Find points where the tangent line to y=f(x)=ex−xy=f(x)=\sqrt{e^{x}-x}y=f(x)=ex−x​ is horizontal and the tangent line at (1,e−1)(1, \sqrt{e-1})(1,e−1​).

Problem 30035

Find the speed of a roller coaster at points B, C, and D, starting from 120 m high with g=10 m/s2g=10 \mathrm{~m/s}^2g=10 m/s2.

Problem 30036

Găsiți valoarea marginală și elasticitatea cererii pentru funcțiile: Q=a+b⋅p Q = a + b \cdot p Q=a+b⋅p, Q=a⋅ep Q = a \cdot e^{p} Q=a⋅ep, Q=apb+p Q = \frac{a p}{b + p} Q=b+pap​, Q=6+8p+3p2 Q = 6 + 8 p + 3 p^{2} Q=6+8p+3p2, Q=3⋅2+p3+p Q = 3 \cdot \frac{2 + p}{3 + p} Q=3⋅3+p2+p​.

Problem 30037

Determine the intervals where the function is increasing, given its max of 6 at x=3x=3x=3 and min of -2 at x=−5x=-5x=−5.

Problem 30038

A paint brush falls from a ladder. How do its velocity and acceleration change? Choose two answers for each.

Problem 30039

Studiați funcția de cerere C(p)=500−2pC(p)=500-2pC(p)=500−2p. Determinați elasticitatea EC(5)E_{C}(5)EC​(5) și analizați elasticitatea cererii.

Problem 30040

Find the stationary points of the function $y=\frac{1}{3} x^{3}-x^{2}-15 x-9$ . Options: $x=5, x=3$ ; $x=5, x=-3$ ; $x=-5, x=3$ ; $x=-5, x=-3$ .

Consider the function $f(x)=\frac{x^{2}}{1+x+x^{2}}$ for $x > -2$ . (a) Does $f$ have a maximum? (YES / NO) (b) Does $f$ have a minimum? (YES / NO) (c) For which $x$ is $f$ decreasing?

Find $f_{y y}$ for the function $z=e^{\ln x}(x^{2}-y^{2})$ .

Find the maximum value of $f(x)=\frac{x^{2}+2 x-3}{x^{2}+1}$ at its stationary point: $-1+\sqrt{5}$ , $1+\sqrt{5}$ , 2, or $2-\sqrt{5}$ .

Find the derivative of $y=e^{2-x} \ln e^{\sqrt{2 x}}$ . Simplify first.

Consider the function $f(x)=\frac{x^{2}}{1+x+x^{2}}$ for $x > -2$ . Does it have a maximum (YES/NO), a minimum (YES/NO), and where is it decreasing?

Evaluate the integral: $\int \frac{18 \tan ^{2} x \sec ^{2} x}{\left(2+\tan ^{3} x\right)^{2}} d x$

Find the values of $x \in \mathbb{R}$ for which $f(x)=\sum_{n=0}^{\infty} e^{-2 n x}$ is defined, and compute $f^{\prime}(x)$ .

Determine the nature of the function $z=-x^{2}+xy-y^{2}-7$ : max $z=-7$ , min $z=0$ , saddle $z=8$ , max $z=0$ .

Find the incorrect statement for extreme values of $z=xy$ with constraint $x^{2}+2y^{2}=1$ : $y= \pm \frac{1}{2}$ , $x= \pm \frac{\sqrt{2}}{2}$ , $z= \pm 2$ , $\lambda= \pm \frac{\sqrt{2}}{4}$ .

Find the derivative of $y=x^{3}-12 x+4$ , find turning points $A$ and $B$ , and the tangent at $C(1,-7)$ in $y=p x+q$ form.

Find the extreme value of $z=49-x^{2}-y^{2}$ subject to $x+3y=10$ . Which statement about the extrema is correct?

Calculate the value of $\int_{a}^{b}\left(e^{x}+\frac{1}{x}\right)$ . Choose the correct expression from the options.

Evaluate the integral $\int_{-1}^{0} \frac{x^{3}}{\sqrt{x^{4}+9}} d x$ .

