Primer parcial 2020-2021, Exámenes de Cálculo diferencial y integral

¡Descarga Primer parcial 2020-2021 y más Exámenes en PDF de Cálculo diferencial y integral solo en Docsity! DEPARTMENT OF MATHEMATICS Integral Calculus Groups 121 Continuous Evaluation I March 24, 2021 Time: 100 minutes Maximum Marks: 10 1. Solve the following inhomogeneous first-order differential equation, supposing t > 0: y′ + coth(t) y = csch3(t) coth2(t) . (1) [1.5] 2. Consider the function α1(x) = ∫ x 0 1 (4 + t2)(t+ 1) dt . (2) • (0.5 points) Using the fundamental theorem of calculus, show that α1 is monotonically increasing over the interval [0, 3]. • (2 points)Consider the function α(x) = α1(x) + I(x− 1) , (3) where I(x − 1) is the step function with jump-discontinuity at x = 1. Compute the Riemann-Stieltjes integral ∫ 3 0 (x2) dα (4) [2.5] 3. Consider the circle of equation x2 + y2 = 9 , (5) and the hyperbola of equation y2 8 − x2 = 1 . (6) Compute the area of the region R of the positive halfplane (y > 0) enclosed between the circle and the hyperbola. [2.5] 4. Consider the improper integral Lk = ∫ 1 0 logk ( 1 x ) dx . (7) • (0.5 points) Show that L1 converges. Group: Name and Last Name: