MATH 116 Midterm Exam-AID Session

Zahra Mahmood Bodla

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About Zahra

A function is a rule that associates exactly one output value to a given input value.

Notation: y = f(x)

Domain: set of allowable input values.Range: set of all possible output values.

The Vertical Line Test: A curve is a function in the x-y plane iff the curve does not intersect a vertical line more than once.

Functions

Examples1. y = x2 + 1Domain: ℝ Range: (1, ∞)

2. x2 + y2 = 9 is not a function as it does not pass the vertical line test

A function f is even if the graph of f is symmetric with respect to the y axis; algebraically f(-x) = f(x).

A function f is odd if the graph of f is symmetric with respect to the origon; algebraically f(-x) = -f(x).

A periodic function is a function that repeats itself after some given period, or cycle; mathematically f(t) = f(t + nT), where n is an integer and T is the period.

ExamplesA function f(x) has graph:

Even periodic extension of f(x):

Odd periodic extension of f(x):

Determine if following functions are even or odd or neithera) f(x)=sqrt(x)b) f(x)=x*abs(x)

Solutions:Substitute –x as input to each function

a) f(-x)=sqrt(-x), we cannot perform any algebraic operation henceforth, this function is neither odd or even.

b) f(-x)=( -x)*abs(-x)f(-x)=-x*abs(x), f(-x)=-f(x), therefore odd

Absolute Value Function: a function that gives the magnitude of its input values, example absolute value of -3 is 3.

Composite FunctionsLet g(x) have domain D1 and Range R1 , and f(x) have domain D2 ⊃ R1 then the composition of f and g is the function f o g defined by; (f o g)(x) = f(g(x))

Example:If g(x)=x2-1, and f(g(x))=sqrt(g(x)), then the domain of f(g(x)) is such that g(x)>=0, which is iff x<=-1 or x>=1. So the domain of f(g(x)) is therefore (-∞,-1]U[1,∞), and the range is [0,∞).

A function is called one to one for any x1 , x2 in the domain of f with x1 not equal to x2 the f(x1) is not equal to f(x2).

y=x2, not one-to-one

y=x3, one-to-one

One to one function

The inverse of a function: If f is one-to-one with domain A and range B. Then its inverse f-1, is defined as

f-1(y)=x iff f(x)=y, with domain B and range A.

Basically an inverse of function takes the output of f and returns the corresponding input.  When finding the inverse of a function:Check if the function is one-to-oneSolve the equation for x in terms of y, f(x)Then interchange them to get f-1(x)

Inverse Property: If f(x) and g(x) are inverses of each other, then f(g(x))=x and g(f(x))=x. i.e. if f(x)=x^3-1, g(x)=(x+1)^(1/3), f and g are inverses of each other.f(g(x))=((x+1)^(1/3))^3-1 = x+1-1 =xConversely you can check g(f(x))=x

Trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most basic and important trig. Functions are: sin(x), cos(x) and tan(x).

Inverse trigonometric functions are the inverse functions of the trigonometric functions, though they do not meet the official definition for inverse functions as their ranges are subsets of the domains of the original functions. Since none of the trigonometric functions are one-to-one, they must be restricted in order to have inverse functions.For example, just as the square root function is defined such that y2 = x, the function y = arcsin(x) is defined so that sin(y) = x. There are multiple numbers y such that sin(y) = x; for example, sin(0) = 0, but also sin(π) = 0, sin(2π) = 0, etc. It follows that the arcsine function is multivalued: arcsin(0) = 0, but also arcsin(0) = π, arcsin(0) = 2π, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x in the domain the expression arcsin(x) will evaluate only to a single value, called its principal value. These properties apply to all the inverse trigonometric functions.

Arcsin x

Arctan x

Arccos x

ExamplesEvaluate each of the following

we use the following restrictions on inverse cosine : The restriction on the   guarantees that we will only get a single value angle and since we can’t get values of x out of cosine that are larger than 1 or smaller than -1 we also can’t plug these values into an inverse trig function. So, using these restrictions on the solution we can see that the answer in this case is The second solution then follows asNote:

Exponential Functions: functions in the form of ax , where “a” is a constant, example f(x)=3x.They always have the property with various constants, such that the domain is (-∞,∞) and the range is (0, ∞). Logarithmic Functions: are the inverse of exponential functions, in the form loga(x), where a is the base constant,

example f(x)=log3(x). Observe from the above definition that the domain of various bases of these functions is (0,∞) and the range is (-∞,∞).

  

        ex: is an unique exponential function, such that the slope of tangent line at point x=0 is equal to 1. It is used so often

that its inverse or corresponding logarithmic function has its own notation, which is ln, f(x)=ln(x). (more on tangent lines later). e is approximately 2.718 correct to three decimal places.

Rules for exponentials and logarithms:ln(a)+ln(b)=ln(ab) a-n=1/an

ln(a)-ln(b)=ln(a/b) a0=1ln(an) =nln(a) loga(a)=1

Hyperbolic Functions: combinations of ex and e-x arise so frequently in nature, that they are given specific names. These are functions that have the same relationship to a hyperbola as trigonometric functions have a relationship to a circle. Definitions:sinh(x)=(ex-e-x)/2 cosh(x)=(ex+e-x)/2  

tanh(x)=sinh(x)/cosh(x)Like trigonometric, there are very useful identities we can use in solving problems involving these functions, which you can check using the function definitions.sinh(-x)=-sinh(x)cosh(-x)=cosh(x)cosh2(x)-sinh2(x)=11-tanh2(x)=sech2(x), divide 3. By cosh2(x)sinh(x+y)=sinh(x)cosh(y)+cosh(x)sinh(y)cosh(x+y)=cosh(x)cosh(y)+sinh(x)sinh(y)

A limit of a function is used to describe the value a function approaches as input approaches some value.In general, we say f(x) has limit L as x approaches a if f(x) can be made arbitrarily close to L by taking x sufficiently close to a limx-

>af(x)=L

Fact: For any polynomial limx->af(x)=f(a)

Limit LawsIf limx->af(x)=F and limx->ag(x)=G1. limx->a (f(x)+g(x))=F+G2. limx->a (f(x)-g(x))=F-G3. limx->a (f(x).g(x))=FG4. limx->a (f(x)/g(x))=F/G if G is not equal 0

Limits and Continuity

One sided Limits Left hand limit: limx->a- f(x)=L means f(x) has limit L as x approaches a from the left.Right hand limit: limx->a+ f(x)=L means f(x) has limit L as x approaches a from the right.limx->a f(x)=L iff limx->a- f(x)=L and limx->a+ f(x)=L.

Infinite LimitsWe can have limits that approach ∞ or -∞, because as they approach x=a, the output gets infinitely large (e.g. limx->0(1/x2)= ∞), these are called vertical asymptotes.  We can also evaluate the limit of functions approaching infinity, these limits are called horizontal asymptotes (e.g. limx->-∞(1/x2)=0 and limx->+∞(1/x2)=0, x2 greatly dominates 1 as x->∞).

Squeeze Theorem: if f(x)≤g(x)≤h(x), and limf(x)=limh(x)=L, then limg(x)=L, when x→a.

ExamplesFind limit of sqrt(x2+1) –x as x→∞

2

22

2

2 2 2 2

2

2

lim 1

1lim 1 ( )1

1 1 1lim( )1

1lim( )11lim( )11

1lim( )1( )* 1 1

1

lim( )11 1

1lim

1lim 1 1

01lim(1 ) lim(1)

01 0 101 1020

x

x

x

x

x

x

x

x

x

x x

x x

x xx xx x

x x x x x xx x

x x

x xx

xx

x

x

x

x

x

What’s the horizontal asymptote of sqrt(x2+1) - x from [0,∞)?Since the limit of sqrt(x2+1) –x as x→∞ was 0, we know the horizontal asymptote is x axis.

Find limx→0 (x2/3*cos(1/x2)). -1≤cos(a)≤1 for any a-1≤cos(1/x2)≤1-x2/3≤x2/3cos(1/x2)≤x2/3

limx→0(-x-2/3)= limx→0(x-2/3)=0Therefore by squeeze theorem limx→0(x2/3cos(1/x2))=0

ContinuityA function f(x) is continuous at x=a if it satisfies limx->af(x)=f(a) i.e

1.limx->af(x) exists2.f is defined at x=a3. limit is equal to the value of the functionA function that does not satisfy one or more of these points is discontinuous at x=a

Types of discontinuitiesInfinite (asymptote), jump, hole

Composition Rule for LimitsIf limx->ag(x)=L and f(y) is continuous at y=L then limx-

>a(fog)(x)= limx->a(f(g(x))= f(limx->ag(x))=f(L)

ExamplesFind limit of

1. limx->0 (sin(x)+cos(x))2. limx->1(x2-1)/(x-1) 

