Chapter 6.2

Notes

Linguistic Mapping Exercises

Knowledge Checks

  1. Show the derivative of \(f(x)=\arcsin(x)\) is \[f'(x)=\frac{1}{\sqrt{1-x^2}}.\]
  2. Compute the derivative of the function \(f(x)=\arcsin\left(\ln\left(\tfrac{1}{\sqrt{1-x^2}}\right)\right).\)
  3. Take \(f(x)=\left(\frac{(\cos(x)-2x)(\exp_2(x)+4)}{\sin(x)+4}\right)^3.\) Determine \(f'\) by using logarithm differentiation.
  4. Take \(f(x)=(x+3)^{\sin(x)}\). Determine \(f^\prime(x).\)
  5. Take \(f\) to be the function that is given by \[f(x,y)=5x^2y^4+4xy+\csc(-8x+3y^2).\] Determine \(f_x(1,2)\) and \(f_y(1,2).\)
  6. Assume that \(y\) is defined implicitly by the equation \[y^2x+\arctan(y+2)=xy.\] Calculate \(\frac{dy}{dx}.\)
  7. The equation \[x^4y-xy^8=-899934390\] implicitly defines \(y\) as a function of \(x\) in an open rectangle around the point \((9,10).\) Determine an equation for the line that is tangent to the solution set to the equation at the point \((9,10)\).
  8. A spherical balloon is being inflated with air at a rate so its volume is increasing at a rate of \(150\) centimeters cubed per second. Determine the rate at which the surface area is changing when the diameter of the balloon is \(65\) centimeters.

Remember, do the knowledge checks before checking the solutions.

Practice Problems

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