DIFFERENTIAL EQUATIONS 2024
TEST QUESTIONS WITH CORRECT
DETAILED ANSWERS UPDATED
Polynomial - Answer-Comes from poly- (meaning "many") and -nomial (in this case
meaning "terms"...so it says "many terms"
What can have constants, variables, and exponents but never division by a variable? -
Answer-A Polynomial
What is a linear polynomial? - Answer-polynomial of degree one
Polynomials of degree one, two or three are respectively - Answer-linear polynomials,
quadratic polynomials and cubic polynomials
What is the degree of the polynomial with only one variable? - Answer-The largest
exponent of that variable
What can have no variable at all? - Answer-A Polynomial (e.g. 22)
What is a differentiable function of one variable in calculus? - Answer-A function whose
derivative exists at each point in its domain. As a result, the graph of a differetiable
function must have a (non-vertical) tangent line at each point it its domain, be relatively
smooth, and cannot contain any break, bends, cusps.
What is a direction field (or slope field/ vector field)? - Answer-A picture of the general
solution to a first order differential equation with the form dy/dx = f(x,y).
What does "differential" represent in calculus? - Answer-Represents the principal part of
the change in a function y = f(x)with respect to changes in the independent variable.
What is the differential of a function equal to? - Answer-The derivative of the function
times the differential of the independent variable: dy = df(x) = f'(x)dx
What Does Delta Mean? - Answer-Has 2 different mathematical meanings for the
uppercase symbol. Both meanings begin with the letter D just like the Greek letter:
Difference or Discriminant.
What is a Greek alphabet with several different mathematical meanings? - Answer-
Delta
, Which delta is more often seen in calculus? - Answer-Lowercase Delta
What letter does both delta cases start with? - Answer-The letter 'D'
Which meaning of Delta is simply the difference, or change, in a certain quantity? -
Answer-Difference
Which meaning of Delta is found in the quadratic formula and determines the number of
solutions a particular quadratic equation has? - Answer-Discriminant
What is the complete set of possible values of the independent variable? - Answer-
DOMAIN of a function
What is the set of all possible x-values which will make a function work, and will output
real y-values? - Answer-Domain
When finding the domain it's important to remember what two things? - Answer-(1) The
denominator (bottom) of a fraction CANNOT BE ZERO. (2) The number under the
square root sign MUST BE POSITIVE.
What is the domain of this function is x >= -7, since x cannot be less than -7? - Answer-
y = sqrt (x + 7);
What is plugging some number less than -7 (like -8 or -10) into the function y = sqrt(x +
7)? - Answer-The only ones that "work" and give me an answer are the ones greater
than or equal to -7.
What is how to find the domain? - Answer-Looking for those values of the independent
variable (usually x) which I'm ALLOWED to use.
What is the complete set of all possible RESULTING VALUES of the dependent
variable (y, usually), after I have substituted the domain? - Answer-RANGE of a function
What is the resulting y-values I get after substituting all the possible x-values? - Answer-
Range
What is the spread of possible y-values (minimum y-value to maximum y-value)? -
Answer-RANGE of a function
What is how to find the range? - Answer-Substitute different x-values into the
expression for y to see what is happening. (Ask myself: Is y always positive? Always
negative? Or maybe not equal to certain values?)
What are those values of a variable that cause a polynomial to evaluate to zero? -
Answer-ROOTS of a polynomial
TEST QUESTIONS WITH CORRECT
DETAILED ANSWERS UPDATED
Polynomial - Answer-Comes from poly- (meaning "many") and -nomial (in this case
meaning "terms"...so it says "many terms"
What can have constants, variables, and exponents but never division by a variable? -
Answer-A Polynomial
What is a linear polynomial? - Answer-polynomial of degree one
Polynomials of degree one, two or three are respectively - Answer-linear polynomials,
quadratic polynomials and cubic polynomials
What is the degree of the polynomial with only one variable? - Answer-The largest
exponent of that variable
What can have no variable at all? - Answer-A Polynomial (e.g. 22)
What is a differentiable function of one variable in calculus? - Answer-A function whose
derivative exists at each point in its domain. As a result, the graph of a differetiable
function must have a (non-vertical) tangent line at each point it its domain, be relatively
smooth, and cannot contain any break, bends, cusps.
What is a direction field (or slope field/ vector field)? - Answer-A picture of the general
solution to a first order differential equation with the form dy/dx = f(x,y).
What does "differential" represent in calculus? - Answer-Represents the principal part of
the change in a function y = f(x)with respect to changes in the independent variable.
What is the differential of a function equal to? - Answer-The derivative of the function
times the differential of the independent variable: dy = df(x) = f'(x)dx
What Does Delta Mean? - Answer-Has 2 different mathematical meanings for the
uppercase symbol. Both meanings begin with the letter D just like the Greek letter:
Difference or Discriminant.
What is a Greek alphabet with several different mathematical meanings? - Answer-
Delta
, Which delta is more often seen in calculus? - Answer-Lowercase Delta
What letter does both delta cases start with? - Answer-The letter 'D'
Which meaning of Delta is simply the difference, or change, in a certain quantity? -
Answer-Difference
Which meaning of Delta is found in the quadratic formula and determines the number of
solutions a particular quadratic equation has? - Answer-Discriminant
What is the complete set of possible values of the independent variable? - Answer-
DOMAIN of a function
What is the set of all possible x-values which will make a function work, and will output
real y-values? - Answer-Domain
When finding the domain it's important to remember what two things? - Answer-(1) The
denominator (bottom) of a fraction CANNOT BE ZERO. (2) The number under the
square root sign MUST BE POSITIVE.
What is the domain of this function is x >= -7, since x cannot be less than -7? - Answer-
y = sqrt (x + 7);
What is plugging some number less than -7 (like -8 or -10) into the function y = sqrt(x +
7)? - Answer-The only ones that "work" and give me an answer are the ones greater
than or equal to -7.
What is how to find the domain? - Answer-Looking for those values of the independent
variable (usually x) which I'm ALLOWED to use.
What is the complete set of all possible RESULTING VALUES of the dependent
variable (y, usually), after I have substituted the domain? - Answer-RANGE of a function
What is the resulting y-values I get after substituting all the possible x-values? - Answer-
Range
What is the spread of possible y-values (minimum y-value to maximum y-value)? -
Answer-RANGE of a function
What is how to find the range? - Answer-Substitute different x-values into the
expression for y to see what is happening. (Ask myself: Is y always positive? Always
negative? Or maybe not equal to certain values?)
What are those values of a variable that cause a polynomial to evaluate to zero? -
Answer-ROOTS of a polynomial
