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

2. 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)

3. 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

4. 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

5. 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)

6. 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?

7. 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)

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

9. 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

10. 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

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

12. 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

13. 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?

14. 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

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

16. 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

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

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

19. Helpful Links Theorems Quizlet:
Tangent Line Review:
Linear Approximation Review: