Unit 2 - Calculus and Physics

What is a Limit?

How is it read?

A limit allows us to evaluate the output (typically a undef value) of a func at a certain x value

lim / x —> c f(x) = L is read as “as x approaches the value c, the func f(x) will approach L (the limit)“

What MUST you do when determining a limit and why?

you MUST ALWAYS FACTOR the function given in a limit.

• The magic of the limit is by factoring the function

• If we do not, we will get a undefined result 

EX: if we plug 3 into x² - 9 / x - 3 we get 0/0 or undef

• If we factor it into (x-3)(x+3)/ (x-3) we get x+3 and now we get 6.

What is a Derivative and what does it give?

What is it VERY useful for?

A Derivative is a function that gives us the slope of the tangent line to the OG func

—> slope of tangent line at x=c means derivative of f(x) at x=c

• A Derivative is very useful to find the rate of change in smth

What are the 2 ways to get the slope of the tangent line of a graph?

What does this tell us?

we will use the graph of f(x) = x³ and the point x = 2

1. Limits (stinky)

   • we need to create a limit expression and factor it so (y₂ - y₁)/ (x₂ - x₁) which is f(x) - f(2)/ x - 2 and we can factor it to x² + 2x + 4 = 12

2. Derivatives

   • we will take x³ and use the power rule to make it 3x² and put in 2 = 12

basically Derivatives let us find the tangent line to the OG func using the limits of secant lines (approaching x=2)

What are the 3 ways you can write the derivative of f(x) at x = c?

1. f’ (c)

2. M_tan = lim x—>c (M_sec)

3. lim x —> c f(x) - f(c)/ x-c

(use this formula to get tangent line of graph without derivatives)

What is the equation definition for a derivative?

What happens to x-c as x gets closer to c?

look at photo

What is lebnitz notation?

How can f’ (c) be written?

Lebnitz’s notation is dy/dx (“the derivative of y (function output on graph) with respect to x“)

• f’ (c) can be written as dy/dx | x = c

In Physics, what will a secant line and a tangent line find?

secant = avg ____

tangent = instantaneous ______

What should you do when a Q asks for the derivative of a physics term?

Imagine a graph of that term over time and remember the slope.

EX: Derivative of position = velocity (cus velocity is the slope of a dt graph)

What are the 6 Derivative Rules?

(very important to know!)

1. The Derivative of a constant (horizontal line) is 0

2. If y= f(x) = mx+b, then derivative = m (cus linear slope)

3. Power rule: f’(x)= nx^(n-1)

4. If y = af(x) + bg(x), y’ = af’(x) + bg’(x)

5. Product Rule: if y = f(x)g(x), then y’= f’(x)g(x) + g’(x)f(x)

6. Chain Rule: if y = f(g(x)), then y’ = f’(g(x)) x g’(x)

Based on sm Theorem, where are the Maximum and Minimum points on a function that has been differentiated?

—> What does this mean for Physics Qs?

What are the 4 steps to find the min/max values of a graph given a interval?

• The Max/min points will show on either ends of a given interval as well as the point where the derivative slope = 0

—> This means in a Physics Q asking for max/min velocities within a given interval, we test the lowest, highest and point that makes derivative 0.

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EX: particle moving at v(t)= t³ - 27t + 60. What is min/max speeds in the interval 0 <= t <= 5s?

1. V’(t) = 3t² - 27 —> t = +-3

2. reject -3 cus outside of interval (get 2nd derivative ONLY to check concavity)

3. we must test t = 0 = 60m/s, t = 5 = 50 m/s and t = 3 = 6 m/s

4. max = 60 m/s and min = 6 m/s

What are the 3 steps to find the maximum/minimum VALUE of a function not given a interval?

• Remember that the max/min of a function is when the SLOPE OF THE DERIVATIVE IS 0.

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1. Derive the function and set the y to 0 cus slope of derivative will be 0 at max/min

2. once you get your x value, check the shape/Concavity of the OG graph (This is the ONLY time you should use a 2nd Derivative)

3. plug this x value into OG EQUATION and get result

What is a 2nd derivative?

WHEN SHOULD IT BE USED?

