The END! Calculus

 $\frac{\text{d}}{\text{d}x}\left(x^n\right)=$  

•  $nx^{\left(n-1\right)}$  

•  $x^n\ln x$  

•  $x^n$  

•  $nx$  

 $\frac{\text{d}}{\text{d}x}\left[f\left(g\left(x\right)\right)\right]=$  

•  $f'\left(g\left(x\right)\right)g'\left(x\right)$  

•  $f'\left(g'\left(x\right)\right)$  

•  $f'\left(x\right)g'\left(x\right)$  

•  $f'\left(x\right)g\left(x\right)+g'\left(x\right)f\left(x\right)$  

 $\frac{\text{d}}{\text{d}x}\left(\ln x\right)=$  

•  $\frac{1}{x}$  

•  $\frac{1}{x\ln x}$  

•  $\ln x$  

•  $e^x$  

The base of a solid is the region in the first quadrant enclosed by the graph of 𝑦 = 2 − 𝑥² and the coordinate axes. If every cross section of the solid perpendicular to the 𝑦-axis is a square, the volume of the solid is given by...

• $\int_0^2\left(2-x^2\right)^2dx$

• $\int_0^2\sqrt{2-y}dy$

• $\int_0^2\left(2-y\right)dy$

• $\int_0^2\left(2-x^2\right)dx$

Derive y=e^3x

• e^3x
• 3e^3x
• 3x
• 3xe^3x

Find the derivative f(x) = xe^x

• f'(x) = e^x

• f'(x) = xe^x + xe^x

• f'(x) = e^x - xe^x

• f'(x) = e^x + xe^x

Solve the following differential equations:
 $\frac{\text{d}y}{\text{d}x}=3y$  

•  $y=Ce^{3x}$  

•  $y=3x+C$  

•  $\frac{y^2}{2}=3x+C$  

•  $\ln y=x+C$  

What integral would allow you to find the volume of the region bounded by y = 2x² and y = 8 around the line y = 11. 

• Bounds: [0, 2]; π ∫(4x⁴ - 8)dx
• Bounds: [0, 2]; π ∫((11 - 2x²)² - 9)dx
• Bounds: [-2, 2]; π ∫((11 - 2x²)² - 9)dx
• Bounds: [-2, 2]; π ∫(4x⁴ - 8)dx

2. Solve the differential equation

𝑑𝑦/𝑑𝑡 =3𝑡²/𝑦 with initial condition

𝑦(2) = 0.

• 𝑦 = ln(15t)

• 𝑦 = 16t³

• 𝑦 = (2𝑡³ − 16)^1/2

• 𝑦 = (2𝑡³ −16)

Determine the volume of the region bounded by y = x² - 2x and y = x that is rotated about y = 4.

• 5.4
• 30.6
• 96.133
• 108.332