Find the tangent line to the function $y \ln (2 x-y)+x^{2} y-7 x^{2}+16=0$ at the point (2,3).

Find the slope of the tangent line to $y=F(x)$ at $x=3$ , where $F(x)=\int_{2}^{x} e^{1/t} dt$ .

Berechne die Ableitung der Funktion $f(x)=-0,5 x^{2}+4 x$ an der Stelle $x=3$ mit der $h$ -Methode.

Berechne $f^{\prime}(3)$ für die Funktion $f(x)=-0,5x^2+4x$ mit dem Differenzenquotienten.

Evaluate the integral: $\int_{1}^{4} \frac{d y}{2 \sqrt{y}(1+\sqrt{y})^{2}}$

Find the derivative of the function $y=\frac{3x+9}{2-x}$ .

Calculate the integral $\int_{2}^{4} \frac{d x}{x \ln x}$ .

Calculate the limit: $\lim _{x \rightarrow-1^{-}} e^{\frac{-1}{1-x^{2}}}$ . If it doesn't exist, state that.

Find the integral of $\frac{1}{x^{2}}$ with respect to $x$ .

Find the integral of $\frac{x^{3}+5}{x^{2}}$ with respect to $x$ .

Calculate the integral $\int_{1}^{2} \frac{x-4}{x^{2}} d x$ .

Find the derivative: $\frac{d}{d x}\left(\int_{0}^{2 x} \ln \left(t^{3}+1\right) d t\right)$ for $x>0$ .

Calculate the integral from -1 to 3 of the function $x^{2}-2x$ . What is $\int_{-1}^{3}\left(x^{2}-2 x\right) d x$ ?

Find $F^{\prime}(x)$ if $F(x)=\int_{4}^{x^{2}} \sqrt{t} dt$ for $x>0$ .

Determine if the series $\sum_{n=1}^{\infty} \frac{\cos (n)}{n^{2}}$ converges or diverges.

Find the limit: $\lim _{n \rightarrow \infty} \ln \left(\frac{4 n^{2}+a}{2 n^{2}+1}\right) \cdot 2 n^{2}-1$ .

A ball is dropped from a height of $3.5 \mathrm{~m}$ . Calculate the time it takes to hit the sidewalk.

A stone is thrown upward at $20.0 \mathrm{~m/s}$ from a $50.0 \mathrm{~m}$ building. Find its velocity after $5.00 \mathrm{~s}$ .

Find the radius of convergence for the series $\sum_{n=1}^{\infty}\left(\frac{3^{n}(n !)^{2}}{(2 n) !}\right) x^{n}$ .

Find points where the tangent line to $y=f(x)=\sqrt{e^{x}-x}$ is horizontal and the tangent line at $(1, \sqrt{e-1})$ .

Find the speed of a roller coaster at points B, C, and D, starting from 120 m high with $g=10 \mathrm{~m/s}^2$ .

Găsiți valoarea marginală și elasticitatea cererii pentru funcțiile: $Q = a + b \cdot p$ , $Q = a \cdot e^{p}$ , $Q = \frac{a p}{b + p}$ , $Q = 6 + 8 p + 3 p^{2}$ , $Q = 3 \cdot \frac{2 + p}{3 + p}$ .

Determine the intervals where the function is increasing, given its max of 6 at $x=3$ and min of -2 at $x=-5$ .

Studiați funcția de cerere $C(p)=500-2p$ . Determinați elasticitatea $E_{C}(5)$ și analizați elasticitatea cererii.

Find $T_{1}(x), T_{2}(x), \ldots, T_{5}(x)$ for $f(x)=e^{x}$ at $a=1$ .

Gegeben ist die Funktion $f(t) = 2 \cdot 1,15^{t}$ . Zeichne den Graphen für $0 \leq t \leq 8$ und berechne die mittlere Änderungsrate für die Intervalle $[0; 2]$ und $[5; 8]$ .

Find the max at $x=3$ , min at $x=-5$ . Identify intervals of increase/decrease, domain, and range of the function.

