14.8. Instantaneous Rate of Change http://www.ck12.org
- Approximate the slope ofy= 3 x^2 +1 at( 1 , 4 )by using secant lines from the left. Will the actual slope be greater
or less than the estimates? - Evaluate the following limit and explain how it confirms your answer to #3.
limx→ 1
( 3 x (^2) + 1 − 4
x− 1
)
- Approximate the slope ofy=x^3 −2 at( 1 ,− 1 )by using secant lines from the left. Will the actual slope be greater
or less than the estimates? - Evaluate the following limit and explain how it confirms your answer to #5.
limx→ 1
(x (^3) − 2 −(− 1 )
x− 1
)
- Approximate the slope ofy= 2 x^3 −1 at( 1 , 1 )by using secant lines from the left. Will the actual slope be greater
or less than the estimates? - What limit could you evaluate to confirm your answer to #7?
- Sketch a complete cycle of a cosine graph. Estimate the slopes at 0,π 2 ,π,^32 π, 2 π.
- How do the slopes found in the previous question relate to the sine function? What function do you think is the
derivative of the cosine function? - Sketch the liney= 2 x+1. What is the slope at each point on this line? What is the derivative of this function?
- Logan travels by bike at 30 mph for 2 hours. Then she gets in a car and drives 65 mph for 3 hours. Sketch both
the distance vs. time graph and the rate vs. time graph. - Explain what a tangent line is and how it relates to derivatives.
- Why is finding the slope of a tangent line for a point on a function the same as the instantaneous rate of change
at that point? - What do limits have to do with finding the slopes of tangent lines?