calculus equations

Question 1

tangent line

Answer 1

f (x) =‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ (x, y)
f ‘(x) = m
y₂ - y = m(x₂-x)

Question 2

normal line

Answer 2

f (x) =‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ (x, y)
f ‘(x) = m
y₂ - y = (-1/m)(x₂-x)

Question 3

concavity values

Answer 3

concave up ⟹ f ‘‘(x) < 0‎ ‎ ‎ ‎ ‎ (negative)
concave down ⟹ f ‘‘(x) > 0 ‎ ‎ ‎ ‎ (positive)

Question 4

inflection points

Answer 4

f ‘‘(x) = 0

Question 5

stationary points

Answer 5

minimum/maximum
f ‘(x) = 0

Question 6

how to do first derivative test

Answer 6

‎ ‎ ‎ ‎ ‎ ‎ ‎‎ +/-‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ +/-
|————-|————-|
f ‘(x₁)‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ x‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ f ‘(x₂)

Question 7

how to classify from first derivative test

Answer 7

⋂ shaped (+y) with + = left and - = right ⟹ positive
⋃ shaped (-y) with - = left and + = right ⟹ negative