Learn – Calculus – Theorems of Calculus

1 Mean value theorem

1.1 Exercises

1.1.1 Problem set

Determine the intervals where the Mean Value theorem applies on the following functions.

  1. |x|
  2. \sin x + \cot x
  3. |\cot x|
  4. \lfloor x \rfloor. (Here, \lfloor x \rfloor denotes the greatest integer that is less than or equal to x. For example, \lfloor 3.12 \rfloor = 3, \lfloor -2.12 \rfloor = -3 and \lfloor -100 \rfloor = -100)
  5. \begin{cases} x & \mbox{ if }x \le 0 \\ x^2 & \mbox{ if }x > 0\end{cases}
  6. \begin{cases} 0 & \mbox{ if }x \le 0 \\ x^2 & \mbox{ if }x > 0\end{cases}
  7. \begin{cases} x^3 & \mbox{ if }x \le 0 \\ x^2 & \mbox{ if }x > 0\end{cases}

1.1.2 Problem set

For each of the following problems, estimate the function f(x) that satisfies the given condition.

  1. f'(x) = 0 in \left(-\infty, \infty\right)
  2. f'(x) = 0 in \left(-\infty, 0\right)\cup\left(0,\infty\right)
  3. f'(x) = 2 in \left(-\infty, \infty\right)
  4. f'(x) = x in \left(-\infty, \infty\right)

2 Fundamental theorem of calculus

2.1 Exercises

2.1.1 Problem set

  1. Find f(x) if f(x) = \frac{d}{dx}\int\limits_1^x e^{t^2}\, dt.

  2. Find f(x) if f(x) = \frac{d}{dx}\int\limits_{-5}^x \sin^7 t\, dt.

  3. Find f(1) if f(x) = \frac{d}{dx}\int\limits_{-5}^x \frac{1}{t^{24}+1} dt.

  4. Find f(1) if f(x) = \frac{d}{dx}\int\limits_{10}^x \frac{1}{t^{10}+10} dt.

2.1.2 Problem set

  1. Find f(x) if f(x) = \frac{d}{dx}\int\limits_1^{2x} 2^{t^2}\, dt.

  2. Find f(x) if f(x) = \frac{d}{dx}\int\limits_3^{x^2} \frac{1}{t^{8}+3}\, dt.

  3. Find f(x) if f(x) = \frac{d}{dx}\int\limits_1^{\ln x} 2^{t^2}\, dt.

  4. Find f(x) if f(x) = \frac{d}{dx}\int\limits_x^{2\pi} \cos^8 t\, dt.

  5. Find f(x) if f(x) = \frac{d}{dx}\int\limits_{x^2}^{5} \ln(1+3t^2)\, dt.