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Tangent, Limits, and Velocity in Calculus

Learn about tangent lines to curves, limits, and velocity problems in calculus with examples and explanations. Explore how to find equations of tangent lines and determine instantaneous velocities at specific points.

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Tangent, Limits, and Velocity in Calculus

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  1. Chapter 2: Limits 2.1 The Tangent & Velocity Problems

  2. Tangent • A tangent to a curve is a line that touches the curve in exactly one spot. • Should have the same direction as the curve as the point of contact

  3. Example 1 • Find an equation of the tangent line to the parabola y = x2at the point P(1,1). • We need to know the slope of the tangent… • But there’s only one point! • If we choose a point on the parabola that’s nearby, we could compute the slope of a secant line…

  4. Example 1 cont… • Choose x ≠ 1 so that Q ≠ P • If we choose Q(1.5,2.25), we have:

  5. Example 1 cont… • Check out the table: • If we choose several x’s close to 1, the closer Q is to P, the closer x is to 1, and the closer the slope is to 2 • Slope of the tangent line must be 2!

  6. Example 1 cont… • Find an equation of the tangent line to the parabola y = x2at the point P(1,1).

  7. Limits • We say that the slope of the tangent line is the limit of the slopes of the secant lines

  8. Example 2 • The flash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the flash is set off. The data in the table describe the charge Q remaining on the capacitor (measured in microcoulombs) at time t (measured in seconds after the flash goes off). Use the data to draw the graph of this function and estimate the slope of the tangent line at the point where t = 0.04. (the slope of the tangent line represents the electric current flowing from the capacitor to the flash bulb in microamperes)

  9. Velocity • If you watch the speedometer of a car, the needle doesn’t stay still long • Velocity of the car is not constant • The car has definite velocity at each moment • What is its instantaneous velocity??

  10. Example 3 • Suppose that a ball is dropped from the upper observation deck of the CN Tower in Toronto, 450 m above the ground. Find the velocity of the ball after 5 seconds. • Galileo’s law:

  11. Homework • P. 65 • 1, 5, 7

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