END OF YEAR CALCULUS GAME

DERIVATIVES

100

What is a derivative? 

The derivative describes SLOPE 

200

5x²-15x

10x-15

300

Find the derivative: 

g(x)=e^x(5x³+2x+6)

g^'(x)= (e^x)(15x²+2) + (e^x)(5x³+2x+6)

400

Find the derivative: 

g(t)=-2(2t²-1/2t)⁶

g^'(t)=-12(2t²-1/2t)⁵ (4t-1/2)

500

Find the derivative: 

y= -4log₂(3x⁵)

dy/dx = -4(1/ln(2)3x⁵)(15x⁴)

LIMITS

100

What has to be true for a limit to exist? 

The limit must approach the same point from both sides

200

What three things must be true for a limit to be continuous? 

- The limit has to exist

- The limit has to have a y-value 

- The limit has to equal the y-value 

300

Given g(x) = 12-x⁴

Find the linear approximation of g(x) at the point where x=2

Slope: 

12-x⁴ = -4x³ ---------> -4(2)³ = -32

Point: (2,-4)

12-(2)⁴ = -4 

Equation: 

y+4=-32(x-2) -------> y=-32(x-2)-4

400

Find the linear approximation of f(x) = x³-2x+3 at the point where x=2

Slope: 

3x²-2 -----> 3(2)²-2 = 10

Point: (2,7)

(2)³-2(2)+3=7

Equation: 

y-7 = 10(x-2) --------> y=10(x-2)+7

500

From the previous question, use linear approximation to estimate f(2.1)

10(2.1-2)+7 ------> 8

TRIG INTEGRAL

100

f^'(x):

e^x

Se^xdx = e^x+c

200

cosx

Scosxdx = Sinx + c

300

-csc²x

Scsc²xdx = -cotx+c

400

secxtanx

Ssecxtanxdx = secx +c

500

-cscxcotx

Scscxcotx = -cscx +c

ANTIDERIVATIVES

100

S(3x⁴-2x+10/x⁶)dx

S(3/5x⁵-x²-x^-5)+c

200

Sx²(x²-4x+2)dx

S1/3x³(1/3x³-2x²+2x)+c

300

f(x) = 3x²

f(x) = x³ + c

400

f(x) = (2x + 3)/(x² + 2x + 1)

f(x) = ln|x + 1| - 1/(x + 1) + c

500

f(x) = e^x/(1+e^x)

f(x) = 1 + e^x - ln|1+e^x| + c

PRODUCT RULE

100

g(x) = (2x+1)(x³-1)

(2x+1)(3x) + (x³-1)(2)

= 2x³+3x²+3x-2

200

y=(3x²-2x)(x²+3x-4)

(3x²-2x)(2x+3) + (6x-2)(x²+3x-4)

= 27x²-42x+8

300

g(x) = x²(5x³+3)

(x²)(15x²) + (5x³+3)(2x)

= 25x⁴+6x

400

y=x^-2(ex³+3)

(x^-2)(3ex²) + (-2x)(ex³+3)

= 3ex^-4 - 2xex⁴ - 6x

500

f(x) = x² sin(x)

x²sinx + x²sinx

= f'(x) = 2x sin(x) + x² cos(x)