Study Flashcards
Differentiate this problem:
y=ln(x+1 / √x-2
1/ x+1 - 1 /√x-2 • 1 /2√x-2
Do this example:
Differentiate:
f(x)=log(10) (2+sinx)
u=2+sinx and y=log(10) (u)
f’(x)=dy/dx=dy/du • du/dx
f’(x)=1/ u(ln(10)) • cosx
f’(x)=1/ (2+sinx)(ln(10)) • cosx
=cosx/ (2+sinx)(ln(10))
How do you use the squeeze theorem for lim(x^2 (cos(7πx)) = 0
x—>0
Range of cos functions is always -1 to 1, so:
-1 <= cos(7πx) <= 1.
Multiply all by f(x) to get:
-x^2 <= x^2 (cos(7πx)) <= x^2.
All of these have limit 0 as x—>0 so by squeeze theorem, g(x) (the middle function)
approaches 0.
If you get a positive and negative limit for a problem like:
lim 8x+56 / |x+7|
x—>-7
Limit DNE
Do this example:
lim 8-|x| / 8+x
x—>-8
1 because when x is positive, the answer is unsolved and therefore DNE
Why is this undefined?:
g(x)= { 2-x^2 if 1 < x <= 2
{ x-3 if x > 2
Evaluate lim g(x)
x—>2
Because it is equal to two and not equal to two at the same time.
Do these examples:
lim [[x]] lim [[x]] lim [[x]]
x—>-5+ x—>-5 x—>-5.2
lim [[x]] lim [[x]]
x—>n- x—>n+
-5 DNE -6 n-1 n
Do this example:
Find number a such that the limit exists:
lim 3x^2 + ax + a + 6 / x^2 + x -2
x—>-2
Find the denominator number with opposite sign of limit (x+2).
Replace numerator (x) value with limit value and solve only numerator to get a.
Replace a into original equation for new equation and then solve for limit.
If lim f(x) = 0 and lim g(x)
x—>3 x—>3
does lim f(x) / g(x) exist?
x—>3
Yes
When both functions’ limits equal 0, what CAN exist?
The division of the limits of these functions
If g is continuous at a and f is continuous at g(a), then
The composite function (f o g) given by (f o g)(x) = f(g(x)) is continuous at a
The composite function (f o g) given by (f o g)(x) = f(g(x)) is continuous at a when
g is continuous at a and f is continuous at g(a)
Which types of functions are continuous at every number in their domain?
Polynomials, rationals, roots, trigs, inverse trigs, exponentials, logarithmic
Polynomials, rationals, roots, trigs, inverse trigs, exponentials, logarithmic
Functions continuous at every number in their domain
Why is the discontinuity at x=1 of
f(x)={ x+8 if x<0
{ e^x if 0 <= x <= 1
{ 9-x if x > 1
only continuous from the left?
Because there is a jump when x<0 and 0<=x<=1 in which the only actual defined points are at the second function (closed circle)
Do this example:
For what value of the constant c is the function f continuous on
(-∞, ∞)?
f(x)={ cx^2 + 2x if x< 5
{ x^3 - cx if x>= 5
Plug in 5 for x for both equation. Set these derived equations equal to each other and solve for c to get 115/30
All types
CHAR, INT, SMALLINT, VARCHAR, TINYINT, MEDIUMINT, BIGINT, DECIMAL, DATE
Statement to add constraint to column
ALTER TABLE table_name MODIFY COLUMN column_name VARCHAR(20) NOT NULL;
ALTER TABLE table_name MODIFY COLUMN column_name VARCHAR(20) NOT NULL;
Statement to add constraint to column
Make an attribute not negative
UNSIGNED constraint
Statement to return a number to 2 decimal places
SELECT TRUNCATE(num1/num2, 3);
to return to “3” decimal places
SELECT TRUNCATE(num1/num2, 3);
Returns number to 3 decimal places
Return remainder of number
SELECT attribute/number MOD 60;
Will return remainder of division by 60
SELECT attribute/number MOD 60;
Will return remainder of division by 60
Statement to change column value
UPDATE table_name SET column_name = value
UPDATE table_name SET column_name = value
Statement to change column value
Statement to delete specific value from table
DELETE FROM table_name WHERE (condition);
DELETE FROM table_name WHERE (condition);
Statement to delete specific value from table
Statement to set default on attribute
ALTER TABLE table_name ALTER COLUMN column_name SET DEFAULT (value);
Statement to add primary key into table
ALTER TABLE table_name ADD PRIMARY KEY table_name_pk (attribute_name);
Statement to add column to table
ALTER TABLE table_name ADD COLUMN column_name VARCHAR(20);
Statement to get rid of default value
ALTER TABLE table_name ALTER column_name DROP DEFAULT;
Statement to change datatype of column
ALTER TABLE table_name MODIFY COLUMN column_name VARCHAR(20);
ALTER TABLE table_name MODIFY COLUMN column_name VARCHAR(20);
Statement to change datatype of column
Statement to find any values that start with ‘a’
WHERE column_name LIKE ‘a%’
WHERE column_name LIKE ‘a%’
Statement to find any values that start with ‘a’
Statement to find any values that end with ‘a’
WHERE column_name LIKE ‘%a’
WHERE column_name LIKE ‘%a’
Statement to find any values that end with ‘a’
Statement to find any values that have ‘a’ in any position
WHERE column_name LIKE ’%a%’
WHERE column_name LIKE ’%a%’
Statement to find any values that have ‘a’ in any position
Statement that finds any values that have ‘a’ in the second position
WHERE column_name LIKE ‘_a%’
WHERE column_name LIKE ‘_a%’
Statement that finds any values that have ‘a’ in the second position
Statement to find any values that start with ‘a’ and are at least 3 characters in length
WHERE column_name LIKE ‘a__%’
WHERE column_name LIKE ‘a__%’
Statement to find any values that start with ‘a’ and are at least 3 characters in length
Statement to find any values that start with ‘a’ and end ‘o’
WHERE column_name LIKE ‘a%o’
WHERE column_name LIKE ‘a%o’
Statement to find any values that start with ‘a’ and end ‘o’
Statement to drop primary key
ALRER TABLE table_name DROP PRIMARY KEY
Statement to add unique value an existing column
ALTER TABLE table_name ADD UNIQUE (column_name)
ALTER TABLE table_name ADD UNIQUE (column_name)
Statement to add unique value an existing column
Statement to format decimals
DECIMAL(totalSize, decimalPlaces)
Statement to add column after another column
ALTER TABLE table_name ADD COLUMN column_name VARCHAR(20) AFTER column_name
What does the graph of the tangent like do when a function’s graph has a kink?
Vertical asymptotes at x=0 line
Vertical asymptotes at x=0 line for graph of derivative when
Original graph has a kink
Do this example:
Find the points on the curve y=x^4-16x^2 +2 where the tangent like is horizontal
Find derivative and set equal to 0. Solve to get 0, √8, and -√8
What is the derivative of 2^30?
0 because it counts as a constant
What is the derivative of a log:
log(a) (x)
1 / xln(a)
