CALCULUS I

Course Description
Limits, continuity and differentiability with reference to continuous probability. Differentiation by first principles and by rule for 𝑥 𝑛
(integral and fractional n), sums, products, quotients, chain rule, trigonometric, logarithmic and exponential functions of a single
variable. L’Hospital’s rule. Parametric differentiation. Applications: equations of tangent and normal, rates of change and stationary
points. Integration: anti-derivatives and their applications to marginal and total functions eg marginal cost and total cost. Mean value
theorem of differential calculus. Rolle’s theorem.
Course Outline
WK TOPIC/SUB-TOPIC COMMENTS
1. NUMBERS AND FUNCTIONS
i. What is a number
ii. Functions
iii. Inverse functions and implicit functions
2. DERIVATIVES (1)
i. The tangent to a curve
ii. Tangent to a parabola
iii. Instantaneous velocity
iv. Rates of change
v. Examples of rates of change
3. LIMITS AND CONTINUOUS FUNCTIONS
i. Informal definition of limit
ii. Formal definition of limit
iii. Variations on the limit theme
iv. Properties of the limit
v. Examples of limit computations
vi. When limits fail to exist
vii. Limits and inequalities
viii. Continuity
ix. Substitution in limits
x. Two limits in trigonometry
4. DERIVATIVES (2)
i. Derivatives defined
ii. Direct computation of derivatives
iii. Differentiable implies continuous
iv. Some non-differentiable functions

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 v. The differentiation rules
- The sum, product and quotient rules
vi. Differentiating powers of functions
- The product rule with more than one factor
- The power rule
- The power rule for negative integer exponents
- The power rule for rational exponents
- Derivative of 𝑥 𝑛 for integer n
vii. Higher derivatives
viii. Differentiating trigonometric functions
ix. The chain rule
x. The chain rule and composing more than two functions
xi. Implicit differentiation
5. CAT I
6. GRAPH SKETCHING AND MAX-MIN PROBLEMS
i. Tangent and normal lines to a graph
ii. The intermediate value theorem
iii. Finding sign changes of a function
iv. Increasing and decreasing functions
v. Mean Value Theorem
vi. Rolle’s theorem
vii. Examples
- The parabola, 𝑦 = 𝑥 2
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- The hyperbola, 𝑦 = 𝑥
- Graph of a cubic function
- A function whose tangent turns up and down infinitely often
near the origin
7. MAXIMA AND MINIMA
viii. Must there always be a maximum?
ix. Examples: functions with and without maxima or minima
x. General method for sketching the graph of a function
xi. Convexity, Concavity and the Second Derivative
xii. Optimization Problems
8. EXPONENTIALS AND LOGARITHMS
i. Exponents
ii. Logarithms
iii. Properties of logarithms
iv. Graphs of exponential functions and logarithms

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 v. The derivative of 𝑎 𝑥 and the definition of e
vi. Derivatives of Logarithms
vii. Limits involving exponentials and logarithms
viii. Exponential growth and decay
ix. L’Hospital’s Rule
9. CAT 2
10. THE INTEGRAL
i. Area under a Graph
ii. When f changes its sign
iii. The Fundamental Theorem of Calculus
iv. The indefinite integral
v. Properties of the Integral
vi. The definite integral as a function of its integration bounds
vii. Methods of integration
11. APPLICATIONS OF THE INTEGRAL
i. Areas between graphs
ii. Cavalieri's principle and volumes of solids
iii. Examples of volumes of solids of revolution
iv. Volumes by cylindrical shells
v. Distance from velocity, velocity from acceleration
vi. The length of a curve
vii. Examples of length computations
viii. Work done by a force
ix. Work done by an electric current
Exam
Course Assessment
Continuous Assessment Tests (30%)
End of Semester Examination (70%)