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Calculus Worksheet: Differentiation and Applications

Worksheet
Grade 8
English

Calculus Worksheet: Differentiation and Applications

Basic Derivative Problems

Instructions: For each function below, find the derivative

. Show all your work.
  1. 

  1. 

  1. 

  1. 


Intermediate Problems: Product Rule & Quotient Rule

Instructions: Differentiate each function. Clearly indicate where you use the product or quotient rule.
  1. 

  1. 

  1. 


Application Problems

Instructions: Read each scenario carefully and solve the problem. Show all steps and include units in your final answer.
  1. A balloon rises vertically such that its height
    
    (in meters) after
    
    seconds is given by
    
    . Find the instantaneous rate of ascent when
    
    seconds.

  1. Find the dimensions of a rectangle with perimeter 40 units that has the maximum possible area. Show how you set up and solve the optimization problem.


Challenge Problem: Graph Interpretation

Instructions: Consider the following graph of the function

(sketch not to scale):
[Image: A sketch of a function with a local maximum at

, a local minimum at

, and an inflection point at

. The curve rises to a peak at

, descends, flattens at

, dips to a trough at

, then rises again.]
Based on the graph, answer the following:
  1. At which points is
    
    ?
  1. For which intervals is
    
    ?
  1. Where is the slope of the tangent greatest?
  1. At what
    
    -value does the graph have an inflection point?