Limits and Continuity
The value a function approaches as x approaches a specific number.
What is a limit?
The derivative represents this rate of change.
What is the instantaneous rate of change?
If f′(x)>0f′(x)>0, the function is doing this.
What is increasing?
An integral represents this quantity on a graph.
What is area under a curve?
This rule is used to differentiate a product of two functions
What is the Product Rule?
If a function has no breaks, holes, or jumps at a point, it is said to be this.
What is continuous?
Find the derivative:
f(x)= 3x
What is 3?
If f′′(x)<0, the graph is this.
What is concave down?
Evaluate:
∫3x^2 dx
What is x^3+C?
Differentiate:
y=e^x
What is e^x?
Evaluate:
limx→3(2x+1)
What is 7?
The derivative of a constant is always this.
What is 0?
A point where f′(x)=0f′(x)=0 or does not exist.
What is a critical point?
The Fundamental Theorem of Calculus connects derivatives and this.
What are integrals?
The notation used to represent a limit.
Differentiate:
f(x)= 7x^4 − 3x + 8
What is 28x^3 - 3
Differentiate:
f(x)= sin(x)
What is cos(x)?
The derivative is used to find this line that touches a curve at exactly one point.
What is a tangent line?
Evaluate:
∫02 x dx
What is 2?
This theorem guarantees that a continuous function takes on every value between f(a)f(a) and f(b)f(b).
What is the Intermediate Value Theorem?
This theorem states that if a function is continuous on [a,b] and differentiable on (a,b), then there exists a point where the instantaneous rate of change equals the average rate of change.
What is the Mean Value Theorem?
Differentiate:
y=(3x^2+1)(x−4)
What is 6x(x−4) + (3x^2+1)?
Differentiate:
y= (x^2 + 3) / (x - 1)
What is 2x(x-1) - (x^2+3) / (x-1)^2
The process of approximating area using rectangles is called this.
What is a Riemann Sum?
If a function is differentiable at a point, it must also be this.
What is continuous?