Math 1111 Final Exam Formula Sheet PDF

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Formula Sheet Difference of Two Squares: x2 − y2 = (x + y)(x − y)

Difference of Two Cubes: x3 − y3 = (x − y)(x2 + xy + y2 ) Sum of Two Cubes: x3 + y3 = (x + y)(x2 − xy + y2 )

Quadratic Formula: x=

√ −b± b2 −4ac 2a

is/are the solution(s) for ax2 + bx + c = 0

Discriminant: b2 − 4ac is the discriminant for ax2 + bx + c = 0 Distance Formula: The distance between (x1 , y1 ) and (x2 , y2 ) is

√

(x2 − x1 )2 + (y2 − y1 )2

Midpoint Formula: The midpoint between (x1 , y1 ) and (x2 , y2 ) is

(x

1 +x2

2

2 , y1 +y 2

)

Equation of a Circle (Standard Form): (x − h)2 + (y − k )2 = r2 • (h,k) is the center of the circle • r is the radius of the circle Slope of a Line: m=

y2 −y1 x2 −x1

where (x1 , y1 ) and (x2 , y2 ) are points on the line

2

Equations of a Line: Slope Intercept Form: y = mx + b • m - slope of the line • b - y-intercept Point-Slope Form: y − y1 = m(x − x1 ) • m - slope of the line • (x1 , y1 ) is a point on the line Average Rates of Change: The average rate of change of the function f (x) from x = a to x = b is given by

f (b)−f (a) b−a

Transformations of Functions: Vertical Shifts (assuming k is positive) • g(x) = f (x) + k shifts the graph of f (x) up k units • g(x) = f (x) − k shifts the graph of f (x) down k units Horizontal Shifts (assuming k is positive) • g(x) = f (x + k) shifts the graph of f (x) left k units • g(x) = f (x − k) shifts the graph of f (x) right k units Reflections • g(x) = −f (x) reflects the graph of f (x) over the x-axis • g(x) = f (−x) reflects the graph of f (x) over the y-axis Stretches and Compressions (assuming k > 1 and 0 < a < 1 is positive) • • • •

g(x) = kf (x) stretches the graph of f (x) vertically by a factor of k g(x) = af (x) compresses the graph of f (x) vertically by a factor of a g(x) = f (kx) compresses the graph of f (x) horizontally by a factor of g(x) = f (ax) stretches the graph of f (x) horizontally by a factor of 1a

1 k

Quadratic Function Forms • Standard form: f (x) = ax2 + bx + c • Vertex form: f (x) = a(x−h)2 +k where (h, k) is the vertex of the quadratic function Vertex of a Quadratic Function in Standard Form: ( ) b2 , c − Vertex = −b 2a 4a

3

Horizontal Asymptotes: Let f (x) =

p(x) q(x)

where p(x) = an xn + an−1 xn−1 + · · · a1 x + a0 and q(x) = bm xm + bm−1 xm−1 + · · · b1 x + b0

• If m < n, then f (x) does not have a horizontal asymptote. • If m > n, then f (x) has a horizontal asymptote at y = 0 • If m = n, then f (x) has a horizontal asymptote at y = bamn Properties of Exponents: • xa xb = xa+b • (xa )b = xab

• x−n = • x1/n =

1 xn

√ n x

Properties and Laws of Logarithms: • y = logb (x) if and only if by = x • blogb (x) = x for all x > 0 • logb (bx) = x for all x > 0 • logb (x) =

logA (x) logA (b)

• logb (xy) = log b (x) + logb (y) ( ) • logb xy = logb (x) − log b (y)

• r logb (x) = log b (xr )

4

Compound Interest Formula: ( )nt A(t) = P 1 + nr • • • • •

A(t) - total amount after t years P - initial amount r - annual interest rate (written as a decimal) n - number of times compounded per year t - time (in years)

Compounded Continuously Formula: A(t) = P ert • • • •

A(t) - total amount at time t P - initial amount r - growth or decay rate (written as a decimal and negative if decaying) t - time

Newton’s Law of Cooling: To (t) = Ts + (Ti − Ts )ert To (t) - temperature of the object at time t Ts - temperature of the object’s surroundings Ti - initial temperature of object r - the continuous rate of cooling of the object (written as a decimal and negative to indicate the cooling process) • t - time

• • • •

Savings Plan Formula (Regular Deposits):  [(  1+ A(t) = Pm   • • • • •

] r )nt −1  n  (r )  n

A(t) - total amount after t years Pm - regular contribution amount r - annual interest rate (written as a decimal) n - number of times compounded per year t - time (in years)

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Half Life: A(t) = P e • • • •

ln(1/2) tH

t

A(t) - amount of substance at time t P - initial amount of substance tH - half-life of the substance t - time

pH Scale: pH = − log (H + ) • pH - an object’s pH level • H + - an object’s concentration of hydrogen ions measured in moles per liter Earthquake Richter Scale: R = log

( A) S

• R - the magnitude of an earthquake • A - amplitude of the earthquake wave (also known as the intensity) • S - the amplitude of the smallest detectable wave...