Differentiation, Implicit Differentiation and Related Rates

Air is being pumped into a spherical balloon at a rate of 5 cm³/min. Determine the rate at which the radius of the balloon is increasing when the diameter of the balloon is 20 cm.

• $0.18\$ cm/min

• $\frac{2}{18\pi}$ cm/min

• $\frac{1}{80\pi}$ cm/min

Tags

CCSS.HSA.CED.A.2

CCSS.HSG.GMD.A.3

A 15 foot ladder is resting against the wall. The bottom is initially 10 feet away from the wall and is being pushed towards the wall at a rate of 
  $\frac{1}{4}$  ft/sec. How fast is the top of the ladder moving up the wall 12 seconds after we start pushing?

• 0.1319 ft/sec

• 0.1425 ft/sec

• 0.1387 ft/sec

Tags

CCSS.HSA.CED.A.2

CCSS.HSG.SRT.C.6

CCSS.HSG.SRT.C.8

A tank of water in the shape of a cone is leaking water at a constant rate of  $2\ ft^3$  /hour
. The base radius of the tank is 5 ft and the height of the tank is 14 ft. At what rate is the depth of the water in the tank changing when the depth of the water is 6 ft?

• - 0.1386 ft/hr

• -1.3411 ft/hr

• -0.1436 ft/hr

Tags

CCSS.HSA.CED.A.2

CCSS.HSG.GMD.A.3

A spherical balloon is inflated so that its volume is increasing at the rate of 20  $ft^3$  /min. How fast is the surface area of the balloon increasing when the radius is 4 feet?

• 12 ft/min

• 10 ft/min

• 21 ft/min

Tags

CCSS.HSA.CED.A.2

CCSS.HSG.GMD.A.3

Differentiate the following  $y=x^2\ln\ x+2e^{3x}$  

•  $\frac{dy}{dx}=\frac{x}{2x}+6e$  

•  $\frac{dy}{dx}=x\ 2\ \ln\ x+6e^{3x}$  

•  $\frac{dy}{dx}=x+2x\ \ln\ x+6e^{3x}$  

Differentiate  $y=\left(x\ \tan\ e^x\right)^3$  

•  $3\left(xe^x\sec^2e^x+\tan e^x\right)\left(x\ \tan e^x\right)^2$  

•  $3xe^x\sec^2+3\tan e^x\left(x\tan e^x\right)$  

•  $\left(3xe^x\sec^2e^x\right)\left(3x\ \tan\ e^x\right)^2$  

Find the  $\frac{dy}{dx}$  for $x^3+y^3=3xy^2$  by using Implicit Differentiation

•  $\frac{2y^2-x^2}{2y-2xy}$  

•  $\frac{y^2-x^2}{y^2-2xy}$  

•  $\frac{y^2+x^2}{2y-2xy}$  

Using Implicit Differentiation, find  $\frac{\text{d}y}{\text{d}x}$  for  $\sin\ \left(x^2y^2\right)=x$  

•  $\frac{1}{2x^2y\ \cos\left(x^2y^2\right)}-\frac{2y}{x}$  

•  $\frac{2}{2x^2y^2\cos\ \left(x^2y^2\right)}-\frac{y}{x}$  

•  $\frac{1}{2x^2y\cos\left(x^2y^2\right)}-\frac{y}{x}$  

Tags

CCSS.HSA.REI.A.1