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Ch. 6 Review AP Calculus 6.2: Integrals of Reciprocal Functions 6.2: Second Fundamental Theorem of Calculus 6.3: Log Properties (The Big Four) 6.4: Solving Exponential Equations (logs) 6.4: Logarithmic Differentiation (exponential functions) Growth/Decay Problems (using logs to solve)… including Separation of Variables Derivatives/Integrals of Transcendental Functions (trig, exponential, logs) Topics If f(x) = 𝑥 cos 𝑡 2 𝑑𝑡, find f’(x). 5𝑥 2𝑡 𝑒 1 If g(x) = 𝑑𝑡 , find g’(x). Example 8, pg. 276 Second Fundamental Theorem of Calculus 4 8𝑥 − 1 𝑑𝑥 3𝑒 2𝑥 5 − 4𝑒 2𝑥 tan 𝑥 𝑑𝑥 𝑑𝑥 Integrating Reciprocal Functions 2𝑒 𝑙𝑛4𝑥 𝑙𝑛𝑒 𝑥2 3 log 2 ln 𝑥2 𝑠𝑖𝑛𝑥 Simplifying Logs 𝑑 𝑑𝑥 log 5 𝑥 𝑑 𝑥+2 7 𝑑𝑥 Derivatives of Logs/ Logarithmic Differentiation Find f’(x) if 𝑓 𝑥 = (3𝑥+7)5 3 𝑥+2 Derivatives of Logs/ Logarithmic Differentiation Power Rule, Chain Rule Product Rule, Quotient Rule e^x 5^x ln x log 3 𝑥 Differentiation/Integration Methods 𝑠𝑖𝑛𝑥 𝑑𝑥 = −𝑐𝑜𝑠𝑥 + 𝑐 𝑐𝑜𝑠𝑥 𝑑𝑥 = 𝑠𝑖𝑛𝑥 + 𝑐 sec 𝑥 𝑑𝑥 = ln | sec 𝑥 + tan 𝑥| + 𝑐 csc 𝑥 𝑑𝑥 = −ln | csc 𝑥 + cot 𝑥| + 𝑐 tan 𝑥 𝑑𝑥 = ln | sec 𝑥 | + 𝑐 cot 𝑥 𝑑𝑥 = −ln | csc 𝑥 | + 𝑐 Trig Integrals See Population Problem, pg. 269. We now know how to solve this QUICKLY!!! Separation of Variables Know how to substitute given values into R(t) = 𝑎0 𝑒 𝑘𝑡 formula. Be able to recognize derivative (rate of change, instantaneous rate, slope of tangent, etc.) vs. integral (sum, area under curve, total accumulation). Exponential Growth/Decay