Limits & Continuity
What is lim x→2 (3x+1)?
What is 7?
What is the derivative of x^2?
What is 2x?
What does the first derivative tell you about a function?
What is increasing/decreasing behavior?
What is the integral of x^2?
What is (1/3)x^3+C?
What does a removable discontinuity look like on a graph?
What is a hole?
What does it mean for a function to be continuous at a point?
What is the limit exists, the function is defined, and the limit equals the function value?
Differentiate sin(x)
What is cos(x)?
If f′(x)=0, what kind of point could this be?
What is a critical point?
What is the area under the curve f(x)=3 from 0 to 4?
What is 12?
If a function is differentiable, is it always continuous?
What is yes?
What is lim x→0 (sinx)/x?
What is 1?
What rule do you use to differentiate f(x)g(x)?
What is the product rule?
What test can you use to find relative extrema using the second derivative?
What is the second derivative test?
What does the Fundamental Theorem of Calculus Part 1 say?
What is that the derivative of an integral gives you the original function?
A function f is increasing when f′(x)>0
True or False?
What is true?
True or False: A function can be continuous but not differentiable.
What is true?
Find the derivative of ln(x^2+1).
What is 2x/(x^2+1)?
A particle's position is given by s(t)=t^3−3t^2. When is the particle at rest?
What is when s′(t)=0, so t=0 and t=2?
Evaluate the integral from x=1 to 2 of (2x+1)dx.
What is 4.5?
A Riemann sum with equal subintervals uses which values for a right endpoint approximation?
What are the rightmost x-values of each subinterval?
Evaluate lim x→∞ (3x^2 +4)/(5x^2 x-2)
What is 3/5?
Differentiate (x^2 + 1)/x
What is (x^2 - 1)/x^2
A function has a local maximum x=3 and f′(x)=(x−3)2(x+2).
True or False: This supports the local max at x=3.
What is false? (Because derivative doesn’t change sign.)
If f(x)=∫0->x (cos(t2)dt), what is f′(x)?
What is cos(x^2)?
Let fff be a twice-differentiable function such that:
f(2)=4
f′(2)=−3
and f′′(x)>0 for all x in the interval (1,3)
Is the linear approximation of f(x) at x=2 an underestimate or an overestimate of the actual value of f(2.5)?
What is an underestimate?