12.3 Tangent Lines and Velocity Objectives: Find instantaneous rates of change by calculating slopes of tangent lines. Find average and instantaneous velocity.
Tangent Vs. Secant Recall, that a secant is a line that intersects another line twice, while a tangent line intersects another line in exactly one point. We can approximate the slope of these lines, using 2 points. The closer and closer the points are, the better the approximation is, and the closer the line is to being tangent.
The slope of this tangent line is the rate of change in the slope of the curve at that instant. We will use limits to define the slope of a tangent line more precisely.
To define the slope of the tangent line y = f(x) at the point (x, f(x)), find the slope of the secant line through this point and one other point on the curve. Let the x – coordinate of the 2nd point be x + h, for some small value of h. The corresponding y – coordinate for this point is then f(x + h), as shown in the figure.
The slope of the secant line through these 2 points is given by: This expression is called the difference quotient.
As the second point approaches the first, or as h0, the secant line approaches the tangent line at (x, f(x)). We define the slope of the tangent line at x, which represents the instantaneous rate of change of the function at that point, by finding the limits of the slopes of the secant lines at h 0.
Example 1 A. Find the slope of the line tangent to the graph of y = x2 + 1 at (2, 5).
Example 1 B. Find the slope of the line tangent to the graph y = x2 – 3 at the point (–2, 1).
Example 2 A. Find an equation for the slope of the graph of y = x2 + 2x at any point.
Example 2 B. Find an equation for the slope of the graph at any point of y = x2 + 3x – 2.
Example 3 A. As part of a physics experiment, a ball is catapulted upward. The height of the ball is h(t) = –16t 2 + 95t + 15, where t is in seconds and the height of the ball is measured in feet. What was the ball’s average velocity between t = 1 and t = 2?
Example 3 B. A baseball is thrown upward into the air. The height of the baseball is h(t) = –16t2 + 85t + 6, where t is in seconds and the height of the baseball is measured in feet. What was the baseball’s average velocity between t = 1 and t = 3?
Example 4 A. Tourists standing on a 300-foot-tall viewing tower often drop coins into the fountain below. The height of a coin falling from the tower after t seconds is given by h(t) = 300 – 16t 2. Find the instantaneous velocity v(t) of the coin at 2 seconds.
Example 4 B. Molly’s father is building her a tree house. He accidentally drops his hammer out one of the windows in the house. The height of the hammer falling from the tree house is given by h(t) = 90 – 16t 2. Find the instantaneous velocity v(t) of the hammer at 1 second.
Example 5 A. The distance a bumblebee flies along a path is given by p(t) = 12t – 6t 3 + 1, where t is given in seconds and the distance of the bumblebee from its starting point is given in inches. Find the equation for the instantaneous velocity v(t) of the bumblebee at any point in time.
Example 5 B. The distance an ant crawls along a path is given by c(t) = 14t – 10t3 + 2, where t is given in seconds and the distance of the ant from its starting point is given in inches. Find an equation for the instantaneous velocity v(t) of the ant at any point in time.