DP Mathematics HL Questionbank

• Syllabus

9.4

Sub sections and their related questions

The integral as a limit of a sum; lower and upper Riemann sums.

• 12N.3ca.hl.TZ0.3c: Let \(\sum\limits_{n = 1}^\infty  {\frac{1}{{{n^2}}}}  = L\) . The diagram below shows the graph...
• 08M.3ca.hl.TZ1.4: (a)     Using the diagram, show that...
• 13M.3ca.hl.TZ0.5a: Figure 1 shows part of the graph of \(y = \frac{1}{x}\) together with line segments parallel to...
• 13M.3ca.hl.TZ0.5b: An improved upper bound can be found by considering Figure 2 which again shows part of the graph...
• 11N.3ca.hl.TZ0.4b: (i)     Show, by means of a diagram, that...
• 15N.3ca.hl.TZ0.4b: Use the inequality in part (a) to find a lower and upper bound for \(\pi \).
• 18M.3ca.hl.TZ0.3a: Find the value of \(\int\limits_4^\infty  {\frac{1}{{{x^3}}}{\text{d}}x} \).
• 18M.3ca.hl.TZ0.3b: Illustrate graphically the...
• 18M.3ca.hl.TZ0.3c: Hence write down a lower bound for \(\sum\limits_{n = 4}^\infty  {\frac{1}{{{n^3}}}} \).
• 18M.3ca.hl.TZ0.3d: Find an upper bound for \(\sum\limits_{n = 4}^\infty  {\frac{1}{{{n^3}}}} \).

Fundamental theorem of calculus.

• 16M.3ca.hl.TZ0.2a: Write down \(f'(x)\).
• 16M.3ca.hl.TZ0.2b: By differentiating \(f({x^2})\), obtain an expression for the derivative of...
• 16M.3ca.hl.TZ0.2c: Hence obtain an expression for the derivative of...
• 18M.3ca.hl.TZ0.3a: Find the value of \(\int\limits_4^\infty  {\frac{1}{{{x^3}}}{\text{d}}x} \).
• 18M.3ca.hl.TZ0.3b: Illustrate graphically the...
• 18M.3ca.hl.TZ0.3c: Hence write down a lower bound for \(\sum\limits_{n = 4}^\infty  {\frac{1}{{{n^3}}}} \).
• 18M.3ca.hl.TZ0.3d: Find an upper bound for \(\sum\limits_{n = 4}^\infty  {\frac{1}{{{n^3}}}} \).

Improper integrals of the type \(\int\limits_a^\infty {f\left( x \right){\text{d}}} x\) .

• 12M.3ca.hl.TZ0.5a: Find the set of values of k for which the improper integral...
• 12N.3ca.hl.TZ0.3a: Prove that \(\mathop {\lim }\limits_{H \to \infty } \int_a^H {\frac{1}{{{x^2}}}{\text{d}}x} \)...
• 12N.3ca.hl.TZ0.3d: Hence show that...
• 12N.3ca.hl.TZ0.3e: You are given that \(L = \frac{{{\pi ^2}}}{6}\). By taking k = 4 , use the upper bound and lower...
• 08M.3ca.hl.TZ1.2: (a)     Using l’Hopital’s Rule, show that...
• 08M.3ca.hl.TZ2.2: Find the exact value of \(\int_0^\infty  {\frac{{{\text{d}}x}}{{(x + 2)(2x + 1)}}} \).
• 14M.3ca.hl.TZ0.1d: Find the value of the improper integral...
• 16M.3ca.hl.TZ0.2a: Write down \(f'(x)\).
• 16M.3ca.hl.TZ0.2b: By differentiating \(f({x^2})\), obtain an expression for the derivative of...
• 16M.3ca.hl.TZ0.2c: Hence obtain an expression for the derivative of...
• 18M.3ca.hl.TZ0.3a: Find the value of \(\int\limits_4^\infty  {\frac{1}{{{x^3}}}{\text{d}}x} \).
• 18M.3ca.hl.TZ0.3b: Illustrate graphically the...
• 18M.3ca.hl.TZ0.3c: Hence write down a lower bound for \(\sum\limits_{n = 4}^\infty  {\frac{1}{{{n^3}}}} \).
• 18M.3ca.hl.TZ0.3d: Find an upper bound for \(\sum\limits_{n = 4}^\infty  {\frac{1}{{{n^3}}}} \).