Calculus Jeopardy - Abhilash Krishna Raj

Integrals

100

∫(x^2 + 5)dx

x^3/3 + 5x +C

200

∫(2x^4 + 4x^2) dx

2x^5/5 + 4x^3/3 +C

300

-3∫ 3 (-x^2 + 9)dx

36

400

1∫ 0 (-7x^4 - 7x^3 + x^2 -1)dx

-19/60

500

-pi∫pi (-sin^2(x)-cos(x))dx

((cos(x)-2)sin(x)-x)/2 + C

Derivatives

100

d/dx (x^2 - 2)

2x

200

d/dx (5x^3 + 8x^2 + 1)

15x^2 + 16x

300

d/dx (csc(x))

-cot(x)csc(x)

400

d/dx (2cot(x) - 7csc(x))

7cot(x)csc(x) - 2csc^2(x)

500

d/dx (5tan(x) + sin(x) - 7x^7 - 3x^4)

5sec^2(x) + cos(x) - 49x^6 - 12x^3

Limits

100

lim_x->∞ (x^2)

∞

200

lim_x->∞ (tan(x))

DNE/Diverges

300

lim_x->0+(|x|/x)

1

400

lim_x->-pi/6(sin^-1(x)/3)

-1

500

lim_x->0(-1/(3x^3 + 2)

-0.5

Applications of Derivatives

100

Relative minimum of 3x^3 + 5x^2 on the domain [-1,1]

x=0, (0,0)

200

How many local extrema are located on the graph of 3sin^2(x) - cos(x), on the domain (0, 2pi)

3

300

Determine the concavity of the function 3x^3 + 8x^2 + 4 at x=0

concave up

400

What is the tangent line for the function -4x^3 + 7x at x=0

y=7x

500

How many inflection points are in the function (-8x^3 - 5x^2)/(4x^4 + 1)

5

Theorems

100

If a function is continuous on [a, b], and if k is any number between f(a) and f(b), then there must be a value, x = c, where a < c < b, such that f(c) = k. 

IVT

200

If f is a function continuous on a closed interval [a, b] and that the derivative exists on (a, b). Then there exists a c in (a, b) for which f(b) - f(a) = f'(c)(b - a) .

MVT

300

d/dx a∫x f(x)dx = f(x)

Fundamental Theorem of Calculus P1

400

f'(x) = lim_h->0(f(x + h) - f(x))/h

Limit formula of Differentiation

500

a∫b f(x)dx = F(b) - F(a)

Fundamental Theorem of Calculus P2