• Let’s think through the idea that derivative = slope of the tangent line
  • Lets start with: slope of a line
  • Slope is:
    • The steepness of a line
    • ^{y2 – y1}⁄_{x2 – x1}
    • ^rise⁄_run
    • m in y=mx+b
  • Instead of calling it ‘slope’, we can call it ‘derivative’
    • The slope of the line y=2x+3 is 2
    • The derivative of y=2x+3 is 2
  • Derivative = f'(x) = y’ = d/dx
    • The derivative of y=2x+3 is y’=2
  • We want to find the ‘slope’ / ‘derivative’ of a curve
    • We only know how to get the slope of a line!! What can we do?
    • ONE IDEA to piece together slope of a curve:
      • Let’s “smooth out” the curve and make it into a line by just connecting a straight line through the end points, and get the slope of that line
        • We can get the slope of the curve ^rise⁄_run a.k.a. ^{y2 – y1}⁄_{x2 – x1}
          • (Decide which mindset to use based on the information you have)
      • If we do not have a defined point #2, we create it as x+h
      • This is the “average rate of change” of the curve. You get a rough average using this method.
    • ANOTHER IDEA to piece together the slope of a curve:
      • Let’s drawn the line tangent to the slope, and get the slope of the tangent line
        • This is interesting! We can do the same approach as above with making two end points on the tangent line to get the slope of the tangent line. However, the tangent line only touches the slope at one point. In this case we actually found the slope at one specific point! This is the instantaneous rate of change of the curve at that exact moment.
    • derivative = slope of tangent = slope of line = rate of change =  ^rise⁄_run = f'(x)
  • If a line or curve is decreasing, then the ‘slope’/’derivative’ is negative
  • If a line or curve is increasing, then the ‘slope’/’derivative’ is positive
  • The ‘slope’/’derivative’ of a horizontal line or horizontal tangent line is 0
  • The derivative of a line looks like a horizontal line at y=slope=m
    • Line is y=mx+b
    • Slope is m
    • Derivative = slope
    • y’ = slope
    • y’ = m
    • The graph of y’ = m looks like a horizontal line at y=m

   

 

Old Stuff (feels so long ago!)

4.1.2

Pythagorean Trig. Identities

1. cos²(θ) + sin²(θ) = 1
2. tan²(θ) + 1 = sec²(θ)
3. cot²(θ) + 1 = csc²(θ)

CoFunction Trig. Identities

1. sin(π/2 – θ) = cos(θ)	cos(π/2 – θ) = sin(θ)
2. sec(π/2 – θ) = csc(θ)	csc(π/2 – θ) = sec(θ)
3. tan(π/2 – θ) = cot(θ)	cot(π/2 – θ) = tan(θ)

Reciprocal Trig. Identities

1. csc(θ) = 1/sin(θ)	sin(θ) = 1/csc(θ)
2. sec(θ) = 1/cos(θ)	cos(θ) = 1/sec(θ)
3. cot(θ) = 1/tan(θ)	tan(θ) = 1/cot(θ)

Quotient Trig. Identities

1. tan(θ) = sin(θ)/cos(θ)

cot(θ) = cos(θ)/sin(θ)

Odd / Even Trig. Identities

1. sin(-θ) = -sin(θ)	csc(-θ) = -csc(θ)
2. cos(-θ) = cos(θ)	sec(-θ) = sec(θ)
3. tan(-θ) = -tan(θ)	cot(-θ) = -cot(θ)