http://www.ck12.org Chapter 14. Concepts of Calculus
Rate vs. Time: (this is the graph of the derivative of the original function shown above)
3.
- The slope of the line between( 0. 7 , 0. 73 )and( 1 , 1 )is:m 1 =^00.^7. 73 −− 11 ≈ 2. 19
- The slope of the line between( 0. 8 , 0. 83 )and( 1 , 1 )is:m 2 =^00.^8. 83 −− 11 ≈ 2. 44
- The slope of the line between( 0. 9 , 0. 93 )and( 1 , 1 )is:m 3 =^00.^9. 93 −− 11 ≈ 2. 71
- The slope of the line between( 0. 95 , 0. 953 )and( 1 , 1 )is:m 1 =^00.^95. 953 −− 11 ≈ 2. 8525
- The slope of the line between( 0. 975 , 0. 9753 )and( 1 , 1 )is:m 1 =^00 ..^9759753 −− 11 ≈ 2. 925625
The slope at( 1 , 1 )will be slightly greater than the estimates because of the way the slope curves. The slope at( 1 , 1 )
appears to be about 3.
Practice
- Approximate the slope ofy=x^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 #1.
limx→ 1
(x (^2) − 1
x− 1