Derivatives and Differentiability (3.1-3.5)

What is the quadratic formula?

(-b±√b²-4ac)/2a

What do you do when you are trying to find the derivative of a function that involves a large fraction and you may have to foil the top?

1. After foiling, see if anything can cancel 2. If you have an x-value with the same power on the top and the bottom, then you will need to split apart the fraction 3. For example, for (x³-1)/x³ you would split the fraction up to be x³/x³ - 1/x³ 4. Make the x-value over the x-value equal one, and then make the x value come to the top of the fraction by making the exponent negative 5. Continue to do the steps to find the derivative and then make just the x-value and its negative exponent go on the bottom of a fraction for your answer, Ex. 3x⁻⁴ = 3/x⁴

What are the four situations when a function might not have a derivative?

1. Corners 2. Cusps 3. Vertical Tangent Line 4. Points of Discontinuity (Jumps, removable, infinite, and oscillating)

How do you sketch the derivative of a function?

1. Find out where the slopes are positive and negative 2. Find out where the slopes are equal to 0 (x-intercepts on the f(x) graph) 3. Find out the slopes of specific points on the f(x) graph and then you have points on the f'(x) graph

What do you do when you asked to find the tangent line for a curve at a specific point that has a specific slope?

1. Find the derivative of the curve 2. Set the derivative equal to the slope 3. Take your x value(s) and plug them into the original equation to get the y-value(s) 4. Create an equation using point-slope form

What do you do when asked to find the horizontal tangents of an equation?

1. Find the derivative of the equation 2. Set the derivative equal to zero 3. Take your x-value(s) and plug them into their original equation to get the y-value(s) 4. Create an equation using point-slope form

What do you do when asked to find the body's acceleration each time the velocity is zero and given a position equation?

1. Find the velocity and acceleration equations by taking the derivatives 2. set the velocity equation equal to zero 3. plug those x-values into the acceleration equation to get the speeds

How do you check graphically if you got the correct derivative?

1. Graph the original equation 2. Graph the nDer(

On a velocity graph, what is the trick to know where a particle is speeding up or slowing down?

1. If the line is approaching the x-axis then it is slowing down 2. If the line is going away from the x-axis then it is speeding up

How do you find the derivative of a function like (x² + 5x -1)/x²

1. Make everything else over x² 2. Make everything not a fraction (negative exponents) 3. Derive 4. Change back to fractions for ones with negative exponents

What do you do when asked to sketch the graph of a continuous function of f with f(0) = 1 and are given f'(x){2, x<2 and -1, x>2 (2 and -1 being the slopes of f(x))

1. Plot the point given from the value plugged in (0,1) 2. Sketch the appropriate slopes on the left and right hand side based on the signs

When asked to write a function for an equation of a shape as a value of something involving the circumference and area of a circle, what do you do?

1. Solve for the r-value 2. plug into the other equation 3. Break apart the fraction, then derive it, then multiply it back

When asked to write a function for an equation of a shape as a value of something involving an equilateral triangle, what do you do?

1. Solve for the triangle if it is a special triangle 2. split the bottom up into a two fractions, s/2 and s/2 3. plug in the values, split up the fraction, derive, then multiply

What do you do when you have something like 1 + 1/x + 1/x² + 1/x³ and you are trying to find the derivative?

1. change the fractions with exponents on the bottom to variables with negative exponents 2. derive them 3. turn them back into fractions

What do you do when asked to find something like d⁷²⁵/dx⁷²⁵(sinx)?

1. find out the pattern of the trig function being derived and find out how many times it takes to be derived to get back to the original function 2. then set it up to be something like d/dx sinx = cosx when 4k+1 (1 there because the number is divisible by 5) 3. set the 4k+1 equal to 725 and now you know that at that number of derivative you will have that specific cosx

What do you do when trying to derive a function and you have something like ³√x⁴ on top of a fraction?

1. give the variable a fraction as an exponent 2. quotient rule 3. keep negative powers on top if there are any

What do you do when you are asked to find the derivative of something like this? 1+cosx/sinx

1. make them two seperate fractions (1/sinx)(cosx/sinx) 2. use the trig identities to simplify them 3. derive

What do you do when you have a fraction involving two smaller fractions and you are trying to find the derivative?

1. multiply by the reciprocal of the denominator after combining all terms 2. quotient rule

How do you find the equation of the tangent and normal line of a function involving a trigonometric equation and you are given a x-value?

