Limit Definition

d/dx = lim(∆x→0) f(∆x+x) – f(x) / ∆x

Finding points where tangent line to function is horizontal. (Horizontal tangent line to a function)

1. Take derivative

2. Derivative=0

3. Solve for x

4. Plug x into original for y

2. Derivative=0

3. Solve for x

4. Plug x into original for y

Equation of line normal to function at given point.

1. Take derivative

2. Plug given x value into derivative (for slope)

3. Negative reciprocal of slope

4. Plug into point-slope form

2. Plug given x value into derivative (for slope)

3. Negative reciprocal of slope

4. Plug into point-slope form

Evaluating Trig Functions: Sin

sin0 = 0

sin(π/6) = 1/2

sin(π/4) = √2/2

sin(π/3) = √3/2

sin(π/2) = 1

sin(π/6) = 1/2

sin(π/4) = √2/2

sin(π/3) = √3/2

sin(π/2) = 1

Evaluating Trig Functions: Cos

cos0 = 1

cos(π/6) = √3/2

cos(π/4) = √2/2

cos(π/3) = 1/2

cos(π/2) = 0

cos(π/6) = √3/2

cos(π/4) = √2/2

cos(π/3) = 1/2

cos(π/2) = 0

Evaluating Trig Functions: Tan

tan0 = 0

tan(π/6) = √3/3

tan(π/4) = 1

tan(π/3) = √3

tan(π/2) = U

tan(π/6) = √3/3

tan(π/4) = 1

tan(π/3) = √3

tan(π/2) = U

Evaluating Trig Functions: Csc

csc0 = U

csc(π/6) = 2

csc(π/4) = √2

csc(π/3) = 2/√3

csc(π/2) = 1

csc(π/6) = 2

csc(π/4) = √2

csc(π/3) = 2/√3

csc(π/2) = 1

Evaluating Trig Functions: Sec

sec0 = 1

sec(π/6) = 2/√3

sec(π/4) = √2

sec(π/3) = 2

sec(π/2) = U

sec(π/6) = 2/√3

sec(π/4) = √2

sec(π/3) = 2

sec(π/2) = U

Evaluating Trig Functions: Cot

cot0 = U

cot(π/6) = √3

cot(π/4) = 1

cot(π/3) = √3/3

cot(π/2) = 0

cot(π/6) = √3

cot(π/4) = 1

cot(π/3) = √3/3

cot(π/2) = 0

Find value of k so function f(x) is tangent to a line.

1. Find derivatives of both equations

2. Set derivatives = (slopes are =)

2. Set derivatives = (slopes are =)

3. Set originals = (graphs intersect)

4. Decide which equation is easier to solve for either k

5. Substitute k value into other equation (only x left)

6. Solve for x

7. Plug x value into equation to find k

Functions aren’t differential if…

-Vertical tangent lines

-Sharp turns

-Sharp turns

Derivative of: sin(x)

cos(x)

Derivative of: cos(x)

-sin(x)

Derivative of: tan(x)

sec²(x)

Derivative of: sec(x)

sec(x)tan(x)

Derivative of: cot(x)

-csc²(x)

Derivative of: csc(x)

-csc(x)cot(x)

Finding d²y/dx²

1. Solve for dy/dx

2. Take derivative of dy/dx

3. Plug in dy/dx into derivative of dy/dx (possible fraction buster)

2. Take derivative of dy/dx

3. Plug in dy/dx into derivative of dy/dx (possible fraction buster)

Average rate of change/average velocity

f(b)-f(a) / b-a

Instantaneous rate of change/instantaneous velocity

f'(x)

What values does average velocity = instantaneous velocity.

1. Set avg. & inst. equations =

2. Plug in equations as Y1 & Y2 into graphing calculator

3. Set window using given interval

4. Graph → calc → intersections

5. Make sure x values are within interval

2. Plug in equations as Y1 & Y2 into graphing calculator

3. Set window using given interval

4. Graph → calc → intersections

5. Make sure x values are within interval

Volume of Sphere

V = (4/3)πr³

Surface Area of Sphere

SA = 4πr²

Volume of Right Circular Cylinder (& General Prisms)

V = Bh

Volume of Square Pyramid

V = (1/3)Bh

Volume of Right Circular Cone

V = (1/3)πr²h