Chapter 1 to 4 - Quick Review Problems 1. Use a graphing calculator to graph f() x x2/3 . State why Rolle’s Theorem does not apply to f on the interval 1,1. a) f is not continuous on b) is not defined on the entire interval c) ff( 1) (1) d) is not differentiable at x 0 e) Rolle’s Theorem does apply 2. The graph of f ' is shown below. Estimate the open intervals in which is increasing or decreasing. y 1 –1 1 2 3 x a) Increasing ,1and 3, ; decreasing (1, 3) –1 b) Increasing (0,2); decreasing ,0 and 2, –2 c) Increasing , –3 d) Increasing ,0 and 2, ; decreasing (0,2) e) Increasing 0.5,0.5 and 2.8, ; decreasing , 0.5 and 0.5,2.8 2 3. Given that f( x ) x 12 x 28 has a relative maximum at x 6 , choose the correct statement. a) f is negative on the interval ( ,6) b) f is positive on the interval , c) f is negative on the interval 6, d) f is positive on the interval 6, e) None of these 4. Let fx be a polynomial function such that f ( 2) 5 , f ( 2) 0 , and f ( 2) 3 . The point ( 2,5) is a(n)________________________ the graph of f . a) Relative maximum b) Relative minimum c) Intercept d) Point of Inflection (e) Absolute minimum 1 1 5. Use a graphing calculator to graph fx . Use the graph to determine the open intervals (x 1)2 where the graph of the function is concave upward or concave downward. a) Concave downward: , b) Concave downward: ,1 ; Concave upward: ( 1, ) c) Concave downward: ,1 and ( 1, ) d) Concave upward: ( , 1) and e) Concave upward: ; Concave downward: 6. Find all extrema, if any, in the interval 0,2 if f x sin x x . Write as ordered pairs. 7. A differentiable function has only one critical number: x 3. Identify the relative extrema of f if f 4 1 and f 21 . 2 8. Find all points of inflection of the function f x 52 x45 x . 2 9. State why the Mean Value Theorem does not apply to the function fx x 12 on the interval 3,0 . Give your answer as a complete sentence. 2 AP Calculus Notes 5.1 Anti-derivatives and Indefinite Integration Exploration: For each of the following derivatives, find the original function f (x) . 1) F' ( x ) 2 x 2) F'2( x ) 6 x 3) F' ( x ) cos x 1 5 4) F'(x) ex 3 5) F'(x) 6) F'(x) 1 x2 2 5x The function F is an anti-derivative of f on an interval I if F' ()() x f x for all x . F is an anti-derivative rather than the anti-derivative because any constant C would work. Explanation: Find the derivative of f (x) x2 , f (x) x2 5, and f (x) x2 e . Because of this, you can represent all anti-derivatives of f( x ) 2 x by :___________ The constant C is called the constant of integration. The function represented by F is the general anti- derivative of f , and F() x x2 C is the general solution of the differential equation F' ( x ) 2 x . Notation for Anti-derivatives: The operation of finding all solutions of this equation is called anti-differentiation or indefinite integration and is denoted by the integral sign . The general solution is: y f()() x dx F x C 3 The expression f() x dx is read as the anti-derivative of f with respect to x. So, dx serves to identify x as the variable of integration. The term indefinite integral is a synonym for anti-derivative. Basic Integration Rules The inverse nature of integration and differentiation can be used to obtain: F' ()() x dx F x C d If f()() x dx F x C then f()() x dx f x dx Exploration: What is the anti-derivative of each of the following? Try to develop the basic power rule for integration: a) f (x) x2 b) f (x) x3 c) f (x) x4 So, the Power Rule for Integration is: xndx _______________ Ex. 1: Integrate each of the following polynomial functions: a) (x 2) dx b) dx c) (3x42 5 x x ) dx 4 Differentiation Formula Integration Formula d kx dx d kf(x) dx d f (x) g(x) dx d n x dx d sin x dx d cos x dx d tan x dx d sec x dx d csc x dx d cot x dx d x e dx d x a dx d ln x dx 5 The most important step in integration is rewriting the integral in a form that fits the basic integration rules. Ex. 2: Rewrite each of the following before integrating: Original Integral Rewrite