1. plug in the x-value into the original equation to get the y-value 2. find the derivative of the function 3. plug in the x-value to get the slope at that point 4. use point-slope form to create the equation 5. flip and change the sign of the slope for the normal equation

How do you find the equation of the horizontal tangent of a function involving a function with trigonometric values?

1. set the derivative of the function equal to zero 2. change the trig values using trigonometric identities 3. get a fraction and then split them out into two fractions being multiplied by each other 4. with 1/sin²x (2cosx-1) you divide the zero by the 1/sin²x 5. then add the 1, then divide by 2 6. find the positive value on the unit circle that gives you the x-value of 1/2

When asked to describe the motion of something and asked at what times it changes directions what do you do?

1. set up a number line 2. write in the values you got for finding when it stands still 3. plug in numbers that are between the values into the velocity equation to get either positive or negative values

What do you do when you have an extremely large function on the top of a fraction with a small function on the bottom and you are trying to find the derivative?

1. split the fraction up into pluses and minuses with all smaller fractions having the same denominator 2. simplify the fractions (subtract power on top by power on bottom, may get negative exponents) 4. derive 5. turn values with negative exponents back into fractions

What is the equation for area of an equilateral triangle?

1/2bh

What is the equation for circumference of a circle?

2πr

With a 30, 60, 90 triangle, what values correspond to each angle?

30: x 60: x√3 90: 2x

With a 45, 45, 90 triangle, what values correspond to each angle?

45: x 90: x√2

What does local linearity state?

A function is differentiable at a point if it looks linear when zoomed in upon.

With position, velocity, and acceleration, what are derivatives of each other?

Acceleration is the derivative of velocity, and velocity is the derivative of position.

What do you do when an original equation has something like x³/3 or x²/2 and you are trying to find the derivative?

Break apart the fraction by making it a fraction times a fraction. For example, for x³/3, make it 1/3 × x³. Then finding the derivative will be much easier.

If a function has a derivative - meaning the slope exists - the function is said to be what?

Differentiable

What do you do when you have a variable with a fraction as an exponent in front of a larger function?

Distribute it with the other function

What are the free fall constants for earth in English and Metric units.

English units: g = 32 ft/sec² Metric units: g = 9.8 m/sec²

What is a trick you can do when solving for the derivative for a function and you need to cancel something?

Factor out a negative

When asked to find out how fast water is running out of something at a specific time, what do you do? How do you find the average rate?

For a specific time you have to find the IROC using dy/dx or finding the lim h→0. To find the average rate you have to get two points and you get them by plugging in how much water there is at time zero and how much water there is at time ten. You plug the values into the equation.

What do you do when you have three functions being multiplied by each other and you have to find the derivative?

Group the first two things together and then do the product rule.

What is the Derivative of a Constant Function rule?

If a function has a constant value (c), then the derivative is equal to 0. Ex. y=3, y'=0

What is the Power Rule for Negative Integers Powers of x?

If n is a negative integer in x^n, then the derivative is nx^n-1.

What is the Power Rule for Positive Integers Powers of x?

If n is a positive integer in x^n then the derivative is nx^n-1. Ex. y=x⁵, y'=5x⁴

What is the Constant Multiple Rule?

If there is a constant value being multiplied by a number with an exponent, then the derivative is the constant value multiplied by the derivative of the number with an exponent. Ex. y=6x⁵, y'=6×5x⁴=30x⁴

What is the Quotient Rule?

If two functions are being divided, then the derivative is the bottom function times the derivative of the top function minus the top function times by the derivative of the bottom function, all over the bottom function squared.

What is the Product Rule?

If two functions are being multiplied by each other than the derivative is the first function times the derivative of the second function plus the second function times the derivative of the first function.

What is the Sum and Difference Rule?

If you have two different functions that are being added or subtracted then the derivative is the derivative of the first function ± (same sign) the derivative of the second function. Ex. y=x³+x⁷, y'=3x²+7x⁶

What does a cusp look like when you zoom in far enough?

It turns into a vertical line

What does it mean when it asks for you to find the first four derivatives of a function?

Keep finding the derivative of the equation four times.

If you are trying to derive a function with something like ³√x⁴, what do you do?

Make it an x with a fraction as an exponent. Number in root on bottom, and power of variable on top. Ex. x^4/3

If you have a number with a negative exponent, what do you do?

