Calculus v2.1
The left hand limit of f(x) is 2, the right hand limit is 9, what is the limit?
If both the right- and left-hand limits are not equal, then the limit does not exist.
When is the chain rule used?
On composite functions
Dx(tan(x)) =
sec²(x)
If f(x) is undefined at c, how can you find the limit?
- By creating a table and inputting values close to c - By graphing f(x), - By using algebra to simplify the function
Dx(cos(x)) =
- sin(x)
Dx(csc(x)) =
-csc(x)cot(x)
Dx(cot(x)) =
-csc²(x)
[lim x→±∞] x/1 + x^k = where k is an integer greater than 0
0
What are the steps for finding the derivative of f(x) = x⁻²
1.) Apply power rule... -2x⁻²⁻¹ 2.) Apply negative exponent rule (x⁻ⁿ = 1/xⁿ)... -2/x⁻³
What are the steps for applying the chain rule?
1.) Determine inner and outer functions, the inner function is usually (but not always) found inside of parenthesis. It is the function that is performed first. 2.) Take the derivative of both functions 3.) Write the derivative of the first function, evaluated at the second function. Then multiply by the derivative of the second function.
What are the steps for implicit differentiation?
1.) Differentiate both sides, differentiating x terms as normal 2.) Differentiate the Y ONLY terms and add y' next to each 3.) Differentiate x and y terms using product/quotient rule, add y' next to each y term differentiated 4.) Isolate y' terms 5.) Solve for y'
How do you find stationary points?
1.) Find f' 2.) Set f' = 0 and solve
How do you evaluate the area under a curve on a simple definite integral?
1.) Find the antiderivative of the function (F(x)) 2.) Evaluate at lower limit 3.) Evaluate at upper limit 4.) Area = F(upper) - F(lower)
How do you use the mean value theorem for integrals?
1.) Find the average value of the function using integrals 2.) Set average value equal to the function and then solve for x
How do you find the local maximum and minimum?
1.) Find the critical points 2.) Evaluate f at the critical points, highest is max, lowest is min
How does Newtons method work?
1.) Make a guess for x by plugging it in to f(x) and seeing if = 0 2.) Find the x intercept of the tangent line at that point using x intercept = x - f(x)/f'(x) 3.) Repeat using the x axis / tangent line intersection as the new "guess" input until sufficiently close to 0
What are the steps for solving a basic differential equation when the slope and a point are given?
1.) Set y = the slope and integrate 2.) Plug in the point to solve for the C 3.) Solve for C 4.) Put solution for C into integrated equation
What is the first derivative test?
1.) Take derivative 2.) Set derivative =0 to find critical points 3.) Determine where f is increasing/decreasing 4.) If f'(x) increasing on one side of the point, and decreasing on the other, the point is a local extreme 5.) Plug points into f(x) to determine extremes
How do you find the *equation of the tangent line* to a curve at a given point?
1.) Take the derivative of f(x) 2.) Input the given x value into f'(x) to find the slope 3.) Put into point slope form y - y₁ = m(x - x₁) 4.) Solve for y to put into slope intercept form
How do you use the mean value theorem?
1.) Take the derivative of the function 2.) Find Δy/Δx (slope of secant line between the interval points) 3.) Set secant line slope equal to the derivative 4.) Solve for x 5.) Make sure that all solutions to x are within defined interval
How do you find where x is increasing or decreasing on an interval
1.) take the derivative of f(x) 2.) Use the first derivative test to find the zeros of f'(x) 3.) Use test points to test *in between the intervals* to see where f(x) is positive or negative
Dx(sec(x)) =
1/cos(x) OR sec(x)tan(x)
The x intercept of the tangent line (for Newton's method) is equal to
= x - f(x)/f'(x)
What is a removable discontinuity? What does it look like on a graph?
A discontinuity that can be factored out of the function. It is usually a hole in the graph.
What is a non-removable discontinuity? What does it look like on a graph?
A discontinuity that cannot be factored out of the function. It is usually a jump or asymptote in the graph.
What is a composite function?
A function which is containing another function within it
What is the second derivative test?
A method for determining local extrema and concavity points of a function 1.) Do the first derivative test to find stationary points (c) 2.) Take the second derivative 3.) If f''(c) < 0 then f(c) is a local maximum of f, If f''(c) > 0 then f(c) is a local minimum of f *note that >0 is min and <0 is max* 4.) Set f''(x) = 0 and solve to determine concavity using points between zeros
What is an inflection point
A point where f is concave up on one side of c, and concave down on the other side
What is a singular point?
A point where the graph of f has a sharp corner, vertical tangent, or jump
What is a secant line?
A secant line is a straight line which intersects a curve at two points, whose slope is the *average rate of change* between x₁ and x₂
What is a tangent line?
A secant line is a straight line which touches a curve at one and only one point, whose slope is the *instantaneous rate of change* at this point.
What are definite integrals used for?
Calculating the area under a curve
What is a useful trick for finding the limit as x approaches infinity of a rational function
Divide the numerator and denominator by x to the highest power of x that appears in the denominator, then solve the limit
Critical points may be
End points, stationary points, singular points
(T or F) For a function to be continuous on a *closed interval [a,b]*, the endpoints must have a defined limit
False, a only needs to be right continuous, and b only needs to be left continuous [a,b]
What does Newtons method for?
Finding the roots (zeros) of a function
What is the constant function rule?
