DIFFERENTIAL CALCULUS

Which term refers to a relation that assigns exactly one element of set Y to each element of set X?

Function

Which best describes the domain of a function?

The set of all values for which the function is defined

What is the range of a function?

The set of all possible output values

Find the domain of f(x)=ln(x²−5x+6).

(-∞, -2) ∪ (3, +∞)

Find the domain of (x²−5x+6)/(x²+5x+4).

(-∞, -4) ∪ (-1, 2] ∪ (3, +∞)

What is the value a function approaches as x approaches a number?

Limit

What is required for a limit to exist?

Left-hand and right-hand limits must be equal

1^∞ is what kind of form?

Indeterminate form

lim (x→3) (x²−5x+6)/(x−3)

1

lim (x→5) (x²−5x+6)/(x−5)

DNE

lim (x→1) (2−x)tan(πx/2)

∞

lim (x→π/4) sin(2(x−π/4))/(x−π/4)

2

lim x→5⁺ of f(x)

25

lim (x→1.5) floor(x)

1

Conditions for continuity at x=a

All of these

Limit exists but ≠ function value

Removable discontinuity

Left and right limits differ

Jump discontinuity

Function grows unbounded

Infinite discontinuity

Value of k for continuity at x=2

1.5

a and b for continuity

a=3, b=-4

What does the derivative represent?

Slope of the tangent line

If derivative > 0

Increasing

Derivative changes + to -

Local maximum

Critical point definition

f'=0 or f' undefined

Derivative of y = x^x

x^x(1 + ln x)

f(x)=x³−5x+2, f′(2)

7

y′ of x² + xy − y² = 4

(2x − y)/(2y + x)

Average ROC from 1 to 4

13 m/s

Instantaneous ROC at t=4

22 m/s

Tangent line at x=1

y = 5x + 1

Max volume from sheet

486

Maximize x²y³ (x+y=10)

6

Min of 2x+3y (xy=6)

12

Max area with 100m fencing

1250 m²

dA/dt at r=5

20π

Ladder problem dy/dt

-0.75

Kite ds/dt

8

Balloon dy/dt

100

Find r when dA/dt=2 dr/dt

2/π

Second derivative meaning

Concavity

If f''>0

Concave up

Inflection point condition

f''=0 and changes sign

Inflection point value

-5

Second derivative of (x+1)(x−3)³

12(x−3)(x−1)

Third derivative of ln(x)

2/x³

What is a parametric equation?

x and y expressed in terms of a parameter

Slope dy/dx for parametric equations

(dy/dt)/(dx/dt)

dy/dx for x=t²+1, y=t³−t

(3t²−1)/(2t)

Second derivative d²y/dx² for parametric

(6t²+2)/(8t³)

Speed at t=1

5√5

Partial derivative meaning

Rate of change holding other variables constant

Notation ∂f/∂x

∂f/∂x

Mixed partial example

∂²f/∂x∂y

Clairaut theorem

∂²f/∂x∂y = ∂²f/∂y∂x

∂f/∂x at (2,1)

8

y′ of x³ − xy + y³ = 3

(3x² + y)/(x − 3y²)

∂²f/∂x²

2y

∂²f/∂x∂y

2x + 6y

∂²f/∂y²

6x − 6y