Limits
Let's start easy...
f(x) = x2 - 6x + 12
What is the limit of the function f(x) as x approaches 5?
7
Let's start with the simple chain rule.
Differentiate the following
y = 3x5
dy/dx = 15x4
Let's start with the fundamentals.
What does it mean when a function is continuous at a point?
It means that the function at that point has no jumps, holes, or gaps in its graph and that the left-hand and right-hand limits at that point are equal.
You got this!
Write the equation of the line tangent to f(x) = 2x2 - 5 at the point where x = 2.
y - 3 = 8(x - 2)
Let's start fresh, no literally this is one of the first things we learned in the unit.
When is the horizontal asymptote of a rational function y = 0?
When the greatest exponent in the numerator is less than the greatest exponent in the denominator.
How about something a little harder?
g(x) = (x2 - 9) / (x2 + 9x + 18)
What is the limit of the function g(x) as x approaches -3?
-2
Oh come on... this one is a freebie
Differentiate the following
y = 5x2 - 3x + 1
dy/dx = 10x - 3
I hope you got remember the foundations of continuity and differentiability ;)
What are the two categories of discontinuities?
Removable and Nonremovable
Have fun with this one...
Let f be a differentiable function with f(1)=4 and f'(1)=-4. Let the function g(x)=(2x-2)f(x). Write the equation of the line tangent to the graph of g at the point where x=1.
y = 8(x - 1)
How about this one?
How do you find the vertical asymptote(s) of a rational function?
Set the denominator equal to 0 and solve for x(s).
Okay, the real challenges start now.
What does it mean when the limit of a function is infinity or negative infinity?
The limit does not exist.
Hehehe I dare you to try this one.
Differentiate the following
y = (5x + 2) / (x2 + 1)
dy/dx = (-5x2 - 4x + 5) / (x2 + 1)2
Last concept question hehe
What does it mean when a function is differentiable at a point?
It means the function is continuous at that point and the left-hand and right-hand limits of the derivatives at that point are equal.
Chain rule again hehe
Write the equation of the line tangent to f(x) = 3(4x2 - 5)2 at the point where x = 1.
y - 3 = -48(x - 1)
Now let's apply the definitions!
y = (x + 7) / (3x + 6)
Find the vertical asymptote of this function.
x = -2
Hope you can get this one :)
f(x) = 2x2 + 3 g(x) = 5x3 + 8
What is the limit of (1/3)[f(x)g(x)] as x approaches 2?
176
Let's see how good you know your trigonometry.
Differentiate the following
y = cos2(2x)
dy/dx = -4cos(2x)sin(2x)
And time to apply your knowledge!
Determine if this function is differentiable at c.
f(x) = |x2 - 4| at c = 2.
f(x) is not differentiable at c because the left-hand and right-hand limits are not equal.
Well look who we have here: e.
The function f is given by f(x) = -e3x - 3. What is the equation of the line tangent to the graph of f when x = ln2?
y + 11 = -24(x - ln2)
Hmmm I wonder what the answer is.
y = 2(x + 3)(x + 7) / (x + 1)(x + 7)
Find the horizontal asymptote(s) of the function
y = 2
Wow! Going for the big points huh? Try answering this!
What is the definition of a limit?
The limit of a function is a value that f(x) gets closer to as x approaches some number.
Have fun with this one...
Find dy/dx using implicit differentiation
sin(x2y) = y2
dy/dx = 2xycos(x2y) / (2y - x2cos(x2y))
Ooh this is tricky.
f(x) = {x2 - 4x - 15, x ≤ -2
-8x - 19, x > -2
Determine whether the function is differentiable, continuous, both, or neither.
The function is continuous and differentiable
Is trigonometry your friend or enemy?
Given f(x) = 2cot2(x), write the equation of the line tangent to y = f(x) when x = 5π / 3.
π = pi
y - (2/3) = (16√3 / 9) (x - 5π / 3)
Fun fact: Linear asymptotes are the title of this column
y = (x3 - 9) / (5x2 + x + 2)
Find the equations for any linear asymptotes
Vertical Asymptotes: none
Horizontal Asymptotes: none