University:
University of North Dakota
Course:
Math 165 | Calculus I
Academic year:

2024
Views:

300

Pages:

2
Author:

Drevzh

Workshop #2 1. A ball is thrown down from the top of a building. The distance (in feet) it has traveled after t seconds is f (t) = 16t2 + 20t. (a) Find the average velocity of the ball between t = 2 and t = 5. (b) Find the average velocity of the ball between t = 2 and t = 4. (c) Find the average velocity of the ball between t = 2 and t = 3. (d) Find the average velocity of the ball between t = 2 and t = 2.5. (e) Find the average velocity of the ball between t = 2 and t = 2.1. (f) Find the average velocity of the ball between t = 2 and t = 2.01. (g) From your answers to the above, what do you expect will be the instantaneous velocity of the ball at time t = 2 seconds? (h) Find an expression for the average velocity of the ball between t = 2 and t = 2 + h. (i) Simplify the expression you found in part 1h, and determine what happens to this simplified expression when h is very close to zero. 2. The graph of f is shown below. (a) Find an expression for the slope of the line between (0, f (0)) and (2, f (2)). (b) Find an expression for the slope of the line between (a, f (a)) and (2, f (2)). (c) Find an expression for the slope of the line between (2, 4) and (b, f (b)), where b is an arbitrary number. (d) Find an expression for the slope of the line between (2, 4) and (2 + h, f (2 + h)). 3. Consider the function f (x) = 3x2 − 2x + 1. (a) Find the slope of the line that passes through (2, 9) and (2.5, f (2.5)). (b) Find the slope of the line that passes through (2, 9) and (2.1, f (2.1)). (c) Find an expression for the slope of the line that passes through (2, 9) and (2 + h, f (2 + h)). (d) Simplify the expression you found in part 3c, and determine what happens to this simplified expression when h is very close to zero. (e) Looking at your answer to part 3d, what information do you now know about the line that is tangent to the graph of y = f (x) at x = 2? (f) Find the equation of the line that is tangent to the graph of f at (2, 9).