Studied by 0 people

0.0(0)

Get a hint

Hint

1

What do the difference quotients f(a+h)−f(a)/h and f(x)−f(a)/x−a express?

They express the average rate of change of a function over an interval

New cards

2

How can the instantaneous rate of change of a function at x=ax = ax=a be expressed?

It can be expressed by limh→0 f(a+h)−f(a)/h or limx→a f(x)−f(a)/x−a provided the limit exists. These are equivalent forms of the definition of the derivative and are denoted f′(a).

New cards

3

How is the derivative of f defined?

The derivative of f is the function whose value at x is limh→0 f(x+h)−f(x)/h, provided this limit exists.

New cards

4

What are the notations for the derivative of y=f(x)

Notations for the derivative include dy/dx, f′(x), and y′.

New cards

5

In what forms can the derivative be represented?

The derivative can be represented graphically, numerically, analytically, and verbally

New cards

6

What is the derivative of a function at a point?

The derivative of a function at a point is the slope of the line tangent to a graph of the function at that point.

New cards

7

How can the derivative at a point be estimated?

The derivative at a point can be estimated from information given in tables or graphs

New cards

8

How can technology be used in relation to derivatives?

Technology can be used to calculate or estimate the value of a derivative of a function at a point.

New cards

9

What is the relationship between differentiability and continuity?

If a function is differentiable at a point, then it is continuous at that point. If a point is not in the domain of f, then it is not in the domain of f′.

New cards

10

Can a continuous function fail to be differentiable at a point in its domain?

Yes, a continuous function may fail to be differentiable at a point in its domain.

New cards

11

How can the derivative be calculated for functions of the form f(x) = x^r

Direct application of the definition of the derivative and specific rules can be used to calculate the derivative.

New cards

12

How can sums, differences, and constant multiples of functions be differentiated?

Sums, differences, and constant multiples of functions can be differentiated using derivative rules.

New cards

13

How can the power rule be used in combination with other properties?

What specific rules can be used to find the derivatives for sine, cosine, exponential, and logarithmic functions?

New cards

14

What specific rules can be used to find the derivatives for sine, cosine, exponential, and logarithmic functions?

Specific rules for these functions include the derivatives of sinx, cosx, e^x, and lnx.

New cards

15

How can recognizing an expression for the definition of the derivative of a function whose derivative is known be useful?

It offers a strategy for determining a limit.

New cards

16

How can derivatives of products of differentiable functions be found?

Derivatives of products of differentiable functions can be found using the product rule

New cards

17

How can rearranging tangent, cotangent, secant, and cosecant functions using identities help in differentiation?

It allows differentiation using derivative rules.

New cards