Slippery Slopes
The derivative of 2x^32.
What is 64x^31?
int_0^1 3x^2+2x+4 dx
What is 6?
frac{dy}{dx} if
y^2+2xy=1
What is
-frac{y}{y+x}?
If lim_{x->0^-} f(x)=1 and lim_{x->0^+} f(x)=2 , then at x=0, f(x) has this kind of discontinuity.
What is a jump discontinuity?
This mathematician, frequently considered the father of calculus, is most widely known for his theory of gravitation.
Who is Sir Isaac Newton?
The derivative of
e^{2x^2}.
What is
4xe^{2x^2}?
int frac{x+1}{x^2+2x}dx
What is
1/2ln|x^2+2x|+C?
At the moment when the radius of a sphere ( V=4/3pir^3 ) is 5 feet, the volume of the sphere is increasing at a rate of 100pi feet/second. This is the rate at which the radius is increasing.
What is 1 foot/second?
What I look for on the graph of f'(x) to determine whether f(x) is increasing at x=a.
What is above the x-axis (positive)?
This mathematician, who invented the dy/dx notation for derivatives, is best known for his feud with another important figure in the history of calculus.
Who is Wilhelm Gottfried Leibniz?
The slope of the line tangent to cos(pix) at x=0.5.
What is
-pi?
int frac{x+1}{x^2+x}dx
What is
ln|x|+C?
If the velocity of a particle is given by v(t)=3t^2+2 , and the particle's position at t=0 is 1, then this is the position of the particle at t=2.
What is 13?
The concavity of f(x) = -x^3+2x^2-3x at x=1.
What is concave down?
Who is Pythagoras?
lim_{x->0} frac{2x}{cosx-e^x}.
What is -2?
If f(0)=3, f(2)=5, f(4)=8, and f(6)=20, this is the approximation for int_0^6 f(x)dx using the Trapezoidal Rule with 3 subintervals of equal length.
What is 49?
{dw}/dx if w=cosz , z=2y , and y=e^{x^2} .
What is
-4xe^{x^2}sin(2e^{x^2})?
If f is continuous, f(1)=2, and f(2)=3, then there is some c, with 1<c<2, so that f(c)=2.5. This is the name of the theorem that guarantees this.
What is the Intermediate Value Theorem?
This mathematician, whose greatest contribution to math was writing a textbook, is most famous for a technique named after him which he himself did not discover.
Who is Guillaume de L'Hospital?
The slope of the constant polar function r=2 at theta=pi/4.
What is -1?
When the region between f(x)=x^2 and g(x)=4x-3 is revolved about the x-axis, this is the resulting integral expression.
What is
int_1^3 (4x+3)^2-(x^2)^2dx?
f(x)=4x^4+3x^2 and g is f's inverse. If g(1)=1, this is g'(1).
What is 22?
If f is differentiable, f(1)=2, and f(2)=3, then there is some c, with 1<c<2, so that f'(c)=1. This is the name of the theorem that guarantees this.
What is the Mean Value Theorem?
This Ancient Greek mathematician, who began the tradition of proof-based mathematics, is best known for his book "The Elements." His name is popularly used in the context of spaces with unusual geometry.
Who is Euclid?