Shortcut Methods
Shortcut Methods and Tricks for Numericals on Continuity, Differentiability, and Applications of Derivatives
Continuity
 To determine the continuity of a function at a point, check whether the limit of the function at the point exists and is equal to the function’s value at the point.
 For rational functions, check for any undefined points (where the denominator becomes zero) and determine if the function can be redefined at these points to make it continuous.
Differentiability
 To determine the differentiability of a function at a point, check whether the limit of the difference quotient exists at that point.
 Use the definition of the derivative to calculate the derivative directly, if possible.
 For standard functions (e.g., polynomials, trigonometric functions), use known derivative formulas.
 For composite functions, use the chain rule to calculate the derivative.
Applications of Derivatives
 Rate of change: Calculate the derivative of the function and evaluate it at the given point to find the rate of change.
 Slope of a tangent line: Calculate the derivative of the function and evaluate it at the given point to find the slope of the tangent line.
 Graph analysis: Find the critical points (where the derivative is zero or undefined) and inflection points (where the concavity changes) by analyzing the derivative.
 Optimization: To find maximum and minimum values, find the critical points and determine whether they correspond to maxima or minima by analyzing the second derivative.
Typical Numericals
Continuity
 Find the points of discontinuity of the function 𝑓(𝑥) = (𝑥 + 2)/(𝑥^2  4𝑥  5).
Differentiability
 Determine whether the function 𝑓(𝑥) = 𝑥 is differentiable at x = 0.
Applications of Derivatives

Find the rate of change of the function 𝑓(𝑥) = 3𝑥^2 + 2𝑥 + 1 at x = 2.

Find the slope of the tangent line to the curve 𝑦 = 𝑥^3  2𝑥^2 + 3𝑥 at the point (1, 1).

Find the critical points and determine the intervals of increase/decrease for the function 𝑓(𝑥) = 𝑥^3  3𝑥^2 + 2𝑥  5.

Find the maximum and minimum values of the function 𝑓(𝑥) = 𝑥^2 + 4𝑥 + 3 on the interval [0, 4].