IB Math · Topic practice · IB Mathematics AA HL · Unit 5
Calculus (AHL) A mixed exercise set, like a textbook: every problem type this unit covers, shuffled together and getting harder as you go. Work on paper, reveal to check — and hit
All levels Level 1 · Basics Level 2 · Standard Level 3 · Exam
1. Let f ( x ) = ( x 2 + 1 ) ( x 3 + 1 ) f(x) = (x^2 + 1)(x^3 + 1) f ( x ) = ( x 2 + 1 ) ( x 3 + 1 ) . Find f ′ ( 2 ) f'(2) f ′ ( 2 ) .
2. Let f ( x ) = x + 1 x 2 + 1 f(x) = \dfrac{x + 1}{x^2 + 1} f ( x ) = x 2 + 1 x + 1 . Find f ′ ( 2 ) f'(2) f ′ ( 2 ) .
3. f ( x ) = ( 2 x − 5 ) 2 f(x) = (2x - 5)^{2} f ( x ) = ( 2 x − 5 ) 2 . Find f ′ ( 0 ) f'(0) f ′ ( 0 ) .
4. The point ( 1 , 1 ) (1, 1) ( 1 , 1 ) lies on the curve x 2 + y 2 + 2 x = 4 x^2 + y^2 + 2x = 4 x 2 + y 2 + 2 x = 4 . Use implicit differentiation to find d y d x \dfrac{dy}{dx} d x d y at this point.
5. Using the substitution u = x 2 + 1 u = x^2 + 1 u = x 2 + 1 or otherwise, evaluate ∫ 0 1 2 x ( x 2 + 1 ) 2 d x \displaystyle\int_0^{1} 2x\,(x^2 + 1)^2\,dx ∫ 0 1 2 x ( x 2 + 1 ) 2 d x .
6. Find the area enclosed between y = 1 x y = 1x y = 1 x and y = 4 x 2 y = 4x^2 y = 4 x 2 .
7. Let f ( x ) = ( x 2 + 2 ) ( x 3 − 2 ) f(x) = (x^2 + 2)(x^3 - 2) f ( x ) = ( x 2 + 2 ) ( x 3 − 2 ) . Find f ′ ( 2 ) f'(2) f ′ ( 2 ) .
8. Let f ( x ) = x − 2 x 2 + 2 f(x) = \dfrac{x - 2}{x^2 + 2} f ( x ) = x 2 + 2 x − 2 . Find f ′ ( 1 ) f'(1) f ′ ( 1 ) .
9. f ( x ) = ( 2 x − 4 ) 4 f(x) = (2x - 4)^{4} f ( x ) = ( 2 x − 4 ) 4 . Find f ′ ( 0 ) f'(0) f ′ ( 0 ) .
10. The point ( 1 , − 3 ) (1, -3) ( 1 , − 3 ) lies on the curve x 2 + y 2 + 4 x = 14 x^2 + y^2 + 4x = 14 x 2 + y 2 + 4 x = 14 . Use implicit differentiation to find d y d x \dfrac{dy}{dx} d x d y at this point.
11. Using the substitution u = x 2 + 3 u = x^2 + 3 u = x 2 + 3 or otherwise, evaluate ∫ 0 4 2 x ( x 2 + 3 ) 2 d x \displaystyle\int_0^{4} 2x\,(x^2 + 3)^2\,dx ∫ 0 4 2 x ( x 2 + 3 ) 2 d x .
12. Find the area enclosed between y = 6 x y = 6x y = 6 x and y = x 2 y = x^2 y = x 2 .
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