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Calculus is at the core of advanced mathematics, covering concepts such as limits, derivatives and integrals to describe changes and motion. Calculus Homework Bank offers... Load more Hide more

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(b) \( \lim _{x \rightarrow-\infty} \frac{8}{e^{x}+8}= \) [NOTE: If needed, enter INF for \( \infty \) and MINF for \( -\infty \).]
United States Jan 19, 2025
26. Estude, quanto à continuidade, cada uma das funções nos pontos indicados: 26.1 \( f(x)=2 x^{3}-x+3 \); para \( x=-2 \) e \( x=10 \) 26.2 \( f(x)=\frac{x^{2}-4}{x-4} \), para \( x=0 \) e \( x=2 \) \( 26.3 f(x)=\left\{\begin{array}{l}x+1 \text { se } x<2 \\ \frac{x}{2}, \text { se } x \geq 2\end{array}\right. \), para \( x=-2 \) \( 26.4 f(x)=\left\{\begin{array}{l}3 x+1 \text { se } x \leq 0 \\ \frac{\sqrt{x}}{x}, \text { se } x>0\end{array}\right. \), para \( x=0 \) \( 26.5 f(x)=\left\{\begin{array}{l}\frac{x-1}{\sqrt{x-1}} \text { se } 1<x \leq 2 \\ 0, \text { se } x \leq 1 \\ -x+1, \text { se } x>2\end{array}\right. \), para \( x=1 \) e \( x=2 \) \( 26.6 f(x)=\left\{\begin{array}{c}\frac{x^{2}-3 x+2}{x-1} \text { se } x \neq 1 \\ -1, \text { se } x=1\end{array}\right. \), para \( x=1 \)
Sao Tome and Principe Jan 19, 2025
25. Determine \( r \) if \( \sum_{n=1}^{\infty} 2 r^{n-1}=12 \). (5)
South Africa Jan 19, 2025
Al aplicar la Teorema Fundamental del Cálculo, si \( F(x) = \int_{0}^{x} (3t^2 - 4) dt \), determina \( F'(2) \).
Venezuela Jan 19, 2025
\( \left. \begin{array} { l }{ V _ { 2 } = \pi \int _ { 9 / 2 } ^ { 5 } ( f ( x ) ^ { 2 } - g ( x ) ^ { 2 } ) d y } \\ { = \pi \int _ { 9 / 2 } ^ { 5 } ( ( 1 / 2 ) ^ { 2 } - ( x - 9 / 2 ) ^ { 2 } ) d x } \end{array} \right. \)
United States Jan 19, 2025
2. Suppose that \( \mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})=2 \). Find \( \mathbf{v} \cdot(\mathbf{u} \times \mathbf{w}) \)
United States Jan 19, 2025
Data la funzione di velocità \( v(t) = 12t^2 - 4t \), calcola l'accelerazione dell'oggetto al tempo \( t = 3 \).
Italy Jan 19, 2025
Exercise: 1. Evaluate (a) \( \lim _{x \rightarrow 0} \frac{2 \sin x-\sin 2 x}{2 e^{x}-2-x^{2}} \) (b) \( \lim _{x \rightarrow \infty} \frac{7 x-28}{x^{3}} \) (c) \( \lim _{x \rightarrow 1} \frac{x^{2}-x-2}{x^{2}-1} \) 2. Evaluate (a) \( \lim _{x \rightarrow 1} x^{\frac{1}{x-1}} \). (b) \( \lim _{x \rightarrow 0} x^{\sin x} \) (c) \( \lim _{x \rightarrow 0} x^{x} \).
Kenya Jan 19, 2025
4. A wooden golf tee has approximately the dimensions of the solid obtained by revolving about the \( x \)-axis the region bounded by the graphs of \( f \) and \( g \), where \( f(x)=\left\{\begin{array}{c}\frac{1}{2} x, 0 \leq x \leq \frac{1}{2} \\ \frac{1}{4}, \frac{1}{2} \leq x \leq \frac{7}{2} \\ \frac{1}{4}\left[1+\left(x-\frac{7}{2}\right)^{2}\right], \frac{7}{2} \leq x \leq \frac{9}{2} \\ \frac{1}{2}, \frac{9}{2} \leq x \leq 5\end{array}\right. \) and \( g(x)=\left\{\begin{array}{c}0,0 \leq x \leq \frac{9}{2} \\ x-\frac{9}{2}, \frac{9}{2} \leq x \leq 5\end{array}\right. \) are measured in centimeters. Determine how much wood goes into a golf tee.
United States Jan 19, 2025
use the iterative method to find a real root of the following equations correct to four sig. fig. \( \begin{array}{ll}\text { (1) } \cos x=3 x-1 & \text { (2) } \sin x=10(x-1) \\ \text { (3) } e^{x}=\cot x & \text { (4) } 1+x^{2}=x^{3} \\ \text { (5) } \sin ^{2} x=x^{2}-1 & \text { (6) } x=(5-x)^{1 / 3}\end{array} \)
Nigeria Jan 19, 2025
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Exploring Calculus: From Differentiation to Real-World Uses

What is Calculus?

Calculus is a branch of mathematics that focuses on studying the change of quantities. It involves two main concepts:


Differentiation: This deals with finding the rate of change of a function, or the slope of the curve at any given point.

Integration: This concerns calculating the total accumulation, like finding the area under a curve or the volume of a solid.


In simpler terms, calculus helps us understand how things change and accumulate, which is essential for analyzing anything from motion and growth to complex scientific phenomena.

Who Invented Calculus?

The development of calculus is credited to two mathematicians: Isaac Newton and Gottfried Wilhelm Leibniz. Though both independently formulated the core principles of calculus in the late 17th century, a significant dispute arose over who was the true inventor. Newton developed his version in the 1660s but kept it largely unpublished, while Leibniz published his work in 1684, making his method more widely known. The debate between these two continues to be a notable chapter in the history of mathematics.

What is Calculus Used For?

Calculus is used in virtually every field that requires mathematical modeling and analysis. Some of the key applications include:


Physics: Calculating forces, motion, and energy changes.

Engineering: Designing structures, analyzing electrical circuits, and optimizing systems.

Economics and Business: Modeling economic trends, maximizing profits, and predicting market behavior.

Medicine: Modeling disease spread, analyzing medical tests, and optimizing treatment plans.

Space Exploration: Calculating orbital dynamics, distances, and satellite trajectories.

Weather Forecasting: Modeling weather patterns and predicting climate changes.

Music and Acoustics: Analyzing sound waves and optimizing audio performance.


Essentially, calculus is the tool that enables us to analyze and solve problems in real-world scenarios involving change and accumulation.

How Can UpStudy Calculus Homework Bank Help You?

UpStudy’s Calculus Homework Bank is a comprehensive tool designed to help students master calculus concepts by providing real homework questions that others have encountered. Here's how it can help you:


Real-Life Examples: By browsing actual homework questions, you gain insight into the kinds of problems that students typically struggle with, helping you prepare for exams and assignments.

Practice Makes Perfect: With a vast collection of problems, you can practice a wide range of calculus topics, from differentiation to integration, ensuring you're well-prepared for any challenge.

Step-by-Step Solutions: The Homework Bank doesn't just provide questions—it also includes detailed, step-by-step solutions, guiding you through the process and helping you understand how to approach complex problems.

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