2.1: The Derivative and Tangent Line Problem

Question 1

Definition of Derivative, f’(x): The definition of f at any x is given by

Answer 1

f’(x) = lim Δx ->0 (f(x + Δx) - f(x) / Δx)

Question 2

Notation for Derivative: Given y = f(x)

Answer 2

dy/dx = y’ = f’(x) = lim Δx->0 (f(x + Δx) - f(x) / Δx)

Question 3

Slope of the Tangent Line is given by the Derivative

Answer 3

m(tan) = f’(x) = lim Δx->0 (f(x + Δx) - f(x) / Δx)

Question 4

Point Slope form

Answer 4

y - y1 = m(x - x1), point = (x1, y1), slope = m

Question 5

Slope Intercept form

Answer 5

y = mx + b, y-intercept = (0,b), slope = m

Question 6

Alternate form of derivative (formula)

Answer 6

f’(c) = lim x->c (f(x) - f(c) / x - c)

Question 7

One-sided derivatives (the derivative from the left)

Answer 7

f’_(c) = lim x->c^- (f(x) - f(c) / x - c)

Question 8

One-sided derivatives (the derivative from the right)

Answer 8

f’+(c) = lim x->c^+ (f(x) - f(c) / x - c)