Bozeman Public Schools Mathematics Curriculum Calculus Process Standards: Throughout all content standards described below, students use appropriate technology and engage in the mathematical processes of problem solving, reasoning, communication, connections and representations. When taught in high school, calculus should be presented with the same level of depth and rigor as are entry-level college and university calculus courses. Calculus is a widely applied area of mathematics and involves a beautiful intrinsic theory. 1.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions. This knowledge includes one-sided limits, infinite limits, and limits at infinity. Students know the definition of convergence and divergence of a function as the domain variable approaches either a number or infinity: 1.1 Students prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions. lim x "3 x 2 + 3x x 2 + 6x + 9 is: a) -3 (b) -1 (c) 1 (d) 3 (e) nonexistent lim b # x x "b x # b is 1.2 Students use graphical calculators to verify and estimate limits. x -0.3-0.2-0.1 0.1.2.3 f(x) 2.018 2.008 2.002 2 2.002 2.008 2.018 g(x) 1 1 1 2 2 2 2 h(x) 1.971 1.987 1.997 Undefined 1.997 1.987 1.971 The table above gives the values of three functions f, g, and h near x = 0. Based on the values given, for which of the functions does it appear that the limit as x approaches zero is 2? 1
1.3 Students prove and use special limits, such as the limits of (sin(x))/x and (1 cos (x))/x as x tends to 0. { sin(x) x " 0 Consider the function f(x) = x k x = 0 In order for f(x) to be continuous at x = 0, the value of k must be: (a) 0 (b) 1 (c) -1 (d) π (e) a number greater than 1 2.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function. If f(x) = x 2 " 9 x + 3 is continuous at x = -3, then f)-3) must be? 3.0 Students demonstrate an understanding and the application of the Intermediate Value Theorem and the Extreme Value Theorem. If f(x) = x 3 x + 3 and if c is the only real number such that f = 0, tghen c is between: (a) -2 and -1 (b) -1 and 0 (c) 0 and 1 (d) 1 and 2 (e) 2 and 3 4.0 Students demonstrate an understanding of the formal definition of the derivative of a function at a point and the notion of differentiability: 4.1 Students demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function. At the point of intersection of f(x) = cos(x) and g(x) = 1 x 2, the tangent lines are: (a) the same line (b) parallel lines (c) perpendicular lines (d) intersecting but not perpendicular lines (e) none of these 4.2 Students demonstrate an understanding of the interpretation of the derivative as an instantaneous rate of change. Students can use derivatives to solve a variety of problems from physics, chemistry, economics, and so forth that involve the rate of change of a function. 2
A cylindrical tank is initially filled with water to a depth of 16 feet. A valve in the bottom is opened and the water runs out. The depth, h, of the water in the tank decreases at a rate proportional to the square from of the depth: That is: dh = "k h where k is a constant and 0 < k < 1. dt a) What is the solution to the differential equation in terms of k. b) After the valve is opened, the water falls to a depth of 12.25 feet in 8 hours. Find the value of k. c) How many hours after the valve was first opened will the tank be completely empty. 4.3 Students understand the relation between differentiability and continuity. Let f(x) = { 1+ e"x 0 # x # 5 1+ e x"10 5 < x #10 Which of the following statements is true? I. f(x) is continuous for all values of x in the interval [0,10]. II. f (x), the derivative of f(x), is continuous for all values of x in the interval [0,10]. III. The graph of f(x) is concave upwards for all values of x in the interval [0,10]. (a) I only (b) II only (c) III only (d) I and III only (e) I, II, and III 4.4 Students develop derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions. If f(x) = Arctan( 1 ), then f (x) = x 5.0 Students know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions. Let f and g be differentiable functions such that f(1) = 4, g(1) = 3, f (3) = -5 f (1) = -4, g (1) = -3, g (3) = 2 If h(x) = f(g(x)), find h (1) 6.0 Students find derivatives of defined functions and use implicit differentiation in a wide variety of problems in physics, chemistry, economics, and so forth. 3
