CALCULUS. Berkant Ustaoğlu CRYPTOLOUNGE.NET

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1 CALCULUS Berkant Ustaoğlu CRYPTOLOUNGE.NET

2 Secant 1 Definition Let f be defined over an interval I containing u. If x u and x I then f (x) f (u) Q = x u is the difference quotient. Also if h 0, such that x = u + h I then Q = f (x) f (u) x u = f (u + h) f (u) u + h u is the difference quotient with increment h = f (u + h) f (u) h

3 Secants and tangent 2

4 Secant computations 3 P = (u, f (u)) = slope(x) = x f (x) slope ( 1, 1 ), f (x) = x2 2 2, f (x) f (u) x u = x x 1 x f (x) slope

5 Tangents 4 Definition (tangent line) Let f (x) be a function defined on an open interval I containing u. The tangent line to f (x) at the point P(u, f (u)) is the line that passes through P and has slope m tan = lim h 0 f (u + h) f (u) h = lim x u f (x) f (u) x u

6 Example 5 Find tangent to f (x) = x 2 at x = 3 using lim x 3 f (x) f (3) x 3 and = lim h 0 f (3 + h) f (3) h =

7 Derivative at a point 6 Definition (derivative at a point) Let f (x) be a function defined at a point u. The derivative of f at u denoted by f (u) is given by f (u) = lim h 0 f (u + h) f (u) h = lim x u f (x) f (u) x u

8 Example 7 for f (x) = 3x 2 4x + 1 find f (2) = lim x 2 f (x) f (2) x 2 for s(x) = 2 x = 2x find s (12) = lim h 0 s(12 + h) s(12) h

9 Examples 8 Find f 1 (0) and f 2 (0) if any of them exists, where f 1 (x) = x 1 3 f 2 (x) = x 2 3

10 Instantaneous rate of change 9 Definition The instantaneous rate of change of a function f (x) at a value u is its derivative f (u).

11 Example 10 Recall for for s(x) = 2 x = 2x find s (12) = lim h 0 s(12 + h) s(12) h for any value of x what is s (x)? = 8 7

12 Derivative 11 Definition Let f (x) be a function. The derivative of f (x), denoted by f (x) is the function whose domain consists of those values of x such that the following limit exists f (x) = lim h 0 f (x + h) f (x) h

13 Differentiable function 12 f (x) is differentiable at u if f (u) exists f (x) is differentiable on I if f (x) is differentiable for all u I f (x) is differentiable if f (x) is differentiable for all values in its domain

14 Examples 13 Find f (x) for f 1 (x) = 3 f 2 (x) = x 5 f 3 (x) = x 2 2x f 4 (x) = x

15 f (x) vs f (x) 14

16 Continuity vs differentiability 15 Theorem If f (x) is differentiable at u then f (x) is continuous at u.

17 Examples 16 f 1 (x) = x f 2 (x) = x 1 3 f 3 (x) = f 4 (x) = { ( x sin 1 ) x if x 0 0 if x = 0 { x 2 sin ( 1 x) if x 0 0 if x = 0

18 Example 17 Find b and c such that { 1 10 f (x) = x2 + bx + c if x < 10 1 x + 5 if x is continuous and differentiable

19 Higher order derivatives 18 f (x) = 2 x 3 5 x 2 11 x + 7 f (x) = 6 x 2 10 x 11 f (x) = 12 x 10. f (n) (x) = [ f (n 1)]

20 Constant rule 19 Theorem If f (x) = c, then f (x) = 0. d d x c = 0

21 Power rule 20 Theorem If f (x) = x n, then f (x) = nx n 1. d d x xn = nx n 1

22 Examples 21 d (x + h) 3 x 3 d x x3 = lim h 0 h d (x + h) x d x x 1 2 = lim h 0 h

23 Linearity 22 Theorem [αf (x) ± βg(x)] = αf (x) ± βg (x) d d [αf (x) ± βg(x)] = α d x d x f (x) ± β d d x g(x)

24 Examples 23 Find derivatives of d d x 3x3 = d ( 2x ) = d x Find tangent line to f (x) = x 2 4x + 6 at x = 1

25 Product rule 24 Theorem [f (x)g(x)] = f (x)g(x) + f (x)g (x) d d d [f (x)g(x)] = [f (x)] g(x) + f (x) d x d x d x [g(x)]

26 Product rule 25 v u v u v v uv uv u u

27 Examples 26 Let f (2) = 2, f (2) = 4, g(2) = 1 and g (2) = 6. Find j (2) where j(x) = f (x)g(x) Find j (x) if j(x) = ( x ) ( 3x 3 5x )

28 Quotient rule 27 Theorem Let f (x) and g(x) be differentiable functions then ( f g ) (x) = f (x)g(x) f (x)g (x) (g(x)) 2

29 Examples 28 Find k (x) if k(x) = 5x2 4x + 3

30 Combining rules 29 Find k (x) if k(x) = 3f (x) + x 2 g(x) Find k (x) if k(x) = f (x)g(x)j(x) Find k (x) if k(x) = 2x3 f (x) 3x + 2

31 Average vs Instantaneous 30 average f (a + h) f (a) h instantaneous f (a) = lim h 0 f (a + h) f (a) h f (a + h) f (a) + f (a)h

32 Example 31 Given f (3) = 2, f (3) = 5 estimate f (3.2)

33 Physics 32 Definition Let s(t) be the position of an object at time t The velocity of the object at time t is v(t) = s (t) The speed of the object at time t is v(t) The acceleration of the object at time t is a(t) = v (t) = s (t)

34 Example 33 s(t) = t 3 9t t + 4, t 0 v(t) =? When is particle at rest? When is particle at moving to the left and when to the right?

