MTH251
Sample Exam 2
Fall 2018
Name:
Date:

Show all work if full or partial credit is desired. You may use a graphing calculator or Desmos. If Desmos is used, then the device must be in airplane-mode (i.e., no wifi, cellular, or Bluetooth connections). No notes, books or websites allowed.

Part 1 No Calculator

  1. `f(x)=3`,   so `f'(x)=`
  2. `g(x)=x`,   so `d/(dx)[g(x)]=`
  3. `q(t)=t^2`,   so `d/(dt)[q(t)]=`
  4. `d/(d theta)[sin(theta)]=`
  5. Use the limit definition of the derivative to find the slope of the tangent line to the curve `r(x)=3/x` at `x=-2`.
    1. `r(-2+h)=`
    2. `r(-2+h)-r(-2)=`
    3. `(r(-2+h)-r(-2))/h=`
    4. `lim_{h to 0}(r(-2+h)-r(-2))/h=`
    5. So, `r'(-2)=`

Part 2 Calculator Allowed

Find the following derivatives.

  1. `d/dx[2x^2-3x+5]=`
  2. `d/dx[(x+2)^2]=`
  3. `d/dx[3 sqrt(x)]=`
  4. `d/dx[(x^2+3x+6)(4x^2+5)]=`
  5. `d/dx[(x^2+1)/(x+1)]=`
  6. `d/dx[2 sin(x) cos(x)]=`
  7. `d/dx[sqrt(1-x^2)]=`
  8. `d/dx[sin^2(3x-pi)]=`
  9. `d/dx[(e^(3x)-e^(-3x))/2]=`
  10. `d/dx[arcsin(x)]=`
  11. `d/dx[ln(3x)]=`
  12. `d/dx[ln(sin(x))]=`
  13. `d/dx[3*2^x]=`
  14. `d/dx[sinh(x)]=`

Fun with Derivatives

  1. Calculate the slope of the tangent line (rate of change) to `f(x)=3/x`   at   `x=-2` by finding the derivative function.
    1. `d/dx[f(x)]=`
    2. `f'(-2)=`
    3. The equation of the tangent line is
    4. Sketch a graph of `f(x)` on the interval `[-4,0]` along with the tangent line when `x=-2`.
      Be sure to label and scale the axes, and use a ruler if necessary.
  2. A ball is dropped from a height of 30 meters. Its height above ground (in meters) t seconds later is given by `h(t)=−4.9t^2+30`.
    1. At what time does the ball hit the ground?
    2. What is the instantaneous velocity of the ball when it hits the ground?
    3. What is the average velocity during its fall?
  3. Use implicit differentiation to find `dy/dx` without first solving for `y`.
    `x^4+y^4=16`
  4. Find the slope of the tangent line to the curve `4x^2+xy+3y^3=30` at the point `(1,2)`.
  5. Use logarithmic differentiation to find `dy/dx`   if   `y=(1+x)^(1/x)`.
  6. During the first couple weeks of a new flu outbreak, the disease spreads according to the equation `I(t)=2300*e^(0.054t)`, where `I(t)` is the number of infected people `t` days after the outbreak was first identified.
    Find the rate at which the infected population is growing after 8 days and include the appropriate units.