Philadelphia University Real Analysis II (250411) Faculty of Science
advertisement
Philadelphia University Faculty of Science Department of Basic Sciences & Mathematics Fall 2014/ 2015 Real Analysis II (250411) First Exam: Thursday 20\11/2014 Time: 50 min. Dr. Khaled Hyasat Name: Student's Number: Grade: ( / 20) Subject: Differentiation, the Mean value theorem Keywords: continuous, differentiable, MVT Q.1 Prove the following Theorem" If f: I→R has a derivative at c I , then f is continuous at c"? ( / 5 points) …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… Q.2 Theorem: let f: I→R be differentiable on interval I, then f is decreasing on I if and only if f 0 , for all x I , Use the mean value theorem to prove this? ( / 5 points) …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… Q.3 True or False, if false give the true: ( / 5 points) (i) The function f(x) = x is an example of a function that satisfies this statement: "Continuity at a point c is not a sufficient condition for the derivative to exist at c". ( T/F ) …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… (ii) If f1 f 2 ... f n f , then ( f n )(c) (n 1)( f (c)) n f (c) . ( T /F ) …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… (iii) The geometric view of the MVT is that there is some point on the curve y=f(x) at which the tangent line is parallel to the line segment through the points (a, f(a)) and (b, f(b)). ( T /F ) …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… Q.4 Use The Mean Value Theorem to prove that: sin x sin y x y For all x, y in R. ( / 5 points) …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………... …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………... Good Luck Q.2 State whether the following equation is an exact, if not, find an integrating factor that, when multiplied by it become exact? Keyword: (Exact Equations and Integrating Factors) ( x 2) sin ydx x cos ydy 0 ( / 5 points) …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… Good Luck Q2. Find the Fourier Series for the following function 1 , - 6 x 0 f (x) , 0 x 6 1 , f(x 12) f(x) ( / 5 points) Solution:………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… Q3. Find the solution of the following heat conduction problem: 64u xx ut 0 x 80, ( / 5 points) u(0, t) 0 , u(80, t ) 0 , t 0, u(x , 0 ) x , 0 x 80 Solution:………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ………………………………………………………………………………………………………….... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... Q4. Find the temperature u(x ,t) in a metal rod of length 30 cm that is insulated on the ends as well as on the sides and whose initial temperature distribution is u(x,t) = 10 for 0 < x < 30 . Formulate the heat conduction problem: ( / 5 points) Solution:………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ………………………………………………………………………………………………………….... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………… Good Luck ...…………………………………………………………......................................................................... .................................................................................................................................................................... .................................................................................................................................................................... ..............................................................................................................................Good Luck .................................................................................................