Why is acceleration a 2nd derivative of pos/disp?

The 2nd derivative is the derivative of a derivative. 

• It tells us the curvature/concavity of the function’s graph

• 2nd derivative is ONLY used to check CONCAVITY of graph (which tells us if a value is a:

  • Max if neg 2nd derivative (cus graph curves down)

  • Min if pos 2nd derivative (cus graph curves up)

Acceleration is a 2nd derivative of position/displacemet cus its the slope of the slope (accel is the slope of velocity)

if accel > 0, then position graph will CURVE up (velocity will be linear rise )

if accel < 0 then pos graph will Curve down

What does the 2nd derivative > 0 or <0 tell us?

• if f’’(x) > 0, then graph will be concave (curved) up with the tangent line below the graph.

• if f’’(x) < 0, then graph will be concave (curved) down with the tangent line above the graph.

1. What is something VERY important to know about a Derivative?

2. In a position vs time graph what does a Derivative find?

3. In a question giving rate (circle radius Q), what should you do with derivatives?

1. Since a Derivative is the slope of a tangent limit to a OG function, it finds the RATE of the OG function or the slope

2. In a pos vs time graph, a Derivative will find VELOCITY = slope of graph

3. Since derivative is a rate, you can express this rate already as a derivative and put it in with another equation that HAS BEEN DERIVED!!😱

How do you know when to differentiate?

You can tell when to differentiate based on vocab

• if eq given for position but asking for velocity

• if asking for a instantaneous rate

• asking for rate/ slope

• asking for max/min of graph

What is the 1 case where we have to differentiate twice?

• what should you also remember?

1. when the 1st derivative isn’t factorable (can only derive again if Q is asking abt shape of graph. IF NOT, then DONT derive twice and LEAVE IT)

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—> if you are doing 2nd derivative to find max value or need to plug value found in 2nd derivative, PLUG INTO THE 1st Derivative and NOT the OG eq

Can we do derivatives in the denominator of a fraction?

NOOOOOOOOOO!!!

→ we MUST bring the var up and then do it

f’(x) = 1/ x² —> f”(x) = x^-2 = -2x^-3 = - 2/x³

A curve exists such that the slope of the tangent to the curve has the same value as the y- coordinate at that point.

—> What two things can you relate together to form a equation?

We can say f’(x) = f(x)

we can do this because f’(x) is the derivative = slope of tangent line of f(x)

we also know f(x) gives us the y-value cus of pre-calc knowledge 🤯

What is the derivative of √2 ?

Explain why…..

the derivative of √2 is ZERO because it is a CONSTANT (cus no variables 🤯)

—> ALWAYS LOOK for a variable in a equation before differentiating. this will show if you can quickly put 0 for it.

Let f(x) be a function where f’(x) < 0 for all x >0. If x₁ and x₂ are in the domain of f(x) and 0<x₁<x₂, than what inequality best describes the relationship between f(x₁) and f(x₂)?

The relationship of f(x₁) and f(x₂) is f(x₁) > f(x₂)

• This is because the question told us the derivative (slope of tangent line) is negative for all positive x values, meaning we are in quadrant 1.

• We also know x₁<x₂

• we can conclude that the line goes down the more right we go

What is a

• Antiderivative/ Indefinite Integral

• Definite Integral

• Antiderivative/ Indefinite Integral = a function whos derivative gives the original function (basically its derivative gives what was in the integral…🙄)

• Definite Integral = used to find area under the curve

What is VERY important to remember about antiderivatives?

—> Based on this, what MUST you always include in an integral?

• Any constant can be added to the antiderivative cus a constant = 0 when differentiating. EX: (x⁴ + 3 is also an antiderivative of f(x) = 4x³)

• We must ALWAYS add the “constant of Integration“ (c) into our integral (x + c)

What is the notation of a integral?

What can be pulled out?

How do you Recheck an Integral? (Important!)

• always have dx where “x“ can be changed for whatever variable ferreira uses f(a) = da

• any constant values affecting a function can be pulled out

To RECHECK an integral, you can take the derivative of the antiderivative answer you got and see if its answer is the same as the OG expression in the integral

What should you always be asking for each term of a function when finding the antiderivative in a integral?

