DPST1012 Chap.9 Limits, Derivatives and Their Applications
Limits, Derivatives and Their Applications
Limits and the meaning of the derivative
Calculus begins in Week 6 with Topic 5, the introduction to differentiation, and continues in Week 8 with its applications; the Week 8 test covers the first five topics. The syllabus starts with the limit operator, continuity, computations with limits, and limits at infinity with their asymptotes.
The derivative is then defined from first principles as the limit of secant gradients, which gives it its geometric meaning: the gradient of the tangent to the curve at a point, or the instantaneous rate of change.
The differentiation rules
After one or two first-principles calculations, the rules take over.
The power rule, the product rule, the quotient rule and the chain rule, together with the derivatives of sine, cosine, the exponential and the natural logarithm, cover every function in the course.
The syllabus also asks what differentiability means: a function with a corner, such as the absolute value at zero, is continuous there but has no derivative.
Applications in Week 8
Stationary points are where the derivative is zero, and they are classified by the sign change of the first derivative or by the sign of the second derivative.
The second derivative also describes concavity, and a point of inflection is where the concavity changes. Optimisation problems build a function from a description, find its stationary points and check the ends of the domain. Curve sketching combines intercepts, stationary points, concavity and, for rational functions, vertical and horizontal asymptotes.
What this chapter covers
- 01
Limits and continuity
- 02
Limits at infinity and asymptotes
- 03
First principles and the tangent gradient
- 04
Product, quotient and chain rules
- 05
Derivatives of trigonometric, exponential and log functions
- 06
Stationary points and their classification
- 07
Concavity and points of inflection
- 08
Optimisation and curve sketching
Worked example · free
Find, classify and use stationary points
- 2f′(x) = 6x² − 18x + 12 = 6(x − 1)(x − 2), so the stationary points are at x = 1 and x = 2.
- 1f(1) = 2 − 9 + 12 − 4 = 1 and f(2) = 16 − 36 + 24 − 4 = 0.
- 2f″(x) = 12x − 18: f″(1) = −6 < 0 gives a maximum at (1, 1); f″(2) = 6 > 0 gives a minimum at (2, 0).
- 1f″(x) = 0 at x = 1.5 and changes sign there; f(1.5) = 6.75 − 20.25 + 18 − 4 = 0.5, so the inflection is (1.5, 0.5).
Key terms
- Limit
- The value a function approaches as its input approaches a given number, whether or not the function reaches it there.
- Derivative
- The instantaneous rate of change of a function, equal to the gradient of its tangent and defined as a limit.
- Chain Rule
- The rule that the derivative of f(g(x)) is f′(g(x)) times g′(x), used for functions inside functions.
- Stationary Point
- A point on a curve where the derivative is zero, so the tangent is horizontal.
- Point of Inflection
- A point where a curve changes concavity, so the second derivative changes sign there.
Limits, Derivatives and Their Applications FAQ
Do I need to differentiate from first principles in exams?
The syllabus lists the first-principles definition, so expect to use it on simple polynomials. Show the expansion, cancel the h, and only then let h approach zero; the rules are used for everything else.
How do I choose between the product and the chain rule?
Use the product rule when two functions of x are multiplied, such as x² times e to the 3x. Use the chain rule when one function sits inside another, such as the fifth power of 2x + 1. Some questions need both.
What if the second derivative is zero at a stationary point?
Then the second derivative test gives no answer, and you check the sign of the first derivative on either side. The point may be a maximum, a minimum or a horizontal point of inflection.
Why check the endpoints in an optimisation problem?
On a closed interval the largest or smallest value can occur at an end rather than at a turning point. Compare the function's value at every stationary point and at both ends before giving the answer.
How do I find a horizontal asymptote of a rational function?
Compare the highest powers on top and bottom. Equal powers give the ratio of the leading coefficients, a higher power below gives the line y = 0, and a higher power above means there is no horizontal asymptote at all.
Exam move
Keep a page of rules and add one fully worked example per rule; each week differentiate ten mixed functions, then classify the stationary points of one cubic by both tests. Finish every optimisation with a sentence that answers the question in its own units.
Working through Limits, Derivatives and Their Applications in DPST1012? Sia is AskSia’s AI Mathematics tutor — ask any DPST1012 Limits, Derivatives and Their Applications question and get a clear, step-by-step explanation grounded in how DPST1012 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
