AP Calculus BC Class Starter January 22, 2018

Transcription

1 January 22, Given the function, find the following. (a) Evaluate f(4). (b) The definition of the derivative can be written two ways, as indicated below. Find both forms and evaluate the derivative at x = Given f(x) = x 6, find a formula for f and sketch its graph. 1

2 January 23, Each limit below represents a derivative, f'(a). Find f(x) and a. Then, find f'(x) and f'(a). (a) (b) (c) 2. Let n be the tangent line to the parabola y = x 2 at the point (1,1). The angle of inclination of n is the angle that n makes with the positive direction of the x axis. Calculate to the nearest degree. 2

3 January 24, Find A and B in so that. 2. The position of an object at time t seconds is given by meters. Find the object s acceleration each time the velocity is zero. 3. Evaluate: 4. Find a quadratic function such that f(2) = 5, f (2) = 3, and f (2) = Given that, if the rate of change of f at x = c is twice its rate of change at x = 1, then find c. 3

4 January 25, Find the values of A, B, and C such that the equation y = Ax 2 + Bx + C satisfies the differential equation y'' + 6y' 3y = 4x Given that and, evaluate: 3. Given the curve: (a) Find the tangents to the curve at the points where the slope is 4. (b) What is the smallest slope of the curve? At what value of x does the curve have this slope? 4. Let. Find f'(0) and f''(0). 5. An apple farmer currently has 156 trees yielding an average of 12 bushels of apples per tree. He is expanding his farm at a rate of 13 trees per year, while improved husbandry is improving his annual yield by 1.5 bushels per tree. What is the current rate of increase of his total annual production of apples? Use appropriate units. 4

5 January 26, 2018 For today's class starter you will work Free Response Question Number 6 from the 2017 AP Calculus AB Exam. You should pick the question up from the front table. This question must be completed without the use of a calculator! 5

6 1. If,, and, find. 2. Evaluate the following limit: 3. A particle moves along the x axis with velocity at time t 0 given by v(t) = 1 + e 1 t. (a) Find the acceleration of the particle at t = 3. (b) Is the speed of the particle increasing at time t = 3? Justify your answer. (c) Find all values of t at which the particle changes direction. Justify your answer. 4. For what values of r does the function satisfy the equation 6

7 AP Calculus January 31, Find values for a and b that make f differentiable everywhere: 2. Evaluate. 3. If, and is positive for all, what is? 4. The number of gallons of water in a tank m minutes after the tank has started to drain is given by the equation. (a) Find the average rate at which the water is draining over the first 10 minutes. (b) Find the value of t for which the average rate of change equals the instantaneous rate of change on [0, 10]. (c) Find G (8) and interpret its meaning using correct units. 7

8 Extra Problems 1. Given functions and, define h(x) as. Which function has the greatest rate of change at x = 1? 2. Find the equation of the normal line to where x = At what points on the curve is the equation of the tangent line parallel to? 4. The position of a particle is given by the equation, where s is measured in feet and t in seconds. (a) Find the velocity and acceleration of the particle at time t. (b) In the first 5 seconds, when is the particle at rest? (c) In the first 5 seconds, when is the particle moving to the right? (d) What is the total distance traveled by the particle in the first 4 seconds? (e) In the first 5 seconds, when is the particle speeding up? 5. Suppose the position of a particle moving on the x axis is given by the function particle equal to its average velocity?. For what value of t on [1, 4] is the instantaneous velocity of the 6. An ant moves along the x axis so that at time t its position is given by for values of t in the interval [ 1, 1]. a) Find an expression for the velocity of the ant at any given time t. b) Find an expression for the acceleration at any given time t. c) Determine the values of t for which the ant is moving to the right. Justify. d) Determine the values of t for which the ant changes direction. Justify your answer. 8