Fie funcția $R(p)=p \cdot C(p)$ . Arătați că: a) dacă cererea e elastică, $R$ e descrescătoare; b) dacă e inelastică, $R$ e crescătoare. Indiciu: $R^{\prime}(p)=C(p)(1-E_{c}(p))$ .

Calculate the integral $\int_{\pi / 4}^{\pi / 2} \cot t \, dt$ .

Find the average rate of change of $g(x)=-5 x^{3}-2$ from $x=-4$ to $x=3$ .

Calculați derivatele parțiale pentru funcțiile date în punctele $(1,1)$ și $(1,1,1)$ . Răspunsuri incluse.

Gegeben ist die Funktion $f(t)=2 \cdot 1,15^{t}$ . a) Zeichne den Graphen für $0 \leq t \leq 8$ . b) Berechne die mittlere Änderungsrate für die Intervalle $[0 ; 2]$ und $[5 ; 8]$ .

Find the average rate of change of $g(x)=7x^{3}+\frac{5}{x^{2}}$ on the interval $[-2,1]$ .

Find the value of c for which $\lim _{x \rightarrow 2} f(x)$ exists, where $f(x)=\begin{cases} \frac{1}{2} x+c, & x<2 \\ -x+9, & x>2 \end{cases}$ .

Evaluate the integral: $\int \frac{\sin \sqrt{\theta}}{\sqrt{\theta \cos ^{3} \sqrt{\theta}}} d \theta$

Bestimmen Sie die Ableitung der Funktion $y=\frac{1}{3}(x^{3}-6 x^{2}+1)$ .

Bestimmen Sie die erste Ableitung von $y=(4 x-2)(x-7)$ .

Find the simplified difference quotient for $f(x)=x^{2}$ and complete the table for $x=5$ and various $h$ values. $\frac{f(x+h)-f(x)}{h}=$

Find the difference quotient for $f(x)=x^{2}$ and fill in the table for $x=5$ and various $h$ values.

(a) Simplify the difference quotient for $f(x)=x^{2}$ . (b) Complete the table for $x=5$ and various $h$ values.

(a) Simplify the difference quotient for $f(x)=-5x^{2}$ and (b) complete the table for $x=3$ and various $h$ values.

(a) Simplify the difference quotient for $f(x)=-5x+2$ . (b) Complete the table for $x=4$ and various $h$ values.

Find the tangent line equation for $f(x)=\frac{4}{x}$ at the point $\left(-8,-\frac{1}{2}\right)$ . Use $y=\square$ .

Find the tangent line to $f(x)=x^{3}$ at the point $(2,8)$ . The equation is $y=\square$ .

Finde die Tangentengleichung der Funktion $f(x) = x^2 - 5x + 4 + 2\ln(x)$ bei $x = 1$ .

Find the tangent line equation for $f(x)=-3-3 x^{2}$ at the point $(-5,-78)$ . It's $y=\square$ in terms of $\mathrm{x}$ .

Find the tangent line to $f(x)=\frac{2}{x}$ at the point $\left(9, \frac{2}{9}\right)$ . The equation is $y=\square$ .

Find the tangent line equation to $f(x)=4-2 x^{2}$ at the point $(4,-28)$ . The equation is $y=\square$ .

Find the tangent line equation for $f(x)=2 x^{3}$ at the point $(-2,-16)$ . The equation is $y=\square$ .

Find the derivative $f^{\prime}(x)$ using $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ for $f(x)=2x^2$ and calculate $f^{\prime}(6)$ .

Calculaţi derivatele parţiale de ordinul 1, 2 şi 3 pentru funcţiile: (a) $f(x, y)=2 x^{2} y^{4}$ la $a=(-2,2)$ ; (b) $f(x, y)=2 \sqrt{x} y$ la $a=(1,1)$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{5}{3-x}$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=4 x^{2}$ .

Find the derivative $g^{\prime}(t)$ of the function $g(t)=\frac{9}{t^{4}}$ .

Find the derivative $H^{\prime}(x)$ of the function $H(x)=\frac{3}{\sqrt{x+1}}$ .

Find the derivative $g^{\prime}(t)$ of the function $g(t)=\frac{2}{t^{3}}$ . What is $g^{\prime}(t)=?$