DIFFERENTIABILITY
GIVE US A SINE
THE SHAPE OF THINGS
RULES RULES RULES
POTPOURRI

### 100

The function x+|3x-21|-3 is not differentiable at this point.
What is 7?

### 100

This function is the derivative of (sin(x))^3.
What is 3( sin(x) )^2 cos(x) ?

### 100

This line through the point ( a, f(a) ) has slope equal to - 1 / f ' (a).
What is the normal line to the graph of f at the point (a, f(a))?

### 100

If a rectangle has sides with lengths given by the functions f and g, then this expression is the derivative of the area in terms of f, g, and their derivatives.
f' g + f g'

### 100

By 1711 these two intellectuals were feuding over who invented "the method of fluxions and fluents."
Who are Isaac Newton and Gottfried Leibniz?

### 200

This limit is the definition of the derivative of f at a point x.
What is the limit as h goes to zero of ( f( x+h )-f( x ) ) / h ?

### 200

The antiderivative of 5 / (1 + 25 x^2).
What is arctan(5x) + c ?

### 200

A constant 2 ft radius cylinder is growing in height at a rate of 2ft/min also growing in volume this fast.
What is 8 pi ft^3 / min ?

### 200

If f and g are differentiable functions with g(0)=f'(0)=5 and f(0)=g'(0)=-3 then this quantity is equal to the derivative of the product fg at 0.
What is 34?

### 200

Physicists would describe this as the derivative of momentum with respect to time.
What is force?

### 300

The derivative f' displays this phenomenon at x if the function f has a sharp point at x.
What is a jump discontinuity?

### 300

This is the value of tan^2( arcsin( 1/5 ) ) expressed as a ratio of integers.
What is 1/24?

### 300

The line tangent to the unit circle at the point ( - sqrt(99) / 10 , 1 / 10 ) has slope equal to this quantity.
What is the square root of 99?

### 300

If f' (0) = g(0) = 4 and f(0)= g ' (0) = 6 then this number is the derivative of the quotient f / g.
What is -5 / 4?

### 300

The domain of the derivative of the function that divides its input quantity in half before rounding down to the nearest integer is this set.
What is the set of all real numbers except the even integers?

### 400

A strictly decreasing function has a derivative with this sign.
What is negative?

### 400

The domain of arcsin( x^2-3 ).
What is the union of the intervals (-2, -sqrt(2) ) U ( sqrt(2) , 2 )?

### 400

The amount of area between the x axis, the curve y=x^4, and the vertical line x=5.
What is 625?

### 400

This is the next iteration if Newton's method is used to find a root to x^2-x-1 with the initial value 2.
What is 5/3?

### 400

This function is the derivative of the hyperbolic cosine.
What is the hyperbolic sine?

### 500

If a function is differentiable, then it must also have this reliable property.
What is continuity?

### 500

This function is the second derivative of f(x) = cos(x) / ( 1 - sin(x) )
What is cos(x) / ( 1 - sin(x) )^2 ?

### 500

The lines tangent to the curve x^4 + y^4 = 1 at the points ( 2^(-1/4) , 2^(-1/4) ) and ( -2^(-1/4) , 2^(-1/4) ) meet at this point in the plane.
What is the point ( 2^(3/4) , 0 ) ?

### 500

This is the statement of the Mean Value Theorem.
What is if f is differentiable on [a,b] then there is some point c in (a,b) such that f'(c) = (f(b)-f(a))/(b-a) ?

### 500

Even before the differential calculus had a rigorous foundation, infinitesimals and limits were used by many mathematicians, such as the work of this seventeenth-century German mathematician, famous for his Laws of Planetary Motion.
Who is Johannes Kepler?