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
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Find the rate of change of the function 𝑓(𝑥) = 3𝑥^2 + 2𝑥 + 1 at x = 2.
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Find the slope of the tangent line to the curve 𝑦 = 𝑥^3 - 2𝑥^2 + 3𝑥 at the point (1, 1).
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Find the critical points and determine the intervals of increase/decrease for the function 𝑓(𝑥) = 𝑥^3 - 3𝑥^2 + 2𝑥 - 5.
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Find the maximum and minimum values of the function 𝑓(𝑥) = -𝑥^2 + 4𝑥 + 3 on the interval [0, 4].
