Deflnitions, statements and proofs that may appear on the flrst midterm Math 221 The material for the flrst midterm includes prerequisites, everything we covered about limits, everything we covered about continuity, and the Tuesday September 22 lecture?s material on derivatives (basically, just the deflnition, using this deflnition to compute derivatives of simple functions, and tangent lines). Most problems on the test will be computational. In addition to more compu- tational problems, a few questions on the test may ask you to state a deflnition or theorem, or to prove some property. The list of the deflnitions and properties that may be on the test appears below. Not all these questions will appear on the test, of course. Deflnitions and statements: 1. Deflne what does it mean that the function f(x) is continuous at x = a: 2. Deflne the derivative of a function f(x) at x = a: 3. State the limit laws for addition, subtraction, multiplication and division. 4. State Sandwich Theorem. 5. State Intermediate Value Theorem for continuous functions. 6. Write the equation of a tangent line to the graph y = f(x) of a difierentiable function f at a point (a;f(a)): Proofs: 1. ?¡? proof that the limit exists and is equal to something for simple functions. 2. Prove that if f(x) is difierentiable at x = a; then it is continuous at x = a: You may use without proof the limit laws (for the limits of sum, difierence, product and quotient), but you should state which property are you using in each step. 1
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