Annapolis High School



AP CALCULUS REVIEW PACKET #2Name:____________________________________Part 1: Basic RulesIf fx=x+324x-7, then f'x= If fx=cos26x, then f'π4=If fx=e1x2, then f'x= If fx=x2+344x+7, then f'x=If fx=sinx2, then f'π2=If fx=e2x3, then f'x= If fx=x+324x-7 , then f'x=If fx=e-5x, then f'x= If fx=cos(4x), then f'π16=Part 2: Interpreting Derivatives from a tableThe polynomial function fhas selected values of its first derivativef' is gvein in the table table.Which of the following statements must be true?X2345f’(x)-1-1/402f is increasing on the interval (2, 4)f is decreasing on the interval (4,5) f has a local minimum at x = 4The graph of f has a point of inflection at x = 4The graph changes concavity in the interval (3, 5)The polynomial function f has selected values of its second derivative f'' given in the table.Which of the following statements must be true?X2345f’’(x)6-50-3f is increasing on the interval (2, 4)f is decreasing on the interval (3,5) f has a local minimum at x = 4The graph of f has a point of inflection at x = 4The graph changes concavity in the interval (2, 4)The polynomial function fhas selected values of its first derivativef' is given in the table table.Which of the following statements must be true?X0123f’(x)-10-26f has a critical value at x = 1f is increasing on the interval (0,2) f has a local minimum at x = 1The graph of f has a point of inflection at x = 4The graph changes concavity in the interval (0,2 )The polynomial function f has selected values of its second derivative f'' given in the table.Which of the following statements must be true?X0123f’’(x)-1026f is increasing on the interval (1, 3)f is decreasing on the interval (0, 1) f has a local minimum at x = 1The graph of f has a point of inflection at x = 1The graph changes concavity in the interval (1, 3)Part 3: Interpreting GraphsThe graph of f', the derivative of f, is shown for 0≤x≤9.left10870800On what intervals if f increasing?On what intervals if f decreasing?At what x value(s) does f have a local maximum?At what x value(s) does f have a local minimum?The graph of f', the derivative of f, is shown for -3≤x≤3.-9969510795000On what intervals if f increasing?On what intervals if f decreasing?At what x value(s) does f have a local maximum?At what x value(s) does f have a local minimum?The graph of f', the derivative of f, is shown for a≤x≤h.-15049517018000On what intervals if f increasing?On what intervals if f decreasing?At what x value(s) does f have a local maximum?At what x value(s) does f have a local minimum?Part 4: Implicit Differentiation161451-6985000146050-38072001630346540500 Part 5: Tangent/Normal Lines In the xy-plane, the line 3x + y = k, where k is a constant, is tangent to the graph of y=x2+5x+10. What is the value of k? -4(B)6(C) -6(D)2(E)0What is the slope of the line tangent to the curve y=arcsin(3x) at the point at which x=13? Undefined (B) 1(C) -3/2(D) 3/2 (E) 0In the xy-plane, the line x – y = k, where k is a constant, is tangent to the graph of y=3x2-5x. What is the value of k? 1(B)0(C) -3(D)3(E)-2What is the slope of the line tangent to the curve y=arctan2x at the point at which x=-12 0 (B) 1(C)?(D) -? (E)23 In the xy-plane, the line 2x + y = k, where k is a constant, is tangent to the graph of y=-x2+6x-8. What is the value of k? 4(B)8(C) -4(D)-8(E)-2What is the slope of the line tangent to the curve y=arctan4x at the point at which x=-12 -15 (B) 15(C) -45 (D) -43 (E)4522536153175000The graph of a function f is shown above. Which of the following could be the graph of f', the derivative of f?FRQ’sThe function f is defined by fx=32x-x2 for -2≤x≤2Find f'xWrite an equation for the line tangent to the graph of f at x = -1.Let g be the function defined by gx= fx for -2≤x≤1x+11 for 1<x≤2. Is g continuous at x = 1? Consider the close curve in the xy-plane given byx2+xy+y2=27Find dydxWrite an equation for the line normal to the curve at the point (1, 2)Find the coordinate points of the points on the curve where the line is tangent to the curve is vertical.Find the coordinate points of the points on the curve where the line is tangent to the curve is horizontal.The function f is defined by fx=400-x2 for -20≤x≤ 20Find f'xWrite an equation for the line tangent to the graph of f at x = -16.Let g be the function defined by gx= fx for -20≤x≤12x+4 for 12<x≤20.Is g continuous at x = 12?Consider the close curve in the xy-plane given byx2+4y-y4=12Find dydxWrite an equation for the line normal to the curve at the point (2, -1)Find the coordinate points of the points on the curve where the line is tangent to the curve is vertical.Is it possible for this curve to have a horizontal tangent at points where it intersects the x-axis? Let f be a function define on the closed interval 0≤x≤3.5 with f1=1. The graph of f', the derivative of f, is depicted below.For 0≤x≤3.5 , find all the values x at which f has a relative minimum. JUSTIFY your answer.For 0≤x≤3.5 , find all the values of x at which the graph of f has a point of inflection. JUSTIFY your answer.Find all intervals on which the graph of f is concave down and also has a negative slope. Explain your reasoning.CAN’T DO THE 25 on your copy! Do this one instead.What is the slope of the line tangent to the curve y=arctan4x at the point at which x=-12 -15 (B) 15(C) -45 (D) -43 (E)45Let f be a function define on the closed interval -3≤x≤9 with f0=1. The graph of f', the derivative of f, is depicted below.For -3≤x≤9 , find all the values x at which f has a relative maximum. JUSTIFY your answer.For -3≤x≤9 , find all the values x at which f has a relative minimum. JUSTIFY your answer.For -3≤x≤9 , find all the values of x at which the graph of f has a point of inflection. JUSTIFY your answer.Find all intervals on which the graph of f is increasing. Explain your reasoning.Find all intervals on which the graph of f is concave up and also has a negative slope. Explain your reasoning.Find all intervals on which the graph of f is concave down and also has a positive slope. Explain your reasoning. ................
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