Tangent Lines and Theorems Notes Sofia Alba, Isha Birla, Ashley Irwin

Finding Tangent Lines 1. Plug in the given x value into the original function to get y(x) 2. Take the derivative of the original function to get y’(x) and plug in the given x value 3. Substitute your values into the equation TANGENT EQUATION: (y-y₁)=m(x-x₁)

Strategy Create a mini chart with all the values you will be needing to create the tangent line, such as: Then once you have filled it out, just plug in the values into the tangent equation f(x) f’(x) x

Find the line tangent to f(x)= 15- 2x2 at x=1 Example #1 Find the line tangent to f(x)= 15- 2x2 at x=1

Tangent Line Equation: (y-13)=-4(x-1) Solution f(1) f’(1) x 13 -4 1 f’(x)= -4x Tangent Line Equation: (y-13)=-4(x-1)

Example #2 For a function f we are given that f(8)=1 and that f’(8)=2, what is the equation of the line that is tangent to the function f at x=8?

Tangent Line Equation: (y-1)=2(x-8) Solution f(8) f’(8) x 1 2 8 Tangent Line Equation: (y-1)=2(x-8)

Theorems: IVT Definition: If f is continuous over [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)=k

Theorems: MVT Definition: If f is continuous over [a,b] and differentiable over (a,b), then there is some point c between a and b such that f’(c)= f(b)-f(a) b-a

Theorems: Rolle’s Definition: If f is continuous over [a,b] and differentiable over (a,b) and f(a)=f(b), then there is at least one number c in (a,b) such that f’(c)=0

Theorems: EVT Definition: If f is continuous over a closed interval, then f has a maximum and minimum value in that interval

Strategy Pay attention to the wording!! IVT is the theorem NOT concerning derivatives MVT is about finding a value for which the instantaneous equals the average Rolle’s has asks you to find a value for which f’(c) equals 0 (**R0lle’s) Last strategy--Remember all the components of the theorems, because you can’t prove the theorem unless all the components are true: IVT: Continuity over a CLOSED interval MVT: Continuity over a CLOSED interval and differentiability over an OPEN interval Rolle’s Theorem: Same as MVT components but f(a) must also equal f(b) **THEOREMS CAN ONLY APPLY IF ALL OF THESE ARE TRUE

Is there a solution to x5 - 2x3 - 2 = 0 between x=0 and x=2? Example #1 Is there a solution to x5 - 2x3 - 2 = 0 between x=0 and x=2?

Solution The function is continuous over the closed interval, therefore IVT applies f(2) = 12 and f(0) = -2 Because 0 is a value in between f(a) and f(b), there is a value c on the interval (a,b) in which f(c)=0

Example #2 Find the value of c on the interval [0.5,2] that satisfies the MVT for f(x)= (x+1) / x

Solution The function is continuous ON THE INTERVAL [0.5, 2] and differentiable on the interval (0.5, 2), therefore MVT applies f’(x)= f’(c)= -1 f(2)-f(0.5) -1 -1 c² x² = -1 2-0.5 -1 is not in the domain = -1 C = 1, -1 c²

Example 3 Prove Rolle’s theorem is valid for f(x)= -x3+2x2+x-6 on the interval [-1,2]

Solution f(x) is continuous on [-1,2] and differentiable on (-1,2) f(-1)= -4 & f(2)= -4, therefore Rolle’s theorem is valid f’(x)= -3x2+4x+1 f’(c)= -3c2+4c+1 = 0 *have to use quadratic formula* c= 2±7 BOTH + & - VALUES INCLUDED IN DOMAIN 3

Helpful Links Theorems Quizlet: https://quizlet.com/174868503/calculus-theorems-flash-cards/ Tangent Line Review: http://gato-docs.its.txstate.edu/slac/Subject/Math/Calculus/Findting-the-Equation-of-a-Tangent-Line/Finding Linear Approximation Review: http://tutorial.math.lamar.edu/Classes/CalcI/LinearApproximations.aspx