Math 31A 2010 03 02 MATH 31A DISCUSSION JED YANG 1 Theorems 1 1 Comparison Theorem If g x f x on an interval a b then Z b Z b f x dx g x dx a a 1 2 Fundamental Theorem of Calculus I Assume that f x is continuous on a b and let F x be an antiderivative of f x on a b Then Z b f x dx F b F a a 1 3 Fundamental Theorem of Calculus II Assume that f x is continuous on a b Let Z x f t dt A x a Then A is an antiderivative of f that is A x f x or equivalently Z x d f t dt f x dx a Furthermore A x satisfies the initial condition A a 0 2 Fundamental Theorem of Calculus R 2 1 Exercise 5 3 39 Write the integral 0 cos x dx as a sum of integrals without absolute values and evaluate Solution Notice that cos x is nonnegative on 0 2 and nonpositive on 2 R 2 R 4 2 As such the integral in question is 0 cos x dx 2 cos x dx sin x 0 sin x 2 1 0 0 1 2 2 2 Exercise 5 3 52 Apply the Comparison Theorem to the inequality sin x x 3 2 valid for x 0 to prove 1 x2 cos x 1 Apply it again to prove x x6 sin x x for x 0 Solution On 0 t for some t 0 we have sin x x By the Comparison Theorem Rt Rt 2 we get 0 sin x dx 0 x dx This gives cos t 1 t2 2 hence 1 t2 cos t Since cos is even cos t cos t satisfies the same inequality Applying this again Rt Rt 3 2 we get 0 1 x2 dx 0 cos x dx yielding t t3 sin t as desired Math 31A Yang 2 2 3 Exercise 5 4 40 Find the smallest positive critical point of Z x F x cos t3 2 dt 0 and determine whether it is a local min or max Solution As usual to find critical point we take derivative and set to 0 By FTC2 we get F x cos x3 2 The smallest positive zero is when x3 2 2 So x 2 2 3 is the smallest positive critical point Since F goes from positive to negative it is a local max