Tangent Line Approximations and Theorems

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Presentation transcript:

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

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

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

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

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

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)

Thanks! Any questions?