Calculus 1 Final Cheat Sheet, Cheat Sheet of Calculus

Download Calculus 1 Final Cheat Sheet and more Cheat Sheet Calculus in PDF only on Docsity! Final Exam Study Guide for Calculus I 1 General Comments The final exam will be a 2.5 hour CUMULATIVE exam. This means that in principle, you are responsible for everything we have covered this term. As a study aid, I have listed below the major definitions, theorems, proofs, deriva- tions, and ideas that you are responsible for. I have NOT individually listed the various differentiation formulas and techniques that we have learned: you are nevertheless responsible for them. In particular, you should be able to differentiate any function that I put in front of you, which means that you need to know (for example) the product rule, the chain rule, how to differ- entiate exponential functions, etc., etc. Moreover, you need to be able to use implicit and logarithmic differentiation if necessary. Finally, you should be able to compute derivatives and integrals directly from the definitions if asked to do so. As I mentioned in class, I good way of checking your overall comprehension of the differential calculus is to carefully practice using the DIADECIS method to sketch the graphs of functions. Since you never know ahead of time what differentiation technique you will have to use, or what limits you will need to compute, these problems will force you to review all the computational techniques that we have learned. Definitions: 1. Limits (all variations, including one-sided, at infinity, etc.): a) Provisional, intuitive versions b) Precise versions (with , δ,M,N , etc.) 2. Continuity (f is continuous at a, f is continuous, etc.) 3. The derivative of f at a 1 4. Differentiability (f is differentiable at a, f is differentiable, etc.) 5. Antiderivative of a function 6. The definite integral of a continuous function f on an interval [a, b] Theorems: 1. The Intermediate Value Theorem 2. The Extreme Value Theorem 3. Fermat’s Theorem concerning local extrema 4. The Mean Value Theorem 5. Basic properties of the integral (pp. 373-75 of the text) 6. The Fundamental Theorem of Calculus (I and II) Proofs: 1. Fermat’s Theorem 2. Consequences of the Mean Value Theorem: a) f ′(x) = 0 for all x ∈ (a, b) implies f is constant on (a, b) b) Increasing / Decreasing Test 3. The Fundamental Theorem of Calculus II Derivations: 1. Use implicit differentiation to derive the formulas for the derivatives of the inverse trig functions (I will provide you with any non-obvious trig identities that you might need) 2. Use implicit differentiation to derive the formula for the derivative of loga(x) 2