Calculus Battle

Algebra/Trig

100

Factor 4x² -4x - 3

(2x - 3)(2x + 1)

200

What is the tangent at the angle 765^o ?

1

300

Find the horizontal and vertical asymptotes of: 

( x - 7 ) / (x² - 3x + 2)

HA: y = 0

VA: x = 2, x = 1

400

State the 3 Pythagorean identities.

sin²x + cos²x = 1

1 + cot²x = csc²x

tan²x + 1 = sec²x

500

Which trigonometric functions are even, and which are odd?

Cosine and Secant are even, the rest are odd.

Delta/Epsilon

100

What is chosen first, the epsilon or the delta?

The epsilon.

200

Given f(x) = x, and epsilon = 1, find a delta to satisfy the epsilon/delta definition for the limit as x approaches 2.

delta = 1

300

Given the limit as x approaches 1 of the function 5x - 3 is 2, when epsilon is any number greater than zero, what is delta?

delta = epsilon/5

400

Given epsilon > 0, and f(x) = k, what do we choose delta to be as x approaches any x₀?

delta can be anything because the constant function f(x) = k gives us a true statement. k - k = 0 < epsilon, is always true. 

500

What is the relationship between limits, continuity, and derivatives? 

all differentiable functions are continuous, all continuous functions have a limit.

Limits

100

When are you allowed to use L'Hopital's rule?

When the limit is indeterminant. 0/0 or infinity/infinity

200

Find the limit as x -> infinity of the following 

f(x) = (3x-2) / (1 - 7x)

-3/7

300

What are the 3 requirements for the function to be continuous at a point x₀? 

The function must be defined, the limit must exist, and the limit must equal f(x₀).

400

Find the limit as x -> 0 of 

(x - sin x) / x^3

1/6

500

Find the limit as x -> 0 of 

(e^x - 1) / (cos(x) - 1)

DNE

Chain Rule

100

Find the second derivative of the following 

y = 9 tan(x/3)

y'' = 2sec(x/3) sec(x/3) tan(x/3)

200

Find the 65^th derivative of y = e^-2x

 -2⁶⁵ e^-2x

300

What is the derivative (with respect to x) of a^u, where u is a function of x?

(d/dx) a^u = a^u ln(a) (du/dx)

400

Find the derivative 

y = exp(sec²(tan(2x)))

exp(sec²(tan(2x))) (2sec²x)(tan(x)tan(2x) + sec²(2x))

500

Find the derivative 

y = x^sin(2x)

x^sin(2x) (2cos(2x) ln(x) + sin(2x)/x)

Challenge

100

Using epsilons and deltas, describe when a number L is not the limit of f(x).

When we choose an epsilon so that there does not exist a corresponding delta so that 

|x - x₀| < delta --> |f(x) - L | < epsilon 

200

Why does sin(1/x) have no limit as x approaches 0?

Oscillating 

300

What is the intermediate value theorem and how is it used?

We must have a continuous function on an interval [a, b] in the x direction that maps to [f(a), f(b)]. If y is in the interval [f(a), f(b)], then there must exist an x so that f(x) = y.

We can use it to show when functions do and don't cross the x-axis (aka have roots).

400

Find the limit as x -> 0 of 

(x + x cos(x)) / (sin (x) cos(x))

1

500

Find the limit as x -> infinity of 

(1 - 1/x)^x

1/e