Limits and Derivatives
Derivative introduced as rate of change both as that of distance function and geometrically.
Intuitive idea of limit
Limits of −
Definition of derivative, relate it to slope of tangent of a curve, derivative of sum, difference, product and quotient of functions.
The derivative of polynomial and trigonometric functions.
Applications of Derivatives
Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normal, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool)
Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations)
SUMMARY
Limits and Derivatives
1. We say lim x→a– f(x) is the expected value of f at x = a given the values of f near x to the left of a. This value is called the left hand limit of f at a.
2. We say lim x→a+ f(x) + is the expected value of f at x = a given the values of f near x to the right of a. This value is called the right hand limit of f(x) at a.
3. If the right and left hand limits coincide, we call that common value as the limit of f(x) at x = a and denote it by lim x→a f(x).
4. The expected value of the function as dictated by the points to the left of a point defines the left hand limit of the function at that point. Similarly the right hand limit.
5. Limit of a function at a point is the common value of the left and right hand limits, if they coincide.
6. For a function f and a real number a, lim x→a f(x) and f (a) may not be same (In fact, one may be defined and not the other one).
7. For functions u and v the following holds :
(u ± v)' = u' ± v'
(uv)' = u'v + uv'.
8. Following are some of the standard derivative -
d/dx (sin x) = cos x
d/dx (cos x) = -sin x
Applications of Derivatives
1. First Derivative Test Let f be a function defined on an open interval I. Let f be continuous at a critical point c in I. Then
(i) If f ′(x) changes sign from positive to negative as x increases through c, i.e., if f ′(x) > 0 at every point sufficiently close to and to the left of c, and f ′(x) < 0 at every point sufficiently close to and to the right of c, then c is a point of local maxima.
(ii) If f ′(x) changes sign from negative to positive as x increases through c, i.e., if f ′(x) < 0 at every point sufficiently close to and to the left of c, and f ′(x) > 0 at every point sufficiently close to and to the right of c, then c is a point of local minima.
(iii) If f ′(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima. Infact, such a point is called point of inflexion.
2. Second Derivative Test Let f be a function defined on an interval I and c ∈ I. Let f be twice differentiable at c. Then
(i) x = c is a point of local maxima if f ′(c) = 0 and f ″(c) < 0 The values f (c) is local maximum value of f .
(ii) x = c is a point of local minima if f ′(c) = 0 and f ″(c) > 0 In this case, f (c) is local minimum value of f .
(iii) The test fails if f ′(c) = 0 and f ″(c) = 0. In this case, we go back to the first derivative test and find whether c is a point of maxima, minima or a point of inflexion.
3. Working rule for finding absolute maxima and/or absolute minima
Step 1: Find all critical points of f in the interval, i.e., find points x where either f ′(x) = 0 or f is not differentiable.
Step 2:Take the end points of the interval.
Step 3: At all these points (listed in Step 1 and 2), calculate the values of f .
Step 4: Identify the maximum and minimum values of f out of the values calculated in Step 3. This maximum value will be the absolute maximum value of f and the minimum value will be the absolute minimum value of f .
4. A point c in the domain of a function f at which either f ′(c) = 0 or f is not differentiable is called a critical point of f.
5. Let y = f(x), ∆x be a small increment in x and ∆y be the increment in y corresponding to the increment in x, i.e., ∆y = f(x + ∆x) – f(x).
