Copyright © Cengage Learning. All rights reserved. 2 Differentiation Copyright © Cengage Learning. All rights reserved.

Higher-Order Derivatives 2.6 Higher-Order Derivatives Copyright © Cengage Learning. All rights reserved.

Objectives Find higher-order derivatives. Find and use a position function to determine the velocity and acceleration of a moving object.

Second, Third, and Higher-Order Derivatives

Second, Third, and Higher-Order Derivatives The “standard” derivative f  is often called the first derivative of f. The derivative of f  is the second derivative of f and is denoted by f . The derivative of f  is the third derivative of f and is denoted by f .

Second, Third, and Higher-Order Derivatives By continuing this process, you obtain higher-order derivatives of f. Higher-order derivatives are denoted as follows.

Example 2 – Finding Higher-Order Derivatives Find the value of g(2) for the function Solution: Begin by differentiating three times.

Example 2 – Solution cont’d Then, evaluate the third derivative of g at t = 2.

Second, Third, and Higher-Order Derivatives Note that with each successive differentiation, the degree of the polynomial drops by one. Eventually, higher-order derivatives of polynomial functions degenerate to a constant function.

Second, Third, and Higher-Order Derivatives Specifically, the nth-order derivative of an nth-degree polynomial function is the constant function where Each derivative of order higher than n is the zero function.

Example 3 – Finding Higher-Order Derivatives Find the first four derivatives of Solution:

Example 1 – Solution cont’d

Acceleration

Acceleration You have seen that the velocity of a free-falling object (neglecting air resistance) is given by the derivative of its position function. In other words, the rate of change of the position with respect to time is defined to be the velocity. In a similar way, the rate of change of the velocity with respect to time is defined to be the acceleration of the object.

Acceleration To find the position, velocity, or acceleration at a particular time t, substitute the given value of t into the appropriate function, as illustrated in Example 4.

Example 4 – Finding Acceleration A ball is thrown upward from the top of a 160-foot cliff, as shown in Figure 2.32. The initial velocity of the ball is 48 feet per second, which implies that the position function is where the time t is measured in seconds. Find the height, velocity, and acceleration of the ball at t = 3. Figure 2.32

Example 4 – Solution Begin by differentiating to find the velocity and acceleration functions.

Example 4 – Solution cont’d To find the height, velocity, and acceleration at t = 3, substitute t = 3 into each of the functions above.

Acceleration In Example 4, notice that the acceleration of the ball is –32 feet per second squared at any time t. This constant acceleration is due to the gravitational force of Earth and is called the acceleration due to gravity. Note that the negative value indicates that the ball is being pulled down—toward Earth.

Acceleration The following position function neglects air resistance, which is appropriate because the moon has no atmosphere—and no air resistance. This means that the position function for any free-falling object on the moon is given by where s is the height (in feet), t is the time (in seconds), v0 is the initial velocity (in feet per second), and h0 is the initial height (in feet).

Acceleration This position function is valid for all objects, whether heavy ones such as hammers or light ones such as feathers.

Example 6 – Finding Velocity and Acceleration The velocity v (in feet per second) of a certain automobile starting from rest is where t is the time (in seconds). The positions of the automobile at 10-second intervals are shown in Figure 2.33. Find the velocity and acceleration of the automobile at 10-second intervals from t = 0 to t = 60. Figure 2.33

Example 6 – Solution To find the acceleration function, differentiate the velocity function.

Example 6 – Solution cont’d In the table, note that the acceleration approaches zero as the velocity levels off. This observation should agree with your experience—when riding in an accelerating automobile, you do not feel the velocity, but you do feel the acceleration. In other words, you feel changes in velocity.