Make it be part of a fraction but on the bottom, where it is positive.

What can you do to roots in order to make it easier to find the derivative?

Make the thing being rooted, have a fraction

How do you type in the NDER for your calculator?

Math, 8 (nDeriv(, d/dx(function)x=value

When trying to find the derivative of a function and you have square roots in the function how do you get rid of them?

Multiply by the conjugate over the conjugate. Ex. the conjugate of (√x+h)-(√x) is (√x+h)+(√x)

When trying to find the derivative of a function and you have fractions in the function how do you get rid of them?

Multiply by the denominator of the "little" fractions/ denominator of the "little" fractions.

When you are given an equation to plug into your calculator to find if there are any points of differentiability, what do you do with absolute values or roots?

Set the stuff in the absolute values or roots equal to 0 and then solve. There is most likely going to be discontinuity at that point.

What is the definition of speed?

The absolute value of velocity. Speed = abs(v(t)) = abs(ds/dt)

When comparing a right-hand and left-hand derivatives, what has to happen in order for the point P (1,1) to be differentiable?

The lim x→1⁻ f'(x) has to equal the lim x→1⁺ f'(x)

If you want to find out when a particle is at rest, what equation do you set equal to zero?

The velocity function

What is the relationship between the f(x) graph and the f'(x) graph?

The y-coordinates of the f'(x) graph are the slopes of the f(x) graph at any specific value of x. The slopes of f(x) are the y-coordinates of the f'(x) graph.

When asked to write a function for an equation of a shape as a value of something involving a square inscribed in a circle, what do you do?

Use pythagorean theorem to get your value

What is the equation for volume of a cube?

V=s³

What do you do when you are doing something like the product rule or quotient rule and you have trigonometric values that are being multiplied together, what is something you can do to simplify it?

When there are identities that have things in common. Ex. 2tanxcosx = 2sinx/cosx×cosx/1 = 2sinx

When do you change negative exponents back to fractions for you answer?

When they are not part of a fraction.

When given a velocity graph and asked to graph the speed graph of it, what do you do?

You reflect the negative portion of the velocity graph about the x-axis.

What does the notation dA/ds mean?

You're taking the derivative of A with respect to s.

What is the notation for acceleration?

a(t) = v'(t) = s"(t) = d²t/d²v a(t) = dv/dt = d²s/dt²

What are the derivatives of the trig functions?

d/dx sinx = cosx d/dx cos x = -sinx d/dx tan x = sec²x d/dx cscx = -cscxcotx d/dx secx = secxtanx d/dx cot x = -csc²x

What is the notation for evaluating a rate of change of a function at a specific value? Ex. rate of change of A at r=1 and r=8

dA/dr(line)r=1

If f has a derivative at x=a, then what?

f is continuous at x=a

What is f'(x)?

f prime of x

If a and b are any two points in an interval on which f is differentiable, then what?

f' takes on every value between f'(a) and f'(b).

What formula does the calculator use? The Symmetric Difference Quotient

f'(a) = lim h→0 (f(x+h)-f(x-h))/2h

What is the equation for find the derivative at a point?

f'(a) = lim x→a (f(x)-f(a))/x-a (leave the a in the equation until something can be canceled, then plug in x, then plug in the value of a)

What is the equation for finding the derivative?

f'(x) = lim h→0 (f(x+h)-f(x))/h

What is the notation for jerk?

j(t) = da/dt = d³s/dt³

How do you find instantaneous velocity?

lim ∆→0 (f(t+∆t)-f(t))/∆t

What is the notation for position?

s(t)

What are all of the trig identities as fractions?

sin = 1/csc cos = 1/sec tan = 1/cot csc = 1/sin sec = 1/cos cot = 1/tan

On the unit circle, what do all of the trigonometric functions represent?

sin = y-value cos = x-value tan = y-value/x-value csc = 1/y-value sec = 1/x-value cot = x-value/y-value

What are the pythagorean identities?

sin²x + cos²x = 1 1 + cot²x = csc²x tan²x + 1 = sec²x

What is another way to write cot and tan using sin and cos?

tan = sin/cos cot = cos/sin

What is the notation for velocity?

v(t) = s'(t) = ds/dt = s'

What are the other ways to notate the derivative of a function?

y' dy/dx df/dx d/dx f(x)

What is the equation for area of a circle?

πr²