Lesson 6 Applications of Differential Calculus

Transcription

1 Lesson 6 Applications of Differential Calculus The line tangent to the graph of a function f at ( 0, f ( 0 is the line passing through the point whose slope is f ( 0. y = f ( The slope-point equation of the line is: y f ( 0 = f ( 0 ( 0 f ( o + h The tangent line is the geometry of the f ( 0 derivative. It is the line that kisses f at ( 0, f ( 0. Each point on f ( has a tangent line. 0 o + h Find the equation of the tangent line to y = f ( = at (,. y f ( = f ( ( y = ( y = y = E. : Find the equation of the tangent line to y = f ( = e at ( 0,. Clearly, f ( = f ( 0 + f ( 0 ( 0 + ε f ( f ( 0 = f ( 0 ( 0 + ε That is, more generally, we can estimate the change in y as being equal to the derivative times the change in. Rule: y = f ( 0 + ε, when is small E. : If s = f ( t and f ( t = v = 50 miles / hr, then t =. implies that s = 50. = 5 miles. So, if you are driving at 50 miles per hour, in the net / 0 of an hour you will drive 5 miles, as long as your speed does not change.

2 Taylor s Theorem: Taylor's theorem gives an approimation of a differentiable function near a given point by polynomials, whose coefficients depend only on the derivatives of the function at that point. Taylor s theorem lets one approimate a function f, around 0, by a quadratic function. The approimation improves as 0. Polynomial functions are mathematically pleasant to work with, hence the motive for approimation. f!! n ( f ( 0 + f ( 0 ( 0 + f ( 0 ( f ( 0 ( 0 In finance, we usually only go to the second term so that: f ( f ( 0 + f ( 0 ( 0 + f ( 0 ( 0 E. 3: Given that y = f ( = and that 0 =, estimate the new value of y when =. f ( = f ( = 4 f ( = f ( = 4 f ( = f ( = f ( f f ( = 9 f ( + f (( + (( n! n Note that f ( 3 = 9. E. 4: Given the following data: S Stock price C Call price dc/ds Call delta d C / ds Call gamma...07 Using Taylor s Theorem to two terms, if the price of stock is 50, what is the new price of the call if the stock goes up point? f ( = 7.65 E. 5: Given that s = f ( t and you are driving at a velocity of f ( t = v = 50 mi / hr, and your speed is increasing at a rate of acceleration f ( t = a = 0 mi / hr. Then t =. implies s = = 5.05 miles

3 E. 6: If S o = 45 and the stock decreases by ½ point, then what is the change in the value of the call option? Newton s Method Newton s method uses derivatives and lines tangent to the graph of the function to solve comple problems. The value of the derivative at any is the slope of the line tangent to the graph of the function at (, f (. Consider the following graph of a function y = g(. To solve g( = 0, where n is the solution, we start with an estimate 0 close to n. 0 is called the initial guess. Net consider the tangent line to g at 0. The equation of the tangent line is: y g( 0 = g ( 0 ( 0 The tangent line intersects the -ais at where is clearly closer to n than 0. It follows then that: g( 0 = 0 g ( 0 Newton s method is this recursive procedure, summarized thusly to find the root(s of g( :. Make an initial guess, 0. Proceed through the recursive procedure.

4 g( + = g ( 3. At some guess,, the solution will be close enough to 0 to accept as an approimation to the root. Different initial guess values may lead to different roots. A unique guess value is necessary for each root. E. : Betty and Bob Bo Company make open topped boes from a sheet of cardboard that is 6 feet by 8 feet by cutting an identical square from each corner. The volume V depends on the length of the cut as per V = f ( = ( 6 ( 8 0 < < 3 V = If the volume of the bo must be 0 cu.ft, then what should be? Clearly we must solve the equation: = 0 Using Newton s Method: g ( = g ( = g ( g ( g ( /g ( - g( / g ( We accept.6 as our approimate solution. Indeed a TI-8 calculator gives as the approimate solution. E. : Find the point of intersection in the first quadrant of y = e and y =. Use 0 = as the initial guess. g( = e + g ( = e +

5 g ( g ( g ( g ( g( g ( We accept as the solution. E. 3: Consider a ½ year bond with equation = + + 3; 0 < R + R ( + R ( + R Note the price is 00 and semi-annual coupons are 5. What is the semiannual yield to maturity R? Let = whence 00 = R 3 So g ( = = 0 must be solved. Note g ( = Using Newton s Method with 0 =.90 we get: ~ R g( g ( g ( g ( g( / g ( We accept ~ =.954. Now ~ = + R ~ so + R ~ = ~ whence = ~ = =. 05 The bond yields 0% per annum compounded semiannually Maima and Minima Here are some observations about the relationship of the graph of f and the value of f The tangent line provides the key insight! The tangent line has positive slope. i.e. f > 0. Therefore, the function f is increasing.

