Curve sketching
Calculus can be applied in the typical day to day life of every person, which is why, it is important to study the different concepts under this branch of mathematics. Last chapter, we have discussed everything you need to know about Derivatives. We learned its definition, and we also learned how to find the derivative of a given function.
In this chapter, we will put to use our knowledge and practice from the previous lesson by discussing the applications of differential calculus. There are various real life applications of deriving functions. One example is that it can be used to compute for the rate of measure. It can also be used to find errors and make approximations. It can also be used to identify whether or not a function is increasing or decreasing. Derivation can also be used to compute for the maxima and minima of a given mathematical model.
This chapter has seven sections. In the first part of the chapter, we will look at the position velocity acceleration. Based from our previous lesson, we know that whenever we try to find the derivative of a function, we are actually looking for the instantaneous rate of change of that function. We will use this knowledge to study about the relationship of the position, velocity and the acceleration of a particle. We will also have a number of exercises that will deal with linear motion to help us understand how derivatives can help us for this particular topic.
In the second part of this chapter, we will look at maximization, minimization, and optimization. Optimization is simply the process where you find a value that would maximize or minimize the value of a certain function. We will have a more in depth discussion on the different applications of optimization in the fourth section of this chapter.
For the third part, we will study how to sketch different curves. We will also learn how to identify and locate an inflection point and intervals of concavity. The inflection point is basically a point in a curve where the curvature changes its direction. These two are used in the First Derivative Test and Second derivative test. We will also look into the definition of impossible limits and the L’Hospital’s Rule.
For the last three parts Chapter 3, we will discuss about related rates, Mean Value Theorem and linear approximation or tangent line approximation. The discussion about these three topics will help you solve word problems.
In this chapter, we will put to use our knowledge and practice from the previous lesson by discussing the applications of differential calculus. There are various real life applications of deriving functions. One example is that it can be used to compute for the rate of measure. It can also be used to find errors and make approximations. It can also be used to identify whether or not a function is increasing or decreasing. Derivation can also be used to compute for the maxima and minima of a given mathematical model.
This chapter has seven sections. In the first part of the chapter, we will look at the position velocity acceleration. Based from our previous lesson, we know that whenever we try to find the derivative of a function, we are actually looking for the instantaneous rate of change of that function. We will use this knowledge to study about the relationship of the position, velocity and the acceleration of a particle. We will also have a number of exercises that will deal with linear motion to help us understand how derivatives can help us for this particular topic.
In the second part of this chapter, we will look at maximization, minimization, and optimization. Optimization is simply the process where you find a value that would maximize or minimize the value of a certain function. We will have a more in depth discussion on the different applications of optimization in the fourth section of this chapter.
For the third part, we will study how to sketch different curves. We will also learn how to identify and locate an inflection point and intervals of concavity. The inflection point is basically a point in a curve where the curvature changes its direction. These two are used in the First Derivative Test and Second derivative test. We will also look into the definition of impossible limits and the L’Hospital’s Rule.
For the last three parts Chapter 3, we will discuss about related rates, Mean Value Theorem and linear approximation or tangent line approximation. The discussion about these three topics will help you solve word problems.
Curve sketching
In this section we will expand our knowledge on the connection between derivatives and the shape of a graph. By following the “5Steps Approach”, we will quantify the characteristics of the function with application of derivatives, which will enable us to sketch the graph of a function.
Lessons
Notes:
Guidelines for Curve Sketching
a) domain
b) Intercepts
yintercept: set x=0 and evaluate y.
xintercept: set y=0 and solve for x. (skip this step if the equation is difficult to solve)
c) Asymptotes
vertical asymptotes:
for rational functions, vertical asymptotes can be located by equating the denominator to 0 after canceling any common factors.
horizontal asymptotes:
evaluate $lim_{x \to \infty } f(x)$ to determine the rightend behavior;
evaluate $lim_{x \to \infty } f(x)$ to determine the leftend behavior.
slant asymptotes:
for rational functions, slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator.
d) Compute$f' (x)$
find the critical numbers:
• use the First Derivative Test to find: intervals of increase/decrease and local extrema.
e) Compute$f'' (x)$
• inflection points occur where the direction of concavity changes.
find possible inflection points by equating the$f'' (x)$ to 0.
•Concavity Test:
•inflection points occur where the direction of concavity changes.

1.
Use the guidelines to sketch the graph of:
$f(x)=\frac{x^38}{x^3+8}$