Calculus

Problem 29001

Find suitable $\delta$ values for $\lim _{x \rightarrow 2} 4 x+2=10$ with $\varepsilon=0.2$ . Options: $\delta=0.05$ , $\delta=0.0161$ , $\delta=0.1$ , $\delta=0.0025$ .

See Solution

Problem 29002

Finde die x-Werte, an denen die Ableitung von $f(x)=-\frac{12}{x}$ gleich 3 ist.

See Solution

Problem 29003

Find the area between $y=\frac{1}{x^{2}}$ , $y=-x^{2}$ , $x=1$ , and $x=2$ . Choices: A. $\frac{14}{6}$ B. $\frac{17}{6}$ C. 3 D. $\frac{19}{6}$

See Solution

Problem 29004

Find the definite integral for the area between $y=x$ and $y=5x-x^{3}$ .

See Solution

Problem 29005

Find the area between the curves $y=\frac{1}{x^{2}}$ , $y=-x^{2}$ , from $x=1$ to $x=2$ .

See Solution

Problem 29006

Find the definite integral for the area between $y=x$ and $y=5x-x^{3}$ .

See Solution

Problem 29007

Find $\delta$ for $f(x)=3x-1$ as $x \to 3$ with $\varepsilon=0.05$ so that $|f(x)-L|<\varepsilon$ .

See Solution

Problem 29008

Finde die Stellen, an denen die Ableitung der Funktionen $f(x)=-\frac{12}{x}$ und $f(x)=\frac{1}{3} x^{3}+3,5 x^{2}+9 x-11$ gleich 3 ist.

See Solution

Problem 29009

Find the definite integral for the area between the curves $y=x$ and $y=5x-x^{3}$ .

See Solution

Problem 29010

Differentiate the function $(1+2 x^{2} e^{-x})$ .

See Solution

Problem 29011

Determine if the series converges: $\sum_{n=1}^{\infty} \frac{\arctan (n)}{n^{2}}$

See Solution

Problem 29012

Überprüfen Sie die Produktregel für drei Faktoren an der Funktion $f(x)=x^{2} \cdot x^{3} \cdot x^{4}$ .

See Solution

Problem 29013

Differentiate the expression $(2 x+4)$ .

See Solution

Problem 29014

Calculate the limit: $\lim _{x \rightarrow-\infty} e^{x} \sin \left(x^{2}\right)$ .

See Solution

Problem 29015

Find the derivative of the function $(2x + 4)e^{x}$ .

See Solution

Problem 29016

Eine Firma modelliert Verkaufszahlen eines neuen Smartphones mit $f_{k}(t)=k \cdot(t-15) \cdot e^{-0,01 t}+15 k$ .
a) Bestimmen Sie $k$ , sodass täglich mindestens 750 Smartphones verkauft werden. b) Zeigen Sie, dass der Zeitpunkt der maximalen Verkaufszahl unabhängig von $k$ ist. c) Beweisen Sie, dass Verkaufszahlen nach dem Maximum sinken. d) Berechnen Sie, wann die Verkaufszahlen am stärksten sinken. e) Bestimmen Sie die maximale und langfristige Verkaufszahl für Modelle mit $k=100$ und $k=200$ .

See Solution

Problem 29017

Untersuchen Sie, ob $x_{0}$ eine Wendestelle für die Funktionen a) $f(x)=\frac{1}{6} x^{3}+x^{2}-1$ (bei $x_{0}=0$ ), b) $f(x)=2 x^{3}-6 x^{2}$ (bei $x_{0}=1$ ), c) $f(x)=x^{5}+x^{3}$ (bei $x_{0}=0$ ) ist.

See Solution

Problem 29018

Estimate the limit: $\lim _{x \rightarrow 2^{-}} \frac{\left|x^{2}-4\right|}{x-2}$ .

See Solution

Problem 29019

Finde die Wendepunkte der Funktionen durch die dritte Ableitung: a) $f(x)=2 x^{3}-3 x^{2}-1$ b) $f(x)=\frac{1}{3} x^{3}+3 x^{2}-x$ c) $f(x)=x^{3}-6 x^{2}+12 x-7$ d) $f(x)=\frac{1}{4} x^{4}-\frac{3}{2} x^{2}$ e) $f(x)=\frac{1}{2} x^{4}-4 x^{3}$ f) $f(x)=\frac{1}{4} x^{5}+\frac{5}{2} x^{2}$

See Solution

Problem 29020

Find the average rate of change of $f(x)=50(1.021)^{x}$ from $x=7$ to $x=10$ .

See Solution

Problem 29021

Find when the particle at $x(t)=t^{3}-48t^{2}$ moves left for $t \geq 0$ .

See Solution

Problem 29022

Find $y^{(7)}$ for $y=-\sin x$ . Which option matches: (A) $y$ , (B) $\frac{d y}{d x}$ , (C) $\frac{d^{2} y}{d x^{2}}$ , (D) $\frac{d^{3} y}{d y^{3}}$ , (E) None?

See Solution

Problem 29023

Find the derivative of $f(x)=-x^{2}+x$ . Which option represents it correctly? A, B, C, D, or E?

