First principles derivation + convergence & divergence: notes and practice questions
- Kinematics relates displacement (), velocity (), and acceleration () via differentiation and integration.
- Differentiating displacement gives velocity; differentiating velocity gives acceleration.
- Integrating acceleration gives velocity; integrating velocity gives displacement. Constants of integration are found using initial conditions.
- Velocity is the rate of change of displacement; speed is its magnitude.
- Acceleration is the rate of change of velocity.
- An object speeds up when velocity and acceleration have the same sign, and slows down when they have opposite signs.
- Definite integrals calculate changes in displacement and total distance travelled.
How it is examined
Continuity and differentiability are understood informally and never tested, so a question asking a student to test them is out of syllabus. First principles is polynomials only, so differentiating from first principles is out of syllabus even though it looks like a natural HL question. Higher derivatives usually appear as the th derivative of a simple function proved by induction. 5 to 7 marks.
The first principles definition is given.
- Informal understanding of continuity and differentiability of a function at a point.
- Understanding of limits (convergence and divergence).
- Definition of derivative from first principles .
- Higher derivatives.
In examinations, students will not be asked to test for continuity and differentiability.
Linking questions
- Links to other subjects: theory of the firm (economics).
- Enrichment: the fundamental theorem of calculus.
Practice questions
5 questions · 5 mediumQuestion 1
MediumPaper 1 · no calculator5 marksFind .
Hence, evaluate .
You should use one of the double angle identities to remove the squared cosine.
Plug in the boundaries into your previous answer
Question 2
MediumPaper 1 · no calculator5 marksFind .
Hence, evaluate .
You should use one of the double angle identities to remove the squared cosine.
Plug in the boundaries into your previous answer
Question 3
MediumPaper 1 · no calculator13 marksLet , for .
Express in partial fractions.
Hence, show that is a decreasing function.
Hence, find the exact value of . Give your answer in the form , where is a rational number.
Start by factoring the denominator of the rational function. Then, set up the identity for the partial fraction decomposition and solve for the unknown constants by substituting convenient values for x or by equating coefficients.
To determine if a function is decreasing, you need to analyze its first derivative. Differentiate the partial fraction form of g(x) and examine the sign of g'(x) for all x in its domain.
Integrate the partial fraction representation of g(x) term-by-term. Recall that the integral of 1/(ax+b) is (1/a)ln|ax+b|. After finding the antiderivative, apply the Fundamental Theorem of Calculus by substituting the limits of integration. Finally, use the laws of logarithms to combine the terms into the required form.
Question 4
MediumPaper 1 · no calculator9 marksConsider the function defined by .
(a) By differentiating from first principles, show that .
(b) Hence, find an equation for the normal to the graph of at .
Recall the definition of the derivative from first principles: . You will need to expand .
First, find the coordinates of the point on the curve at . Then, use the derivative found in part (a) to find the gradient of the tangent at this point. The gradient of the normal is the negative reciprocal of the tangent's gradient. Finally, use the point-gradient formula to find the equation of the line.
Question 5
MediumPaper 1 · no calculator7 marksDetermine whether each of the following sequences converges as . If a sequence converges, find its limit.
To find the limit of a rational function as , you can divide both the numerator and the denominator by the highest power of present in the denominator.
Try dividing the numerator and denominator by . Remember that for , can be written as .
Compare the degree of the polynomial in the numerator with the degree of the polynomial in the denominator. What does this imply about the behaviour of the fraction for very large values of ?
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Where marks are lost
- Using your own wrong value after failing a "show that".
- Using an alternative method after "Hence".
