Derivatives

First Derivative

Velocity

Second Derivative

Acceleration

Kinematic Variables

• Position
• Time
• Velocity
• Acceleration

Kinematic Equations

Assumes constant acceleration.

• $$V = V_0 + at$$
• $$x = x_0 + \frac{1}{2}(V_0 + V)t$$
• $$V^2 = V_0^2 + 2a(x - x_0)$$
• $$x = x_0 + V_0t + \frac{1}{2}at^2$$

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!

• $$[X^n]' = nx^{n-1}$$
• $$[\sin(kx)]' = k\cos(kx)$$
• $$[\cos(kx)]' = -k\sin(kx)$$
• $$[e^{kx}]' = ke^{kx}$$

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.

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

To Know

Area of a Cylinder

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

Volume of a Cone

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

• h is the height of the cone.