See Solution

Problem 29024

Find the acceleration of the particle at $t=2$ for $s(t)=\sqrt{t^{3}+1}$ . Options: (A) $3 \mathrm{feet} / \mathrm{sec}^{2}$ , (B) $2 / 3 \mathrm{foot} / \mathrm{sec}^{2}$ , (C) $-1 / 108 \mathrm{foot} / \mathrm{sec}^{2}$ , (D) $-1 / 9 \mathrm{foot} / \mathrm{sec}^{2}$ , (E) None.

See Solution

Problem 29025

Calculate the integral from 0 to 2 of $x^{2}$ with respect to $x$ .

See Solution

Problem 29026

Find the average velocity of the position function $s(t)=-4 t^{2}-5 t+9$ between $t=5$ and $t=5+h$ as a simplified expression in $h$ .

See Solution

Problem 29027

Find the average rate of change of $f(x)=50(1.021)^{x}$ from $x=7$ to $x=10$ .

See Solution

Problem 29028

Estimate the rate of change of $f(x)=\frac{x}{x-2}$ at $x=2$ . Choose: a) 20000 b) 2 c) 0 d) Not continuous.

See Solution

Problem 29029

Find the average rate of change of $f(x) = 3(2)^{2x}$ on the interval $0 \leq x \leq 2$ .

See Solution

Problem 29030

Find the average rate of change of $f(x) = 3(2)^{2x}$ over the interval $0 \leq x \leq 2$ .

See Solution

Problem 29031

Find the instantaneous rate of change of $f(x)=130000(1.06)^{x}$ at $x=5$ . Options: a) \$10000/year b) \$8000/year c) \$12000/year d) \$14000/year

See Solution

Problem 29032

Evaluate $\lim _{x \rightarrow 0^{+}} \frac{\tan ^{3}(2 x)}{x^{4}}$ and justify your answer without using indeterminate forms.

See Solution

Problem 29033

Calculate the integral $\int_{1}^{2} \frac{1}{x} \, dx$ .

See Solution

Problem 29034

Find the average rate of change of height $h(t)=-4.9 t^{2}+150$ from $t=4$ to $t=5$ . Choices: a) $-44.1$ m/s b) $-19.6$ m/s c) $-24.5$ m/s d) $24.5$ m/s.

See Solution

Problem 29035

Calculate the integral from 1 to 2 of the function $\frac{1}{x}$ .

See Solution

Problem 29036

Find the instantaneous rate of change of height for $h(t)=-5 t^{2}+30$ at $t=1$ second. Options: a) $-5 \mathrm{~m/s}$ b) $-20 \mathrm{~m/s}$ c) $5 \mathrm{~m/s}$ d) $-10 \mathrm{~m/s}$ .

See Solution

Problem 29037

Evaluate the integral from 0 to π: $\int_{0}^{\pi} \sin x \, dx$

See Solution

Problem 29038

Find the function where the instantaneous rate of change is closest to 5 at $x=2$ : a) $f(x)=x^{2}$ , b) $h(x)=x^{2}+0.5 x+1$ , c) $j(x)=1.1^{x}$ , d) $g(x)=x^{2}+1$ .

See Solution

Problem 29039

Find the limits: a. $\lim _{x \rightarrow 3}(2 f(x)+g(-x))$ and b. $\lim _{x \rightarrow-3}\left(\frac{g(x)}{f(-x)}\right)$ given limits of $f$ and $g$ .

See Solution

Problem 29040

Find the instantaneous rate of change of height $h(t)=-16 \cos \left(\frac{2 \pi t}{32}\right)+18$ at $t=19$ s using $\Delta t=0.001$ . Round to one decimal. Options: a) $-2.5$ m/s b) $0.13$ m/s c) $-1.7$ m/s d) $0.67$ m/s.

See Solution

Problem 29041

Find the slope of the tangent to the function $f(x)=-3 x^{2}+11 x-6$ at the point $P(-4,-98)$ . a) 35 b) 42 c) -29 d) 17

See Solution

Problem 29042

Find the rate of change of coffee temperature at $t=9$ hours using $T(t)=60(0.5)^{(t/8)}+23$ . Round to two decimals.

See Solution

Problem 29043

Find the instantaneous rate of change of the home value $f(x)=130000(1.06)^{x}$ after 5 years. Options: a) \$8000/year b) \$10,000/year c) \$12,000/year d) \$14,000/year

See Solution

Problem 29044

Find the average rate of change of $f(x)=3(2)^{2 x}$ from $x=0$ to $x=2$ . What is $\frac{\Delta f}{\Delta x}$ ?

See Solution

Problem 29045

Find the instantaneous rate of change of height $h(t)=-16 \cos \left(\frac{2 \pi t}{32}\right)+18$ at $t=19$ s. Use $\Delta t=0.001$ s. Options: a) -2.5 m/s b) 0.13 m/s c) -1.7 m/s d) 0.67 m/s

See Solution

Problem 29046

Find the general antiderivative of $f(r)=\frac{7}{1+r^{2}}-\sqrt[5]{r^{4}}$ and verify your result.

See Solution

Problem 29047

Find the instantaneous rate of change of height at $t=1$ for $h(t)=-5t^2+30$ . Options: a) $-5$ m/s b) $-20$ m/s c) $5$ m/s d) $-10$ m/s.

See Solution

Problem 29048

Find the function $f(x)$ given its derivative $f^{\prime}(x)=\frac{(x+1)}{\sqrt{x}}$ and $f(1)=9$ .

