Calculus Volume 1 · free preview
Antiderivatives
An antiderivative of f is a function whose derivative is f, all antiderivatives differ by a constant, and the basic integration formulas are read backward from the differentiation rules, opening the door to the integral chapters.
1. Reversing differentiation
Reversing differentiation
2. The basic formulas
The basic formulas
3. Initial value problems
Initial value problems
4. Checking and the road ahead
Checking and the road ahead
5. Definition of antiderivative
Which statement defines an antiderivative of the function f on an interval?
6. Why the constant of integration
Why does the general antiderivative of f always include an arbitrary constant C?
7. Power rule integral
What is the indefinite integral of x^3 with respect to x?
8. The exceptional power
The power rule integral excludes n = -1. Which formula covers the integral of 1/x?
9. Initial value determination
The general antiderivative of 2x is x^2 + C. Which condition determines C as 3?
10. Linearity of integration
Which property justifies integrating a sum like (3x^2 + 2x) term by term as the integral of 3x^2 plus the integral of 2x?
11. Integral of cosine
What function plus C has derivative cos x? Answer with the function name like sin x.
12. Integral of e to the 5x
The integral of e^(5x) is e^(5x) over a constant, plus C. What is that constant? Answer with a number.
13. Antiderivative at a point
F of x = x^2 + C is the antiderivative of 2x. If F of 2 = 9, what number is C? Answer with a number.
14. Number of basic formulas
This section's basic formula list covers powers, 1/x, exponentials e^x and b^x, and the trigonometric and inverse trigonometric entries sin, cos, sec squared, and 1/(1 + x^2). How many distinct formulas are listed? Answer with a number.
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