Chapter 9.3, Problem 52E

### Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

Chapter
Section

### Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340
Textbook Problem

# Receptivity In learning theory, receptivity is defined as the ability of students to understand a complex concept. Receptivity is highest when the topic is introduced and tends to decrease as time passes in a lecture. Suppose that the receptivity of a group of students in a mathematics class is given by g ( t ) = − 02 t 2 + 3.1 t + 32 where t is minutes after the lecture begins.(a) At what rate is receptivity changing 10 minutes after the lecture begins?(b) Write a sentence that explains the meaning of your answer in part (a).

(a)

To determine

To calculate: The instantaneous rate of change of receptivity after 10 minutes if the receptivity in a mathematics class is given by

g(t)=0.2t2+3.1t+32

Where t is the time.

Explanation

Given Information:

The receptivity in a mathematics class is given by

g(t)=âˆ’0.2t2+3.1t+32

Where t is the time.

Formula used:

If a function f(x) is defined in the interval [x,x+h], then its instantaneous rate of change is given by the derivative as

fâ€²(x)=limhâ†’0f(x+h)âˆ’f(x)h

Calculation:

Consider the provided statement,

The receptivity in a mathematics class is given by

g(t)=âˆ’0.2t2+3.1t+32

Where t is the time.

Substitute t=t+h in f(x) to get,

g(t+h)=âˆ’0.2(t+h)2+3.1(t+h)+32=âˆ’0.2(t2+h2+2th)+3.1(t+h)+32=âˆ’0.2t2âˆ’0.2h2âˆ’0

(b)

To determine

The meaning of the instantaneous rate of change of receptivity obtained after 10 minutes if the receptivity in a mathematics class is given by

g(t)=0.2t2+3.1t+32

Where t is the time.

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