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Continuity and differentiability
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- 2. Continuity A function iscontinuous at a fixed point if we can draw the graph of the function around that point without lifting the pen from the plane of the paper.
- 3. Continuity of theFunction
- 4. Discontinuity of theFunction Graph of the function cannot be drawn without lifting the pen.
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- 16. Differentiation y changeswith respect to the change in the independent input x. This rate of change is called the derivative of y with respDifferentiation is a method to compute the rate at which a dependent output ect to x. The process of finding a derivative is called differentiation We also use the phrase differentiate f (x) with respect to x to mean find f ′(x).
- 17. Derivatives The derivativeis a measure of how a function changes as its input changes a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity
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- 23. Differentiation & Integration The process of finding a derivative is called differentiation Reverse process of differentiation is called antidifferentiation. Antidifferentiation is the same as integration Differentiation and integration constitute the two fundamental operations in single-variable calculus.
- 24. Differentiation from first Principles Differentiatinga linear function Consider the straight line y = 3x + 2
- 25. Differentiation from first principles Differentiating a linear function Consider the straight line y = 3x + 2
- 26. The gradient ofthe straight line is the same as the rate of change of y with respect to x.
- 27. Differentiation of y= 𝒙 𝟐 The graph of y = 𝑥2 P is the point (x, y). Q is a nearby point. is a small increment in x. change in y is written as Coordinates of Q are
- 28. Differentiation of y= 𝒙 𝟐
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- 30. The derivative off(x) = cos x.
- 31. Derivatives of certainFunctions The following table gives a list of derivatives of certain standard functions.
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- 35. Derivatives of Composite Function Eg. f (x) = 2𝑥 + 1 3 Solution
- 36. Rules of Derivatives (u ± v)′ = u′ ± v′ (uv)′ = u′v+ uv′ (Leibnitz or product rule) 𝑢 𝑣 ’= 𝑢′ 𝑣 −𝑣′ 𝑢 𝑣2 , wherever v ≠ 0 (Quotientrule).
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- 38. Chain Rule Letf be a real valued function Which is a composite of two functions u and v Suppose t = u(x) Suppose t = u(x) and if both dt dx and dv dt exist then
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- 42. Explicit and ImplicitFunction y = 𝑓(𝑥) is not given directly Explicit Function x - y - 𝜋 = 0 It is easy to solve and rewritten as y = x - 𝜋 Implicit Function X + sin xy – y = 0 It gives function implicitily.
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- 46. Derivatives of InverseTrigonometric Ratios f (x) = sin−1 𝑥 Let y = sin−1 𝑥 then x = sin y Differentiating both sides w.r.t. x we get 1 = cos y 𝑑𝑦 𝑑𝑥 ⇒ 𝑑𝑦 𝑑𝑥 = 1 𝐶𝑜𝑠 𝑦 = 1 1−𝑥2
- 47. Derivatives of InverseTrigonometric Ratios f (x) = tan−1 𝑥 Then x = tan y Differentiating both sides w.r.t. x we get 1 = sec2 y dy dx dy dx = 1 sec2 y = 1 1+tan2 y = 1 (1+(tan( tan−1 𝑥)2 Type equation here. = 1 1+𝑥2
- 48. Derivatives of InverseTrigonometric Ratios Derivatives of other Trigonometric Functions are tabulated
- 49. 49 CALCULUS Derivatives of inversefunctions (The Inverse Function Theorem)
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- 51. Exponential Function Curvesgets steeper as the power of x increases. Higher the degree greater is the growth. Steeper the curve faster is the rate of growth. Such a faster growth is Exponential Function.
- 52. Differentiation of Exponentialand Logarithmic Function If 𝑓(x) = 𝑒 𝑥 then𝑓’(x) = 𝑒 𝑥 If 𝑓(x) = log x then 𝑓’(x) = 1 𝑥
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- 56. Derivatives of Parametric Function Neither Explicit nor implicit. Relation between two variable expressed via third variable.
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- 58. Second Order Derivatives The second order derivative is
- 59. Rolle’s Theorem Ris continuous on a,b Differentiable such that There exists some c in between Such that 𝑓’(c)= 0
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- 65. Mean Value Theorem R is continuous on a,b Differentiable such that There exists some c in between Such that
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