6 The tangent line has negative slope. i.e. f > 0. Therefore, the function f is decreasing. p The tangent line is horizontal at point p, f ( p = 0. The slope to left is + and the slope to the right is. Therefore, p is a local maimum. The tangent line is horizontal at point p, f ( p = 0. The slope to left is and the slope to the right is +. Therefore, p is a local minimum. p Definition: 0 is called a critical point of f if and only if f ( = 0. Maima and minima occur at critical points, end points, or points where cusps or corners occur ( f not defined at these points E. Revisited: Consider again Betty and Bob s Bo Company. Let s find the cut ma that results in the maimum volume V ma. Recall that: And, V = V = The critical points are the solutions to V = 0. By the quadratic formula the solutions are and.3. But 0 < < 3, so =..3 is the only critical point. Clearly, <.3 =.3 >.3

7 f ( 0 = 48 f (.3 = 0 f ( = -6 Just plug in various values of from the appropriate intervals to confirm the signs of f. Clearly =.3 is an absolute maimum point. Indeed v ma = f (.3 = or The graph of f is smiling, therefore the second derivative f > 0. or The graph of f is frowning, therefore the second derivative f < 0. Definition: A point on the graph of f where f changes from one concavity to another is called an inflection point. They occur at the solutions of f = 0. 3 E. 5: Let s analyze the graph of y = f ( = First we make a partial table of values y Net we make an f analysis. f ( = f ( = 3( f ( = 0 if and only if = or = 4 These are critical points. < < < < f Positive 0 Negative 0 Positive f Increase Ma Decrease Min increase So = is a local maima and = 4 is a local minima Net we make an f analysis. f ( = 6 8

8 f ( = 6( 3 f ( = 0 if and only if = 3 < < f Negative 0 Positive f Frowning Infle Smiling So = 3 is an inflection point, frowning to the right, smiling to the left.. Partial Derivatives Consider a right circular cylinder, where the volume, V, given by V = Π R L f R, L, is a function of independent variables. How does V change when R and L are changing? The answer is V V Δ V Δ R + Δ L R L where V = f R R and V = f L L are the partial derivatives. = ( Rule: If z = f (, y, then to find treat y as a constant and find the derivative of z with respect to. To find treat as a constant and find the derivative with respect to y. E. : If (, z f y e + = = y = e = e + y + y ( and ( y then 3 z = f, y = 5 y 0 y 5. Find z and z. E. : (

9 Definition If z = f(, y, then z = f, = is called the nd partial derivative, z = f yy, = is the nd partial derivative, and z = f y, = is the mied partial derivative. As a matter of fact, for most functions f not affect the result! E. 3: z = ln( y + y ln( = ln( y + y ( z y = y = = ( y + yln z ( = + ln y z y y = ln ( y + = + y = f that is, the order of differentiation does y, y, 3 E. 4: z = 5 y 0 y 5. Find the second partial derivatives and the mied partial derivative.

10 Proect 6. Find the equation of the tangent line to f at the stated. a. f ( = ln( ; = b. f ( = ; = c. f ( = ;= d. f ( = 3 ;=0. Approimate the function f about the given 0 by Taylor s Theorem with a third degree polynomial a. f ( = e, 0 = 0 b. f ( = ln( ; 0 = c. f ( = e ; 0 = 0 d. f ( = ; = 0 3. A stock option position with value,w=f(, where is the stock price has the associated information. X W 4 0 dw d (delta dwd (gamma Using Taylor s Theorem: a. If increases from 50 to 50½ approimate the new value of W. How much is the profit or loss? b. If decreases from 50 to 49 approimate the new value of W. How much is the profit or loss? 4. Use Newton s Method to solve the equation: + 3 =, 000, Consider a year bond with equation: = R ( + R ( + R ( + R 3 4 ; 0 < R

11 Using the substitution = as per lesson and Newton s Method solve for R as + R per table. 6. For the equation e =, 000 use Newton s Method with an initial guess solution of 0 = 5 to find the net approimate solution. 7. A rectangle is inscribed as per the diagram: y y=4-4 (,y Find the dimensions of the rectangle of maimum area. What is this area? Justify all of your work by calculus. 8. y y = (,y D (,0 Find the point on the curve y = that is closest to the point (,0. What is this minimum distance,d min? Justify all your work by calculus. (Do you recall the distance formula? 3 9. Analyze as per the lesson the graph of y = Analyze as per the lesson the graph of y = 6.