Differential calculus. Background mathematics review

Transcription

1 Differential calculus Background mathematics review David Miller

2 Differential calculus First derivative Background mathematics review David Miller

3 First derivative For some function y The (first) derivative is the slope gradient or rate of change of y as we change If for some small infinitesimal change in, called d y changes by some small infinitesimal amount dy y the first derivative is written dy y d The ratio notation on the right is Leibniz notation

4 First derivative The derivative at some specific point can be written dy y d The value of the derivative is the slope of the tangent line the dashed line in the figure at that point Equal in value to the tangent y y dy d y 3 dy d 3

5 First derivative y Looking at the slope y y y y of the orange line as we reduce the orange line slope becomes closer to the slope of the black tangent line

6 First derivative In the limit as becomes very small i.e., in the limit as tends to zero lim 0 this ratio becomes the (first) derivative dy d lim 0 y y

7 Sign of first derivative If y increases as we increase dy d sloping up to the right 0 y dy d 0 dy d 0 If y decreases as we increase dy d sloping down to the right 0 3

8 Derivative of a power y d d n n n dy d

9 Derivative of a power The derivative of a straight line is a constant The straight line has a constant slope y dy d

10 Derivative of a power The derivative does not depend on the height All these lines have the same slope y dy d

11 Derivative of a power The derivative does not depend on the height All these lines have the same slope y 0.5 dy d

12 Derivative of a power The derivative does not depend on the height All these lines have the same slope y 0.5 dy d

13 Derivative of a power The derivative of a constant y is not changing with y 0.5 is zero dy d

14 Derivative of an eponential 3 d d ep ep 3 y ep dy ep d

15 Derivative of a logarithm d d ln y ln dy d 3 0 3

16 Derivatives of sine and cosine d sin d cos y sin dy cos d

17 Derivatives of sine and cosine d d cos sin y cos dy d sin

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19 Differential calculus Second derivative Background mathematics review David Miller

20 Second derivative The second derivative is The derivative of the derivative d y d dy y d d d The rate of change of the derivative or slope

21 Second derivative The slope at / is, for small dy y0 y d / And similarly at / dy y y0 d So / d y d dy lim d d d 0 dy d dy d / /

22 Second derivative The slope at / is, for small dy y0 y d And similarly at / dy y y d So / / d y d dy 0 lim d d d y y y y

23 Second derivative The slope at / is, for small dy y0 y d And similarly at / dy y y d So / / d y d dy 0 lim d d d 0 0 y y y

24 Sign of second derivative Going from a positive first derivative To a negative first derivative Gives a negative second derivative Going from a negative first derivative To a positive first derivative Gives a positive second derivative y d y 0 d dy dy 0 d 0 d d y d 3 4 0

25 Sign of second derivative Any region where the first derivative is decreasing with increasing Has a negative second derivative Any region where the first derivative is increasing with increasing Has a positive second derivative y d y d 0 d y d 0

26 Sign of second derivative Points where the derivative is neither increasing or decreasing i.e., second derivative is changing sign correspond to zero second derivative Known as inflection points y d d y 0 d y d 0

27 Curvature The second derivative can be thought of as the curvature of a function Large positive curvature

28 Curvature The second derivative can be thought of as the curvature of a function Small positive curvature

29 Curvature The second derivative can be thought of as the curvature of a function Large negative curvature

30 Curvature The second derivative can be thought of as the curvature of a function Small negative curvature

31 Curvature The value of the curvature does not depend on the height of the function All these curves have the same curvature

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33 Differential calculus Linearity and differentiation rules Background mathematics review David Miller

34 Linearity linear superposition For two functions u and v The derivative of the sum is the sum of the derivatives d u v du dv d d d Eample f ln Split into u v ln So du d dv d So du v f d

35 Linearity multiplying by a constant For a function u The derivative of a constant a times a function is a times the derivative d du au a d d Eample f a a Split into So So / du d a / u / du a fa d

36 Linearity An operation or function f is linear if f y z f y f z and linear superposition or additivity condition f a a f multiplication by a constant (or formally homogeneity of degree one ) condition

37 Eample of nonlinear operation The function does not represent a linear operation But f f yz yz y z yz f y f z y z So for this function f y is not in general equal to f f y

38 Product rule For two functions u and v The derivative of the product is d uvu dv v du d d d Eample f sin Split into u v sin So du d dv cos d So duv f cos sin d

39 Quotient rule For two functions u and v The derivative of the ratio or quotient is du dv v u d u d d d v v Eample f Split into 3 3 u v So So du 3 d dv d d u 3 3 d v

40 Quotient rule For two functions u and v The derivative of the ratio or quotient is du dv v u d u d d d v v Eample f Split into 3 3 u v So So du 3 d dv d d u 3 3 d v

41 Quotient rule For two functions u and v The derivative of the ratio or quotient is du dv v u d u d d d v v Eample f Split into 3 3 u v So So du 3 d dv d d u 4 3 d v

42 Quotient rule For two functions u and v The derivative of the ratio or quotient is du dv v u d u d d d v v Eample f Split into 3 3 u v So So du 3 d dv d 4 d u 3 d v

43 Chain rule For two functions f y and The derivative of the function of a function Can be split into a product d f g df dg d dg d Eample h Split into g g f y y

44 Chain rule For two functions f y and The derivative of the function of a function Can be split into a product d f g df dg d dg d Eample h Split into g f g g So dg df g g d dg So dh 4 d g

45 Chain rule For two functions f y and The derivative of the function of a function Can be split into a product d f g df dg d dg d Eample h ep a Split into g a f g ep g So dg df g epg d a dg So dh epa a a epa d g

46 Chain rule For two functions f y and The derivative of the function of a function Can be split into a product d f g df dg d dg d Eample h ep a Split into g a f g ep g So dg df g a epg d dg So dh ep a a aep a d g

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