Questions with Answers on the Second Fundamental Theorem of Calculus

Questions with detailed solutions on the second theorem of calculus are presented.

The second fundamental theorem of calculus states that if f is a continuous function on an interval I containing a and

F(x) = ax f(t) dt


then F '(x) = f(x) for each value of x in the interval I.

Question 1:

Approximate F'(Pi/2) to 3 decimal places if

F(x) = 3x sin(t 2) dt

Solution to Question 1:

  • Since sin(t 2) is continuous for all real numbers, the second fundamental theorem may be used to calculate F'(x) as follows

    F '(x) = sin(x 2)

  • which gives

    F '(Pi/2) = sin( (Pi/2) 2 ) = 0.624 (3 decimal places)


Question 2:

Let

F(x) = 0x 5 / (3 + 2 e t) dt


b) Calculate F'(0)

b) Show that F(1) < F(4)

Solution to Question 2:

  • a) 5 / (3 + 2 e t) is continuous, hence the use of the above theorem gives

    F'(x) = 5 / (3 + 2 e x)

  • which gives

    F'(0) = 5 / (3 + 2 e 0) = 1

  • b) A closer look at F'(x) reveals that F'(x) is positive for all real values of x and therefore function F is an increasing one. Hence since 4 > 1 then

    F(4) > F(1) or F(1) < F(4)


Question 3:

Let

F(x) = -1x2 1 / (1 + t 2) dt


Find F'(x)

Solution to Question 3:

  • Let u = x2. F is now given by

    F(u) = -1u 1 / (1 + t 2) dt


  • Use the second fundamental theorem to obtain

    dF/du = F'(u) = 1 / (1 + u 2)

  • We now use the chain rule of differentiation to write

    F'(x) = dF/dx = dF/du . du/dx = 2x * 1 / (1 + x 4)


Question 4:

Let

F(x) = u(x)v(x) f(t) dt


where f is continuous everywhere and u and v are continuous functions of x. Express F'(x) in terms of u', v', u, v and f.

Solution to Question 4:

  • Let a be a real number and write the given integral as the sum (difference) of two integrals as follows

    F(x) = u(x)v(x) f(t) dt
    = u(x)a f(t) dt + av(x) f(t) dt
    = - au(x) f(t) dt + av(x) f(t) dt


  • Since F(x) is the sum of two functions F1 = - au(x) f(t) dt and F2 = av(x) f(t) dt, then F'(x) is given by

    F(x) = F'1(x) + F'2(x)

  • We now use the second fundamental theorem to write

    dF1/du = - f(u)

    dF2/dv = f(v)

  • We now use the chain rule of differentiation to write

    F'1(x) = dF1/du * du/dx = - f(u) * du/dx

    F'2(x) = dF2/dv * dv/dx = f(v) * dv/dx

  • Finally F'(x) is given by

    F'(x) = F'1(x) + F'2(x)

    = v'(x) * f(v(x)) - u'(x) * f(u(x))


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