Section 4.1

• Know the definitions of antiderivative and indefinite integral, as well as the notation.
• Know the formulas in Table 4.1 (p.277).
• Know the rules for indefinite integration on page 278, and how to use them.
• You should know trig identity (2) on page 41. You do not need to know the other trig identities like the angle sum and double angle formulas (p.42) and the law of cosines (p.43). If you need them, we will supply them to you.

• Know how to find a function F(x) given its derivative f(x) and its value y=F(a) at some point x=a.
• Be able to do problems similar to the examples, such as finding position x(t) given acceleration a(t), initial position x(0), and initial velocity v(0).
• Skip Sketching Solution Curves, Mathematical Modeling, and Computer Simulation.

• Know how to run the Chain Rule backward: be able to find antiderivatives by substitution.
• Know the Power Rule for antiderivatives and remember the specific formulas for sin, cos, sec², csc², secˇtan, and cscˇcot (table at the bottom of page 293).

• Understand how to estimate areas with finite sums.
• Know how to apply this to distance traveled, volumes, and average function values.

• Understand the notation and terminology for summations (Sigma notation), Riemann sums, and definite integrals. You must know the algebra rules on page 310, but not the sum formulas on page  311.
• Know the definition of the definite integral as a limit of Riemann sums when it exists--and that this means as ||P|| goes to zero and for any choice of c_k. You do not need to know the precise epsilon-delta definition on page 313.
• Know Theorem 1 and what it means, and how the definite integral relates to area under the curve for a nonnegative function as defined on page 318.
• Know how to do problems like Example 9 on page 318 (you will be given the relevant sum formulas on page 311 if necessary).

• Know all the rules for definite integrals on page 324 and how to use them. You do not need to know their proofs.
• Know how to compute area between a curve and the x-axis using the definite integral.
• Know the definition on page 328 of the average of a function. Know the Mean Value Theorem for integrals, but not its proof.

• Know both parts of the Fundamental Theorem, what they mean, and how to use them. These are the two most important theorems in Calculus I! You must understand what both parts mean, but you do not need to know their proofs.
• Know how to take derivatives of definite integrals whose limits of integration are functions of x. Review Examples 1-3 on pages 334-335.
• Know how to evaluate a definite integral using an indefinite integral. Review Examples 4 and 5 on pages 336-337.

• Know how to use substitution in definite integrals. You can use either of the methods outlined on page 343, but you must consistently follow only one method in a given problem.

• Know how to make a uniform partition of [a,b] into n subintervals by setting h=(b-a)/n and x_k=a+kh, k=0,1,2,...,n.
• Know how to apply the Trapezoidal Rule and Simpson's Rule to estimate definite integrals.
• You do not need to memorize the formulas for these rules and their errors. We will supply them if needed--but you should understand them and know how to apply them.

• Know the definition and significance of what it means for a function to be one-to-one, and know the horizontal line test on page 450.
• Know: (i) the definition of the inverse of a function; (ii) when the inverse exists; (iii) how the domain and range of the inverse function are related to the domain and range of the original function; and (iv) how to compute the inverse of a function.
• Know how to compute the derivative of the inverse of a function directly or using Theorem 1. Be sure you know (1) well, especially where the derivatives are to be evaluated (a and f^-1(a)).

• Know the definition of ln(x), its derivative, and its range and domain.
• Learn how to apply the properties of natural logarithms given in Table 6.1. Review Examples 3 and 4 on page 460. You do not need to know the proofs of these properties.
• Know what logarithmic differentiation is and how to use it. Review the steps outlined on page 463.
• Know what the integral of 1/u is and how to use it, as in Examples 6 and 7 on page 464. Know the integrals of tan and cot.