AP Calculus Unit 2 Differentiation - Definition & Fundamental Properties

University:
Whitman College
Course:
MATH-125 | Calculus I
Academic year:

2022
Views:

220

Pages:

2
Author:

hakulap1d

Unit 2: Differentiation-Definition & Fundamental Properties ● Derivative: The slope of a tangent line made at a certain point on the function. ○ ■ When trying to find the derivative of a function, plug in the function value, replacing x with x+h, and solve. ● Tangent line: A line that touches a function at one single point ○ The tangent line describes the slope of the function at one point ● Differentiable: A function is differentiable if the derivative for that function exists for every value of x. ● Second Derivative (F”(x)): Take the derivative of a function twice. ○ This means taking the derivative of a derivative. ○ Shows us the concavity of the function - whether the graph opens up or down ● When does a derivative not exist? ○ Cusp: a sharp point ■ This could be at the point of an absolute value function (abs(x)) ○ A point where the function is not continuous ○ Vertical Tangents ■ Ex. The left and right points at the edge of a circle ● Derivative Shortcuts ○ When finding the derivative of a function simply: ■ Multiply the exponent value by the X value ■ Subtract the exponent by one ● Sum and Difference Trigonometric Identities ● Product and Quotient Rule: Used for more complex derivatives ○ Remember: we cannot factor when taking the derivative. ● Derivative of ex ○ d/dx [ex] = ex ○ d/dx [eu] = eu * u’ ● Derivative of Natural Log (lnx) ○ d/dx [lnx] = 1/x ○ d/dx [lnu] = u’/u ● Derivative vs. Continuity ○ A function is continuous if it has no breaking points - where f(x) exists. ○ A differentiable function must be continuous. ○ A continuous function does not need to be differentiable ■ Example: cusp, vertical tangent, or a bend.