Analyze correctly and solve properly application problems concerning the derivatives to include writing equation of tangent/normal line, curve tracing ( including all types of algebraic curves and cusps), optimization problems, rate of change and related-rates problems (time-rate problems).
APPLICATIONS of the DERIVATIVES
LESSON 4: RELATED RATES and TIME RATES PROBLEMS
OBJECTIVE: At the end of the lesson, the student would be able to illustrate and solve related rate problems including time rate problems.
Recall: The Derivative of a function When the concept of the derivative was introduced in the earlier discussion, it was defined as follows: DEFINITION: The derivative of at point P on the curve is equal to the slope of the tangent line at P; thus the derivative of the function with respect to x at any x in its domain is defined as: ) ( x f y ) ( x f y 0 0 ( ) ( ) lim lim x x dy y f x x f x dx x x provided the limit exists.
)) ( , ( 1 1 x f x P )) ( , ( 2 2 x f x Q ) ( x f y x x x x x x 1 2 1 2 y secant line x y In the given figure, we note that the line connecting points P and Q is a secant line of the curve with slope 𝑚 = Δ? Δ? , which gives the average change of y per unit change in x . However, as Q gets closer and closer to P , ∆? decreases and approaches zero.
)) ( , ( 1 1 x f x P )) ( , ( 2 2 x f x Q ) ( x f y x x x x x x 1 2 1 2 y secant line x y The closer Q gets to P, the smaller ∆x becomes (approaching 0) and the ratio Δy Δx , which gives the average change of y per unit change in x , would define the instantaneous change in y per unit change in x ; that is lim ∆?→0 ∆? ∆? = instantaneous rate of change of y with respect to x.
)) ( , ( 1 1 x f x P )) ( , ( 2 2 x f x Q ) ( x f y x x x x x x 1 2 1 2 y secant line x y The derivative of a function y = f(x) is the instantaneous rate of change of y with respect to x.
A function may be single-variable, ? = 𝑓(?) ,or multi-variable, ? = 𝑓(?, ?, … ) . Regardless of the function being single-variable or multi-variable, at times, there is a need to investigate how the function would change in relation to changes in its independent variable/variables or how fast the dependent variable would change at the particular instant the independent variable or variables assume a particular value ( instantaneous rate of change of the dependent variable with respect to its independent variables; general related-rate problems). Also, there are many functions in which the concern is on knowing how fast the variables ( both dependent and independent variables) are changing with respect to time (time-rate problems). It is not necessary to express each of these variables directly as function of time. For example, we are given an equation involving the variables x and y, and that both x and y are functions of the third variable t, where t denotes time.
Both sides of the equation can be differentiated with respect to time t , applying implicit differentiation and the chain rule
- Fall '19
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