Integrate Simplify 1 a) dx x3 1 b) dx 2 x c) 2t(3t 2 9t 1)dt d) (t22 1) dt x3 3 e) dx x2 sin x 2 f) 3 x( x 4) dx g. dx h. sec2 csc2 d cos2 x 6 Initial Conditions and Particular Solutions: There are several anti-derivatives for a function, depending on C. In many applications of integration, you are given enough information to determine a particular solution. To do this you need only know the value of y f (x) for one value of x . (This information is called an initial condition). Ex. 3: The function y 3x2 1dx has only one curve passing through the point (2,4) . Find the particular solution that satisfies this condition. Ex. 4: Find the general solution of F'(x) ex and find the particular solution that satisfies the initial condition F(0) 3. Ex. 5: Find the general solution of f "(x) sin x 2 and find the particular solution that satisfies the initial condition f '(0) 3 and f () 3. 7 Ex. 6: A ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet and shown in the figure. Remember: To go from position to velocity to acceleration – To go from acceleration to velocity to position – a) Find the position function giving the height s as a function of time t. b) What is the speed of the ball when it hits the ground? c) After how many seconds after launch is the ball back at its initial height? 8 Ex. 7: A particle, starting at the origin, moves along the x-axis and it’s velocity is modeled by the equation v(t) 6t 2 30t 24 where t is in seconds and v(t) is meters per second. a) How is the velocity changing at any time t? b) What is the particle’s speed at 3 seconds? c) What is the particle’s position when the acceleration is 6m/ s2 ? d) When is the particle changing directions? e) When is the particle furthest to the left? 9 Ex. 8: A missile is accelerating at a rate of 4t m/ s 2 from a position at rest in a silo 750m below ground. How high above the ground will the missile be after 6 seconds. Ex. 9: The motion of a grizzly bear stalking its prey, walking left and right of a fixed point in ft/s, can be modeled by the motion of a particle moving left and right along the x-axis, according to the 1 acceleration equation a( t ) sin( t ) . If the bear’s velocity is 1 ft/s when t 0 … 3 a) Find the velocity equation. b) How fast was the bear traveling when t 7 ? c) In what direction is the bear traveling when t 5? 10 AP Calculus I 5.1 Quiz Review 6 1. (6x2 7x 5)dx 2. 23x 4 x3 dx 3. dx 11x2 4. 2x (3 x 1) dx 5. (2x 5)2 dx 6. x(x 3 x)dx 9x4 x2 5x 7. (3sec xtan x 4sin x)dx 8. 2csc2 x cos x dx 9. dx 3x 10. Find the original function f (x) given f'( x ) 4 x 2 and the condition f (4) 1. 11 11. A particle moves along the x-axis with velocity given by v( t ) 3 t2 6 t . If the particle is at position x 2 at time t 0, what is the position of the particle at time t 1? [2008 AP MC#7] 12. A cannonball is shot upward from the ground with an initial velocity of 30m/ s . The acceleration is 9.8m/ s2 . a) What is the height and velocity function of the cannonball? b) What is the maximum height of the cannonball? c) What is the velocity of the cannonball when it hits the ground? 12 AP Calculus I Notes 5.2 Area Under a Curve Area under the curve from [0,4] = __________________________ Area under the curve from [1,3] = __________________________ Area under the curve from [0,6] = __________________________ 13 Area under the curve from [-0.5,4.5] = __________________________ Area under the curve from [1,5] = __________________________ 14 15 In this section we will examine the problem of finding the area of a region in a plane. EX 1: Suppose you have to find the area under the curve y 25 x2 from x = 0 to x = 4. Graph: Method 1: Divide the region into four rectangles, where the left endpoint of each rectangle comes just under the curve, and find the area. Is this an over- or under-approximation of the actual? 16 Graph: Method 2: Divide the region into four rectangles, where the right endpoint of each rectangle comes just under the curve, and find the area.