See Solution

Problem 29049

Find the function $f(x)$ given that $f'(x)=12 x^{3}+\frac{1}{x}$ for $x>0$ and $f(1)=1.$

See Solution

Problem 29050

Find the coffee cooling rate at 9 hours using $T(t)=60(0.5)^{(t/8)}+23$ . Round to two decimals.

See Solution

Problem 29051

Find the average rate of change of $f(x)=3(2)^{2x}$ on the interval $0 \leq x \leq 2$ .

See Solution

Problem 29052

Find the instantaneous rate of change of height $h(t)=-5 t^{2}+30$ at $t=1$ s. Options: a) $-5$ m/s b) $-20$ m/s c) $5$ m/s d) $-10$ m/s.

See Solution

Problem 29053

Estimate the instantaneous rate of change in book sales at 6 weeks given sales data for weeks 1 to 10. Choices: a) -3 b) -13 c) -19.001 d) -8

See Solution

Problem 29054

Find $\lim _{x \rightarrow 2} f(x)$ where $f(x)=\ln x$ for $0<x \leq 2$ and $f(x)=x^{2} \ln 2$ for $2<x \leq 4$ .

See Solution

Problem 29055

If $f$ is continuous with $f(3)=7$ , which statement is true? (A) $\lim _{x \rightarrow 3} f(3 x)=9$ (B) $\lim _{x \rightarrow 3} f(2 x)=14$ (C) $\lim _{x \rightarrow 3} \frac{f(x)-f(3)}{x-3}=7$ (D) $\lim _{x \rightarrow 3} f\left(x^{2}\right)=49$ (E) $\lim _{x \rightarrow 3}(f(x))^{2}=49$

See Solution

Problem 29056

Find $\lim _{x \rightarrow 3} f(x)$ for $f(x)=\left\{\begin{array}{ll}\ln 3 x, & 0<x \leq 3 \\ x \ln 3, & 3<x \leq 4\end{array}\right.$ . Choices: (A) $\ln 9$ , (B) $\ln 27$ , (C) $3 \ln 3$ , (D) $3+\ln 3$ , (E) nonexistent.

See Solution

Problem 29057

Find the average rate of change of height $h(t) = -4.9t^2 + 150$ from $t=4$ s to $t=5$ s. Choices: a) 24.5 m/s b) -44.1 m/s c) -24.5 m/s d) -19.6 m/s

See Solution

Problem 29058

Find the instantaneous rate of change of the home's value $f(x)=130000(1.06)^{x}$ after 5 years. Options: a) \$10000/year b) \$12000/year c) \$8000/year d) \$14000/year.

See Solution

Problem 29059

Approximate $\int_{0}^{12} f(x) d x$ using the methods: a) Left Riemann (3 rects), b) Right Riemann (6 rects), c) Trapezoid (3), d) Midpoint ( $\Delta x=4.52$ ).

See Solution

Problem 29060

Find the function where the instantaneous rate of change is nearest to 5 at $x=2$ : a) $f(x)=x^{2}$ b) $h(x)=x^{2}+0.5 x+1$ c) $j(x)=1.1^{x}$ d) $g(x)=x^{2}+1$

See Solution

Problem 29061

Find the instantaneous rate of change of $g(x)=x^{2}-2x+5$ at $x=3$ . Options: a) 6 b) 8 c) 4 d) 2

See Solution

Problem 29062

Given $S_{N}=2-\frac{3}{N^{2}}$ , find: (a) $\sum_{n=1}^{10} a_{n}$ , $\sum_{n=4}^{16} a_{n}$ ; (b) $a_{3}$ ; (c) $a_{n}$ ; (d) $\sum_{n=1}^{\infty} a_{n}$ .

See Solution

Problem 29063

Find the value of $x$ where the derivative of $h(x)=0.5 x^{2}+x-2$ is closest to 0. a) $x=-1$ b) $x=0$ c) $x=1$ d) $x=2$

See Solution

Problem 29064

Calculate the midpoint Riemann sum for $\int_{1}^{4} f(x) d x$ using 3 subintervals from given $f(x)$ values.

See Solution

Problem 29065

Estimate the value change rate of a car after 5 years using $V(t)=24000(0.95)^{t}$ . Choose from: a) $-\$ 150 /$ year b) $-\$ 1250 /$ year c) $-\$ 950 /$ year d) $\$ 150 /$ year

See Solution

Problem 29066

Find the value of $\int_{-7}^{6} f(x) \, dx$ given areas: $A=10$ , $B=13$ , $C=4$ , $D=16$ .

See Solution

Problem 29067

Approximate $\int_{1}^{17} f(x) dx$ using the trapezoidal sum with 5 intervals from $f(x)$ values: $f(1)=25$ , $f(5)=12$ , $f(8)=8$ , $f(11)=7$ , $f(15)=5$ , $f(17)=13$ .

See Solution

Problem 29068

Find the value of $\int_{-8}^{5} f(x) \, dx$ given areas: A=4, B=17, C=18.

See Solution

Problem 29069

Find the equation of the tangent line to $y=\arctan \left(\frac{x}{2}\right)$ at the point $(0,0)$ .

See Solution

Problem 29070

Find the derivative $\frac{d y}{d x}$ for the equation $x^{3} e^{2 y}=4$ .

