Calc 1 MAT 150 Flashcards
What are the steps for solving a function with a number ?
1) Plug the number in for the varriable being asked to be solved
2) Solve
What are the steps for solving a function with a variable for a domain value?
1) Plug the variable where the domain value is in the eqn
2) solve
What are the steps for solving comp functions of different functions such as (f(g)(x)) ?
1) Plug inside function x values in the x values of the outer function
2) simplify to solve
What are the steps for solving problems where they provide a function, and then they ask to simplify an expression of [f(#+h)- f( same #)]/ h ?
1) Place the # + h in every value in the provided function (not in expression)
2) Simplify and solve
What are the steps for solving DQ problems where you are asked to simplify the DQ for the given function?
1) Plug (x+h) for every value of x in the provided function
2) distribute
3) cancel and solve
What are the steps for solving problems that ask:
A function and an interval of its independent variable are given. The endpoints of the interval are associated with the points P and Q on the graph of the function. Answer parts a and b.
After t seconds, an obj dropped from rest, falls a distance d = kt^2, where d is the measured in feet and min = t = max
a) Sketch a graph of the function and secant line through P and Q.
b) Find the slope of the secant line in part a, and interpret your answer in terms of an average rate of change over the interval. Include units in your answer.
For part a, look for the interval points stated.
For part b:
1) To find slope, make a chart of the lowest and highest values.
2) Plug lowest and highest value of t in the interval into the formula d(t) =kt^2.
3) To calc M sec, calc the same way you would find slope of linear line use change in d(t) over change in t.
4) Solve and play close attention to units being measured and what is being measured
What are the steps for solving problems that ask the following?
Determine whether the graph of the following equation and/or function has symmetry about/wrt the x-, y- axis, or the origin. Check your work by graphing. Select all that apply.
1) Make the f(x) into f(-x)
2) Ask yourself is the the the equation now f(x) or -f(x)
If f(-x) = f(x) the eqn is even and wrt y-axis
If f(-x) = -f(x) the eqn is odd and wrt origin
If f(-x) = f(x) the eqn is
even sym and wrt y-axis
If f(-x) = -f(x) the eqn is
odd sym and wrt origin
Linear Function formula
y=x
Quadratic function parabolas formula
y=x^2
Cubic funct formula
y=x^3
Exponential function formula
Y= b^x
log function formula
y= logb^x
transformation function formula
y= cf(a(x-b)) + d
Horizontal scaling formula
y = f(ax)
scaling
shrink or stretch
shift
left or right or up/down
For horizontal scaling, when a>1 what do you do?
Horizontal shrink
For horizontal scaling, when 0 1
Horizontal stretch
horizontal shift formula
y = f(a(x-b))
horizontal shift right when
b>0
or
f(x-c)
horizontal shift left when
x+b
vertical scaling formula
cf(a(x-b)) by a factor |c|
vert stretch when
c > 1
vert shrink when
0
Vert shift to upward
d > 0
Vert shift downward
d
Expotential function general form
f(x) = b^x where b=base
Natural expo function
f(x) = e^x
Inverse functions are
1-1, so horizontal line test and are functions so vertical line test
What are the steps for finding inverse functions?
1) Replace f(x) with y
2) Interchange x and y
3) Solve for y
4) Replace y with f^-1(x) in new eqn
Log functions are what to expo functions
inverse functions
Natual Log funct
When b=e; ln x = loge^x
Common Log function
When b=10, log10^x = logx
log sum ID
logb^(XY) = logb^x + logb^y
Diff log ID
logb^(x/y) = logb^x - logb^y
Power log ID
logb^(x^d) = d (logb^x)
logb^b^x =
xlogb^b
logb^1 =
0
logb^b =
1
inverse log ID
logb^x = b^x
logb^x =
xlogb^b = x
blogb^x =
x (log is cancelled)
blogb^y =
y (log is cancelled)
What are the steps for solving a problem like this:
Determine which graph is a function that is one-to-one on the interval (x, y) but is not one to one on (interval, interval)
1) Take out the side of the interval they asking for (covr it up)
2) Horizontal and vert line test to validate that it is one to one for that interval
How do you find inverse functions that fraction formed such as ? #/ x2 + or minus some number
1) follow normal steps to get inverse
2) multiply denominator on both side of equation
3) distribute on the side of the equation with the variable
4) divide each side by the variable X
5) subtract or add given the number being added to or subtracted to the y2
6) sq root each side
How do you solve :
Solve the following log eqn: logbx = #
logx^3 + 1/4
1) The number is the power/ y where b is the base
2) cancel each side by placing the b value in front log equation (special id)
3) solve like a normal b^power problem
How do you solve: Without using a calculator, solve the following equation. #^x = some other #
4^x = some number
1) Use Change of Base Rules using ln
2) Mulitply each side of eqn by ln
3) Divide to solve for x
Change of base
logb^ (x) = logc^(x) / logc^(b) where c is constant base like e which is = ln
logb^x = ln x/ ln b
^(x+#)= some other
How do you solve :
Without using a calculator, solve the following equation.
