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In this section we look at some applications of the derivative by focusing on the interpretation of the derivative as the rate of change of a function. These applications include acceleration and velocity in physics, population growth rates in biology, and marginal functions in economics.
- 4.4 The Mean Value Theorem
Corollaries of the Mean Value Theorem. Let’s now look at...
- 3.7 Derivatives of Inverse Functions
Find the rate of change of the angle of elevation after...
- 4.5 Derivatives and The Shape of a Graph
Using the First Derivative Test. Consider a function f f...
- 4.7 Applied Optimization Problems
Write a formula for the quantity to be maximized or...
- 6.9 Calculus of The Hyperbolic Functions
Learning Objectives. 6.9.1 Apply the formulas for...
- 1.2 Basic Classes of Functions
One of the distinguishing features of a line is its slope....
- 3.5 Derivatives of Trigonometric Functions
Learning Objectives. 3.5.1 Find the derivatives of the sine...
- 4.2 Linear Approximations and Differentials
Analysis. Using a calculator, the value of 9.1 9.1 to four...
- 4.4 The Mean Value Theorem
the notation from these examples throughout this course. The collection of all real numbers between two given real numbers form an interval. The following notation is used (a;b) is the set of all real numbers xwhich satisfy a<x<b. [a;b) is the set of all real numbers xwhich satisfy a x<b. (a;b] is the set of all real numbers xwhich satisfy a<x b.
INTRODUCTION TO CALCULUS. MATH 1A. Unit 7: Rate of Change. Lecture. 7.1. Given a function f and a constant h > 0, we can look at the new function. f(x + h) f(x) Df(x) = : h. It is the average rate of change of the function with step size h. When changing x to x + h and then f(x) changes to f(x + h).
Lecture 7: Rate of change. Given a function f and h > 0, we can look at the new function. f(x + h) f(x) Df(x) = : h. It is the rate of change of the function with step size h. When changing x to x + h and then f(x) changes to f(x + h). The quotient Df(x) is "rise over run". In this lecture, we take the.
The derivative, f0(a) is the instantaneous rate of change of y= f(x) with respect to xwhen x= a. When the instantaneous rate of change is large at x 1, the y-vlaues on the curve are changing rapidly and the tangent has a large slope. When the instantaneous rate of change ssmall at x 1, the y-vlaues on the
response change f(x) to f(x + h). In this lecture, we take the limit h important formulas dxxn d = nxn−1, d d dx exp(x) = exp(x), dx sin(x) = cos(x), which we have seen already in a discrete setting. You walk up a snow hill of height f(x) = 30−x2 meters. You walk with a step size of h = 0.5 meters.
The Newtonian approach is presented as one focusing on rates of change of functions of a given independent variable (usually time), while that of Liebniz deals with variables and how iv