The voltage V (volts), current I (amperes), and resistance R(ohms) of an electric circuit are related by the equation V = IR. Suppose that V is increasing at a rate of 1 volt/sec while I is decreasing at the rate of i/3 amp/sec. Let denote time in seconds. Find the rate at which R is changing when V = 12 volts and I = 2 amp. Is R increasing or decreasing? 7.0 Students compute derivatives of higher orders. If y = x e x, then d n y dx n = a) e x (b) e nx (c) (x + n)e x (d) x n e n (e) (x + n 2 )e x 8.0 Students know and can apply Rolle s Theorem, the Mean Value Theorem, and l Hôpital s rule. Let f(x) be a differentiable function defined only on the interval -2 x 10. The table below gives the value of f(x) and its derivative f (x) at several points of the domain. x -2 0 2 4 6 8 10 f(x) 26 27 26 23 18 11 2 g(x) 1 0-1 -2-3 -4-5 The line tangent to the graph of f(x) and parallel to the segment between the endpoints intersects the y-axis at the point a) (0,27) (b) (0,28) (c) (0,31) (d) (0,36) (e) (0,43) 9.0 Students use differentiation to sketch, by hand, graphs of functions. They can identify maxima, minima, inflection points, and intervals in which the function is increasing and decreasing. Sketch, by hand, the following rational function. List all critical points, find the interval(s) where the function is increasing or decreasing, and describe its concavity. 4x f(x) = x 2 " 4 10.0 Students know Newton s method for approximating the zeros of a function. 4
The curve y = tan(x) crosses the line y = 2x between x = 0 and x = " 2. Use Newton s method to find where. 11.0 Students use differentiation to solve optimization (maximum-minimum problems) in a variety of pure and applied contexts. A man in a boat is 24 miles from a straight shore and wishes to reach a point 20 miles down shore. He can travel 5 miles per hour in the boat and 13 miles per hour on land. Find the minimal time for him to reach his destination and where along the shore he should land the boat to arrive as fast as possible. 12.0 Students use differentiation to solve related rate problems in a variety of pure and applied contexts. A customer has asked you to design an open-top rectangular stainless steel vat. It is to have a square base and a volume of 32 ft 3, to be welded from quarter-inch plate, and to weigh no more than necessary. What are the dimensions you recommend? 13.0 Students know the definition of the definite integral by using Riemann sums. They use this definition to approximate integrals. Find the area bounded by the curve y = x 2 e "x, and the x-axis if 1 x 4. Use 6 subintervals with left point rectangles. 14.0 Students apply the definition of the integral to model problems in physics, economics, and so forth, obtaining results in terms of integrals. Solon Container receives 450 drums of plastic pellets every 30 days. The inventory function (drums on hand as a function of days) is I = 450 - t 2 2. Find the average daily inventory. If the holding cost for one drum is 2 cents per day, find the average daily holding cost. 15.0 Students demonstrate knowledge and proof of the fundamental theorem of calculus and use it to interpret integrals as anti-derivatives. 5
For what value of k, k > 0, does k # (4kx " 5k)dx = k 2 0 16.0 Students use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve, and work. Example 1: The region in the first quadrant enclosed by the graphs y = x and y = 2 sin(x) is revolved about the x-axis. The volume of the resulting solid figure is: Example 2: Two skydivers (A and B) are in a helicopter hovering at 6400 ft. Skydiver A jumps and descends for 4 seconds before opening her parachute. The helicopter then climbs to 7000 feet and hovers there. Forty-five seconds after A leaves the aircraft, B jumps and descends for 13 seconds before opening her parachute. Both skydivers descend at 16 ft/sec with their parachute open. Assume that the skydivers fall freely (no effective air resistance) before their parachutes open. a) At what altitude does A s parachute open? b) At what altitude does B s parachute open? c) Which skydiver lands first? 17.0 Students compute, by hand, the integrals of a wide variety of functions by using techniques of integration, such as substitution, and integration by parts. They can also combine these techniques when appropriate. Find the volume of the solid generated by revolving the region bounded by the x- axis and the curve y = x sin(x), 0 x π bout the line x = π. 18.0 Students know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals. 1 # dx = 4 " x 2 (a) Arcsin x 2 + C (b) 2 4 " x 2 + C (c) Arcsin(x) + C (d) ) 4 " x 2 + C (e) 1 2 Arc sin( x 2 ) + C 19.0 Students compute, by hand, the integrals of rational functions by combining the techniques in standard 17.0 with the algebraic techniques of partial fractions and completing the square. 6
2" 3 + 5" 2 + 8" + 4 # d" = (" 2 + 2" + 2) 2 20.0 Students compute the integrals of trigonometric functions by using the techniques noted above. cos(x) # sin 2 (x) + sin(x) " 6 dx 21.0 Students understand the algorithms involved in Simpson s rule and the trapezoid method. They use calculators or computers or both to approximate integrals numerically. A brief calculation shows that if 0 x 1, then the second derivative of f(x) = 1+ x 4 lies between 0 and 8. Based on this, about how many subdivisions would you need to estimate the integral of f from 0 to 1 with an error no greater than 10-3 in absolute value using the trapezoidal rule? 7