3. sin(x)+cos(x) is continuous everywhere, so the limit is sin(0)+cos(0)=0+1=1

4. We cannot substitute x=1, because the function is not continuous at 1We have,limx->1(x2-1)/(x-1)=limx->1((x+1)(x-1))/(x-1) ``divide out x-1``=limx->1(x+1), x+1 is continuous at x=1=1+1=2

Heavyside FunctionThe Heavyside function is defined as,

Heaviside functions are often called step functions.  Here is some alternate notation for Heaviside functions.  We can think of the Heaviside function as a switch that is off until t = c at which point it turns on and takes a value of 1.  So what if we want a switch that will turn on and takes some other value, say 4, or -7? Heaviside functions can only take values of 0 or 1, but we can use them to get other kinds of switches.  For instance 4uc(t) is a switch that is off until t = c and then turns on and takes a value of 4.  Likewise, -7uc(t) will be a switch that will take a value of -7 when it turns on.

Derivatives: The derivative of f(x) is defined by the function f`(x), which is defined as the limit limx->a(f(x)-f(a))/(x-a)

if we let h=x-a, then we get this limit in a different form, but expressing the same thing and sometimes easier to use.f`(x)=Limh->0(f(x+h)-f(x))/h

The derivative represents an infinitesimal change in the function with respect to x in this context. Hence the derivative is a function that can be used evaluate the instantaneous rate of change at any point x on the function f(x). (e.g. the derivative of x2 is 2x, which can be obtained using the definition). Derivatives are particular important motion, the derivative of the position of an object gives its velocity, and the derivative of its velocity gives its acceleration.

Differentiation

ExamplesGet derivative of f(x)=abs(x) at x=0 using the definition above. f’(0)=Limh->0(abs(0+h)-abs(0))/hf’(0)=Limh->0(abs(h))/hWhen dealing with absolute values for input, we have consider when the input is positive and when it is negative When h<0,abs(h)=(-h)So Limh->0-(-h/h)= Limh->0-(-1)=-1When h>0,abs(h)=hSo Limh->0+(h/h)= Limh->0(1)=1Since the left limit and right limit are not equal, we know that abs(x) has no derivative at x=0.

Derivative Of Tan(x) Using The Definition

0

0

0

0

tan( ) tan( )lim

sin( ) sin( )cos( ) cos( )lim

sin( ) cos( ) sin( ) cos( )cos( ) cos( )lim

(sin( )cos( ) sin( ) cos( ))cos( ) sin( )(cos( )cos( ) sin( )sin( ))cos( )cos( )lim

lim

h

h

h

h

h

x h xh

x h xx h x

hx h x x x h

x h xh

x h h x x x x h x hx h xh

2 2

0

2 2

0

2 2

0

0

cos( )sin( )cos( ) sin( ) cos ( ) sin( ) cos( ) cos( ) sin ( )sin( ))cos( ) cos( )

sin( ) cos ( ) sin ( )sin( )cos( ) cos( )lim

sin( )(cos ( ) sin ( ))cos( ) cos( )lim

sin( )(1)limcos( ) co

h

h

h

x x h h x x x h x hx h xh

h x x hx h xh

h x xx h xh

hx h

0 0

2

2

s( )sin( ) 1lim *lim

cos( ) cos( )11*( )

cos( 0)cos( )1

cos ( )sec ( )

h h

x hhh x h x

x x

xx

Rules for DifferentiationDerivative of a constant function.The derivative of f(x) = c where c is a constant is given by f '(x) = 0

Derivative of a power function (power rule).The derivative of f(x) = x r where r is a constant real number is given by f '(x) = r x r - 1

Derivative of a function multiplied by a constant.The derivative of f(x) = c g(x) is given by f '(x) = c g '(x)

Product Rule

Quotient Rule

Chain Rule

( )* ( ) ( ) ( ) ( ) ( )d d df x g x f x g x g x f xdx dx dx

2

( ) ( ) ( ) ( )( )( ) ( )

