Home › Subjects › A-Level H2 Maths › Differentiation
A-Level Differentiation
What the A-Level syllabus expects for Differentiation, and how to practise it.
What the syllabus expects
- Reading off the graph what f'(x) > 0, f'(x) = 0, f'(x) < 0 and f''(x) > 0, f''(x) < 0 signify
- Connecting the graph of y = f'(x) to that of y = f(x)
- Differentiating simple functions given implicitly or parametrically
- Classifying stationary points (local maxima, local minima and points of inflexion) analytically in straightforward cases with the first or second derivative test
- Pinpointing maximum and minimum points with a graphing calculator or graphing software
- Estimating the value of a derivative at a point using a graphing calculator or graphing software
- Problems on tangents and normals to curves, including curves defined implicitly or parametrically
- Problems seeking local maxima and minima
- Problems about connected rates of change
- Non-stationary points of inflexion, and obtaining the second derivative of a parametrically defined function
How it's examined
Questions on this topic most often ask you to find, determine, solve, evaluate. About 8% of the past-paper style questions in Rae's bank for this subject sit in this topic.
Worked examples
Example 1 (5 marks)
The curve C has equation y = (x^2 + 12)/sqrt(x) for x > 0. By differentiating, demonstrate that C possesses a stationary point at (2, 16/sqrt(2)), and establish whether it is a maximum, a minimum or a point of inflexion.
Show the worked answer
y = (x² + 12)/x^(1/2) = x^(3/2) + 12 x^(-1/2). dy/dx = (3/2)x^(1/2) - 6 x^(-3/2). Set = 0: (3/2)x^(1/2) = 6 x^(-3/2) => (3/2)x² = 6 => x² = 4 => x = 2 (x>0). Then y = (4+12)/sqrt2 = 16/sqrt2, so the stationary point is (2, 16/sqrt2). d^2y/dx² = (3/4)x^(-1/2) + 9 x^(-5/2). At x=2 both terms are positive, so d^2y/dx² > 0 => minimum.
Example 2 (3 marks)
The watch retailer also wishes to model its yearly promotion spending, C thousand dollars per year. For t years with t >= 0, the model is C = t^3 - 12t^2 + (k + 36)t, in which k is a positive constant. Determine the set of k-values for which C increases with t.
Show the worked answer
C = t³ - 12t² + (k + 36)t. Then dC/dt = 3t² - 24t + (k + 36). C increases with t on t >= 0 provided dC/dt >= 0 for all t >= 0. The quadratic 3t² - 24t + (k + 36) has its minimum at t = -(-24)/(2*3) = 4, which lies in the domain t >= 0. Minimum value: 3(4)² - 24(4) + (k + 36) = 48 - 96 + k + 36 = k - 12. Require k - 12 >= 0, i.e. k >= 12. (Given k > 0, this is the binding condition.)
Example 3 (3 marks)
A bakery's monthly revenue R (in thousands of dollars) is described by the model R = 3000 - 6000/(3 + 0.5t), where t denotes the number of years elapsed since 2015. Using differentiation, determine the value of dR/dt when t = 2, and interpret what this quantity tells you about the bakery's revenue in the given situation.
Show the worked answer
R = 3000 - 6000(3 + 0.5t)⁻¹. Differentiate: dR/dt = -6000 * (-1)(3 + 0.5t)⁻² * 0.5 = 3000/(3 + 0.5t)². At t = 2: 3 + 0.5(2) = 4, so dR/dt = 3000/4² = 3000/16 = 187.5. Since R is in thousands of dollars, this is +187.5 thousand dollars per year, i.e. $187,500 per year. Positive value means revenue is increasing.
More worked questions on this topic
- With p = pi/3, the tangent Mp and normal Np meet the x-axis at Q and R. Find the area of triang (4 marks)
- A cubic curve goes through (2, 3) and (−3, −22). Determine its equation given that it has a sta (4 marks)
- The curve has a stationary point at (pa, qa) for positive constants p and q. Find the exact p a (4 marks)
- A cubic f(x) = ax^3 + bx^2 + cx + d has a turning point at x = -1 and passes through (2, 10) an (4 marks)
- For the right hexagonal pyramid with V²=(3/4)(a²x⁴ − x⁶) and fixed edge a, find, in terms of a, (4 marks)
- A 3.12 m ladder slides down a vertical wall while its foot moves along the floor at a steady 0. (4 marks)
- For the right hexagonal pyramid with V²=(3/4)(a²x⁴ − x⁶), the volume is greatest when x²=(2/3)a (4 marks)
- The curve y = ax + b + c/(x^2 - 1) has real constants a, b, c. It crosses the x-axis at x = 2, (4 marks)
More A-Level H2 Maths topics
Functions · Graphs and their transformations · Equations and inequalities · Sequences and series · Vectors in two and three dimensions: basic properties · Scalar and vector products · all of A-Level H2 Maths