U3L2: Sec.3.1 Higher Order Derivatives, Velocity and Acceleration

Transcription

1 U3L1: Review of Prerequisite Skills for Unit # 3 (Derivatives & their Applications) Graphing polynomial and simple rational functions Working with circles in standard position Solving polynomial equations Finding the equations of tangents and normals Using the common formulas: circle: circumference and area Cylinder: SA & Vol. Homework: p. 116 # 1 8, 9 omit d (do without technology) U3L: Sec.3.1 Higher Order Derivatives, Velocity and Acceleration Consider a down hill skier. What is the excitement of the sport? The position, s of a skier is described as a function of time, t. Therefore, s = f(t). Recall that the change in position, or velocity, v, at any time, t is the. ie: v(t) = Note: speed is the absolute value of v(t). There is no. If v(t) > 0 we say that the object is moving to the of the origin. If v(t) < 0 we say that the object is moving to the of the origin. If v(t) = 0 than the object. But, the velocity changes as well. The rate of change in the velocity, is the, a, which is the derivative of the. d s a(t) =.Also represented by. This is called the. dt Eg.1: Find the second derivative of each of the following expressions. a) y 3 = x x b) 5 y = x c) 8 y = x d) y = x Page 1 of 14

2 When the exponent is a positive integer: i) Find ( 9) y of Ie: y = x 4 y ' = 4x y '' = 4 y y 3 ( 3)( x ) ( 3 ) = 4 ( 3 )( )( x ) ( 4 ) = ( 4 )( 3 )( )( 1 ) = 4! 9 y = x ii) Find the 11 th derivative of 11 y = 13x e) y xy = 3 **implicit differentiation To complete the second derivative, we now need to use the quotient rule. Page of 14

3 Eg.: A particle moves along a straight line with the equation of motion ( t) = t 3 15t + 63t 49, t 0 Solution: s. a) Find the velocity and acceleration functions at time t. b) Find the intervals of time in which the object is moving in a positive direction and a negative direction. Ie: find the turning point, also known as the critical point. V(t) = 0 c) Determine when the particle is speeding up (accelerating) or slowing down (decelerating). a) b) v(t) = 0 is at rest when t = and t =. a(t) = 0 is a constant velocity when t =. v(t) a(t) v(t)a(t) t = sec. t = sec. t = sec. t = sec. V(t) > 0 when t < 3 and t > 7 (moving to the right, away from the origin) V(t) < 0 when 3 < t < 7 (moving to the left, towards the origin) c) a(t) > 0 when t > 5 a(t) < 0 when t < 5 How do we know when the acceleration is speeding up or slowing down. The object is accelerating if [a(t)][v(t)] = positive. ie: both negative or both positive The object is decelerating if [a(t)][v(t)] = negative. ie: opposite signs Therefore the object is accelerating when 3 < t< 5 and when t > 7 Therefore the object is decelerating when t < 3 and 5 < t < 7. Homework: (p. 17 #, 3def, 4, 5, 8-1, 14) Page 3 of 14

4 U3L3: Sec.3. Minimum and Maximum on an Interval (Extreme Values) Grade 11: Determine the vertex of y = x 6x This function is in standard form. We need to complete the square to get it to vertex form. Show using algebra tiles and with algorithm. Method 1: Method : Therefore the vertex is (, ). The minimum value of the function would be when x =. The min./max. value is the y-portion of the vertex and where this min./max. occurs is the x-portion of the vertex. Calculus: Determine the max./min. value of f ( x) = x 6x + 19 and state where it occurs. Since the min./max. value occurs at the vertex of the function which is a turning point, the slope of the tangent (ie: the derivative) will be zero. f (x) = 0 at a maximum or minimum. Therefore, this is where it occurs, thus So the max./min. occurs at x = 3 and is 10 but how do we find out whether it s a maximum or minimum without relying on previous parabola knowledge. Ie: leading coefficient is + or -. At a minimum, f (x) < 0 to the left of f (x) = 0 and f (x) > 0 to the right and vice versa for the maximum. Draw this scenario. Create a First Derivative Chart Interval f '( x) = x 6 x < 3 x = 3 x > 3 Therefore as you approach x = 3 from below it is (downward slope) and from above it is (upward slope) so f(x) has a of 10 at x = 3. Page 4 of 14

5 Local vs. Absolute Maximum/Minimum 3 Eg.1: Determine the maximum or minimum values of f ( x) x x = when x ε[,1]. At a max./min. f (x) = 0 Therefore x = and x = f( ) = f( ) = f( ) = f( ) = Interval f '( x) = 3x x x < 0 x = 0 0 < x < 3 x = 3 x > 3 Therefore f(x) has a local maximum at x = 0 and at x = 1 (the endpoint) and a local minimum at x = /3 and an absolute minimum at x = - (the endpoint). Page 5 of 14

