7.1 - MATH 1081  Monday, April 4   ...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 1081  Monday, April 4    Chapter 7 – Section 1    ANTIDERIVATIVES  Homework #11 (due 4/11):              Clicker Check­in:  Choose any letter to check in now.    Section 7.1 #16, 20, 54, 60  Section 7.2 #18, 24, 32, 40, 42  Just when you thought you were getting all these derivatives  under control, we are going to flip things on their head…    Suppose that we know that     f ' ( x ) = 2 x .    Can we figure out what the original  f ( x )  was?    Is there only one possible  f ( x )  for which  f ' ( x ) = 2 x ?    If  F ' ( x ) = f ( x ) , then we call  F ( x ) the antiderivative of  f ( x ) .    That is, the antiderivative of  f ( x )  is the function  F ( x )  that we  would take the derivative of to get  f ( x ) .      Further, there is a theorem which states if both  F ( x ) and  G ( x )   are antiderivatives of  f ( x ) , then  F ( x ) − G ( x ) = C  for some  constant  C .    That is, once we have found one an antiderivative of a function  f ( x ) , we know that up to adding a constant to it, this is the  only function that will work as an antiderivative.  The notation that we will use to indicate that we are finding  the antiderivative of a function is called an integral sign.    It looks like  ∫  (kind of an elongated “S”, which we will see a  connection to in a later section).    Similar to when we found derivatives, we need to know with  respect to which variable we are working.  So this integral sign  ALWAYS comes paired with something of the form  dx ,  indicating that  x  is the variable we are interested in.    So, for example, the antiderivative of the function  2 x , with  respect to the variable  x  is written as ∫ 2 x dx .    The act of finding the function to which this is equal is called  antidifferentiation.  It is also called integration.      We can also refer to  ∫ f ( x ) dx  as an indefinite integral where    f ( x )  is called the integrand and  x  is the variable of integration.  Example:  Evaluate the indefinite integral.          1.   ∫ 9 x 8 − 4 x + 2 dx   ( )                       ⎛ 1/ 2 1 2 u ⎞ 2.   ∫ ⎜ u + − e ⎟ du   ⎝ ⎠ u     3.   ∫ t −2 dt   3 t Rules of Integration:    Rule 1: The Integral of a Constant k .   Rule 2: ∫ k dx = kx + C , The Power Rule where C is any constant 1 n +1 ∫ x dx = n + 1 x + C , ( n ≠ −1) n Rule 3: The Integral of a Constant Multiple of a Function ∫ k ⋅ f ( x ) dx = k ∫ f ( x ) dx , where k is a constant Rule 4: The Sum/Difference Rule ⎣ ⎦ ∫ ⎡ f ( x ) ± g ( x )⎤ dx = ∫ f ( x ) dx ± ∫ g ( x ) dx Rule 5: Integral of the Exponential Function a kx a kx dx = + C, k ≠ 0 ∫ k ( ln a ) Rule 6: Integral of the Function x −1 1 ∫ x dx = ∫ x dx = ln x + C −1 When we are given the derivative of the function and the  coordinates of a single point of the function, we can actually  determine the exact function (not up to some constant).    These are often referred to as initial value problems.  To solve  them, we first find the antiderivative, then use the added  information to determine the value of the constant.    Example:  Suppose that the acceleration of a particle at time  t    is given by the function  a ( t ) = 18t + 8 .  If the velocity  at  t = 1 is  v = 15  and the starting position of the  particle, at time  t = 0  is 3, then find the position  function  s ( t ) .    Clicker Check­out:  Choose any letter to check out now.      Next time – Section 7.2      ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online