See Solution

Problem 29071

Find the equation of the tangent line to $y=\arctan \left(\frac{x}{2}\right)$ at the point $(0,0)$ .

See Solution

Problem 29072

Find the value of $\int_{-4}^{7} f(x) \, dx$ given areas: A=10, B=19, C=6, D=5 (A, C below x-axis; B, D above).

See Solution

Problem 29073

Find the value of $\int_{-3}^{8} f(x) \, dx$ given areas: $A = 10$ , $B = -6$ , $C = 23$ , $D = -4$ .

See Solution

Problem 29074

Calculate $S_{3}, S_{4}, S_{5}$ for the series $S=\sum_{n=6}^{\infty}\left(\frac{1}{n+1}-\frac{1}{n+2}\right)$ .

See Solution

Problem 29075

Find the horizontal and vertical asymptotes of $f(x)=\frac{2 x^{2}-1}{x^{2}+3}$ using limits.

See Solution

Problem 29076

Find the horizontal and vertical asymptotes of $f(x)=\frac{2x+1}{x^{2}-x}$ using limits.

See Solution

Problem 29077

Calculate the integral: $\int \frac{1}{\sqrt{9-x^{2}}} d x$

See Solution

Problem 29078

Evaluate the limits and sums for sequences:
1. $a_n = \frac{2}{n^2 + 2n}$ : find $\lim_{n \to \infty} a_n$ and $\sum_{n=1}^{\infty} a_n$ .
2. $b_n = \ln\left(\frac{n+1}{n}\right)$ : find $\lim_{n \to \infty} b_n$ and $\sum_{n=1}^{\infty} b_n$ .

See Solution

Problem 29079

Find the sum of the series: $\sum_{n=1}^{\infty} \arctan (9 n)$ .

See Solution

Problem 29080

Find $f(t)$ given that $f''(t) = 2 e^{t} + 3 \sin(t)$ , with conditions $f(0) = -6$ and $f(\pi) = -6$ .

See Solution

Problem 29081

Find the smallest $N$ such that $S_{N}=\sum_{k=1}^{N} \frac{1}{k} > 4$ . What is $N$ ?

See Solution

Problem 29082

Find the average rate of change of the population given by $f(t)=0.008 t^{2}+2.1 t+175$ from 1990 to 2010. Round to 1 decimal place.

See Solution

Problem 29083

Find $f(5)$ for the function where $f^{\prime \prime}(x)=9x+4\sin(x)$ , given $f(0)=2$ and $f^{\prime}(0)=2$ .

See Solution

Problem 29084

Explain the difference between $f(a)=L$ and $\lim _{x \rightarrow a} f(x)=L$ . What is this equality called?

See Solution

Problem 29085

What term describes a quantity decreasing at a rate proportional to its current value? A. Negative slope B. Positive slope C. Exponential decay D. Exponential growth

See Solution

Problem 29086

Find the limit: $\lim _{x \rightarrow 0}\left(2-2^{1 / x}\right)$ .

See Solution

Problem 29087

Find consecutive integers for $f(x) = -x^3 + 11x^2 - 36x - 33$ to locate relative extrema: max between $x = \square$ and $x = \square$ ; min between $x = \square$ and $x = \square$ .

See Solution

Problem 29088

A balloon inflates at $100 \pi \mathrm{ft}^{3} / \mathrm{min}$ . Find the radius increase rate at $5 \mathrm{ft}$ and surface area rate.

See Solution

Problem 29089

Evaluate the sequence $b_{n}=\ln \left(\frac{n+1}{n}\right)$ : a. Show $\lim _{n \rightarrow \infty} b_{n}=0$ . b. Find $\sum_{n=1}^{\infty} b_{n}$ .

See Solution

Problem 29090

Find the relative extrema of $f(x) = x^3 + 2x^2 - 3x - 1$ . Fill in: max between $x=\square$ and $x=\square$ , min between $x=\square$ and $x=\square$ .

See Solution

Problem 29091

Model water temperature with $f(x) = 2x^3 - 3x^2 - 0.8x$ . Graph it, find key features, and identify intercepts and extrema.

See Solution

Problem 29092

Find the max and min of $f(x)=\ln(x^{2}+x+1)$ on $[-1,1]$ . Provide exact values, no rounding.

See Solution

Problem 29093

Find the derivative of these functions with respect to $x$ : 1. $y=x^{8}$ , 2. $y=5 x^{9}$ , 3. $y=8 x+9$ , 4. $y=x^{2}-7 x+2$ , 5. $y=x^{3}-3 x^{2}+3 x-1$ , 6. $y=x^{-4}+3 x^{-1}$ , 7. $y=x-\frac{1}{2 x^{2}}+\sqrt{2}$ , 8. $y=2 x^{\frac{1}{2}}+x-\frac{1}{x^{3}}$ , 9. $y=-\sqrt{x}+\frac{4}{x^{2}}-8$ , 10. $y=x^{2}(x+1)$ , 11. $y=3 x^{\frac{1}{3}}\left(x^{2}-x+1\right)$ , 12. $y=\frac{2 x^{3}-x+5}{x^{2}}$ , 13. $y=\frac{x^{2}-2 x+1}{\sqrt{x}}$ , 14. $y=6 x \sqrt{x}+\sqrt[3]{8 x^{2}}$ .

