Calculus Section 2.1 The Tangent and Velocity Problems
· Tangent Line: is a line that touches a curve. A tangent line should have the same direction as the curve at the point of contact.
· Secant Line: is a line that passes through a graph at two points. PQ
· Instantaneous velocity: is the velocity at a specific moment in time. We use a tangent line and limits to calculate instantaneous velocity.
Ex. 1) Find the equation of the tangent line to the parabola at the point .
Graph of
Select a nearby point to create a secant line. Q
Slope at the point
Points approaching from the right
Slope at the point
Points approaching from the left
Point-Slope Form
Use the slope m found in addition to the original point
Ex. 2) If a arrow is shot upward with a velocity of 58 , its height in meters after t seconds is given by .
a) Find the average velocity over the given time intervals.
1) [1, 2]
(the interval lasts 1 second)
2) [1, 1.5]
(the interval last 0.5 second)
3) [1, 1.1]
(the interval last 0.1 second)
4) [1, 1.01]
(the interval lasts 0.01 second)
Difference Quotient:
b) Find the instantaneous velocity after 1 second.
Assignment page 91 #7(a)and(b) then 5
Calculus Section 2.1 The Tangent and Velocity Problems Day 2
Ex. 3) The position of the car is given by the values in the table.
t (seconds) 0 1 2 3 4 5
s (feet) 0 10 32 70 119 178
a) If P is the pointon the graph, find the slopes of the secant line PQ when Q is the point
1) The points are and
2) The points are and
3)
4)
b) Use the values of t that correspond to the points closest to to estimate the instantaneous velocity at .
Assignment page 92 #1 and 2