116Deriv review95 - Saint Mary's College
MATH 116-2006
Derivative & Antiderivative review
This review covers (most of) the first-semester material on limits, derivatives, and antiderivatives that we will call upon for second semester (doesn't include much on graphs, on log & exponential, exponential growth functions, or trig functions - which also will show up quite a bit). This list is intended to remind you of derivative ideas covered so that you can see where you need to find more practice problems - it does not include enough exercises for relearning techniques if you have trouble.
In these notes, the notation Dxf is used for "the derivative of the function f with respect to the variable x" which is also written or (if the variable is understood) f′ .
Review exercises
(should be done by 1/23/2006, but will not be collected) In Greenwall, Ritchey, Lial:
p.148 18, 19, 31, 45,
p.169 27, 33 [Notice the logarithmic scale on the “ratio” axis]
p.185 11, 35, 36
p.221 7, 25, 37
p.231 11, 33, 41
p260 11, 17, 19, 23, 27, 31, 45, 47, 49, 73
p.316 11, 29
p.347 13, 17
p.378 7, 13, 15, 27, 29, 37, 43, 49(a,b)
p.387 5, 13, 21, 33
NOTES
Limits
The limit of a function f(x) as the value of x approaches a number a [written ] is the number L that f(x) gets close to as values of x are chosen closer and closer to a. [Limit may not exist] We have two basic approaches to finding limits:
a.) The “table” approach: Make a table with two rows: values for x on the top – arranged getting closer & closer to a (from both larger & smaller values), corresponding values of f(x) on the bottom (calculated from the rule for f) – if the values on the bottom approach a single value, that is the limit.
b.) The algebraic approach: If the function is given by a formula near a (same formula for values a little larger as for values a little smaller), see if the formula makes sense [gives a number] when x = a. If it does, the number given is the limit. If not, sometimes the formula can be simplified for all values except and this simplified formula will give a value for x = a – the “almost everywhere” rule says this value is the limit.
Derivatives
Derivative is a rate of change (In particular, if N(t) is size at time t , then DtN gives growth rate ) and because slope also represents rate of change, Dxf gives the slope of the tangent to the graph of y = f(x). The derivative is defined by the limit of a difference quotient, but for functions given by “elementary functions” (algebra, exponential/logarithmic functions, trig functions and their combinations) we have some shortcut rules. The “approximate and then take a limit” approach used in the definition of the derivative will reappear this semester in the definition of the integral.
I. Derivative Rules
Rules for finding the derivative from a formula:(u and v represent any functions of the variable x , or any expressions involving the variable x)
A.) The big five (written in the most general form)[there are more - but these are by far the most important]:
1.) Power rule: Dxun = nun-1Dxu
2.) Exponential rule: Dxeu = eu Dxu (For base a : Dxau = au Dxu ln a)
3.) Logarithm rule: Dxln |u| = Dxu (For base a : Dxlogau = Dxu logae)
4.) Sine rule: Dx sin u = cos u Dxu
5.) Cosine rule: Dx cos u = -sin u Dxu
B.) Reduction rules:
1.) Constant coefficient rule: Dx(ku) = k Dx(u) for any constant k
2.) addition/subtraction rule: Dx(u + v) = Dxu + Dxv and Dx(u - v) = Dxu - Dxv
3.) Product rule: Dxuv = v Dxu + u Dxv [or : h0Dhi + hiDh0]
4.) Quotient rule Dx=
5.) Chain rule: Dxf(u) = Duf(u) Dxu
Examples [everything uses the chain rule]:
a.) Dx(8x3e5x) = 8(e5x Dx(x3) + x3Dxe5x) = 8(e5x(3x2) + x3(e5x(5)) ) = [Uses constant coefficient, product, power, exponential rules]
b.) Dte = e Dt(5t2) = e = [Uses exponential, constant coefficient, power rules)
c.) Dx(3x2 + 1)3 = 3(3x2 + 1)2Dx(3x2 + 1) = 3(3x2 + 1)26x = [Uses power, constant coefficient, addition rules]
d.) Dv ln(v3 - 5v) = = =
[Uses logarithm, power, subtraction rules]
e.) Dt 3t sin(5t2- t) = sin(5t2-t) Dt(3t) + 3t (Dt sin(5t2-t)) = sin(5t2-t) (3) + 3t (cos(5t2-t) Dt(5t2-t)) = [uses product, constants coefficient, power, sine, subtraction rules]
Everything uses the chain rule - it's even built in to the big five, in their general forms. Remember this: it is very important in finding antiderivatives
Remember that Dxx = 1, Dx(anything else) ≠ 1 (same for any other variable - Dtt = 1 but Dtx ≠ 1 )
II. Implicit Differentiation:
If A = B, then DxA = DxB (Used with the chain rule to get derivatives, when we can't solve an equation for y - or [usually used with Dt , in this case] for getting rate-of-change relations from relations on quantities )
Example:
If 3x2 + ln y - xy2 = 8, we can get Dxy at any point (x,y) on the graph, by implicit differentiation:
Dx(3x2 + ln y - xy2) = Dx8
3(2x) + Dxy - (y2 + x(2yDxy)) = 0
6x - y2 + Dxy - 2xyDxy = 0
( - 2xy)Dxy = -6x + y2
[Need to know both x and y to get numerical value - important ideas are: 1. taking derivative of both sides of equation; 2. keeping Dxy as a symbol (a new variable) since we don't have a formula for it during the work]
Antiderivatives
A function F(x) is an antiderivative of f(x) if DxF(x) = f(x). The general antiderivative of a function always contains an undetermined constant (represented as C ) so that it represents all antiderivatives at once. To get a particular antiderivative, you need to know the value of the function (antiderivative) at one point.
