Limits/Cont and Derivatives
On the board, draw a function with two jump discontinuities, one vertical asymptote, and one hole.
What is [check if board is correct]?
Find the equation of the line tangent to the graph of y = x^2 + sin(x) when x = pi.
what is y - pi^2 = (2pi - 1)(x - pi)?
This is the derivative of arctan(x^2).
What is 2x/(1+x^4)?
what is the indefinite integral of 5x^3sin(x^4 - 12)?
what is -5/4cos(x^4-12) + C?
These are the names of Mr. Lester's cats.
What are Kenai and Luna?
This is the limit as x --> infinity of (x^2 - 7x + 11)/(3x^2 + 4).
What is 1/3?
State, out loud, the mean value theorem, including the conditions to apply the theorem, with as much mathematical precision as possible.
What is, "if a function is continuous on [a,b] and differentiable on (a,b), there exists an x = c, a < c < b, where f'(c) = (f(b)-f(a))/(b-a)." Can be said in words instead (e.g. instantaneous = average etc.)
What is 1?
This is the solution to the differential equation dP/dt = kP.
What is P(t) = Ce^(kt) (or something similar)?
These are the inventors of calculus.
Who are Newton and Liebniz?
This is the limit definition of the derivative of a function f(x) in terms of h.
what is the limit as h --> 0 of [f(x+h) - f(x)] / h?
Find all points of inflection of f(x) = sin(x).
What are all the multiples of pi?
This is the antiderivative of x^3 - 4/x + sin(x).
What is (1/4)x^4 - 4ln|x| - cos(x) + C?
This is a representation of a differential equation with a bunch of short line segments representing slope values throughout the xy-plane.
What is a slope field?
What is savagely hangs up the phone?
This is the derivative of sec(tan(sin(x))).
What are ln(x) or sqrt(x), most likely?
This is the integral from 0 to 1 of 5x^3 - 11x^2.
What is -29/12?
This word refers to the type of differential equation at which population increases until it reaches a certain carrying capacity.
What is logistic?
This is who actually came up with L'Hopital's Rule?
Who is Johann Bernoulli?
Find the absolute minimum and maximum values of g(x) = (1/3)x^3 - 2x^2 - 21x + 11 on the interval [-8, 8]. No calc
What is a min of -359/3 and a max of 47?
What is finding critical numbers using f'(x) and plugging those x-values into f''(x) to see if f(x) is concave up there (minimum) or concave down there (maximum).
Explain each of the four basic types of Riemann sums precisely -- draw a picture illustrating each.
What is [check the board]
What will you always do on the AP exam when you are given a differential equation and asked to find the solution?
What is separate the variables!!!!!!!!
What is the integral of ln(x)?
what is xln(x) - x + C?