4.08a-d Maclaurin series, improper integrals and volumes of revolution Quiz: OCR Further Maths, Unit 1

The 20 questions

1. What is the coefficient of x^n in the Maclaurin expansion of f(x)?

   • f^(n)(0) / n!
   • f^(n)(x) / n!
   • f^(n)(0) / n
   • f(0) / n!

2. What is the Maclaurin expansion of e^x up to the x^2 term?

   • x + x^2/2
   • 1 + x + x^2/2
   • 1 - x + x^2/2
   • 1 + x + x^2

3. What is the first non-zero odd degree term in the Maclaurin series for cos(x)?

   • x^3/3
   • None
   • -x^3/6
   • x

4. What is the Maclaurin series expansion of ln(1+x) up to x^2?

   • x - x^2
   • x - x^2/2
   • 1 + x - x^2/2
   • x + x^2/2

5. What is the coefficient of x^3 in the Maclaurin expansion of sin(x)?

   • 1/6
   • 1/3
   • -1/3
   • -1/6

6. For what range of x values is the Maclaurin series for ln(1+x) valid?

   • x > -1
   • -1 < x < 1
   • -1 < x <= 1
   • -1 <= x <= 1

7. What is the coefficient of x^3 in the Maclaurin series for (1-x)^(-1)?

   • 3
   • -1
   • -3
   • 1

8. What is the x^2 term in the Maclaurin expansion of e^(2x)?

   • x^2
   • 4x^2
   • x^2/2
   • 2x^2

9. What is the x^2 term in the Maclaurin expansion of cos(2x)?

   • -4x^2
   • -2x^2
   • -x^2/2
   • -x^2

10. Why is the integral from 1 to infinity of x^(-2) dx classed as improper?

    • Undefined integrand
    • Infinite limit
    • Discontinuous integrand
    • Negative area

11. What is the value of the improper integral from 1 to infinity of x^(-2) dx?

    • Infinity
    • 0
    • 1
    • 1/2

12. What is the value of the improper integral from 0 to 1 of x^(-1/2) dx?

    • Infinity
    • 2
    • 1/2
    • 1

13. What word describes an improper integral that does not tend to a finite limit?

    • Divergent
    • Continuous
    • Undefined
    • Convergent

14. What is the outcome of evaluating the improper integral from 1 to infinity of 1/x dx?

    • 1
    • 0
    • Divergent
    • ln(2)

15. What is the formula for the volume of revolution of y=f(x) about the x-axis?

    • pi * integral(y dx)
    • 2*pi * integral(y dx)
    • integral(y^2 dx)
    • pi * integral(y^2 dx)

16. What is the formula for the volume of revolution of x=f(y) about the y-axis?

    • integral(x^2 dy)
    • pi * integral(y^2 dx)
    • 2*pi * integral(x dy)
    • pi * integral(x^2 dy)

17. What is the volume generated when y=x between x=0 and x=1 rotates fully about the x-axis?

    • pi
    • pi/2
    • pi/3
    • 2*pi/3

18. What is the volume generated when y=sqrt(x) from x=0 to x=2 rotates about the x-axis?

    • pi
    • 4*pi
    • 2*pi
    • pi/2

19. What is the volume formula when rotating the area between upper curve y1 and lower y2 about the x-axis?

    • integral(y1^2 - y2^2 dx)
    • pi * integral((y1 - y2)^2 dx)
    • 2*pi * integral(y1 - y2 dx)
    • pi * integral(y1^2 - y2^2 dx)

20. What is the x^3 term in the Maclaurin expansion of arctan(x)?

    • -x^3/6
    • x^3/6
    • x^3/3
    • -x^3/3

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