The antiderivative rules all come from the derivative rules, but using them is not as straightforward because the chain rule kicks in from the beginning. We use [pic] to represent "the general antiderivative of f(x) with x as variable" Since "the general antiderivative" of a derivative is the origional function plus or minus any constant, we have[pic] . The rules are often written in a shorter form in which u represents any function of x , and du represents u′dx (that is, Dxu dx) thus[pic]
A. The big five:
1.) Power rule (the nice case) = + C if n ≠ -1 (covers other negative exponents, though ) (Remember special appearance when n=1 : = + C )
[Short form: [pic]]
2.) Power rule ( the not nice case : sometimes called special fraction rule)
= = = ln + C
[Short form: [pic] ]
3.) Exponential rule = eu + C
[Short form [pic]]
*4.) sine rule: = - cos u + C [New for second semester]
[Short form: [pic]]
*5.) cosine rule: = sin u + C [New for second semester]
[Short form:[pic]]
B. Reduction rules:
1.) Constant coefficient rule: = k for any constant k (Constants can filter out through the integral symbol - though variables cannot)
2.) Sum/difference rules = ±
C. The "change of variables" (also called "u-substitution") technique: The [short form] formulas are interpreted to mean that, for example any un can be integrated if we have du , but not otherwise. We can use the constant coefficient rule to take care of missing constant coefficients (because the adjustments filter out to the front of the antiderivative), but nothing else can be missing (or added). Every antiderivative computation that we can handle at this stage uses one of the "big five" rules, - or a table rule, as we will see later. The "change of variables" technique is based on the chain rule and is used all the time, just as the chain rule is used in every derivative situation.
Examples:
a.) = with u = x2 (because if u = x2 , then u′ is Dx(x2) = 2x so du = 2x dx , which is exactly what we have - notice it's OK that the 2x and the dx are not right next to each other in the integral, as long as they are multiplied) thus we use the exponential rule and get
= = e+ C =
b.) is almost with n = 5, but u has to be (3t2-2t) to give us a match, so du = (6t-2)dt = 2 (3t-1)dt and what we have is (3t-1)dt . Since all that is missing is the 2 (a constant), we can fix this:
= = = + C =
[This uses "nice" power rule]
c.) = =
(with u = ex+x, so that u′ = (ex+1) ) = ln + C =
[Uses the "less nice" power rule. In this case, the absolute value can't be dropped because we don't know whether ex+x is positive or negative]
d.) = = = (with u = 3x2-1, u′ = 6x) = + C = + C =
[This uses the "nice" power rule, even though we have a fraction, because it fits both conditions: 1. the denominator is a power and 2. the numerator is (except for a constant) the deriviative of the base of the power]
We still have to see what antiderivative rules we get from the derivative rules for sine & cosine and from products. The quotient rule does not give any useful antiderivative technique.
If we have a formula r(t) for the rate of change of some quantity N(t), then r(t) = N'(t) and N(t) = = - but we need a value of N at some value of t to get the particular antiderivative.
Example: If a bacterial population grows so that the growth rate (by mass) is given by r(t) = 1.2e.01t μg/hr, and the mass after 4 hours is 190 μg , we can get the formula for the size N(t) by taking N(t) = = e.01t + C , and using the known size (and some arithmetic) to get 190 = N(4) = 120e.01(4) + C so that C = 190-121=69 and
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