"AP Calculus TEST #3 Outline Emphasis on Integral Calculus"
AP Calculus TEST #3 Outline Emphasis on Integral Calculus You should be able to: 1. Determine the derivative of a function using the definition of a derivative (or first principles). 2. Differentiate various functions (constant to a variable, variable to a variable, variable to a constant, trigonometric, inverse trigonometric, logarithmic and implicit) 3. Sketch the derivative of a function. 4. Sketch a function and label critical points (including min, max and inflection points). 5. Use the second derivative test to determine if a critical point is a min or a max. 6. Determine absolute and relative (or local) min or max values for a function. 7. Solve related rate and optimization problems of various types 8. Use differentials to solve problems. 9. Discuss the continuity and differentiability of a function. 10. Use linearization to determine a specific value and be able to explain how this process works. 11. Determine the limit of a function algebraically, numerically or graphically. 12. Determine the equation of a normal or tangent line to a curve at a given point. 13. Calculate the average or the instantaneous growth of a function. 14. Determine the area under a curve from a table of values or a graph and the use of Riemann Sums. 15. Use technology to determine the value of a derivative or integral 16. Interpret derivatives and integrals. 17. Determine the value of an integral (simple ones only) by hand 18. Understand Rolle’s Theorem, the Mean Valve Theorem for derivatives and the Mean Value theorem for Integrals and be able to apply them. 19. Understand and be able to apply the Fundamental Theorem of Calculus (Parts 1 and 2) 20. Solve simple first order differential equations involving motion. Some Practice Questions: 1 The table below shows the amount of water leaking out of a tank at a rate of r(t) litres per hour. Give an upper and lower estimate of the total quantity of water that has leaked out of the tank in 12 hours. Time 1 2 3 4 5 6 7 8 9 10 11 12 (hours) Leakage 8 6 5 5 4 3 2 1 0.5 0.5 0.3 0.2 (litres/hour) 2. Given the velocity, in km per hour, of a car can be modeled by the function: v(t ) = 2t 2 + −4t + 10 . Determine the distance it has traveled during the time interval t ∈ [0,8] using four rectangles and MRAM . Illustrate your answer on the diagram. 3 Use your calculator to evaluate each of the following and illustrate your answer with a diagram. 3 ∫ 7 0 a) −1 5 ∂x b) ∫ (5 − x) ∂x −2 c) ∫ −5 25 − x 2 ∂x 4 ⎧2 x, x < 2 ⎫ d) ∫ f ( x) ∂x ; given that: 0 f ( x) = ⎨ ⎩4, x ≥ 2⎭ ⎬ 4 After investing $1000 at an annual interest rate of 7% compounded continuously for t years, you balance is $B, where B = f(t). i) What are the units of ∂B ? ∂t ii) What is the financial interpretation of ∂B ? ∂t 5 Let P(t) represent the price of a share of stock of a corporation at time t. What does each of the following statements tell us about the signs of the first and second derivatives of P(t)? i) “The price of the stock is rising faster and faster.” ii) “The price of the stock is close to bottoming out.” 6 The rate of depreciation for a new piece of equipment at a factory is given as p(t ) = 50t − 600 for t ∈ [0,10] , where t is measured in years. Write an integral that would represent the total loss of value of the equipment over the first five years and then evaluate the interval. 7 If R(t) is the profit that a company has made, measured in dollars per month, and t is measured in months (0 representing April 1st; t =1 representing May 1 etc.), 6 what does ∫ R(t ) ∂t = 5600 0 mean? 8 Water is leaking out of a barrel at a rate of r (t ) litres per minute. Write a definite integral expressing the total quantity of water which leaks out of the barrel in the first 2 hours. 9 Use part 1 of the Fundamental Theorem to compute each integral exactly. Determine the following integrals by hand. Answers should be exact. 2 3 ∫ (2 x − 3) ∂x ∫ (x − 2) ∂x 2 a) b) 0 0 4 2 2 c) ∫ ( x + 3x) ∂x 0 d) ∫ (4 x − x 1 2 ) ∂x π π 2 4 e) ∫ (2 sin x) ∂x 0 f) ∫ (sec x tan x) ∂x 0 1 3 ∫ (e ∫ (3e −x g) x − e ) ∂x h) 2x − x 2 ) ∂x 0 0 x −3 4 5 6 i) ∫ x ∂x 1 j) ∫ 3x + 4 ∂x 2 10 Suppose that, for a particular population of organisms, the birth rate is given by b(t ) = 410 − 0.3t organisms per month and the death rate is given by 12 d (t ) = 390 + 0.2t organisms per month. Determine ∫ [b(t ) − d (t )] ∂t and explain 0 what it represents. 11 Suppose that the temperature t months into the year is given π by: T (t ) = 64 − 24 cos( t ) in degrees Fahrenheit. Estimate the average 6 temperature over an entire year. Explain why this answer is obvious from the graph of T(t). 12 The linear density of a rod of length 4 m is given by ρ ( x) = 9 + 2 x measured in kilograms per meter, where x is measured in meters from one end of the rod. Find the total mass of the rod. x 1 13 Find the interval on which the curve y = ∫ ∂t is concave upward. 0 1+ t + t 2 3x 5x 1 14 Find the derivatives of : ∫x u + 1 ∂u and 2 ∫ cos(2u ) ∂u cos x t2 1+ u4 x 15 If F ( x) = ∫ f (t ) ∂t , where f (t ) = ∫ ∂u , find F ′′(2) 1 1 u 16 Given the graph of p(x) as shown below, determine h(3), h’(3) and h” (3) if x h( x ) = ∫ p(t ) ∂t −1 x 17 Solve for x given that: x>0 and: ∫ (t 3 − 2t + 3) ∂t = 4 . 0 x 18 Find the linearlization of: h( x) = 3 + ∫ 5 cos(2t ) ∂t at x = 0 0 x 19 Find f(3) if ∫ f (t ) ∂t = 5 0 x +1 20 A dam released 1000 m3 of water at 10m3/min and then another 1000 m3 at 20m3/min. What was the average rate at which the water was released? Give reasons for your answer. 21 A car is 130 m from a stop sign when it begins to decelerate at a rate of t m/s2. If its speed when it begins to decelerate is 20m/s: a) when will it come to a full stop? b) how far from the stop sign will it be when it stops? 22 You are driving along a highway at a steady 60mph (88ft/sec) when you see an accident ahead and slam on the brakes. What constant deceleration is required to stop your car in 242 feet? 23 Suppose that the marginal cost of manufacturing an item when x thousand units ∂c are produced is: = 3x 2 − 12 x + 15 dollars per item. Find the cost function ∂x c(x) if c(0) = 400. 24 The acceleration of a particle moving along a coordinate line is 2 + 6t m/s2. The initial velocity (at time zero) is 4m/s. a) Find the velocity as a function of time t. b) How far does the particle move during the first second of its trip?