35 Population growth 34 Definition If P(t) is the number of entities present in a population at time t, then the population growth rate of P(t) is defined to be P (t).

36 Example: population growth 35 Example The population of a city is tripling every five years. If its current population is what will be the approximate population in 2 years?

37 Economics 36 Definition If C(x) is the cost of producing x items, then the marginal cost is MC(x) = C (x). If R(x) is the revenue obtain from selling x items, then the marginal revenue is MR(x) = R (x). The profit P(x) from selling x items is R(x) C(x). The marginal profit MP(x) = P (x) = R (x) C (x). Note: marginal cost is C (x) C(x + 1) C(x) 1

38 Example: economics 37 Example Assume that the number of barbecue dinners that can be sold x, is related to the price p by p(x) = x 0 x 300 Compare the marginal revenue as a function with its estimate for 100 units.

39 Derivatives of sin x and cos x 38 Theorem sin (x) = cos(x) cos (x) = sin(x)

40 Examples 39 sin x = cos x sin (x) = sin (x) = sin (4) (x) =

41 Examples 40 ( 5x 3 sin x ) = ( cos x 4x 2 ) = tan (x) =

42 Chain rule 41 Theorem Let f (x) and g(x) be functions. For all x in the domain of g(x), for which g(x) is differentiable and f (x) is differentiable at g(x), the derivative of the composite function: i(x) = (f g)(x) = f (g(x)) is given by i (x) = [(f g)(x)] = f (g(x))g (x) Alternatively, if y = y(u) and u = u(x) then d y d x = d y d u d u d x

43 Examples: chain rule 42 [ 1 (3x 2 + 1) 2 ] = [ sin 3 (x) ] [ cos 3 ( 5x 2)] [(2x + 1) 5 (3x 2) 7] = = =

44 Examples: chain rule 43 [k (f (g (x)))] = ( [cos x 2 + x)] = [ ( sin cos 2 + x + ] sin x) =

45 Inverse derivative 44 f (x) = x g(x) = f 1 (x) = 2(x 3) f (x) (2, 5) q p p q (5, 2) f 1 (x)

46 Inverse derivative 45 (2, 5) = (2, f (2)) = (g(5), 5) = (f 1 (5), 5) (5, 2) = (5, g(5)) = (4, f 1 (5)) = (f (2), 5) x = f (g(x)) 1 = f (g(x))g (x) g (x) = 1 f (g(x)) [ f 1 ] (x) = 1 f (f 1 (x)) g (5) = 1 f (g(5)) = 1 f (f 1 (5)) = 1 f (2) = 1 2

47 Rational powers 46 f (x) = x 3 g(x) = 3 x f (x) = x 5 g(x) = 5 x f (x) = x n g(x) = n x f (x) = x p/q = ( x 1/q) p g(x) = p x q g (x) = 1 f (g(x))

48 [arcsin θ] 47 f (x) = sin(x) g (x) = g(x) = arcsin(x) 1 f (g(x)) = 1 cos (g(x)) = 1 1 x 2 1 u θ 1 u 2

49 Examples 48 ( ) arcsin x 1 = arcsin ( x 2 1 ) =

50 Circle 49 y = ± 25 x 2 y 2 + x 2 = 25 y =?

51 Examples 50 Find y if x 3 sin y + y = 4x + 3 Find tangent line at (3, 4) to y 2 + x 2 = 25 Find tangent line at (1, 1) to yx 2 + xy 2 = 2 Find y if y 2 + x 2 = 25

52 Exponential functions 51 Fact: B(x) = b x is continuous Fact: B (0) = lim x 0 b 0+h b 0 h exists Fact: there is unique value e such that E (0) = (e x ) x=0 = 1

53 Exponential derivative 52 Theorem If B(x) = b x then If E(x) = e x then [b x ] = B (0)b x [ e g(x) ] = e g(x) g (x)

54 Examples 53 e sin(x2 +5x) = e x2 x =

55 Logarithmic derivative 54 Theorem If x > 0 and y = ln x then y = [ln x] = 1 x If g(x) > 0 then for h(x) = ln (g(x)) h (x) = 1 g(x) g (x)

56 Examples 55 [ ( )] x 2 sin x ln = 2x + 1 [ ln ( x 2 sin(x 3 + 4x) )] =

57 General derivatives 56 Theorem Let b > 0, b 1 if y = log b x then y = 1 x ln b if y = b x then y = b x ln b

58 Examples 57 [ 3 x ] = 3 x + 1 [log 2 (3x + 1)] = For both find y (1)

59 Logarithmic differentiation 58 Find y if y = x r y = x x y = ( 2x ) tan x y = x x 2 x + x 3