AP Calculus Unit 1

Average ROC

100

This is the average ROC from [0,1].

What is 3?

200

This is the average ROC of f(x) = -x² over the interval [0,10].

What is -10?

300

This is the average ROC of g(x) = -2x +17 over the interval [-10,-7]. 

What is -2?

Limits

100

If   lim _x-->4  g(x) = 7, and   lim _x-->4  h(x) = 2, this is the value of   lim _x-->4 g(x) - h(x).

What is 5?

200

This is the value of  lim _x-->2 (8 - 3x + 12x^2)

What is 50?

300

This is the value of  lim _x-->10   x/(3 - sqrt(x-9))?

What is 5?

Squeeze Theorem and IVT

100

Prove that there is a value x = c between .5 and 1 such that f(c) = 0 for the function f(x) = y=6x³-2x

Because f(.5)<0 and f(1)>0, by the intermediate value theorem, there is a value c between .5 and 1 for which f(c) = 0.

200

Prove that the function f(x) = x³ + x + 7 has a solution.

f(-2)=-3 and f(-1)=5 and 0 is between -3 and 5, therefore by the IVT, this function has a solution.

300

If g(x) ≤ f(x) ≤ h(x) for all x between (3, 5) can squeeze theorem be used with g(x) and h(x) as defined below to find  lim(f(x)) _{x →4} ?

g(x) = -(x-4)² + 5

h(x) = (x² - 2x - 8)/(x² - 7x +12)

No

Continuity and Discontinuity

100

This is the x value of the removable discontinuity for the function f(x) = (x² - x - 6) / (x²-9). 

What is 3?

200

This is the type of discontinuity for the function f(x) = (x² - x - 2) / (x - 2).

What is removable discontinuity?

300

This is the value of x at which the function f is continuous.

What is 1?

Infinite Limits

100

This is the value of lim _x-->∞ ((x²-25)/(x²+2x-15).

What is 1?

200

This is the value of lim _x-->3⁺   9/(x-3)⁵ .

What is ∞? (This means there is a vertical asymptote!)

300

This is the y-value of the horizontal asymptote in the function f(x) = 2^x .

What is 0?