Unit 10: Limits and Derivatives
Contents
1. Understanding Limits
A Limit describes the value that a function approaches as the input approaches some value. It is the foundation of Calculus.
Definition: We say limx \to a f(x) = L if the value of f(x) can be made as close to L as we like by taking x sufficiently close to a.
- Left Hand Limit (LHL): limx \to a- f(x)
- Right Hand Limit (RHL): limx \to a+ f(x)
- Existence: A limit exists only if LHL = RHL.
2. Standard Limit Formulas
These formulas are essential for solving complex limit problems quickly:
2. limx \to 0 (sin x) / (x) = 1
3. limx \to 0 (ex - 1) / (x) = 1
4. limx \to 0 (log(1+x)) / (x) = 1
3. Concept of Derivative
The Derivative of a function measures the sensitivity to change of the function value with respect to a change in its input value. Geometrically, it represents the slope of the tangent to the curve at a point.
First Principle of Derivatives:
4. Rules of Differentiation
To find derivatives without using the first principle every time, we use these standard rules:
- Sum/Difference Rule: (d) / (dx)(u ± v) = (du) / (dx) ± (dv) / (dx)
- Product Rule: (d) / (dx)(uv) = u (dv) / (dx) + v (du) / (dx)
- Quotient Rule: (d) / (dx)((u) / (v)) = (v (du) / (dx) - u (dv) / (dx)) / (v2)
5. Derivatives of Standard Functions
| Function f(x) | Derivative f'(x) |
|---|---|
| xn | nxn-1 |
| sin x | cos x |
| cos x | -sin x |
| tan x | \sec2 x |
| ex | ex |
| log x | 1/x |
6. Exam Focus Enhancements
- The 0/0 Form: If direct substitution in a limit gives 0/0, try to factorize, rationalize, or use standard formulas to eliminate the indeterminacy.
- Quotient Rule Order: In the Quotient Rule, always start with the denominator (v) in the numerator: v · u'. Swapping this will result in a sign error.
- Derivative of a Constant: Never forget that the derivative of a constant (like 5 or π) is always 0.
- Limit vs Function Value: Thinking f(a) must be equal to limx \to a f(x). A function can have a limit at a point where it is not defined.
- sin x vs sin x°: Derivative rules for trig functions assume x is in radians. If in degrees, convert first.
- Missing dx: Forgetting to write dx in the notation (d) / (dx). It indicates which variable you are differentiating with respect to.
Q: What is the physical meaning of a derivative?
A: It represents the Instantaneous Rate of Change. For example, the derivative of displacement with respect to time is velocity.
Q: When does a function NOT have a derivative?
A: A function is not differentiable at points where the graph has a sharp corner (like |x| at x=0), a vertical tangent, or a discontinuity.