3(x-5) = 360
1) Use inverse rule logb^b^x = x, and take log on each side
ex) log3^3(x-5) = log3^(360)
2)Cancel and rewrite and solve for x
x= 5 + log3^(360)
formula for average velocity
Vav = s(b) - s(a)/ b - a
What is the period of a trig f(x) ?
the smallest positive real number such that f(x+P) = f(x) for all x in the domain
Cos, sin, csc, sec theta is what period?
2 pi
COT and tan theta is what period?
Pi
What are the pythagorean IDs for the trig functions?
sin2theta + cos2theta =1
1 + cot2theta = csc2theta
tan2theta +1 = sec2theta
sin =
y
cos =
x
arc =
s
r =
radius
radian/ theta =
S/R
Unit circle
r = 1 & theta = s
sin theta =
y/r
cos theta =
x/r
tan theta =
y/x
cot theta =
x/y
sec theta =
r/x
csc theta =
r/y
how do you convert degrees to radians mulitply qty by
pi/180
how do you convert radian to degree qty by
180/ pi
reciprocal id for trig functs
tan theta= 1/cot theta
sec theta = 1/cos theta
csc theta = 1/ sin theta
Quotient ID for trig function
tan theta = sin theta/ cos theta
cot theta = cos theta/ sin theta
Double- Angle ID for trig functions
sin2theta = 2 sin theta cos theta cos2theta = cos2 theta - sin2 theta
In quad one, all trig functions are
positive or > 0
In quad two, only sin and csc are
positive
In quad 3, tan and cot are only
positive
In quad 4, only cos and sec are
positive
Asymtope pattern for tan theta =
Pi/2 +k(pi)
domain and range for asymtompe for tan
domain : theta | theta cant = pi/2 + kPi
range = R
pattern for cot asymtopes
kpi
domain and range for pattern for cot asymtopes
theta | theta cant equal kpi (domain)
range = R
arcsin x & arccos x D =
[-1, 1]
arctan x and arccot x D=
(-inf, +inf)
arcsin x range
[-pi/2, pi/2]
arccos x range
[0, pi]
arctan x range
(-pi/2, pi/2)
arccot x range
(0, pi)
How to solve equations that look like this:
Evaluate the following expression by drawing a unit circle and the appropriate rt triangle.
sin (-3pi/4)
Memorization of chart
convert xpi/denom to degrees
For tan, use pythagorean therom
How to solve equations that look like this:
Eval:
sin^-1 (1/2)
1) What common angle are they looking for
If it is sin/ cos(1) use the period version of graph
How to solve equations that look like this:
Find the exact value in radians, of the expression
sin^-1 (-1) =
1) The answer must lie inbetween the interval [-1/2, 1/2] since it is sin (depends on trig function based on memory)
2) Use memorization of angles table to solve
How to solve equations that look like this:
Use a right triangle to write the following expression as an algebraic expression. Assume that x is positive and that the given inverse trig function is defined fir the expression in x.
1) Find inside trig function
2) Based on inside trig function, the O, H, A sides will be given of triangle
3) Use find pythrgorean theorem to solve side needed
How to solve equations that look like this:
2 ladders of length, a, lean against opposite walls of any alley with their feet touching, as shown in the figure on the right. One ladder extends h feet up the wall and makes a 75 degree angle with the ground, and the other laddder makes a 45 degree angle with ground. Find width of the alley in terms of a, h , and/or k. Assume the ground is horizontal and perpendicular to both walls. question 17 in hw 3
1) What adj 1 and adj2 and add them together
2) Solve adj1 by = cos45(adj1/a)= special angle x a
3) x2 = sq rootof a2 -h2
distance function
s(t) = -16t2+96t
Vav = m sec =
s(t1)- s(t0)/ t1-t0
instaneous velocity
velocity at a specific point close to/ as it approaches average
unit circle
0 radians = what degrees and what is its point on the unit circle?
0 degrees
1,0
unit circle
pi/6 =
points?
30 degress
√3/2, 1/2
unit circle
pi/4=
pts?
45 degs
√2/2, √2/2
unit circle
π/3 =
pts?
60
1/2, √3/2
unit circle
π/2 =
pts?
90
0,1
unit circle
2π/3 =
pts?
120
-1/2, √3/2
unit circle
3π/4 =
pts=
135
- √2/2, √2/2
unit circle
5π/6 =
pts =
150
√3/2, 1/2
unit circle
π=
pts=
180
-1, 0
unit circle
7π/6
210
- √3/2, -1/2
unit circle
5π/4
225
- √2/2, - √2/2
unit circle
4π/3
240
-1/2, - √3/2
unit circle
3π/2
270
0, -1
unit circle
5π/3
300
1/2, - √3/2
unit circle
7π/4
315
√2/2, - √2/2
unit circle
11π/6
330
√3/2, - 1/2
unit circle
2π
360
1, 0
cos 2 theta =
1 - 2 sin^2 theta
tan2theta=
2 tan theta/ 1- tan^2 theta
