Limit
When does limit exist?
Limit exists if and only if limit from the left side equals the limit from the right side.
What are the conditions to check continuity?
1) f(a) defines
2) Limit as x approaches a of the function f(x) exists
3) Condition 1 = Condition 2
What is a critical point?
A critical point is when f'(x) = 0 or f'(x) = undefined
What is the formula of the definition of derivative ?
f'(x) = lim as h->0 of (f(x+h) - f(x))/h
Evaluate: limit as x approaches 0 of the function: (64x3 - 27)/(4x-3)
9
State the interval of continuity of f(x) = 1/(x-3)
(-oo, 3) U (3, oo)
What is an inflection point?
Derive: f(x) = (3x^3 - 4)/2
f'(x) = (9x^2)/2
Evaluate: limit as x approaches 3/4 of the function: (64x3 - 27)/(4x-3)
27
Is f(x) = 5/(x^2 - 3x -10) continuous at x=-2? If not, state which condition(s) in which it fails.
No. Condition 1 fails
How do we determine the maximum and minimum of f from f'?
If f' changes from + to -, there's a max
If f' changes from - to +, there's a min
Derive using the Product Rule: f(x) = (4x^2+3)(5x-1)
Simplify the answer
f'(x) = 60x^2 - 8x +15
Evaluate: limit as x approaches positive infinity of the function: (3x - 5)/(square root of (5x^2 +1))
3/square root of (5)
Where is the following function not continuous?
f(x) = 2x/(5 - e^(x+3))
f(x) is not continuous at x=ln(5) - 3
Let f'(-1) = 100, f'(1) = -5 , f'(5) = 20
Let the critical points of the function f(x) be a = 0 and b = 2. Find the relative max and min of f(x).
max at x = 0
min at x = 2
Derive: y = sec(x) * cot(x). Simplify your answer.
Hint: the simplified answer should be a product of sec(x) and cot(x) or cot(x) and csc(x)-sec(x)*cot^2(x) or
-cot(x) * csc(x)
Evaluate: limit as x approaches positive infinity of the function: e^(x^2x+1)
positive infinity
Where is the following function not continuous?
f(x) = 1/(x*lnx)
f(x) is not continuous at x = 0 and x = 1
Sketch a graph of f that satisfies the following conditions:
f(-2) = 4, f'(-2) = -1
f(1) = 2, f'(1) = 0
f(2) =1, f'(2) = -2
See graph
Derive: Cube root of [(2x^3/3 - 2)^2 * (4x^2/3)]
See answer