See Solution

Problem 29094

Find the antiderivative $F(t)$ of $f(t)=6 \sec ^{2}(t)-2 t^{3}$ with $F(0)=0$ . Calculate $F(1.3)$ .

See Solution

Problem 29095

Find the limit as $x$ approaches 2 for the expression $-\frac{x^{2}}{2}$ .

See Solution

Problem 29096

Maximize the volume of a box with a square base using $60 \mathrm{~m}^{2}$ of material. Find height $h$ and base length $b$ .

See Solution

Problem 29097

Find the antiderivative $F(x)$ of $f(x)=10 \cos x-4 \sin x$ with $F(0)=4$ . What is $F(x)$ ?

See Solution

Problem 29098

Find the function $f(t)$ given that $f''(t)=2 e^{t}+3 \sin(t)$ , with conditions $f(0)=4$ and $f(\pi)=-1$ .

See Solution

Problem 29099

Find $f(4)$ given that $f^{\prime \prime}(x)=5 x+10 \sin (x)$ , $f(0)=4$ , and $f^{\prime}(0)=4$ .

See Solution

Problem 29100

Given that $f$ is decreasing on $[0, 10]$ and $g$ is decreasing on $[0, 5)$ and increasing on $(5, 10]$ , what is true about $h(x)=f(x)+g(x)$ on $[0, 10]$ ?

See Solution

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Calculus

Problem 29001

Find suitable δ\deltaδ values for lim⁡x→24x+2=10\lim _{x \rightarrow 2} 4 x+2=10limx→2​4x+2=10 with ε=0.2\varepsilon=0.2ε=0.2. Options: δ=0.05\delta=0.05δ=0.05, δ=0.0161\delta=0.0161δ=0.0161, δ=0.1\delta=0.1δ=0.1, δ=0.0025\delta=0.0025δ=0.0025.

Problem 29002

Finde die x-Werte, an denen die Ableitung von f(x)=−12xf(x)=-\frac{12}{x}f(x)=−x12​ gleich 3 ist.

Problem 29003

Find the area between y=1x2y=\frac{1}{x^{2}}y=x21​, y=−x2y=-x^{2}y=−x2, x=1x=1x=1, and x=2x=2x=2. Choices: A. 146\frac{14}{6}614​ B. 176\frac{17}{6}617​ C. 3 D. 196\frac{19}{6}619​

Problem 29004

Find the definite integral for the area between y=xy=xy=x and y=5x−x3y=5x-x^{3}y=5x−x3.

Problem 29005

Find the area between the curves y=1x2y=\frac{1}{x^{2}}y=x21​, y=−x2y=-x^{2}y=−x2, from x=1x=1x=1 to x=2x=2x=2.

Problem 29006

Find the definite integral for the area between y=xy=xy=x and y=5x−x3y=5x-x^{3}y=5x−x3.

Problem 29007

Find δ\deltaδ for f(x)=3x−1f(x)=3x-1f(x)=3x−1 as x→3x \to 3x→3 with ε=0.05\varepsilon=0.05ε=0.05 so that ∣f(x)−L∣<ε|f(x)-L|<\varepsilon∣f(x)−L∣<ε.

Problem 29008

Finde die Stellen, an denen die Ableitung der Funktionen f(x)=−12xf(x)=-\frac{12}{x}f(x)=−x12​ und f(x)=13x3+3,5x2+9x−11f(x)=\frac{1}{3} x^{3}+3,5 x^{2}+9 x-11f(x)=31​x3+3,5x2+9x−11 gleich 3 ist.

Problem 29009

Find the definite integral for the area between the curves y=xy=xy=x and y=5x−x3y=5x-x^{3}y=5x−x3.

Problem 29010

Differentiate the function (1+2x2e−x)(1+2 x^{2} e^{-x})(1+2x2e−x).

Problem 29011

Determine if the series converges: ∑n=1∞arctan⁡(n)n2\sum_{n=1}^{\infty} \frac{\arctan (n)}{n^{2}}n=1∑∞​n2arctan(n)​

Problem 29012

Überprüfen Sie die Produktregel für drei Faktoren an der Funktion f(x)=x2⋅x3⋅x4f(x)=x^{2} \cdot x^{3} \cdot x^{4}f(x)=x2⋅x3⋅x4.

Problem 29013

Differentiate the expression (2x+4)(2 x+4)(2x+4).

Problem 29014

Calculate the limit: lim⁡x→−∞exsin⁡(x2)\lim _{x \rightarrow-\infty} e^{x} \sin \left(x^{2}\right)x→−∞lim​exsin(x2).

Problem 29015

Find the derivative of the function (2x+4)ex(2x + 4)e^{x}(2x+4)ex.

Problem 29016

Problem 29017

Problem 29018

Estimate the limit: lim⁡x→2−∣x2−4∣x−2\lim _{x \rightarrow 2^{-}} \frac{\left|x^{2}-4\right|}{x-2}limx→2−​x−2∣x2−4∣​.

Problem 29019

Problem 29020

Find the average rate of change of f(x)=50(1.021)xf(x)=50(1.021)^{x}f(x)=50(1.021)x from x=7x=7x=7 to x=10x=10x=10.

Problem 29021

Find when the particle at x(t)=t3−48t2x(t)=t^{3}-48t^{2}x(t)=t3−48t2 moves left for t≥0t \geq 0t≥0.

