Edexcel IALGeneralGeneral Applications5 Marks
Given that y = 2^x.
- (a)Show that dy/dx = 2^x ln 2.[2]
- (b)Find the exact value of ∫ from 0 to 2 of (2^x) dx.[3]
Mark scheme
Marking points from the official mark scheme. The mark or mark type is shown in brackets.
- Part (a): M1 for taking natural logarithms of both sides to get ln y = x ln 2 or y = e^(x ln 2).
- Part (a): A1 for implicit differentiation 1/y dy/dx = ln 2 or direct differentiation of e^(kx) or a^x. Then substitute y = 2^x back to get dy/dx = 2^x ln 2 (Shown).
- Part (b): B1 for correct integration: ∫ 2^x dx = 2^x / ln 2 (o.e., e.g. e^(x ln 2) / ln 2).
- Part (b): M1 for applying the limits of integration correctly to their integrated expression.
- Part (b): A1 for (2^2 / ln 2) - (2^0 / ln 2) = (4 / ln 2) - (1 / ln 2) = 3 / ln 2 (exact value required).
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