MATH114 Integration and Differentiation

Mock End-of-Module Test

[2.5ex]

Please write your own name and your workshop tutor’s name clearly at the top of each answer sheet. You have got 40 minutes for the test. Use your own calculators. Clearly indicate where you use statements from the lecture notes.

• Consider the function:

  $$f:[0,10]\to ℝ,f(x)=\sqrt{x}.$$

  • Draw the graph of $f$ making sure to label your coordinate system correctly.

  • Compute the upper and lower approximating step functions of $f$ for 2 bisections.

  • Insert the upper and lower approximating step functions for 2 bisections into the graph of $f$.

  • Compute the integral of these two approximating step functions.

  [15]

• • State the (complete) definition of what it means for a series to be convergent. [10]

  • Consider the series ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n$ where

    $$a_n={(-1)}^n\frac{1}{\sqrt{n}},n\in \mathbb{N}.$$

    Does the series converge or diverge? Prove your answer. [10]

• Consider the function

  $$g:ℝ\to ℝ,g(x)=2\cdot |x|.$$

  Draw the graph of the function (with correct labelling). Using difference quotients with all details, determine where $g$ is differentiable and where not, and compute the derivative in case it exists. [15]

[Total marks: 50]