6 Homework: p. 135 # 1,, 4, 5a, 7, 8, 11, 1 Homework: p. 139 # 1-5 U3L4: Mid-Chapter Review U3L5a: Sec. 3.3 Optimization Problems Day # 1 Optimization is finding the maximum or minimum quantity of a function under certain circumstances or restraints. Steps for solving optimization problems: 1. Understand the problem. a. Identify the quantity that can vary. b. Express the quantity to be optimized by a function with only one variable.. Draw and label a diagram, whenever possible. 3. Determine the domain of the function (ie: restrictions on the variable that we are considering) 4. Find all the extreme values. Eg.1: A carpenter is building an open box with a square base for holding firewood. The box must have a surface area of 8 m. What dimensions will yield a maximum volume? Page 6 of 14

7 Eg.: A piece of cardboard which measures 4 cm by 4 cm is to be made into a box by cutting squares out of the corners and folding up the flaps. What dimensions result in a box with a maximum volume? What is this volume? Homework: p. 146 # 1, 3 9, 10** Page 7 of 14

8 U3L5b: Sec. 3.3 Optimization Problems Day # Eg.1: Kara and James are both training for a marathon. Kara s house is located 35 km north of James house. At 6:00 a.m. on a Monday morning, Kara leaves her house and jogs South at 11 km/h. At the same time, James leaves his house and jogs East at 8 km/h. When are Kara and James closest together, given that they both run for.5 hours? Page 8 of 14

9 Eg.: Find the area of the largest rectangle that can be inscribed in a right triangle with legs adjacent to the right angle of lengths 5 cm and 1 cm. The two sides of the rectangle lie along the legs. Homework: p. 146 # 11a, 1bc, 13**, 14a, 15, 19 Page 9 of 14

10 U3L5c: Sec. 3.3 Optimization Problems Day # 3 Eg.3: Cindy makes a candle holder by inscribing a cylinder in a cone. The height of the cone is 15 cm. The radius is 5 cm. Find the dimensions of the cylinder that will maximize its surface area? 15 C h B 15-h A Radius of cone r E D Radius of cylinder in the cone ACD and ABE are similar triangles because angle A is common to both triangles and angle C and angle B are both 90 therefore angle D and angle E must be equal due to triangle sum. Since the two triangles are similar their sides are proportional. AC CD Ie: AB = AD BE = AE Big vs. Small This allows us to now express the surface area as a function of the radius. Therefore the Surface Area of the cylinder is maximized when the radius is 15/4 cm and the height is 15/4 cm. (3.75 cm) Homework: Sec. 3.3 Extra Practice Handout Page 10 of 14

11 U3L6: Sec.3.4 Optimization Problems in Economics and Science Day # 1 Profit, cost and revenue are quantities whose rates of change are measured in terms of the number of units produced or sold. Economic situations usually involve minimizing costs or maximizing profits. Recall: Cost Function C(x) = (number of items)*(cost per item) Revenue Function R(x) = (number of items)*(price per item) Profit Function P(x) = R(x) C(x) Where x is the number of items. Typically, when price increases, the number of items sold decreases. The variable typically used is the number of price increases. The break-even point is the change in profit from a negative to positive or vice-versa. So at the break-even point, P(x) = 0 or R(x) = C(x). Marginal profit, marginal revenue and marginal cost are the rates of change of profit, revenue, and cost, from selling or producing one more unit. Eg.1: A rectangular piece of land is to be fenced using two kinds of fencing. Two opposite sides will be fenced using standard fencing with a cost of $6/m, while the other require heavy duty fencing that cost $9/m. What are the dimensions of the rectangular lot of greatest area that can be fenced for a cost of $9000. Page 11 of 14

12 Eg.: A railroad between two cities carries passengers per year when the fare is $50. If the fare goes up, the number of passengers will decrease, since more people will drive. It is estimated that each $10 increase in fare will result in 1000 fewer passengers per year. The train can not carry more than passengers per year. a) At what fare will the train have no passengers? b) What fare will maximize revenue? c) When revenue is maximized, how many people will ride the train each year? Homework: p. 151 # 1,, 3ab, 4, 6-8 Page 1 of 14

13 U3L7: Sec.3.4 Optimization Problems in Economics and Science Day # Eg.1: A cylindrical chemical storage tank with a capacity of 1000 m 3 is going to be constructed in a warehouse that is 1 m by 15 m by 11 m in height. The specifications call for the base to be made of sheet steel that costs $100/m, the top to be made of sheet steel that costs $50/m and the wall to be made of sheet steel that costs $80/m. a) Determine whether it is possible for a tank of this capacity to fit in the warehouse. If it is possible, state the restrictions on the radius of the tank. b) If fitting the tank in the warehouse is possible, determine the proportions that meet the conditions and that minimize the cost of the steel for construction. Page 13 of 14

14 Eg.: An offshore oil well, P, is located in the ocean 5 km from the nearest point on the shore, A. A pipeline is to be built to take oil from P to a refinery that is 0 km along the straight shoreline from A. If it costs $100000/km to lay pipe underwater and only $75000/km to lay pipe on land, what route from the oil well to the refinery will be the cheapest? (Give your answer correct to one decimal place) Solution: P 5 km R 0 - x C x A 0 km Homework: p. 153 # 9, 10, 11a, 1, 13, 14, 17, 19** U3L8: Unit 3 Review Day # 1 U3L9: Unit 3 Review Day # U3L10: Unit # 3 Test Page 14 of 14