Calculus Jeopardy

Derivatives

100

5x²+6x-2

10x+6

200

10x⁴+4x³-7x²+9x-2

40x³+12x²-14x+9

300

3x²+sin(4x+3)

6x-4cos(4x+3)

400

4x^-7+6x^-2-e^6x

−6e6x−(12/x^3)−(28/x^8)

500

ln(5x⁴-3x²)+e^3x

3e^3x+((20x3−6x)/(5x4−3x2))

Integrals

100

2x³-3x

(2,6)

592

200

4x⁵+4x³-7x²+9

(0,3)

531

300

8x⁹-6x⁷+4x³+8x²-9

(0,3)

42,444.45

400

3sin(4x²)-ln(4x²)

(π /4, π /2)

-1.3993

500

6xcos(8x³)

(π /4, π /2)

0.1259

Theorems

100

If a function is differentiable, the function must also be _______.

Continuous

200

The Extreme Value Theorem states that if a real-valued function f is continuous on the closed interval [a, b], then f must attain a _______ and a ______, each at least once.

minimum, maximum

300

Product rule for integration

d/dx (f(x)g(x))=f(x)g'(x)+g(x)f'(x)

400

This theorem states that if a function is differentiable across the interval (a,b), there exists a point c at which f'(c)=(f(b)-f(a))/(b-a).

Mean Value Theorem

500

Rolle's Theorem states that if a function is differentiable on the interval (a,b), then there exists a number c that a<c<b ad f'(c)=______.

zero

Area Under a Curve

100

4x²+8x

(1,4)

904.7787

200

x⁴-5x³+7x-2

(0,2)

-22.6195

300

5x⁴+7e^x+9

(0,4)

9017.5462

400

4cos(5x+2)-ln(2x²)

(1,3)

-33.1292

500

sin(4x³)-3x²+7

(0,3)

-35.954