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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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Use \( f(x)=\frac{x}{x+1} \) and \( g(x)=3 x^{2}-2 \) to find and simplify \( (g \circ f)(x) \) \( (g \circ f)(x)=\frac{x^{2}-4 x-2}{x^{2}+2 x+1} \) \( (g \circ f)(x)=\frac{x-2}{x+1} \) \( (g \circ f)(x)=\frac{3 x^{2}-2}{3 x^{2}-1} \) \( (g \circ f)(x)=\frac{-4 x-2}{2 x+1} \)
United States Jan 22, 2025
The area of a rectangular cross-section of a solid is given by the dimensions in terms of a linear function: width = \(x\) and height = \(3 - x\) for \(0 \leq x \leq 3\). Calculate the volume of the solid when these rectangles are stacked from \(x=0\) to \(x=3\).
Australia Jan 22, 2025
Use \( f(x)=\frac{x}{x+1} \) and \( g(x)=3 x^{2}-2 \) to find and simplify \( (g \circ f)(x) \) \( (g \circ f)(x)=\frac{x^{2}-4 x-2}{x^{2}+2 x+1} \) \( (g \circ f)(x)=\frac{x-2}{x+1} \) \( (g \circ f)(x)=\frac{3 x^{2}-2}{3 x^{2}-1} \) \( (g \circ f)(x)=\frac{-4 x-2}{2 x+1} \)
United States Jan 22, 2025
Find and simplify the difference quotient \( \frac{f(x+h)-f(x)}{h} \) for the function \( f(x)=2 x+1 \). 2 \( 2 x+1 \) \( \frac{2 h+2}{h} \) 4
United States Jan 22, 2025
Encuentra el volumen del sólido formado al rotar la región bajo la curva \( y = e^{x} \) desde \( x = 0 \) hasta \( x = 2 \) en torno al eje horizontal que pasa por \( y = -1 \).
Cuba Jan 22, 2025
33. Solve \( \sin \theta=\tan \theta \). (A) \( 200^{\circ} \) (B) \( 90^{\circ} \) (C) \( 60^{\circ} \) (D.) \( 0^{\circ} \) 34. The sum of the first three terms of an Arithmetic progression (A.P.) is 18 . If the first term is 4 . Find their product. (A) 260 (B) 210 (C) 192 (D) 130 35. Evaluate \( \lim _{x \rightarrow 3}\left(\frac{x^{2}-2 x-3}{x-3}\right) \). (A) 4 (B) 3 (C) 2 (D) 0 36. Given that \( \overrightarrow{A B}=5 i+3 j \) and \( \overrightarrow{A C}=2 i+5 j \), find \( \overrightarrow{B C} \). (A) \( 3 i-2 j \) (B) \( -7 i-8 j \) (C) \( -3 i+2 j \) (D) \( 3 i+8 j \) 37. Find the derivative of \( 3 x^{2}+\frac{1}{x^{2}} \). (A) \( 6 x-\frac{1}{2 x^{2}} \) (B) \( 6 x-\frac{2}{x^{3}} \) (C) \( 6 x-\frac{1}{2 x} \) (D) \( 6 x+2 x^{2} \) 38. A body starts from rest and moves in a straight line with a uniform acceleration of \( 5 \mathrm{mls}{ }^{2} \). How far, in \( m \), does it go in 10 seconds? (A) 50 m (B) 250 m (C) 350 m (D) 500 m 39. If \( n \) items are arranged two at a time, the number obtained is 20 . Find the value of \( n \). (A) 40 (B) 15 (C) 10 (D) 5 40. The gradient of point \( P \) on the curve \( y=3 x^{2}-x+3 \) is 5 . Find the coordinates of \( P \). (A) \( (1,5) \) (B) \( (1,7) \) (C) \( (1,13) \) (D) \( (1,17) \) 41 Evaluate \( \frac{1}{1-\sin 60^{\circ}} \) leaving your answer in surd form. (A) \( 4-2 \sqrt{3} \) (B) \( 2-\sqrt{3} \) (C) \( 1-\sqrt{3} \) (D) \( 4+2 \sqrt{3} \) 42. Calculate, correct to one decimal place, the standard deviation of the numbers \( 85,84,83,86 \) and 87 . (A) 1.8 (B) 1.6 (C) \( \cdot 1.4 \) (D) 2.0 43. Find the value of \( x \) at the point of intersection of the curve \( y=x^{2}+2 x-3 \) and the line \( y+x=1 \). (A) \( (1,-2) \) (B) \( \left(0,,^{4}\right) \) (C) \( (1,-4) \) (D) \( (2,3) \) 44. Given that \( n=3 \), evaluate \( \frac{1}{(n-1)!}-\frac{1}{(n+1)!} \) (A) 12 (B) \( 2 \frac{1}{2} \) (C) \( \frac{11}{24} \) (D) 2
Nigeria Jan 22, 2025
Encontre o volume do corpo gerado pela revolução da curva definida por \( y = \sqrt{x} \) entre \( x = 1 \) e \( x = 4 \) em torno do eixo y utilizando o método do washer.
Mozambique Jan 22, 2025
23. Given that \( =x^{2}-3 x+9 \), find the value of \( \frac{d y}{d x} \) at \( (0,-3) \). (A) \( -\frac{1}{2} \) (B) \( -\frac{1}{4} \) (C) \( \frac{1}{4} \) (D) \( \frac{1}{2} \) 24. The mean of four numbers is 5 and the mean of another three numbers is 12 . Find the mean of seven numbers. (A) 10 (B) 9 (C) 8 (D) 7 25. Find the gradient of the normal to the curve \( =x^{3}-x^{2} \), at the point where \( x=2 \). (A) \( \frac{1}{8} \) (B) \( -\frac{1}{8} \) (C) 1 (D) \( -\frac{1}{24} \) 26. Which of the following is not an equation of a circle? (A) \( 3 x^{2}+3 y^{2}+5 x+7 y=5 \) (B) \( x^{2}+y^{2}- \) 27. Find the equation of the straight line that passes through \( (2,-5) \) and perpendicular to the line \( 3 x+2 y-4=0 \). (A) \( 3 y+2 x+5=0 \) (B) \( 3 y-2 x+15=0 \) (C) \( 3 y+2 x-19=0 \) 28. Calculate, correct to one decimal place, the length of the line joining points \( X(3,5) \) and \( (5,1) \). (A) 4.0 (D) \( 3 y-2 x+19=0 \) (B) 4.5 (C) 4.7 (D) 5.0 29. Find the unit vector in the direction of \( -5 i+12 j \). (A) \( \frac{1}{13}(-5 i-12 j) \) (B) \( \frac{1}{13}(5 i-12 j) \) (C) \( \frac{1}{13}(5 i+ \) 30. A curve is given by \( y=x^{2}-4 x-12 \). Find the axis of symmetry of the curve. (A) \( x=2 \) (B) \( y=2 \) (C) \( x=-2 \) (D) \( y=-2 \) 31. Find the coefficient of the 6 th term in the binomial expansion of \( \left(1-\frac{2 x}{3}\right)^{10} \) in ascending powers of \( x \) (A) \( -\frac{896 x^{6}}{9} \) (B) \( -\frac{896 x^{6}}{27} \) (C) \( -\frac{896 x^{5}}{27} \) (D) \( -\frac{896 x^{5}}{9} \) 32. The probabilities that Atta and Tunde will hit a target in a shooting contest are \( \frac{1}{6} \) and \( \frac{1}{9} \) respectively. Find the probability that only one of them will hit the target. (A) \( \frac{1}{54} \) (B) \( \frac{13}{54} \) (C) \( \frac{20}{27} \) (D) \( \frac{41}{54} \)
