Calculus Review Flashcards

Flashcards based on lecture notes covering function behaviors, rates of change, transformations, trigonometry, and polar coordinates.

When f''(x) is negative, the graph of f(x) is described as?

Concave down. Rate of Change is decreasing

When f''(x) is positive, the graph of f(x) is described as?

Concave up. Rate of Change is Increasing

What does a point of inflection indicate on a graph?

A change in concavity.

Describe the rate of change for a linear function.

Constant on any interval.

What is constant for quadratic functions over equal-length input intervals?

2nd differences of output values.

How do output values behave for exponential functions over equal-length input intervals?

Output values are proportional.

How does the average rate of change relate to the slope?

It is the slope of the secant line between two points on the function.

What does the limit notation lim x→∞ f(x) = 3 describe graphically?

The graph has a horizontal asymptote at y = 3.

If m < n in the rational function g(x) = (ax^m + …) / (bx^n + …), what is the limit as x approaches infinity?

0

Describe the vertical changes to a function when |a| > 1 in the transformation g(x) = a f(b(x+h)) + k.

Vertical stretch.

What happens to the graph when 'a' is negative in g(x) = a f(b(x+h)) + k?

Reflection over the x-axis.

In transformations, what type of change occurs when 0 < |b| < 1?

Horizontal stretch.

State the product of powers property of exponents.

a^m * a^n = a^(m+n)

What is the power of a power property of exponents?

(a^m)^n = a^(m*n)

Describe the formula for the nth term of an arithmetic sequence.

an = ak + d(n-k)

What is the formula for the nth term of a geometric sequence?

an = ak * r^(n-k)

State the product rule of logarithms.

logb(mn) = logb(m) + logb(n)

What is the power rule of logarithms?

logb(m)^n = n logb(m)

Numerically, how are inverse functions related?

If f(a) = b, then f⁻¹(b) = a

Graphically, how are inverse functions related?

f⁻¹(x) is the reflection of f(x) over the line y = x.

If a function is strictly increasing or strictly decreasing, what can be said about its invertibility?

It is invertible.

What is the relationship between the domain of f(x) and the range of its inverse f⁻¹(x)?

The domain of f⁻¹(x) is the range of f(x), and vice versa.

In circular trigonometry, what does 'r' represent?

The distance from the origin to a point on the circle.

On the unit circle, what is the value of r?

r = 1

On the unit circle, what does cos(Θ) equal?

x-coordinate of point p

On the unit circle, what does sin(Θ) equal?

y-coordinate of point p

In the function f(x) = a sin(b(x+c)) + d, what does 'd' represent?

Midline

In the function f(x) = a sin(b(x+c)) + d, what does 'a' represent?

Amplitude

Write the pythagorean identity.

sin²θ + cos²θ = 1

Write the double angle formula for sine.

sin(2θ) = 2sinθcosθ

What is the first step in solving trig equations?

Isolate the trig function

How to find all solutions in a trig equation?

Take each initial answer ±2πk, for any integer value of k.

How can rectangular coordinates be converted to polar coordinates?

r² = x² + y² , tan⁻¹(y/x) = Ref. angle. Use Quadrant and Ref. angle to find Θ.

How can polar coordinates be converted to rectangular coordinates?

x = r cos(Θ), y = r sin(Θ)