1992 Paper 1 Q8

Year: 1992
Paper: 1
Question Number: 8

Course: LFM Pure and Mechanics
Section: Integration as Area

Difficulty: 1500.0 Banger: 1500.0

integration fundamental theorem of calculus differential equation exponential function integral equation differentiation under integral sign

Problem

Explain diagrammatically, or otherwise, why \[ \frac{\mathrm{d}}{\mathrm{d}x}\int_{a}^{x}\mathrm{f}(t)\,\mathrm{d}t=\mathrm{f}(x). \] Show that, if \[ \mathrm{f}(x)=\int_{0}^{x}\mathrm{f}(t)\,\mathrm{d}t+1, \] then \(\mathrm{f}(x)=\mathrm{e}^{x}.\) What is the solution of \[ \mathrm{f}(x)=\int_{0}^{x}\mathrm{f}(t)\,\mathrm{d}t? \] Given that \[ \int_{0}^{x}\mathrm{f}(t)\,\mathrm{d}t=\int_{x}^{1}t^{2}\mathrm{f}(t)\,\mathrm{d}t+x-\frac{x^{5}}{5}+C, \] find \(\mathrm{f}(x)\) and show that \(C=-2/15.\)

No solution available for this problem.

Rating Information

Difficulty Rating: 1500.0

Difficulty Comparisons: 0

Banger Rating: 1500.0

Banger Comparisons: 0

Show LaTeX source

Problem source

Explain diagrammatically, or otherwise, why 
\[
\frac{\mathrm{d}}{\mathrm{d}x}\int_{a}^{x}\mathrm{f}(t)\,\mathrm{d}t=\mathrm{f}(x).
\]
Show that, if 
\[
\mathrm{f}(x)=\int_{0}^{x}\mathrm{f}(t)\,\mathrm{d}t+1,
\]
then $\mathrm{f}(x)=\mathrm{e}^{x}.$

What is the solution of 
\[
\mathrm{f}(x)=\int_{0}^{x}\mathrm{f}(t)\,\mathrm{d}t?
\]
Given that 
\[
\int_{0}^{x}\mathrm{f}(t)\,\mathrm{d}t=\int_{x}^{1}t^{2}\mathrm{f}(t)\,\mathrm{d}t+x-\frac{x^{5}}{5}+C,
\]
find $\mathrm{f}(x)$ and show that $C=-2/15.$

Back to List