MATH 2250 - Final Exam with 100- correct answers

Limit Definition of Derivative Mean Value Theorem If f(x) is defined and continuous on [a,b] and differentiable on (a,b) then there is at least one point where f'(c) = f(b)-f(a)/b-a A function is continuous at a point x=a if: a) f(a) is defined b) Limit as x approaches a of f(x) c) Limit as x approaches a of f(x) is equal to f(a) Fundamental Theorem of Calculus (P1) If f is continuous on [a,b] F(x) = ∫f(x) and both are continuous on [a,b] F(x) is differentiable on (a,b) F'(x) = d/dx F(x) = d/dx(∫f(t)dt = f(x) Fundamental Theorem of Calculus (P2) F(x) is the antiderivative of f(x) which is continuous on [a,b] a∫b f(x)dx = F(b) - F(a) or a∫b F'(x)dx = F(b) - F(a) Basically antiderivative of f(x) is the integral of f(x) Intermediate Value Theorem If f(x) is continuous on [a,b] and it changes signs between those points then there is a root Extreme Value Theorem If f(x) is continuous on [a,b] there are numbers c and d such that there exist f(c) is an absolute min and f(d) is an absolute max. First Derivative Test If f'(x) > 0 then the function is increasing and if f'(x) < 0 then the function is decreasing. Can tell us if we have a local max or a local min.