ALWAYS ask “what got differentiated to become this?“

What are the 6 steps to find integrals?

1. Factor and Simplify as much as possible

2. Apply any rules (most common is power rule)

3. Pull out any constant values in front or affecting a function

4. ALWAYS think “how did this become this?“ FOR EVERYTHING

5. DONT FORGET TO +C

6. Simplify

What are the 2 rules for Integrals? (so far)

1. Power Rule

   • for R ≠ -1

   • ∫x^rdx = x^r+1 / r+1 + C

2. Sum/Sub Rule

   • ∫(f(x) +/- g(x))dx = ∫(f(x)dx +/- ∫(g(x)dx

When can we use Integration by substitution?

What are the 5 steps to do Integration by Substitution?

We can only do this when the integral is in the form ∫kf(g(x)) x g’(x)dx

STEPS:

1. Identify the inside function (the one with exponents on the outside of brackets)

2. Say “let u = g(x)“ and then find the derivative of that inside func using LEBNITZ NOTATION and solve for dx

if g(x) = 4x²-1 then with LEBNITZ we say: du/dx = g’(x) = 8x —> dx = du/8x

3. Now replace dx with your solved derivative of inside func and replace the inside function with “u“ and LEAVE the OG g’(x) function (this should cancel out the denominator of dx = du/(derivative of inside func))

4. Apply THE POWER RULE!!!!! (usually when its just u^x du)

5. Get the new EQ (DONT FORGET the +c) and then put g(x) for “u”

What should you do when g’(x) is off from the derivative of the inside function?

(g’(x) should always equal derivative of the inside func cus g’(x) = g’(x))

In this unit, if you are off by a factor of k, where the newly derived inside function g(x) will be bigger than the OG g(x)

• This will also be on the denominator when plugging back into the OG integral

• Basically you just pull out that fraction… 

EX: we have ∫x(3x²-9)² dx, where the derivative of 3x²-9 = 6x  which is 6 times more than x.

—> when plugging back into integral we will do 1/6∫x(u)² du/x = 1/6∫(u)² du = -1/6u + c = -1/6(3x²-9) + c

What is a Definite Integral, and how is it different than an Indefinite Integral?

• What is a Definite Integral used for?

• What should you remember about Area?

A Definite Integral is different from a Indefinite Integral as we define a bound for it and it gives us an actual amount rather than a expression. (53 vs x²+2x+3)

• A Definite integral is used to give us an area under a graph (get displacement from a v-t graph)

• The Area given by an integral can be NEGATIVE 😳 (basically cus we can have stuff going west/south) 

What is the Fundamental Theorem of Calculus?

(tells us how a Definite Integral gets area)

Suppose F(x) is a antiderivative of f(x)

• The area bound by f and the x-axis on a <= x<= b is calculated as

  • Area = _a∫^b f(x)dx = F(b) - F(a)

• This means we first get the antiderivative then plug the value of b and the value of a and substract the attained values.

What are the 4 steps to solving a Definite Integral?

1. Check to see if the function is a composite or not. If its a composite like (2t-1)², then EXPAND IT (do not try the indefinite method)

2. Say “for f(x) = function in derivative, F(x) = antiderivative (use power rule)

3. Then calculate F(a) and F(b)

4. Say “By FTC, area = F(b)- F(a) = add your calc = answer“

What is important to know about calculating avg velocity with derivatives or integrals?

DO NOT use any derived and integrated equation

—> ALWAYS remember: velocity = Δx (change in displacement)/ Δt

—> this means you will use the ORIGINAL position function

cus avg = secant line on OG graph 🤯

What are the 3 steps to calculate the value for ABS value definite integrals?

—> Used when finding term with no direction (distance using disp function, or speed using velocity func)

1. find variable value to make the function = 0

2. split the abs value integral into

   1. integral with neg function with bounds: 0 to variable value to make func = 0

   2. integral with OG function with bounds: variable value to make func = 0 to rightmost bound in Q

tbh the best method for this is the graph the function and then make it abs value and graph it to truely see the area you need to calc