Unit 1
Limit: If f(x) approaches the number L as x approaches (but is not equal to) the number a from
both sides of a, then we say that the limit of f(x) as x approaches a is L. It is a y-value that x
approaches (but doesn’t necessarily touch) as it gets closer to a certain x-value.
● The notation lim f(x) x -> a is read as “The limit, as x approaches a, of f(x)”.
○ lim f(x) x -> a- is read as “The limit, as x approaches a from the left, of f(x)”.
○ lim f(x) x -> a+ is read as “The limit, as x approaches a from the right, of f(x)”.
To find limits using a graph, find the limit from the left and from the right. If they are equal, then
that value is the limit. If they are not equal, the limit does not exist.
● Look at the value the function approaches from each side, not the exact point value if
there is one.
○ However to find f(a) of a function, look for the closed point on the graph where x
is equal to a.
For piecewise functions, plug a into the appropriate function (or both depending on question).
● To graph a piecewise function in Desmos, write out the function then put the constraint in
curly brackets. For example: f(x) = x - 2 {x < 3}
For algebraic limits, plug a into the function.
● Keep in mind for certain algebraic limit questions, such as x^2 - 25 / x + 5 as x
approaches -5, x^2 - 25 is equal to (x + 5)(x - 5). The x + 5 cancels out in the numerator
and denominator leaving only x - 5, which you can plug -5 into which will give you -10.
● Another example is (sqrt x) - 9 / x - 81 as x approaches 81. Here, multiply the numerator
and denominator by the conjugate of (sqrt x) - 9, which (sqrt x) + 9. This will give you x -
81 / (x - 81)((sqrt x) + 9). After cancelling out x - 81 in the numerator and denominator,
you are left with 1 / (sqrt x) + 9, which you can plug 81 into which will give you .
Continuous Function: A function is continuous if:
● There are no holes, gaps or jumps
● f(a) is defined (the function has a value at point a)
● lim x -> a f(x) exists (the limit approaches the same value from both sides)
● lim x -> a f(x) = f(a) (the limit equals the function at that point)
To find the simplified form of the difference quotient for f(x), plug f(x) into f(x + h) - f(x) / h (h is an
undefined constant).
● For example, for f(x) = x^2, the difference quotient would be (x + h)^2 - x^2 / h, which
would factor into x^2 + 2xh + h^2 - x^2 / h, which would become 2xh + h^2 / h, which
would become 2x + h.
● Given a table, you would take the simplified form of the difference quotient and plug in
the given values of x and h.
To find average rate of change, use m = y2 - y1 / x2 - x1
, ● m is slope
● x1 and y1 are the first ordered pair
● x2 and y2 are the second ordered pair
A derivative, f’(x) or d/dx, of a function is the slope of a tangent line. It is also considered a limit.
● A tangent line is a line that touches a curve at a single point.
● Ways to write a derivative: dy/dx, y’, f’(x), d/dx [f(x)], Dx [f(x)]
To find a derivative of a function, simplify the difference quotient for f(x) then find the limit as h
approaches 0 (plug 0 in for h).
● There are certain rules you can use to find derivatives more efficiently.
● A derivative of a constant is zero.
● The limit cannot be infinity when taking a derivative of a function.
A derivative can only exist if:
● The function is continuous
● The function is smooth (no holes, gaps, jumps, sharp points or corners)
● The function does not have a vertical tangent line
For two functions, f(x) and g(x), say lim x -> a f(x) = A and lim x -> a g(x) = B. The lim x -> a (f(x)
+ g(x)) is the same as lim x -> a f(x) + lim x -> a g(x), which equals A + B. This also works for
subtraction.
● Sum/Difference Rule: d/dx [f(x) ± g(x)] = f’(x) + g’(x)
○ The derivative of the sum or difference of two functions is the sum or difference
of their derivatives.
For two functions, f(x) and g(x), say lim x -> a f(x) = A and lim x -> a g(x) = B. The lim x -> a (f(x)
* g(x)) is the same as lim x -> a f(x) * lim x -> a g(x), which equals A * B. This also works for
division, provided the limit of the denominator is non-zero.
x^ = (sqrt x), x^ = (cube root of x), x^m/n = (n root x^m), x^-k = 1 / x^k, nx^-k = n / x^k
Power Rule: For a function x^k, the derivative would be nx^n-1
Constant Multiple Rule: The derivative of a constant times a function is the constant times the
derivative of the function
● Example: Derivative of
○ 7x^3 would be 7(3x^2) = 21x^2**
○ 10x^½ = 5 / sqrt (x)
○ -(11/x^7) = 77/x^8
Product Rule: The derivative of a product of two functions is the derivative of the first function
times the second plus the derivative of the second function times the first.
