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MATH 4389 Final: Calculus_1

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University of Houston
MATH 4389

Calculus 1 Review Limit and Continuity Differentiation Applications of Derivatives Exponential and Logarithmic Functions Integration Must know all rules of differentiation Rate of change, related rates, optimization. Increasing/Decreasing Function, local/absolute extreme values, concavity Gathering info from the graph of f, or f’ or f’’. L’Hospital’s rule. Understand how integrals are defined, know all basic rules of integration, u-sub. Intermediate Value Theorem, Extreme Value Theorem, Rolle’s Theorem, Mean Value Theorem Fundamental Theorem of Calculus Definition of derivative: Equation of tangent line at (a,f(a)) : yf f(x)a(') ) Rules of differentiation: uv '' u' v uv   v uvuu ''   ' 2 vv ) ( ' ) ) ( ( )' ( ' () f  gx fg xgx Newton’s Method Goal: To approximate a solution to f()0 (a root or a zero of the function f(x)). Start with a guess;0 If f 0 , then next guess is: f 0x 0 1 0 f x0) In general:xn n f )n f xn) Gathering information from the derivative function: Example: Given the graph of f '(x. Exponential growth/decay kt P(t) Pe0 Half life: kT  ln(2) kT  ln(2) Doubling time: Riemann Sums Recall – left endpoint, right end point, midpoint, Trapezoid approaches, Upper sum, Lower sum. You need to know how to put them in order depending on whether the function increases/decreases. Example: Given the definite integral, how do these sums compare with it? Know the definition of a definite integral. Note: For a positive function, definite integral gives the area under the curve. If the formula of a function is not given, but the graph is given, you can use the area under that function to find the definite integral. 6 Example: The graph given below belongs to f x). fxd) ? 0 Basic rules of integration TABLE OF INTEGRALS r xr1 1  x dx  C ; r  1  dx  ln x C r 1
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