CK12 Calculus - Single Variable

b. Whatis the instantaneousvelocityof the particleatt= 2 nanoseconds?

Answers

1. a. , b. , c. , d.


2. a. , b. , c. , d.


3. 2x, 12.


4. a. , b.


5. a. 6002.5m, b. 171.5m/sec,c. 31.3 m/sec,d. 343 m/sec


6. a. 39.6 m/sec,b. 118.8 m/sec

The Derivative

LearningObjectives
A studentwill be able to:

• Demonstratean understandingof the derivativeof a functionas a slopeof the tangentline.


• Demonstratean understandingof the derivativeas an instantaneousrate of change.


• Understandthe relationshipbetweencontinuityand differentiability.
  The functionf(x) that we definedin the previoussectionis so importantthat it has its own name.

The DerivativeThe functionfis definedby the new function

wherefis calledthederivativeof f with respectto x. The domain

of f consistsof all the valuesof x for whichthe limit exists.

Basedon the discussionin previoussection,the derivativefrepresentsthe slopeof the tangentline at
pointx. Anotherway of interpretingit is to say that the functiony = f(x) has a derivativef ' whose
valueat x is the instantaneousrate of changeof y with respectto pointx.
Example1:

Find the derivativeof
Solution:
We beginwith the definitionof the derivative,