1. Tangent Line Approximations and Theorems
By Brendon Brown, Rochelle Muñoz, Brianna Andersen

2. Interactive Practice Kahoot
Let’s start! Get out a pencil, a calculator, and a piece of scratch paper!

3. Flipbook 18-20
Tangent Line/Normal Line (overestimate, underestimate), Intermediate Value Theorem, Mean Value Theorem

4. Tangent Line/Normal Line
Take first derivative
Find first derivative when x=c to find the slope
Use your slope, x, and y to create the point slope formula
y-y₁ =m( x-x₁).
The tangent line is an overestimate at maximums/concave down and an underestimate at minimums/concave up
NORMAL LINE:
Find the slope when x=c and take the opposite reciprocal

5. Intermediate Value Theorem
DEFINITION:
A continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.
IN SHORT:
On a continuous function, every
y-value will be hit at least once. Think about drawing a line on a piece of paper without bringing your hand up.
Always state that f(x) is a continuous function! Even if already stated in prompt. IN

6. Mean Value Theorem
If the function f(x) is differentiable on the interval (a,b) and continuous on [a,b], then a value c exists in (a,b) such that f’(c)= [f(b) - f(a)] ÷ (b - a)
If all the conditions are met then there is at least one point where the slope of the tangent line equals the slope of the secant line
MUST state the function is differentiable over the open interval (a,b) (continuity is implied)

7. Thanks!
Any questions?