Explanation Definition used: First derivative test: Suppose c to be a critical number for a function f . Assume that f is continuous on ( a , b ) and differentiable on ( a , b ) except at c , and c is the only critical number of f in ( a , b ) . f ( c ) is a relative maximum of f at c if f ′ ( x ) > 0 in the interval ( a , c ) and f ′ ( x ) < 0 in the interval ( c , b ) . f ( c ) is a relative minimum f at c if f ′ ( x ) < 0 in the interval ( a , c ) and f ′ ( x ) > 0 in the interval ( c , b ) . Relative maximum: A function is said to have a relative maximum f ( c ) at c if f ′ ( x ) > 0 in the interval ( a , c ) and f ′ ( x ) < 0 in the interval ( c , b ) . Relative minimum: A function is said to have a relative maximum f ( c ) at c if f ′ ( x ) < 0 in the interval ( a , c ) and f ′ ( x ) > 0 in the interval ( c , b ) . Calculation: The given activity level function is a ( t ) = 0.008 t 3 − 0.288 t 2 + 2.304 t + 7 . Obtain the derivative of the function as follows. a ′ ( t ) = ( 0.008 t 3 − 0.288 t 2 + 2.304 t + 7 ) ′ = 0.008 ( 3 t 2 ) − 0.288 ( 2 t ) + 2.304 ( 1 ) + 0 = 0.024 t 2 − 0.576 t + 2.304 Set a ′ ( c ) = 0 and find the value of c . 0.024 c 2 − 0.576 c + 2.304 = 0 0.024 ( c 2 − 24 c + 96 ) = 0 c 2 − 24 c + 96 = 0 Apply quadratic formula on the above equation to obtain the values of c. c = 24 ± ( − 24 ) 2 − 4 × 1 × 96 2 × 1 = 24 ± 576 − 384 2 = 24 ± 192 2 = 24 ± 13.86 2 That is, c = 24 + 13.86 2 or c = 24 − 13

Finite Mathematics and Calculus with Applications (10th Edition)

10th Edition

ISBN: 9780321979407

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 13.2, Problem 53E

To determine

To find: The highest and the lowest point of the activity level of a certain type of lizard which is given by the function a(t)=0.008t3−0.288t2+2.304t+7 .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.2, Problem 52E

Chapter 13.2, Problem 54E

Students have asked these similar questions

Arctic Foxes The age/weight relationship of male Arctic foxes caught in Svalbard, Norway, can be estimated by the function --0.020(1-66) M(1) = 3583e where t is the age of the fox in days and M(t) is the weight of the fox in grams. Source: Journal of Mammology. (a) Estimate the weight of a male fox that is 250 days old. (b) Use M(t) to estimate the largest size that a male fox can attain. (Hìnt: Find lim M(t).) (c) Estimate the age of a male fox when it has reached 50% of its maximum weight. (d) Estimate the rate of change in weight of a male Arctic fox that is 250 days old. (Hint: Recall that D,e) = f'(1)e().) (e) Use a graphing calculator to graph M(t) and then describe the growth pattern. (f) Use the table function on a graphing calculator or a spread- sheet to develop a chart that shows the estimated weight and growth rate of male foxes for days 50, 100, 150, 200, 250, and 300.

Chapter 13 Solutions

Finite Mathematics and Calculus with Applications (10th Edition)

Ch. 13.1 - Find where the function is increasing and...Ch. 13.1 - Prob. 2YT Ch. 13.1 - Prob. 3YT Ch. 13.1 - Prob. 4YT Ch. 13.1 - Prob. 1WE Ch. 13.1 - Prob. 2WE Ch. 13.1 - Prob. 3WE Ch. 13.1 - Prob. 4WE Ch. 13.1 - Prob. 5WE Ch. 13.1 - Prob. 6WE

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The highest and the lowest point of the activity level of a certain type of lizard which is given by the function a ( t ) = 0.008 t 3 − 0.288 t 2 + 2.304 t + 7 .

Solution Summary: The author explains how the function a(t) is able to find the highest and lowest activity levels of lizards.

Concept explainers

Videos

To find: The highest and the lowest point of the activity level of a certain type of lizard which is given by the function a(t)=0.008t3−0.288t2+2.304t+7 .

Want to see the full answer?

Chapter 13 Solutions