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6.1: 0.0 Special Symbols
Some symbols
6.2: The Derivative as a Function
6.3: Differentiation Rules
6.4: Product and Quotient Rules
6.5: Derivatives of Exponential and Logarithmic Functions
In this section, we explore derivatives of exponential and logarithmic functions. As we discussed in Introduction to Functions and Graphs, exponential functions play an important role in modeling population growth and the decay of radioactive materials. Logarithmic functions can help rescale large quantities and are particularly helpful for rewriting complicated expressions.
6.6: Limits at Infinity and Asymptotes
We have shown how to use the first and second derivatives of a function to describe the shape of a graph. To graph a function f defined on an unbounded domain, we also need to know the behavior of f as x→±∞ . In this section, we define limits at infinity and show how these limits affect the graph of a function. At the end of this section, we outline a strategy for graphing an arbitrary function ff.
6.7: originalThe Definite Integral
If f(x) is a function defined on an interval [a,b], the definite integral of f from a to b is given by \[∫^b_af(x)dx=\lim_{n→∞} \sum_{i=1}^nf(x^∗_i)Δx,\] provided the limit exists. If this limit exists, the function f(x) is said to be integrable on [a,b], or is an integrable function. The numbers a and b are called the limits of integration; specifically, a is the lower limit and b is the upper limit. The function f(x) is the integrand, and x is the variable of integration.
6.8: original The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus gave us a method to evaluate integrals without using Riemann sums. The drawback of this method, though, is that we must be able to find an antiderivative, and this is not always easy.
6.9: Original Integration Formulas and the Net Change Theorem
The net change theorem states that when a quantity changes, the final value equals the initial value plus the integral of the rate of change. Net change can be a positive number, a negative number, or zero. The area under an even function over a symmetric interval can be calculated by doubling the area over the positive x-axis. For an odd function, the integral over a symmetric interval equals zero, because half the area is negative.
6.E: Applications of Derivatives (ALL Chap 4 Exercises)
These are homework exercises to accompany Chapter 4 of OpenStax's "Calculus" Textmap.
6.E: Derivatives (ALL Chapter 3 Exercises)
These are homework exercises to accompany Chapter 3 of OpenStax's "Calculus" Textmap.
6.E: Integration (Exercises)
These are homework exercises to accompany Chapter 5 of OpenStax's "Calculus" Textmap.
6.E: Limits (Exercises)
These are homework exercises to accompany Chapter 2 of OpenStax's "Calculus" Textmap.
6.E: Open Stax 4.1 - 4.5 Exercises
These are homework exercises to accompany Chapter 4 of OpenStax's "Calculus" Textmap.
6.10: Geogebra Calculus