Continuity & differentiability: notes and practice questions
- Strategy: Visualize, define variables and constraints, construct constraint and optimization equations (in one variable), differentiate, solve f'(x)=0, and justify max/min using sign diagrams or the second derivative test.
- Justification: Use the first derivative test (sign change of f'(x)) or the second derivative test (sign of f''(x)) to confirm stationary points as local maxima or minima.
- Global Extrema: Check endpoints of the domain if applicable, alongside stationary points.
- Quadratic Optimization: For y=ax^2+bx+c, the vertex at x=-b/(2a) gives the max (if a<0) or min (if a>0) without calculus.
- Common Errors: Neglecting domain restrictions (e.g., x>0), failing to check endpoints, and not justifying the nature of stationary points.
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
6 questions · 6 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 calculator7 marksConsider the function defined for all by
Show that exists and state its value.
Hence, find the possible value(s) for such that is continuous at .
To show a limit exists at a point for a piecewise function, you need to calculate the limit from the left and the limit from the right. If they are equal, the limit exists. For the left-hand limit, you will encounter an indeterminate form which can be resolved by factoring the numerator.
For a function to be continuous at a point, the value of the function at that point must be equal to the limit of the function as it approaches that point. Use your result from part (a).
Question 5
MediumPaper 1 · no calculator5 marksConsider the function .
Find the value of the constant such that the function is continuous at .
For a function to be continuous at a point, the limit from the left, the limit from the right, and the function's value at that point must all be equal. You will need to evaluate the limits for the different pieces of the function as approaches 0.
Question 6
MediumPaper 1 · no calculator7 marksA function is defined by
where .
(a) Find an equation relating and , given that is continuous at .
(b) Given further that , find the value of and the value of .
For a function to be continuous at a point, the value of the function approaching from the left must be equal to the value of the function approaching from the right. Set the two pieces of the function equal to each other at the given value of .
First, find the derivative of the relevant piece of the function . Then, use the given condition to find the value of one of the constants. Finally, use your result from part (a) to find the other constant.
No question on this page matches those filters. Try another difficulty or paper.
Every Continuity & differentiability question, marked for you
Every answer is marked mark by mark, IB-style, and the AI tutor helps when you are stuck.
Where marks are lost
- Using your own wrong value after failing a "show that".
- Using an alternative method after "Hence".
