Questions and Answers
What is the integral of tan²(x)?
tan(x) - x
What is the integral of 1/(1+x²)?
arctan(x)
What is the integral of sec(x)?
ln | sec(x) + tan(x) |
What is the integral of csc(x)?
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What is the integral of cot(x)?
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What is Taylor's Remainder Theorem?
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What is the Arc Length Formula?
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What is the Surface Area of Arc Length Formula?
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What is the Length of Polar Curve Formula?
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What is the Area of Polar Curve Formula?
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What is the Taylor Series Equation?
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What is the substitution for a² - x²?
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What is the substitution for a² + x²?
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What is the substitution for x² - a²?
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What does Proj_a_(b) represent?
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What does Comp_a_(b) represent?
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What is the Volume by Shell Formula?
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What is the graph of y = ln(x)?
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What is the graph of y = 1/x?
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What is the graph of y = √x?
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What is the graph of y = e^x?
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What is the procedure for disc and washer about a horizontal line?
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What is the procedure for disc and washer around a vertical line?
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Study Notes
Integral Formulas
- Integral of (\tan^2(x)) results in (\tan(x) - x).
- Integral of (\frac{1}{1+x^2}) yields (\arctan(x)).
- Integral of (\sec(x)) gives (\ln | \sec(x) + \tan(x) |).
- Integral of (\csc(x)) results in (\ln | \csc(x) + \cot(x) |).
- Integral of (\cot(x)) is (\ln | \sin(x) |).
Taylor Series and Remainder
- Taylor's Remainder Theorem assists in understanding the error in Taylor series approximations.
Arc Length and Surface Area
- Arc Length Formula calculates the distance along a curve.
- Surface Area of Arc Length Formula computes the area generated by rotating a curve around an axis.
Polar Curves
- Length of Polar Curve Formula is used to determine the arc length when defined in polar coordinates.
- Area of Polar Curve Formula calculates the area enclosed by a polar curve.
Substitution Techniques
- For (a^2 - x^2), use substitution (x = a \sin(\theta)).
- For (a^2 + x^2), apply substitution (x = a \tan(\theta)).
- For (x^2 - a^2), use substitution (x = a \sec(\theta)).
Projections and Components
- Projection of (b) onto (a) denoted as (Proj_a(b)), refers to the vector component of (b) in the direction of (a).
- Component of (b) in the direction of (a\ is represented by (Comp_a(b)).
Volume and Area Formulas
- Volume by Shell Formula involves defining (r(x)) as the radius and (h(x)) as the height for cylindrical shells.
Basic Function Properties
- The graph of (y = \ln(x)) reflects a logarithmic function.
- The graph of (y = \frac{1}{x}) represents a rectangular hyperbola.
- The graph of (y = \sqrt{x}) is a square root function.
- The graph of (y = e^x) illustrates exponential growth.
Disc and Washer Method
- For discs and washers around a horizontal line, integrate with respect to (x).
- For discs and washers around a vertical line, integrate with respect to (y).
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Description
Test your knowledge of essential formulations and equations from Purdue's MA 162 course. This quiz focuses on integrals and Taylor's Remainder, helping you solidify your understanding of calculus concepts. Perfect for exam preparation!
