1.
Differentiate y = 5x⁴ − 3x² + 7.
[2 marks]
- Final answer: 20x³ − 6x
JD SCIENCE
Pure — differentiation and integration
Indicative marking points. Award marks for equivalent scientific or mathematical wording. Do not issue this sheet with the student worksheet.
2.
Differentiate y = (2x + 1)³.
[3 marks]
- Chain rule
- Final answer: 6(2x+1)²
3.
Differentiate y = x² sin x.
[3 marks]
- Product rule
- Final answer: 2x sin x + x² cos x
4.
Differentiate y = (x² + 1)/(x − 1).
[4 marks]
- Final answer: Quotient rule: [(2x)(x−1) − (x²+1)]/(x−1)²
5.
Find dy/dx if x = 2t + 1, y = t².
[3 marks]
- (dy/dt)/(dx/dt) = 2t / 2
- Final answer: dy/dx = t
6.
Find the gradient of y = x³ − 3x at x = 2.
[2 marks]
- 3x² − 3
- Final answer: 9
7.
Find the equation of the tangent to y = x² at (3, 9).
[3 marks]
- Gradient 6
- Final answer: y − 9 = 6(x − 3)
8.
Find the equation of the normal to y = x² at (3, 9).
[3 marks]
- Final answer: y − 9 = −(1/6)(x − 3)
9.
Find the stationary points of y = x³ − 12x + 1 and determine their nature.
[5 marks]
- d²y/dx² = 6x
- Final answer: (2, −15) min, (−2, 17) max
10.
A rectangle has perimeter 20 cm. Show that the area A = 10x − x² and find the maximum area.
[5 marks]
- dA/dx = 10 − 2x = 0
- Final answer: Maximum 25 cm² when x = 5
11.
Integrate 6x² − 4x + 3.
[2 marks]
- Final answer: 2x³ − 2x² + 3x + c
12.
Find ∫ (2x + 1)⁴ dx.
[3 marks]
- Reverse chain rule
- Final answer: (1/10)(2x+1)⁵ + c
13.
Evaluate ∫₀² (3x² + 1) dx.
[3 marks]
- [x³ + x] from 0 to 2
- Final answer: 10
14.
Find the area between y = x², the x-axis, x = 1 and x = 3.
[3 marks]
- ∫ x² dx
- Final answer: 26/3
15.
Find the area between y = x and y = x² from 0 to 1.
[3 marks]
- ∫(x−x²)dx
- Final answer: 1/6
16.
Use the trapezium rule with 4 strips to estimate ∫₀⁴ (x²) dx. Comment on whether this is an over- or underestimate.
[4 marks]
- (h/2)(y0+2y1+2y2+2y3+y4)
- Final answer: h=1; estimate 22; overestimate because the curve is convex
17.
If dy/dx = 6x² and y = 5 when x = 1, find y.
[3 marks]
- Integrate + constant
- Final answer: y = 2x³ + 3
18.
Differentiate y = e^{3x} and y = ln(5x).
[3 marks]
- ln(5x)=ln5+ln x
- Final answer: 3e^{3x} ; 1/x
19.
Differentiate y = e^{x²}.
[2 marks]
- Chain rule
- Final answer: 2x e^{x²}
20.
Find ∫ 1/(2x) dx, x > 0.
[2 marks]
- Final answer: (1/2) ln|x| + c or (1/2) ln|2x| + c
21.
Implicit differentiation: x² + y² = 25. Find dy/dx at (3, 4).
[3 marks]
- 2x + 2y y' = 0
- Final answer: −3/4
22.
Related rates: a sphere has V = (4/3)πr³. If dr/dt = 2, find dV/dt when r = 3.
[4 marks]
- dV/dt = 4πr² dr/dt
- Final answer: dV/dt = 72π
23.
Find the second derivative of y = sin 2x.
[2 marks]
- Final answer: −4 sin 2x
24.
Explain how to decide whether a stationary point is a max, min or point of inflection using the second derivative.
[3 marks]
- f'' > 0 min
- f'' < 0 max
- f'' = 0 inconclusive
25.
Integration by substitution: ∫ 2x(x² + 1)³ dx.
[3 marks]
- u = x²+1
- Final answer: (1/4)(x²+1)⁴ + c
26.
Find the average value of y = 3x² on [0, 2].
[3 marks]
- (1/2)∫ 3x² dx = 4
- Final answer: 4
27.
A curve is y = 1/x. Find the volume generated when the region from x = 1 to x = 2 is rotated about the x-axis. Leave π in the answer if this is on the spec; otherwise find the area under the curve.
[3 marks]
- Final answer: Area = ln 2; volume π∫ x⁻² dx = π(1 − 1/2) = π/2
28.
State the chain, product and quotient rules in Leibniz or function notation.
[3 marks]
- dy/dx = dy/du × du/dx
- uv' + vu'
- (vu' − uv')/v²
29.
A OCR question gives displacement s = t³ − 6t². Find the velocity and acceleration at t = 1.
[3 marks]
- v=3t²−12t, a=6t−12
- Final answer: v = −9, a = −6
30.
Solve dy/dx = 2y, y(0) = 3, by separating variables.
[4 marks]
- ∫ dy/y = ∫ 2 dx
- Final answer: y = 3e^{2x}
31.
Why do you add + c for an indefinite integral but not for a definite integral?
[2 marks]
- Family of antiderivatives
- Constants cancel in the evaluation
32.
Find the x-coordinate of the point on y = x² + 1 where the tangent is parallel to y = 4x.
[3 marks]
- 2x = 4
- Final answer: x = 2