1. Tangent line to a curve

One of the fundamental themes of calculus is to find the slope of the tangent line to a curve at a point. To see how this can be done, consider the point (2, 4) on the graph of the quadratic function .

1. Find the slope m₁of the line joining (2,4)and (3,9). Is the slope of the tangent line at (2,4) greater than or less than the slope of the line through (2,4) and (3,9)?
2. Find the slope m₂of the line joining (2,4)and (1,1). Is the slope of the tangent line at (2,4) greater than or less than the slope of the line through (2,4) and (1,1)?
3. Find the slope m₃of the line joining (2,4)and (2.1,4.41). Is the slope of the tangent line at (2,4) greater than or less than the slope of the line through (2,4) and (2.1,4.41)?
4. Find the slope m_hof the line joining (2,4)and (2+h, f(2+h))in terms of the nonzero number h.
5. Evaluate the slope formula from part d) for .Compare these values with these in part a) -c).
6. What can you conclude the slope m_tan of the tangent line at (2,4) to be? Explain your answer.

2.Fit a Parabola

a) Find the equation of the unique parabola that passes through the three points given to your group. You must show all the algebra necessary to find the equation.

(1,-4), (4,2),(6,-14)

b) Use substitution to verify that each point satisfies the equation you found.

c) Prints out the graph of this parabola and on the graph locate the three points. Also print out a table that shows the coordinates of the points using your function.

3. Motion due to gravity

For problem below, make use of the equation of motion of a vertically projected object:

s=-1/2gt^2+Vot+So

, where s is the height of the object at time t seconds, g is 32 feet per second and and are the initial velocity and initial height, respectively.

Two friends are in competition. Each stands on the roof of the apartment building where each one lives. At the same time, each friend stands at the edge of the roof and throws up a basketball as hard as possible. Joan's apartment building is 120 feet high, and she throws the basketball at a speed of 20 feet per second. Joe's apartment building is 105 feet high, and he throws the basketball at a speed of 34 feet per second.

a) Write the function that describes the height of Joan's basketball, in terms of time measured in seconds. What kind of function is the height function? How do you know?

b) Display the graph of the function you found from a) above. Copy the graph onto your paper and label its axes.

c) What is the maximum height of the basketball, and when does it reach that height? Show your work and explain your answer.

d) When does the basketball hit the ground? Show your work and explain your answer.

e) Write the function that describes the height of Joe's basketball, in terms of time measured in seconds.

f) Which basketball reaches a greater height? Show work and explain your answer.

g) Is there a time when the basketballs are at the same height? Set up the equation and use algebra, showing your work. Write the conclusion in a complete sentence.

h) Which basketball stays up in the air longer? Show work and explain your answer.