MATH114 Integration and Differentiation

Exercises Week 5

[2.5ex]

Workshop questions that you cannot solve during the workshop should be regarded as additional training material and solved at home. Afterwards please compare with the model solutions.

Workshop

• True or false? Think for 1 minute, then resolve the quiz and discuss briefly. Let $f:I\to ℝ$ be a differentiable function.

  • The anti-derivative of $f$ is unique.

  • The anti-derivative of $f$ is unique up to an additive real constant.

  • The derivative of $f$ is unique up to a an additive real constant.

• In analogy to Ex.3.4.2, compute the indefinite integral of

  $$f:ℝ\to ℝ,f(x)=x^2,$$

  with base point $0$. You may use the formula

  $$\sum _{k=1}^n{k}^2=\frac{n(n+1)(2n+1)}{6}.$$

• Prove Corollary 3.4.5.

• Elaborate Remark 3.4.4: what precise conditions does $f$ have to fulfill? How exactly are indefinite integral and derivative related? Can we iterate this, i.e., integrate 10 times and then differentiate 10 times in order to get the original function back?

• State all the anti-derivatives of the function

  $$f:[0,\mathrm{\infty })\to ℝ,f(x)=x^{1/2}+6\cos (x).$$

  Determine the indefinite integral of $f$ with base point $c=2$.