Pre-Calculus Review - Exam Prep

Flashcards to help study key concepts and formulas for an upcoming calculus exam.

When f(x) is concave down, what is happening to the rate of change?

The rate of change is decreasing.

When f(x) is concave up, what is happening to the rate of change?

The rate of change is increasing.

When f(x) is decreasing, what is the rate of change?

The rate of change is negative.

When f(x) is increasing, what is the rate of change?

The rate of change is positive.

What are extrema?

Maxima or minima (relative or absolute) on a graph.

What is a point of inflection?

A point where the concavity of a function changes.

What is a key characteristic of a linear function's rate of change?

The rate of change is constant over any interval.

What is true of the 2nd differences of output values for a quadratic function over equal-length input intervals?

The 2nd differences are constant.

What behavior do output values exhibit in an exponential function over equal-length input intervals?

Output values are proportional.

What behavior do input values exhibit in a logarithmic function, and what is the effect on the output?

Proportional input values result in a constant change in output values.

How is the average rate of change of f(x) over the interval [a, b] calculated?

It is the slope of the line between the points, calculated as (f(b) - f(a)) / (b - a).

lim x→∞ f(x) = 3 verbally means?

As the input value increases without bound, the output values approach 3; the graph has a horizontal asymptote at y=3.

lim x→-∞ f(x) = ∞ verbally means?

As the input value decreases without bound, the output values increase without bound.

Given f(x) = (x-a)/(x-b), what does f(a) indicate?

A hole in the graph.

Given f(x) = (x-a)/(x-b), what does f(b) indicate?

A vertical asymptote.

In the transformation g(x) = a f(b(x+h)) + k, what does 'a' control?

Vertical stretch (if |a| > 1), compression (if |a| < 1), and reflection (if a is negative).

In the transformation g(x) = a f(b(x+h)) + k, what does 'b' control?

Horizontal stretch (if 0 < |b| < 1), compression (if |b| > 1), and reflection (if b is negative).

What is the formula for a^m * a^n?

a^(m+n)

What is the formula for a^m / a^n?

a^(m-n)

What is the formula for (a^m)^n?

a^(m*n)

What is the formula for (ab)^m?

a^m * b^m

What is the formula for a^0?

1

What is the formula for a^(-n)?

1/a^n

If m < n in the rational functions limit, what is the lim x→±∞ ?

0

If m = n in the rational functions limit, what is the lim x→±∞ ?

a

If m > n in the rational functions limit, what is the lim x→±∞ ?

+/- infinity

What is the arithmetic sequence formula?

An = Ak + d(n-k)

What is the gemoetric sequence formula?

Gn = Gk (r)^n-k

What is the log base b (mn) property?

logb(m) + logb(n)

What is the log base b (m/n) property?

logb(m) - logb(n)

What is the log base b (m)^n property?

n logb(m)

What is the log base b (b) proerty?

1

What is the log base b (1) proerty?

0

What is the change of base formula for logs?

logc (m) / logc (b)

What is f^-1(f(x)) equal to?

x

How do you find the inverse of a function algebraically?

Switch x and y on f(x) and solve for y.

What is the relationship between the domain and range of a function and its inverse?

The domain of f^-1(x) is the range of f(x) and vice-versa.

What does cos(A) equal in Circular Trigonometry?

x

What does sin(A) equal in Circular Trigonometry?

y

What does tan(A) equal in Circular Trigonometry?

y/x

What are the Sine/Cosine Graph Properties?

f(x)= a sin (b (x+c)) + d

What do lal and d equal?

lal = amplitude and d=midline

What is the Pythagorean Identity for sine and cosine?

sin^2 + cos^2=1

What is the Quotient Identity for tangent?

sine / cosine

What is the Reciprocal Identity for tangent?

1/cotangent

State the Double Angle formula for Sine.

sin 2θ = 2 sinθ cosθ

What are the inputs and outputs for inverse trig functions?

Input = ratio of sides; Output = Standard position angle

What is the domain and range of arcsin(x)?

Domain: [-1, 1]; Range: [-π/2, π/2]

What is the domain and range of arccos(x)?

Domain: [-1, 1]; Range: [0, π]

What is the domain and range of arctan(x)?

Domain: (-∞, ∞); Range: (-π/2, π/2)

What do r and theta represent in polar coordinates?

r represents the distance from the origin, and theta represents the angle from the positive x-axis.

How are rectangular coordinates (x, y) converted to polar coordinates (r, θ)?

r^2 = x^2 + y^2 and tan^-1(|y/x|) = Reference angle. Use the quadrant to find θ.