Problem 29022

Find y(7)y^{(7)}y(7) for y=−sin⁡xy=-\sin xy=−sinx. Which option matches: (A) yyy, (B) dydx\frac{d y}{d x}dxdy​, (C) d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​, (D) d3ydy3\frac{d^{3} y}{d y^{3}}dy3d3y​, (E) None?

Problem 29023

Find the derivative of f(x)=−x2+xf(x)=-x^{2}+xf(x)=−x2+x. Which option represents it correctly? A, B, C, D, or E?

Problem 29024

Problem 29025

Calculate the integral from 0 to 2 of x2x^{2}x2 with respect to xxx.

Problem 29026

Find the average velocity of the position function s(t)=−4t2−5t+9s(t)=-4 t^{2}-5 t+9s(t)=−4t2−5t+9 between t=5t=5t=5 and t=5+ht=5+ht=5+h as a simplified expression in hhh.

Problem 29027

Find the average rate of change of f(x)=50(1.021)xf(x)=50(1.021)^{x}f(x)=50(1.021)x from x=7x=7x=7 to x=10x=10x=10.

Problem 29028

Estimate the rate of change of f(x)=xx−2f(x)=\frac{x}{x-2}f(x)=x−2x​ at x=2x=2x=2. Choose: a) 20000 b) 2 c) 0 d) Not continuous.

Problem 29029

Find the average rate of change of f(x)=3(2)2xf(x) = 3(2)^{2x}f(x)=3(2)2x on the interval 0≤x≤20 \leq x \leq 20≤x≤2.

Problem 29030

Find the average rate of change of f(x)=3(2)2xf(x) = 3(2)^{2x}f(x)=3(2)2x over the interval 0≤x≤20 \leq x \leq 20≤x≤2.

Problem 29031

Find the instantaneous rate of change of f(x)=130000(1.06)xf(x)=130000(1.06)^{x}f(x)=130000(1.06)x at x=5x=5x=5. Options: a) \$10000/year b) \$8000/year c) \$12000/year d) \$14000/year

Problem 29032

Evaluate lim⁡x→0+tan⁡3(2x)x4\lim _{x \rightarrow 0^{+}} \frac{\tan ^{3}(2 x)}{x^{4}}limx→0+​x4tan3(2x)​ and justify your answer without using indeterminate forms.

Problem 29033

Calculate the integral ∫121x dx\int_{1}^{2} \frac{1}{x} \, dx∫12​x1​dx.

Problem 29034

Find the average rate of change of height h(t)=−4.9t2+150h(t)=-4.9 t^{2}+150h(t)=−4.9t2+150 from t=4t=4t=4 to t=5t=5t=5. Choices: a) −44.1-44.1−44.1 m/s b) −19.6-19.6−19.6 m/s c) −24.5-24.5−24.5 m/s d) 24.524.524.5 m/s.

Problem 29035

Calculate the integral from 1 to 2 of the function 1x\frac{1}{x}x1​.

Problem 29036

Find the instantaneous rate of change of height for h(t)=−5t2+30h(t)=-5 t^{2}+30h(t)=−5t2+30 at t=1t=1t=1 second. Options: a) −5 m/s-5 \mathrm{~m/s}−5 m/s b) −20 m/s-20 \mathrm{~m/s}−20 m/s c) 5 m/s5 \mathrm{~m/s}5 m/s d) −10 m/s-10 \mathrm{~m/s}−10 m/s.

Problem 29037

Evaluate the integral from 0 to π: ∫0πsin⁡x dx\int_{0}^{\pi} \sin x \, dx∫0π​sinxdx

Problem 29038

Find the function where the instantaneous rate of change is closest to 5 at x=2x=2x=2: a) f(x)=x2f(x)=x^{2}f(x)=x2, b) h(x)=x2+0.5x+1h(x)=x^{2}+0.5 x+1h(x)=x2+0.5x+1, c) j(x)=1.1xj(x)=1.1^{x}j(x)=1.1x, d) g(x)=x2+1g(x)=x^{2}+1g(x)=x2+1.

Problem 29039

Find the limits: a. lim⁡x→3(2f(x)+g(−x))\lim _{x \rightarrow 3}(2 f(x)+g(-x))limx→3​(2f(x)+g(−x)) and b. lim⁡x→−3(g(x)f(−x))\lim _{x \rightarrow-3}\left(\frac{g(x)}{f(-x)}\right)limx→−3​(f(−x)g(x)​) given limits of fff and ggg.

Problem 29040

Problem 29041

Find the slope of the tangent to the function f(x)=−3x2+11x−6f(x)=-3 x^{2}+11 x-6f(x)=−3x2+11x−6 at the point P(−4,−98)P(-4,-98)P(−4,−98). a) 35 b) 42 c) -29 d) 17

Problem 29042

Find the rate of change of coffee temperature at t=9t=9t=9 hours using T(t)=60(0.5)(t/8)+23T(t)=60(0.5)^{(t/8)}+23T(t)=60(0.5)(t/8)+23. Round to two decimals.

Find suitable $\delta$ values for $\lim _{x \rightarrow 2} 4 x+2=10$ with $\varepsilon=0.2$ . Options: $\delta=0.05$ , $\delta=0.0161$ , $\delta=0.1$ , $\delta=0.0025$ .

