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Applications of Differentiation 2 The Extreme Value Theorem If f is continuous on a closed interval[a,b], then f attains an absolute maximum value f (c) and an absolute minimum value )f (d at some numbers c and d in []a,b.Fermat’s Theorem If f has a local maximum or minimum atc, and if )f ' (c exists, then 0f ' (c) = . Critical Number A critical number of a function f is a number cin …

Step-by-Step Examples. Calculus. Applications of Differentiation. Finding a Tangent Line to a Curve. Checking if Differentiable Over an Interval. The Mean Value Theorem. Finding the Inflection Points. Find Where the Function Increases/Decreases. Finding the Critical Points of a Function.

Differentiation allows us to find rates of change. For example, it allows us to find the rate of change of velocity with respect to time (which is ...

Application 1 : Exponential Growth - Population. Let P (t) be a quantity that increases with time t and the rate of increase is proportional to the same quantity P as follows. d P / d t = k P. where d p / d t is the first derivative of P, k > 0 and t is the time. The solution to the above first order differential equation is given by.

point. For example, if there is a stationary point at x = 2, we may choose x = 1.9 and x = 2.1. 2. Calculate for each of these two values of x. 3. By observing the sign change of , we can deduce the nature of the stationary point. 4. If the gradient changes from positive to negative before and after the stationary point, the point is a maximum. 5.

Calculus Examples. Step-by-Step Examples. Calculus. Applications of Differentiation. Finding a Tangent Line to a Curve. Checking if Differentiable Over an Interval. The Mean Value Theorem. Finding the Inflection Points. Find Where the Function Increases/Decreases.

We use the derivative to determine the maximum and minimum values of particular functions (e.g. cost, strength, amount of material used in a ...

Applications Differentiation · Navigation · Rates of Change · Maximum and Minimum Problems · Distance, Velocity and Acceleration Problems.

Example 1 The function f (x) =cosx takes on its (local and absolute) maximum value of 1 infinitely many times, since cos2nπ=1 for any integer n and −1≤cosx ≤1 for all x. Likewise, cos( 2n +1)π=−1 is its minimum value, where n is any integer. Example 2 The graph of the function f (x) =3x4 −16x3 +18x2 −1≤x ≤4 is shown.

Since the line tangent to the graph is given by the derivative, differentiation is useful for finding the linear approximation. If one were to take an ...

05.10.2017 · 5. Curve Sketching Using Differentiation, where we begin to learn how to model the behaviour of variables. 6. More Curve Sketching Using Differentiation. 7. Applied Maximum and Minimum Problems, which is a vital application of differentiation. 8.

Some of the most important applications of differential calculus are optimization problems. In these, we are required to find the optimal (best) way of doing something. EXAMPLES Here are some examples of such problems that we will solve in this chapter. What is the shape of a can that minimizes manufacturing costs?

Newton's Method is an application of derivatives that will allow us to approximate solutions to an equation. There are many equations that ...

If there is no change in the sign of the gradient, the stationary point is a point of inflexion. Example 2. A curve whose equation is has stationary points at ( ...

The graph on the right shows a piecewise-defined continuous function. Both these functions are increasing. y x. 0 y x. 0. Examples of ...

To give an example, derivatives have various important applications in Mathematics such as to find the Rate of Change of a Quantity, to find the Approximation ...

DIFFERENTIAL EQUATIONS The general solution of a differential equation involves an arbitrary constant (or constants), as in Example 2. However, there may be some extra conditions given that will determine the constants and, therefore, uniquely specify the solution.

APPLICATIONS OF DIFFERENTIATION We have already investigated some applications of derivatives. However, ... Example 5 demonstrates that, even when f ’(c) = 0, there need not be a maximum or minimum at c. In other words, the converse of the theorem is false in general.

differential. We can also refer to the first differential as $%(’). The second differential is written as !*"!#* or as $%%(’). Example 3 A curve whose equation is has stationary points at (−0.5, 1.75) and (1, −5). Determine the nature of these points. Solution We …

Rate of Change of A Quantity

APPLICATIONS OF DERIVATIVES Derivatives are everywhere in engineering, physics, biology, economics, and much more. In this chapter we seek to elucidate a number of general ideas which cut across many disciplines. Linearization of a function is the process of approximating a function by a line near some point.