Precalculus: Introduction to Calculus flashcards

Question 1

What does the notation lim (x→a) f(x) = L mean?

Accepted Answer

As x approaches a from both sides, the output f(x) gets close to L.

Question 2

When does a two-sided limit of f(x) at x = a exist?

Accepted Answer

When the limits from the left and from the right at a are equal.

Question 3

How is the left-hand limit of f(x) at a written?

Accepted Answer

lim (x→a⁻) f(x) = L

Question 4

How is the right-hand limit of f(x) at a written?

Accepted Answer

lim (x→a⁺) f(x) = L

Question 5

How can the limit of a polynomial function be found?

Accepted Answer

As the sum of the limits of its individual terms.

Question 6

How does the limit of a function raised to a power relate to the limit of the function?

Accepted Answer

It equals the limit of the function raised to that same power.

Question 7

How does the limit of a root of a function relate to the limit of the function?

Accepted Answer

It equals the same root taken of the function's limit.

Question 8

What is a continuous function, described by its graph?

Accepted Answer

A function whose graph has no holes or breaks.

Question 9

What three conditions make f continuous at x = a?

Accepted Answer

f(a) exists, lim (x→a) f(x) exists, and lim (x→a) f(x) = f(a).

Question 10

What is a jump discontinuity at x = a?

Accepted Answer

Both one-sided limits exist at a but are not equal.

Question 11

What is a removable discontinuity?

Accepted Answer

A point of discontinuity where the function can be redefined to make it continuous.

Question 12

Where are polynomial functions continuous?

Accepted Answer

Everywhere.

Question 13

Where are logarithmic functions continuous?

Accepted Answer

On their domain.

Question 14

What is a secant line?

Accepted Answer

A line through two points on a curve.

Question 15

What does the slope of a secant line through two points of a graph measure?

Accepted Answer

The function's average rate of change between those two points.

Question 16

What is the formula for the average rate of change of f from a to a + h?

Accepted Answer

AROC = (f(a + h) − f(a))/h

Question 17

What is the expression (f(a + h) − f(a))/h called?

Accepted Answer

The difference quotient.

Question 18

What is the limit definition of the derivative of f at x = a?

Accepted Answer

f′(a) = lim (h→0) (f(a + h) − f(a))/h, provided the limit exists.

Question 19

What is another name for the derivative of a function at a point?

Accepted Answer

The instantaneous rate of change.

Question 20

Geometrically, what does f′(a) give?

Accepted Answer

The slope of the tangent line to the graph of f at x = a.

Question 21

When is a function f(x) differentiable at x = a?

Accepted Answer

When f′(a) exists.

Question 22

Which form of a line's equation can be used to write the tangent line to a curve?

Accepted Answer

The point-slope form.

Question 23

If s(t) gives an object's position at time t, what is its instantaneous velocity at t = a?

Accepted Answer

s′(a) = lim (h→0) (s(a + h) − s(a))/h

Precalculus: Introduction to Calculus flashcards

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