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Lab: Secants, Tangents, and Local Linearity
Replacement for Lab 2, Contemporary Calculus, Bartkovich et al.
For the given func ons, you will observe the slopes of secant lines for the graph of f that passes through a
par cular point ( a , f ( a )). You will examine what happens to the slopes when a second point that determines
the secant line, ( a + h f , ( a + h ) ), moves closer to ( a , f ( a )) , or put another way, as h →0.
Func ons to Inves gate
Instruc ons
Follow these steps for each of the six func ons, one func on at a me
- By hand, calculate the slope of the secant line that goes through the point ( a , f ( a )) and
( a + 1, f ( a + 1) ). Graph f ( x )and this secant line in Desmos.
- Graph the func on f ( x )in Desmos. Use the ^ symbol for exponents and abs() for absolute values.
Func on
Point of Interest ( a , f ( a ))
A
f ( x ) = x
2
1
4
1
B
f ( x ) = in
sin s ( x ) + 2 (0, 2 )
C
f ( x ) =
x
4
1
D f ( x ) = √
3
x
E
f ( x ) = ( x )
3
2
x
(1, )and − 1
F
f ( x ) =− 3 x
+ 6 (2, )and
Example
For problem A, find the slope of the secant line to f ( x ) that contains and.
2
1
4
1
2
1
+ 1 f (
2
1
Example
3. Define the value of a using a slider to create a parameter. For A, do this by typing a = 0.5as an input.
Then create the point ( a , f ( a ))by typing ( a , f(a) ) as an input. No ce that as you slide the
value for a , the point will move along the curve f ( x ).
4. Create the point ( a + h , f ( a + h )). Desmos will offer to create a slider for h. Click the blue bu on to
create the slider.
- Modify the slider so that can vary between and by increments of. Do this by clicking on the − 1
number on the far le or right of the slider bar. Edit the inequality for h.
- In your notebook, find the equa on for the secant line through the points ( a , f ( a )) and( a + h f , ( a + h )) . This will be a general equa on for the secant lines that will involve h. Graph this line on Desmos.
Observe what happens to the secant line as you move the slider star ng at h = 1 to h = 0and as you
move the slider star ng at h =− 1 to h = 0.
- In your notebook, create a table that shows the value of h and the slopes produced by the related
secant line for a minimum of 8 values of h. Include both posi ve values and nega ve values of h. You
may need to use smaller values of h than the slider will take to detect a pa ern, or lack of pa ern. If so,
adjust the step of the slider to allow for finer control of h OR type in the desired values of h directly.
- Based on your work in the previous steps, do you think f has a slope at the point( a , f ( a ))?
● If you think f has a slope at ( a , f ( a )), then es mate the value of that slope. Use it to write the
equa on of the line tangent to f at ( a , f ( a )). Graph this tangent line in Desmos with f to
confirm your findings. If not, explain why f fails to have a slope.
● Be sure to record your equa on of the line tangent to every given func on at the specified point of
interest (or explain why they do not exist).
Example
Example
Example
