Calculus 1
Lia Vas
Antiderivative and The Indenite Integral
The idea of employing an inverse function is central when solving any equation:
x + 2 = 4 requires application of the function x 2, the inverse of x + 2, to the right hand side.
Thus x = 4 2 = 2
Calculus 1
Lia Vas
The Limit
behavior of a function near a point
The limit of a function f (x) at a point x = a describes the behavior of the function f (x)
near the point x = a.
If the values of f (x) accumulate near point
y = L when values of x approach
Calculus 1
Lia Vas
Implicit Dierentiation. Logarithmic Dierentiation
Implicit functions. If a function can be expressed as y = f (x) it is said to be in the explicit
form. However, in some cases, the variables x and y can be related with an equation F (x,
Calculus 1
Lia Vas
Innite Limits and Limits at Innity
Horizontal and vertical asymptotes
1
Example 1. Consider the function f (x) = x . This function is dened for every value of x except
x = 0. Although not dened at 0, it still may be relevant to know the
Calculus 1
Lia Vas
The Fundamental Theorem of Calculus.
The Total Change Theorem and the Area Under a Curve.
Example 2 of the previous section we approximated the distance traveled during some time
interval and given velocities at certain points of the in
Calculus 1
Lia Vas
Related Rates
The most important reason for a non-mathematics major to learn mathematics is to be able to
apply it to problems from other disciplines or real life. In this section, we focus on modeling real life
scenarios using function
Calculus 1
Lia Vas
The Substitution
Many integrals cannot be evaluated using the rules we covered so far (and many others cannot
be evaluated even by methods we shall cover in Calculus 2). Still there is a class of integrals which
can be evaluated using a
Calculus 1
Lia Vas
Linear Approximation
The dierential. Consider a function y = f (x) and the two points (x, f (x) and (x+h, f (x+h)
on its graph. Recall that x is sometimes used to denote the dierence h between the x-values and
y is used for the dierence
Calculus 1
Lia Vas
Finding Derivative
More Rules - Product, Quotient and Chain
The Product Rule. Both calculus 1 and 2 courses would be much shorter if the derivative of
a product is a product of the derivatives. However, this is not true.
(f g) = f g
For
Calculus 1
Lia Vas
Increasing/Decreasing Test. Extreme Values and The First
Derivative Test.
Recall that a function f (x) is increasing on an interval if the increase in x-values implies an
increase in y-values for all x-values from that interval.
x1 > x2
Calculus 1
Lia Vas
Areas between Curves
In this section we consider the area between two curves. Let f (x) and g(x) are two continuous
functions dened on the interval [a, b] such that f (x) g(x) for all x in [a, b]. If we consider a partition
of [a, b] wi
Calculus 1
Lia Vas
Integrals of Exponential and Trigonometric Functions.
Integrals Producing Logarithmic Functions.
Integrals of exponential functions. Since the derivative of ex is ex , ex is an antiderivative of
ex . Thus
ex dx = ex + c
x
Recall that th
Calculus 1
Lia Vas
Denite Integral The Left and Right Sums
In this section we turn to the question of nding the area between a given curve and x-axis on
an interval. At this time, this question seems unrelated to our consideration of indenite integrals in
Calculus 1
Lia Vas
The Derivative
The rate of change
Knowing and understanding the concept of derivative will enable you to answer the following
questions. Let us consider a quantity whose size is described by a function.
1. How fast is the quantity chang
Calculus 1
Lia Vas
Higher Derivatives. Dierentiable Functions
The second derivative. The derivative itself can be considered as a function. The instantaneous
rate of change of this function is the second derivative. Thus, the second derivative evaluated a
Calculus 1
Lia Vas
Absolute Extrema and Constrained Optimization
Recall that a function f (x) is said to have a
relative maximum at x = c if f (c) f (x) for
all values of x in some open interval containing c.
However, that does not mean that the value f (
Calculus 1
Lia Vas
Continuous functions. Limits of non-rational functions.
Squeeze Theorem. Calculator issues.
Applications of limits
Continuous Functions. Recall that we referred to a function f (x) as a continuous function
at x = a if its graph has no h
Calculus 1
Lia Vas
Finding and Using Derivative
The shortcuts
We have seen that the formula f (x) = limh0 f (x+h)f (x) is manageable for relatively simple
h
functions like a linear or quadratic. For more complex functions, nding the derivative using this
Calculus 1
Lia Vas
Derivatives of Exponential, Logarithmic and Trigonometric
Functions
Derivative of the inverse function. If f (x) is a one-to-one function (i.e. the graph of f (x)
passes the horizontal line test), then f (x) has the inverse function f 1
Calculus 1
Lia Vas
Concavity and Inection Points. Extreme Values and The
Second Derivative Test.
Consider the following two increasing functions. While they are both increasing, their concavity
distinguishes them.
The rst function is said to be concave up