Finde die x-Werte, an denen die Ableitung von $f(x)=-\frac{12}{x}$ gleich 3 ist.

Find the area between $y=\frac{1}{x^{2}}$ , $y=-x^{2}$ , $x=1$ , and $x=2$ . Choices: A. $\frac{14}{6}$ B. $\frac{17}{6}$ C. 3 D. $\frac{19}{6}$

Find the definite integral for the area between $y=x$ and $y=5x-x^{3}$ .

Find the area between the curves $y=\frac{1}{x^{2}}$ , $y=-x^{2}$ , from $x=1$ to $x=2$ .

Find the definite integral for the area between $y=x$ and $y=5x-x^{3}$ .

Find $\delta$ for $f(x)=3x-1$ as $x \to 3$ with $\varepsilon=0.05$ so that $|f(x)-L|<\varepsilon$ .

Finde die Stellen, an denen die Ableitung der Funktionen $f(x)=-\frac{12}{x}$ und $f(x)=\frac{1}{3} x^{3}+3,5 x^{2}+9 x-11$ gleich 3 ist.

Find the definite integral for the area between the curves $y=x$ and $y=5x-x^{3}$ .

Differentiate the function $(1+2 x^{2} e^{-x})$ .

Determine if the series converges: $\sum_{n=1}^{\infty} \frac{\arctan (n)}{n^{2}}$

Überprüfen Sie die Produktregel für drei Faktoren an der Funktion $f(x)=x^{2} \cdot x^{3} \cdot x^{4}$ .

Differentiate the expression $(2 x+4)$ .

Calculate the limit: $\lim _{x \rightarrow-\infty} e^{x} \sin \left(x^{2}\right)$ .

Find the derivative of the function $(2x + 4)e^{x}$ .

Estimate the limit: $\lim _{x \rightarrow 2^{-}} \frac{\left|x^{2}-4\right|}{x-2}$ .

Find the average rate of change of $f(x)=50(1.021)^{x}$ from $x=7$ to $x=10$ .

Find when the particle at $x(t)=t^{3}-48t^{2}$ moves left for $t \geq 0$ .

Find $y^{(7)}$ for $y=-\sin x$ . Which option matches: (A) $y$ , (B) $\frac{d y}{d x}$ , (C) $\frac{d^{2} y}{d x^{2}}$ , (D) $\frac{d^{3} y}{d y^{3}}$ , (E) None?

Find the derivative of $f(x)=-x^{2}+x$ . Which option represents it correctly? A, B, C, D, or E?

Calculate the integral from 0 to 2 of $x^{2}$ with respect to $x$ .

Find the average velocity of the position function $s(t)=-4 t^{2}-5 t+9$ between $t=5$ and $t=5+h$ as a simplified expression in $h$ .

Find the average rate of change of $f(x)=50(1.021)^{x}$ from $x=7$ to $x=10$ .

Estimate the rate of change of $f(x)=\frac{x}{x-2}$ at $x=2$ . Choose: a) 20000 b) 2 c) 0 d) Not continuous.

Find the average rate of change of $f(x) = 3(2)^{2x}$ on the interval $0 \leq x \leq 2$ .

Find the average rate of change of $f(x) = 3(2)^{2x}$ over the interval $0 \leq x \leq 2$ .

Find the instantaneous rate of change of $f(x)=130000(1.06)^{x}$ at $x=5$ . Options: a) \$10000/year b) \$8000/year c) \$12000/year d) \$14000/year

Evaluate $\lim _{x \rightarrow 0^{+}} \frac{\tan ^{3}(2 x)}{x^{4}}$ and justify your answer without using indeterminate forms.

Calculate the integral $\int_{1}^{2} \frac{1}{x} \, dx$ .

Find the average rate of change of height $h(t)=-4.9 t^{2}+150$ from $t=4$ to $t=5$ . Choices: a) $-44.1$ m/s b) $-19.6$ m/s c) $-24.5$ m/s d) $24.5$ m/s.

Calculate the integral from 1 to 2 of the function $\frac{1}{x}$ .

Find the instantaneous rate of change of height for $h(t)=-5 t^{2}+30$ at $t=1$ second. Options: a) $-5 \mathrm{~m/s}$ b) $-20 \mathrm{~m/s}$ c) $5 \mathrm{~m/s}$ d) $-10 \mathrm{~m/s}$ .

Evaluate the integral from 0 to π: $\int_{0}^{\pi} \sin x \, dx$

Find the function where the instantaneous rate of change is closest to 5 at $x=2$ : a) $f(x)=x^{2}$ , b) $h(x)=x^{2}+0.5 x+1$ , c) $j(x)=1.1^{x}$ , d) $g(x)=x^{2}+1$ .

Find the limits: a. $\lim _{x \rightarrow 3}(2 f(x)+g(-x))$ and b. $\lim _{x \rightarrow-3}\left(\frac{g(x)}{f(-x)}\right)$ given limits of $f$ and $g$ .

Find the slope of the tangent to the function $f(x)=-3 x^{2}+11 x-6$ at the point $P(-4,-98)$ . a) 35 b) 42 c) -29 d) 17

Find the rate of change of coffee temperature at $t=9$ hours using $T(t)=60(0.5)^{(t/8)}+23$ . Round to two decimals.

