## Calculus I - Applications of Derivatives

Chapter 8. Applications of the integral 1. Areas between graphs 2. Exercises 3. Cavalieri’s principle and volumes of solids 4. Examples of volumes of solids of revolution 5. Volumes by cylindrical shells 6. Exercises 7. Distance from velocity, velocity from acceleration 8. The length of a curve 9. Examples of length computations State how you could use formulas for derivatives of the sine and cosine functions to derive this formula. (DO NOT do this derivation.) (b)Use the formula given in part (a) to derive the formula for the derivative of the arctangent function. Chapter 4: Applications of Derivatives. The Shape of a Graph, Part II – In this section we will discuss what the second derivative of a function can tell us about the graph of a function. The second derivative will allow us to determine where the graph of a function is concave up and concave down.

## Applications of Differentiation

In the previous chapter we focused almost exclusively on the computation of derivatives. In this chapter will focus on applications of derivatives. There are many very important applications to derivatives.

These will not be the only applications however. We will also see how derivatives can be used to estimate solutions to equations. Critical Points — In this section we give the definition of critical points.

Critical points will show up in most of the sections in this chapter, so it will be important to understand them and how to find them. We will work a number of examples illustrating how to find them for a wide variety application of differentiation calculus pdf functions. Minimum and Maximum Values — In this section we define absolute or global minimum and maximum values of a function and relative or local minimum and maximum values of a function.

We also give the Extreme Value Theorem and Fermat's Theorem, both of which are very important in the many of the applications we'll see in this chapter.

Finding Absolute Extrema — In this section we discuss how to find the absolute or global minimum and maximum values of a function. In other words, we will be finding the largest and smallest values that a function will have. The Shape of a Graph, application of differentiation calculus pdf, Part I — In this section we will discuss what the first derivative of a function can tell us about the graph of a function.

The first derivative will allow us to application of differentiation calculus pdf the relative or local minimum and maximum values of a function and where a function will be increasing and decreasing.

We will also give the First Derivative test which will allow us to classify critical points as relative minimums, relative maximums or neither a minimum or a maximum. The Shape of a Graph, application of differentiation calculus pdf, Part II — In this section we will discuss what the second derivative of a function can tell us about the graph of a function. The second derivative will allow us to determine where the graph of a function is concave up and concave down.

The second derivative will also allow us to application of differentiation calculus pdf any inflection points i. We will also give the Second Derivative Test that will give an alternative method for identifying some critical points but application of differentiation calculus pdf all as relative minimums or relative maximums, application of differentiation calculus pdf. With the Application of differentiation calculus pdf Value Theorem we will prove a couple of very nice facts, application of differentiation calculus pdf, one of which will be very useful in the next chapter.

We will discuss several methods for determining the absolute minimum or maximum of the function. Examples in this section tend to center around geometric objects such as squares, boxes, cylinders, etc. More Optimization Problems — In this section we will continue working optimization problems. The examples in this section tend to be a little more involved and will often involve situations that will be more easily described with a sketch as opposed to the 'simple' geometric objects we looked at in the previous section.

Linear Approximations — In this section we discuss using the derivative to compute a linear approximation to a function. We can use the linear approximation to a function to approximate values of the function at certain points. While it might not seem like a useful thing to do with when we have the function there really are reasons that one might want to do this.

We give two ways this can be useful in the examples. Differentials — In this section we will compute the differential for a function.

We will give an application of differentials in this section. However, one of the more important uses of differentials will come in the next chapter and unfortunately we will not be able to discuss it until then.

Newton's Method is an application of derivatives will allow us to approximate solutions to an equation. There are many equations that cannot be solved directly and with this method we can get approximations to the solutions to many of those equations. Business Applications — In this section we will give a cursory discussion of some basic applications of derivatives to the business field.

Note that this section is only intended to introduce these concepts and not teach you everything about them. Notes Quick Nav Download. Notes Practice Problems Assignment Problems. You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode.

If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. Calculus (differentiation and integration) was developed to improve this understanding. Differentiation and integration can help us solve many types of real-world problems. We use the derivative to determine the maximum and minimum values of particular functions (e.g. cost, strength, amount of material used in a building, profit, loss, etc.). Chapter 4: Applications of Derivatives. The Shape of a Graph, Part II – In this section we will discuss what the second derivative of a function can tell us about the graph of a function. The second derivative will allow us to determine where the graph of a function is concave up and concave down. 2. Applications of Diﬀerentiation 2A. Approximation 2A-1 d √ a + bx = b f(x) ≈ √ a + b x by formula. dx 2 √ a+ bx ⇒ 2 √ a By algebra: √ a + bx = a 1 + bx ≈ a(1 + bx), same as above. a 2a 2A-2 D(1) = −b 1 b x; OR: 1 = 1/a a + bx (a + bx)2 ⇒ f(x) ≈ a − a2 a + bx 1 + .