Nigeria Jan 22, 2025
2. Find the derivatives of the following functions. \( \begin{array}{ll}\text { a) } y=\left(5-2 x^{2}\right)^{6} & \text { b) } y=\frac{1}{(\sqrt{x}-x)^{3}} \\ \text { c) } y=\frac{3}{2 \sqrt{x+1}} & \text { d) } y=(4 x-3)(1-2 x)^{4} \\ \text { e) } y=\frac{2 x-3}{x^{2}-5} & \text { f) } y=\left(x^{3}+1\right) \sqrt{2 x+7}\end{array} \)
Malaysia Jan 22, 2025
1. If \( \frac{5}{\sqrt{2}}-\frac{\sqrt{8}}{8}=m \sqrt{2} \), find the value of \( m \). (A) \( \frac{12}{5} \) (B) \( \frac{9}{4} \) (C) \( \frac{7}{3} \) (D) \( \frac{5}{2} \) 2. Given that \( f: x \rightarrow \sqrt{x} \) and \( g: x \rightarrow 25-x^{2} \), find the value of \( \log (3) \). (A) 4 (B) 3 3. If \( \sqrt{x}-\frac{6}{\sqrt{x}}=1 \), find the value of \( x \). (A) 9 (B) 16 (C) 25 (D) 36 4. If \( 5 x+7=P(x+3)+Q(x-1) \), find the value of \( P \). (A) 3 (B) 2 (C) 1 (D) \( 0 \cdot\left(4 A^{\circ}\right)= \) 5. If \( y=x \sin x \), find \( \frac{d y}{d x} \). (A) \( \sin x-x \cos x \) (B) \( \sin x-\cos x \) (C) \( \sin x+x \cos x \) (D) \( \sin x+\cos x \) 6. A binary operation \( * \) is defined on the set of real numbers, \( R \) by \( * y=\frac{y^{2}-x^{2}}{2 x y}, x, y \neq 0 \), where \( x \) ar (A) \( \frac{13}{12} \) (B) \( \frac{5}{12} \) (C) \( -\frac{5}{12} \) (D) \( =\frac{13}{12} \) 7. Calculate the value of \( \lambda \) for which vectors \( (5 \lambda i+2 j) \) and \( (4 i-3 j) \) are perpendicular. (A) \( \frac{1}{2} \) (B) (C) \( \frac{10}{3} \) (D) \( \frac{3}{10} \) 8. If \( h(x)=x^{2}+p x+2 \) is divided by \( (x+3) \), the remainder is 5 , find \( p \). (A) 1 (B) 2 (C) 3 (D) 4 9. If the nth term of a linear sequence (A.P.) is \( (5 n-2) \), find the sum of the first 12 terms of \( t \) l sequence. (A) 120 (B) 183 (C) 366 (D) 732 10. Find the stationary point of the curve \( =3 x^{2}-2 x^{3} \). (A) \( (1,0) \) (B) \( (-1,0) \) (C) \( (1,1) \) (D) \( (-1,-1) \) 11. Solve: \( 2^{2 x}-5\left(2^{x}\right)+4=0 \) (A) \( x=0 \) and 1 (B) \( x=1 \) and 2 (C) \( x=0 \) and 2 (D) \( x=2 \) and 12. Express \( 75^{\circ} \) in radians leaving your answer in terms of \( \pi \). (A) \( \frac{3}{4} \pi \) (B) \( \frac{5}{6} \pi \) (C) \( \frac{5}{12} \pi \) (D) \( \frac{7}{6} \pi \) 13. A body of mass 42 kg increases its speed from \( 15 \mathrm{~m} / \mathrm{s} \) to \( 43 \mathrm{~m} / \mathrm{s} \) in 12 seconds. Find the force acting c the body. (A) 52 N (B) 98 N (C) 150 N (D) 203 N
Nigeria Jan 22, 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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