Derivatives

First Derivative

Velocity

Second Derivative

Acceleration

Kinematic Variables

• Position
• Time
• Velocity
• Acceleration

Kinematic Equations

Assumes constant acceleration.

• V = V0 + at
• x = x0 + .5(V0 + V)t
• V^2 = V0^2 + 2a(x - x0)
• x = x0 + V0t + .5at^2

Errors

Error

How far the measured value is from the actual value

Measured value - true value

Percent Error

((experimental value - true value) / true value) * 100

Keep the sign! It tells you whether you are too high or too low.

Uncertainty

How far off the measured value could be from the true value.

Like a scale fluctuating in the 10ths place, stop after including the first uncertain digit.

When using a ruler, measure 1 decimal place past the tick marks.

Write down everything you are certain of plus one estimate.

When recording measurements, record all certain measurements, plus 1 uncertain digit.

• These are all significant.

Rules of Significant Figures

Can underline to denote the last significant figure instead of rounding.

Hold off on rounding until the end.

When a problem includes multiple math operators, do them sequentially and apply the appropriate sig fig rule.

Multiplication/Division

Keep the same number of significant digits as the term with the fewest.

Addition/Subtraction

Round off to the # of decimal places as the term with the fewest.

Counting Significant Figures

• Zeroes before non-zeroes are not significant.
• Zeroes between non-zeroes are significant.
• Zeroes after non-zeros are significant if the number has a decimal.

Exact numbers have an infinite number of significant figures. Such as using the constant 2 as the denominator for an average - infinite sig figs.

For constants you should always use enough digits (sa. pi) to not limit the significant figures of your answer.

Derivatives to Know

Don't forget the chain rule!

• (X^n)' = nx^(n-1)
• (sin(kx))' = kcos(kx)
• (cos(kx))' = -ksin(kx)
• (e^(kx))' = ke^(kx)

Integrals to Know

An integral from a to b denotes a sum of the area from a to b. Where f(x)dx is the integrand.

• $x^ndx = (x^(n+1))/(n+1) + C
• $coskxdx = (sin(kx))/k + C
• $sinkxdx = -(cos(kx))/k + C
• $e^(kx)dx = (e^(kx))/k + C

To Know

Area of a Cylinder

dV = pir^2dy where dy is the height of a disk.

Volume of a Cone

V = $[0>h]pir^2dy

• h is the height of the cone.