d dg x f x f x g xd f x dx dxdx g x g x

' '( ( )) ( )* ( ( ))d f g x g x f g xdx

ExamplesA particle moves on a vertical line so that its coordinate at time t is s(t)=t3-12t+3, t≥0.a)Find velocity and acceleration functionsb)When is the particle not moving, moving upward and when is it moving downward?c)Find the distance travelled in the time 0≤t≤3d)When is the particle speeding up? When is it slowing down? Solution:Velocity describes the rate of change of the position of the particle, using power rule v(t)=3t2-12Acceleration describes the rate at which the particle is speeding or slowing down, using power rule a(t)=6t The particle is not moving when it is has no rate of change in its position, so v(t)=0; 3t2-12=0; t2-4=0; t2=4; t=2, we can’t have t<0, so t≠-2The particle is moving upward when v(t)>0, the rate of change in position is positive 3t2-12>0, t>2 (isolate for like above)The particle is moving downward when v(t)<0, 3t2-12<0, t<2, but time must remain positive so we have 0<t<2.Therefore the particle is not moving when t=2, moving upward when t>2 and moving downward when t>0 and t<2. Distance downward: s(0)-s(2)=3-(-13)=16 (when object moving down)Distance upward: s(3)-s(2)= -6-(-13)=7 (when object is moving up)Therefore the total distance particle travelled was 23 units. The particle is slowing up when a(t)*v(t)≤0 and speeding up when a(t)*v(t)≥0. So the particle speeds up when a(t) and v(t) have the same signs and slows down when a(t) and v(t) have the opposite signs.When 0<t<2, a(t) by observation is positive, so the particle is slowing down on 0≤t<2, because v(t)<0. Conversely, the particle is speeding up on t>2 because v(t)>0 and a(t) on that interval.

Implicit Differentiation: is a method that consists of differentiating both sides of a function and then finding

Examples Find the derivative of arcsin(x)

dydx

2 2

2 2

2

2

2

arcsin( )sin( )

(sin( )) ( )

cos( ) 1

1cos( )

cos ( ) sin ( ) 1

cos ( ) 1 sin ( )

cos( ) 1 sin ( )1

1 sin ( )sin( )

1

1

y xy x

d dy xdx dx

dyydx

dydx y

y y

y y

y ydydx yy x

dydx x

Differentiability and ContinuityIf f’(a) exists then f(x) is continuous at x=a i.e. if a function is differentiable it is continuous. Corollary: If f(x) is discontinuous at x=a then f’(a) does not exist.

Examplef(x)=|x|From graph we can see f(x) is continuous everywhere.However, using the definition of derivative, derivative at x=0

Hence no derivative exists at x=0

Derivatives of Trigonometric and Inverse Trigonometric Functions

Example

Derivatives Of Exponential And Log Functions

ExamplesDerivate e(y/x)+sin(x+y2)=0

Logarithmic DifferentiationThe method of logarithmic differentiation ,in calculus, uses the properties of logarithmic functions to differentiate complicated functions and functions where the usual formulas of differentiation do not apply

Examplesy = x sin x

ln y = ln [ x sin x ]ln y = sin x ln xy ' / y = cos x ln x + sin x (1/x)y ' = [ cos x ln x + (1/x) sin x ] y y ' = [ cos x ln x + (1/x) sin x ] x sin x

Derivatives Of Hyperbolic Functions

Rolle’s Theorem: If a function f(x) satisfies1.f(x) is continuous for a≤x≤b2.f’(x) exists for a<x<b3.f(a)=f(b)then there exists at least one point c with a<c<b such that f’(c)=0

Mean Value Theorem: If a function f(x) satisfies4.f(x) is continuous for a≤x≤b5.f’(x) exists for a<x<bthen there exists at least one point c with a<c<b such that f’(c)=(f(b)-f(a))/(b-a)

Newton’s Method: The idea of the method is as follows: one starts with an initial guess which is reasonably close to the true root, then the function is approximated by its tangent line (which can be computed using the tools of calculus), and one computes the x-intercept of this tangent line (which is easily done with elementary algebra). This x-intercept will typically be a better approximation to the function's root than the original guess, and the method can be iterated.If xn is an approximation a solution of f(x)=0 and if f’(xn )≠0 the next approximation is given by,

                                                         

Increasing and Decreasing FunctionsA function f(x) is increasing on an interval I if for all x1> x2 in I, f(x1)> f(x2)A function f(x) is decreasing on an interval I if for all x1> x2 in I, f(x1)< f(x2) Increasing/ Decreasing TestA function is increasing on an interval I if f’(x)≥0 for all x in I and f’(x)=0 at a finite number of pointsA function is decreasing on an interval I if f’(x)≦0 for all x in I and f’(x)=0 at a finite number of points

Critical Point: A critical point of a function is a point in the domain of the function where f’(x)=0 or f’(x) does not existRelative Maximum: A function has a relative maximum f(x0) at x=x0 if there is an open interval I such that f(x)≤f(x0) for all x in I Relative Minimum: A function has a relative minimum f(x0) at x=x0 if there is an open interval I such that f(x)≥f(x0) for all x in I

Good Luck On Your Midterm!

Zahra Mahmood Bodla