Continuity and Differentiability
Continuity and DifferentiabilityContinuous Function
ContinuityA function is continuous 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.
Continuity of the Function
Discontinuity of the FunctionGraph of the function cannot be
drawn without lifting the pen.
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CALCULUSSimple limit problems
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CALCULUSTrigonometric limits
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Differentiationy changes with 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).
DerivativesThe derivative is 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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CALCULUSDerivatives of exponential functions
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Problem
Differentiation & IntegrationThe 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.
Differentiation from first Principles Differentiating a linear
function
Consider the straight line y = 3x + 2
Differentiation from first principles
Differentiating a linear functionConsider the straight line y =
3x + 2
The gradient of the straight line is the same as the rate of
change of y with respect to x.
Differentiating f(x) = sin x
The derivative of f(x) = cos x.
Derivatives of certain Functions
The following table gives a list of derivatives of certain
standard functions.
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CALCULUSDerivatives of trigonometric functions
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Derivatives of Composite Function
Rules of Derivatives
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Chain Rule
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CALCULUSChain Rule problems
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Explicit and Implicit Function
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CALCULUSImplicit differentiation
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Derivatives of Inverse Trigonometric Ratios
Derivatives of Inverse Trigonometric Ratios
Derivatives of Inverse Trigonometric Ratios
Derivatives of other Trigonometric Functions are tabulated
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CALCULUSDerivatives of inverse functions(The Inverse Function
Theorem)
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Exponential FunctionCurves gets 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.
Differentiation of Exponential and Logarithmic Function
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CALCULUSDerivatives of logarithmic functions
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Derivatives of Parametric FunctionNeither Explicit nor
implicit.Relation between two variable expressed via third
variable.
Some standard Derivatives
Second Order DerivativesThe second order derivative is
Rolles TheoremR is continuous on a,bDifferentiable such
thatThere exists some c in betweenSuch that (c)= 0
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Mean Value TheoremR is continuous on a,bDifferentiable such that
There exists some c in betweenSuch that
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