Find the instantaneous rate of change of the home value $f(x)=130000(1.06)^{x}$ after 5 years. Options: a) \$8000/year b) \$10,000/year c) \$12,000/year d) \$14,000/year

Find the average rate of change of $f(x)=3(2)^{2 x}$ from $x=0$ to $x=2$ . What is $\frac{\Delta f}{\Delta x}$ ?

Find the instantaneous rate of change of height $h(t)=-16 \cos \left(\frac{2 \pi t}{32}\right)+18$ at $t=19$ s. Use $\Delta t=0.001$ s. Options: a) -2.5 m/s b) 0.13 m/s c) -1.7 m/s d) 0.67 m/s

Find the general antiderivative of $f(r)=\frac{7}{1+r^{2}}-\sqrt[5]{r^{4}}$ and verify your result.

Find the instantaneous rate of change of height at $t=1$ for $h(t)=-5t^2+30$ . Options: a) $-5$ m/s b) $-20$ m/s c) $5$ m/s d) $-10$ m/s.

Find the function $f(x)$ given its derivative $f^{\prime}(x)=\frac{(x+1)}{\sqrt{x}}$ and $f(1)=9$ .

Find the function $f(x)$ given that $f'(x)=12 x^{3}+\frac{1}{x}$ for $x>0$ and $f(1)=1.$

Find the coffee cooling rate at 9 hours using $T(t)=60(0.5)^{(t/8)}+23$ . Round to two decimals.

Find the average rate of change of $f(x)=3(2)^{2x}$ on the interval $0 \leq x \leq 2$ .

Find the instantaneous rate of change of height $h(t)=-5 t^{2}+30$ at $t=1$ s. Options: a) $-5$ m/s b) $-20$ m/s c) $5$ m/s d) $-10$ m/s.

Find $\lim _{x \rightarrow 2} f(x)$ where $f(x)=\ln x$ for $0<x \leq 2$ and $f(x)=x^{2} \ln 2$ for $2<x \leq 4$ .

Find the average rate of change of height $h(t) = -4.9t^2 + 150$ from $t=4$ s to $t=5$ s. Choices: a) 24.5 m/s b) -44.1 m/s c) -24.5 m/s d) -19.6 m/s

Find the instantaneous rate of change of the home's value $f(x)=130000(1.06)^{x}$ after 5 years. Options: a) \$10000/year b) \$12000/year c) \$8000/year d) \$14000/year.

Approximate $\int_{0}^{12} f(x) d x$ using the methods: a) Left Riemann (3 rects), b) Right Riemann (6 rects), c) Trapezoid (3), d) Midpoint ( $\Delta x=4.52$ ).

Find the function where the instantaneous rate of change is nearest to 5 at $x=2$ : a) $f(x)=x^{2}$ b) $h(x)=x^{2}+0.5 x+1$ c) $j(x)=1.1^{x}$ d) $g(x)=x^{2}+1$

Find the instantaneous rate of change of $g(x)=x^{2}-2x+5$ at $x=3$ . Options: a) 6 b) 8 c) 4 d) 2

Given $S_{N}=2-\frac{3}{N^{2}}$ , find: (a) $\sum_{n=1}^{10} a_{n}$ , $\sum_{n=4}^{16} a_{n}$ ; (b) $a_{3}$ ; (c) $a_{n}$ ; (d) $\sum_{n=1}^{\infty} a_{n}$ .

Find the value of $x$ where the derivative of $h(x)=0.5 x^{2}+x-2$ is closest to 0. a) $x=-1$ b) $x=0$ c) $x=1$ d) $x=2$

Calculate the midpoint Riemann sum for $\int_{1}^{4} f(x) d x$ using 3 subintervals from given $f(x)$ values.

Estimate the value change rate of a car after 5 years using $V(t)=24000(0.95)^{t}$ . Choose from: a) $-\$ 150 /$ year b) $-\$ 1250 /$ year c) $-\$ 950 /$ year d) $\$ 150 /$ year

Find the value of $\int_{-7}^{6} f(x) \, dx$ given areas: $A=10$ , $B=13$ , $C=4$ , $D=16$ .

Approximate $\int_{1}^{17} f(x) dx$ using the trapezoidal sum with 5 intervals from $f(x)$ values: $f(1)=25$ , $f(5)=12$ , $f(8)=8$ , $f(11)=7$ , $f(15)=5$ , $f(17)=13$ .

Find the value of $\int_{-8}^{5} f(x) \, dx$ given areas: A=4, B=17, C=18.

Find the equation of the tangent line to $y=\arctan \left(\frac{x}{2}\right)$ at the point $(0,0)$ .

Find the derivative $\frac{d y}{d x}$ for the equation $x^{3} e^{2 y}=4$ .

Find the equation of the tangent line to $y=\arctan \left(\frac{x}{2}\right)$ at the point $(0,0)$ .

Find the value of $\int_{-4}^{7} f(x) \, dx$ given areas: A=10, B=19, C=6, D=5 (A, C below x-axis; B, D above).

Find the value of $\int_{-3}^{8} f(x) \, dx$ given areas: $A = 10$ , $B = -6$ , $C = 23$ , $D = -4$ .

Calculate $S_{3}, S_{4}, S_{5}$ for the series $S=\sum_{n=6}^{\infty}\left(\frac{1}{n+1}-\frac{1}{n+2}\right)$ .