If f(x) = k where k is *constant* then f'(x) = 0 since there is *no slope*
What is the identity function rule?
If f(x) = x then f'(x) = 1
What is the constant multiple rule?
If the function has a constant multiple k, then... f'k*(x) = k*f'(x) meaning that *the constant can be passed across the derivative operator as it does not change the slope*
How can you use the... lim h→0 [f(x + h) - f(x)/h] formula to find the derivative?
Leave the x variable in the function and perform the function as usual, factoring h terms out. The resulting function is the derivative.
A - symbol above c is a ________ hand limit, meaning x is near but to the __________ of c.
Left hand limit Values to the left (smaller) than c
If the derivative of a function is positive on one side of a critical point, and negative on the other, the critical point is a:
Local extreme
Does f(x) have to be defined at c in order for the limit to exist?
No
A + symbol above c is a ________ hand limit, meaning x is near but to the __________ of c.
Right hand limit Values to the right (larger) than c
What is u-substitution?
Some magical integration trick that undoes the chain rule
Explain and formulate the chain rule
The derivative of a composite function is the derivative of the outer function, evaluated at the inner function, times the derivative of the outer function (f ∘ g)'(x) = f'(g(x))*g'(x)
Explain the fundamental theorem of calculus
The derivative of a definite integral of F(x) is equal to F(x) evaluated at the upper limit, subtract F(x) evaluated at the lower limit if variable.
When given a function where the input is time and the output is distance, the first derivative will give ____________ and the second derivative will give ____________.
The first derivative will give velocity, the second derivative will give acceleration.
How do you find the horizontal asymptote of a function?
The horizontal asymptote is the limit of the function as x approaches ∞ Hint: use highest power of x (HPOX) trick
When deriving a composite function requires more than one chain rule call, always begin with...
The innermost functions
Summarize the mains limit theorem
The limit may be processed on the terms either before OR after the (+, -, *, /, ^) is performed
How do you find the vertical asymptote of a function?
There is often a vertical asymptote at a point where the denominator is equal to 0
When does the second derivative test fail?
When f''(c) = 0 at a stationary point
When and how do you use the chain rule in the FTC?
When the function is composite, the final answer(s) once x(upper/lower) have been input into the function, must be multiplied by the derivative of x(upper/lower) respectively.
How do you find the slope of the tangent line to a curve, at a given point.
Where a = x₁ of the given point
When you can see that two or more rules will be needed to take the derivative, it is wise to...
Write out the steps and rules you will use in order, before starting
How do you find the differential of a function?
You legit just ****ing multiply the derivative by dy
When using the... lim h→0 [f(x + h) - f(x)/h] formula, what should your goal be? How do you do it?
Your goal should be to factor an h from the numerator and denominator so that the limit can be defined at h = 0 You do this by performing the function calls in the numerator and combining like terms. If the non-h terms are canceled out, it is easy to factor an h.
Quotient rule: (f/g)'(x) =
[f'(x)g(x) - g'(x)f(x)]/g²(x)
What is the special trig limit involving *cos*
[lim t→0] 1 - cos(t)/t = 0
What is the special trig limit involving *sin*
[lim t→0] sin(t)/t = 1
When integrating, be sure to add the:
arbitrary constant
Dx(sin(x)) =
cos(x)
Sum and difference rule: (f ± g)'(x) =
f'(x) ± g'(x) the derivative of the sum/diff of two functions is equal to the sum/diff of the derivatives
f(x) is continuous at c if...
f(c) = [lim x→c] f(x)
f(x + ∆x) =
f(x) + f'(x)*∆x
Knowing that f'(x) is a function whose y values are associated with the slope of f(x), it is safe to assume that the largest part of the f'(x) is the part where...
f(x) has the steepest slope
Product rule: (f * g)'(x) =
f(x)g'(x) + g(x)f'(x) The derivative of the product of two functions is equal to the first function times the derivative of the second function, plus the second function time the derivative of the first
Use the _________________ rule to find an antiderivative
generalized power rule
What is a stationary point?
if f'(c) = 0, c is stationary point (tangent line is horizontal)
What is the power rule?
if f(x) = xⁿ, then... f'(x) = n*xⁿ⁻¹
When y''(c) is equal to zero, c is a possible
inflection point
How do you find the average value of a function?
let u = upper limit, l = lower limit 1/u-l * ∫f(x)dx = average
What is the alternative formula to... lim h→0 [f(x + h) - f(x)/h] What is the goal when using this one?
lim x→c [f(x) - f(c)/(x-c)] The goal is to factor out (x-c)
The ______ and _________ rules can be very useful when evaluating derivatives of trig functions
product and quotient, because the power rule does not work on trig functions
Explain the general power rule theorem
r = the exponent you add one to r, and then divide x by the new r.
Dividing by a fraction is the equivalent of multiplying by...
the reciprocal of the fraction
The limit of a trig function as it approaches x is equal to
the same trig function evaluated at x
What is the mean value theorem
there exists some c in the interval where Δy/Δx = f'(x)
Intuitive definition of a limit
to say that the limit as x approaches c, of f(x) is equal to L, is to say that *when x is near but different from c, then f(x) is near L*
What points are candidates for points of inflection?
where f''(x) = 0 or is undefined
The slope of a line is equal to
Δy/Δx = y's per x
∆x = ∆y =
∆x = x₂ - x₁ ∆y = f(x₂) - f(x₁)