Limit: If f(x) approaches the number L as x approaches (but is not equal to) the number a from
both sides of a, then we say that the limit of f(x) as x approaches a is L. It is a y-value that x
approaches (but doesn’t necessarily touch) as it gets closer to a certain x-value.
● The notation lim f(x) x -> a is read as “The limit, as x approaches a, of f(x)”.
○ lim f(x) x -> a- is read as “The limit, as x approaches a from the left, of f(x)”.
○ lim f(x) x -> a+ is read as “The limit, as x approaches a from the right, of f(x)”.
To find limits using a graph, find the limit from the left and from the right. If they are equal, then
that value is the limit. If they are not equal, the limit does not exist.
● Look at the value the function approaches from each side, not the exact point value if
there is one.
○ However to find f(a) of a function, look for the closed point on the graph where x
is equal to a.
For piecewise functions, plug a into the appropriate function (or both depending on question).
● To graph a piecewise function in Desmos, write out the function then put the constraint in
curly brackets. For example: f(x) = x - 2 {x < 3}
For algebraic limits, plug a into the function.
● Keep in mind for certain algebraic limit questions, such as x^2 - 25 / x + 5 as x
approaches -5, x^2 - 25 is equal to (x + 5)(x - 5). The x + 5 cancels out in the numerator
and denominator leaving only x - 5, which you can plug -5 into which will give you -10.
● Another example is (sqrt x) - 9 / x - 81 as x approaches 81. Here, multiply the numerator
and denominator by the conjugate of (sqrt x) - 9, which (sqrt x) + 9. This will give you x -
81 / (x - 81)((sqrt x) + 9). After cancelling out x - 81 in the numerator and denominator,
you are left with 1 / (sqrt x) + 9, which you can plug 81 into which will give you .
Continuous Function: A function is continuous if:
● There are no holes, gaps or jumps
● f(a) is defined (the function has a value at point a)
● lim x -> a f(x) exists (the limit approaches the same value from both sides)
● lim x -> a f(x) = f(a) (the limit equals the function at that point)
To find the simplified form of the difference quotient for f(x), plug f(x) into f(x + h) - f(x) / h (h is an
undefined constant).
● For example, for f(x) = x^2, the difference quotient would be (x + h)^2 - x^2 / h, which
would factor into x^2 + 2xh + h^2 - x^2 / h, which would become 2xh + h^2 / h, which
would become 2x + h.
● Given a table, you would take the simplified form of the difference quotient and plug in
the given values of x and h.
To find average rate of change, use m = y2 - y1 / x2 - x1
, ● m is slope
● x1 and y1 are the first ordered pair
● x2 and y2 are the second ordered pair
A derivative, f’(x) or d/dx, of a function is the slope of a tangent line. It is also considered a limit.
● A tangent line is a line that touches a curve at a single point.
● Ways to write a derivative: dy/dx, y’, f’(x), d/dx [f(x)], Dx [f(x)]
To find a derivative of a function, simplify the difference quotient for f(x) then find the limit as h
approaches 0 (plug 0 in for h).
● There are certain rules you can use to find derivatives more efficiently.
● A derivative of a constant is zero.
● The limit cannot be infinity when taking a derivative of a function.
A derivative can only exist if:
● The function is continuous
● The function is smooth (no holes, gaps, jumps, sharp points or corners)
● The function does not have a vertical tangent line
For two functions, f(x) and g(x), say lim x -> a f(x) = A and lim x -> a g(x) = B. The lim x -> a (f(x)
+ g(x)) is the same as lim x -> a f(x) + lim x -> a g(x), which equals A + B. This also works for
subtraction.
● Sum/Difference Rule: d/dx [f(x) ± g(x)] = f’(x) + g’(x)
○ The derivative of the sum or difference of two functions is the sum or difference
of their derivatives.
For two functions, f(x) and g(x), say lim x -> a f(x) = A and lim x -> a g(x) = B. The lim x -> a (f(x)
* g(x)) is the same as lim x -> a f(x) * lim x -> a g(x), which equals A * B. This also works for
division, provided the limit of the denominator is non-zero.
x^ = (sqrt x), x^ = (cube root of x), x^m/n = (n root x^m), x^-k = 1 / x^k, nx^-k = n / x^k
Power Rule: For a function x^k, the derivative would be nx^n-1
Constant Multiple Rule: The derivative of a constant times a function is the constant times the
derivative of the function
● Example: Derivative of
○ 7x^3 would be 7(3x^2) = 21x^2**
○ 10x^½ = 5 / sqrt (x)
○ -(11/x^7) = 77/x^8
Product Rule: The derivative of a product of two functions is the derivative of the first function
times the second plus the derivative of the second function times the first.
