Ch. 3 (Finney)

Type in the equation for y1 and then press MATH 8

What buttons do you press on your calculator to find the derivative of a function at a specific x-value?

perpendicular

What does NORMAL line mean?

2nd deriative

What does d²y / dx² mean ?

Slope of the tangent line

What does the "derivative" refer to when looking at a graph?

WHERE the object is at a certain time

What does the position equation tell you about an object?

0

What is the derivative of a constant?

-sin x

What is the derivative of cos x ?

-csc²x

What is the derivative of cot x ?

-csc x cot x

What is the derivative of csc x ?

f'(x) + g'(x)

What is the derivative of f(x) + g(x) ?

(f' × g - f × g') / g²

What is the derivative of f(x) / g(x) ?

nxⁿ⁻¹ (move the power out front and subtract 1 from the power)

What is the derivative of f(x) = xⁿ ?

f' × g + f × g'

What is the derivative of f(x)×g(x)?

sec x tan x

What is the derivative of sec x ?

cos x

What is the derivative of sin x ?

sec²x

What is the derivative of tan x ?

AVERAGE = slope between two endpoints INSTANTANEOUS = derivative at a point

What is the difference between average and instantaneous rates of change?

1) Find the derivative of the function (if not already given) 2) Plug the given x-value into the derivative to find the slope 3) Use the slope and the point to write the equation of the line.

What steps should you take to find the equation of a tangent line?

when velocity and acceleration are different signs

When is speed DECREASING?

when velocity and acceleration are the same sign

When is speed INCREASING?

sharp turn or cusp

A graph with a _________ will NOT have a derivative at that point.

What does it mean to "use the linear approximation of f(x) to estimate f(3.2)"?

Find the equation of the tangent line using a point and the slope(derivative) and then plug in 3.2 for x.

Use the slope formula for two points on both sides of the x-value in question

How do you estimate the slope of a curve at an x-value when given x-values around the actual point?

1) Find the derivative and set the numerator equal to 0 2) Plug your answer into the original equation to find the other coordinate

How do you find points of horizontal tangency (where there is a horizontal tangent line)?

1) Find the derivative and set the denominator equal to 0 2) Plug your answer into the original equation to find the other coordinate.

How do you find points of vertical tangency (where there is a vertical tangent line)?

absolute value of velocity

How is speed calculated?

continuous

If a function has a derivative at an x-value, then it must be __________ at that x-value.

take the 2nd derivative of position

If given the position function, how do you find the acceleration function?

take the derivative of position

If given the position function, how do you find the velocity function ?

take the derivative of velocity

If given the velocity function, how do you find the acceleration function?

m∧(5) (x)

If you have a function called m(x) and you want someone to take the 5th derivative, what notation should you use?