Final exercises

1. If t is a real number such that 0<t<2, calculate .
2. Given the function , find the points where it gets a relative maximum. (The integral function is supposed to be extended so that it is everywhere continuous).
3. Solve the following inequality: .
4. f is a function continuous together with its first and second derivative; the tangents to the graph of f in the numbers a and b create an angle whose measure is, respectively, of π/3 and π/4 with he positive x-axis. Calculate the following two integrals: .
5. Given the function , find the intervals where the function is increasing and those where the function is convex.
6. Given the function , find the values of the numbers a and b such that f(0)=0 and . Graph the function.
7. Calculate .
8. Given the function , calculate its integral function with starting point -1. What can we say about the derivative of this integral function?
9. By direct calculation of the integral find the derivative of the function . Find again the same derivative without calculating the integral.
10. Find the domain of the following functions:
    • .
    • .
    • .
    • .
    • .
11. Prove that if f is an integrable function over [-a,a] and f(-x)=-f(x) than . Give a geometrical interpretation of this result.
12. The integral of a function is unchanged if we alter the values of a fucntion in a finite number of points. However this is no more true if we alter the function in a countably infinite number of points. Proof this with an example (sugg.: use Dirichlet's function).