Find the equation of the line tangent to the graph of
at
We can use
[link] , but as we have seen, the results are the same if we use
[link] .
We now know that the slope of the tangent line is
To find the equation of the tangent line, we also need a point on the line. We know that
Since the tangent line passes through the point
we can use the point-slope equation of a line to find the equation of the tangent line. Thus the tangent line has the equation
The graphs of
and
are shown in
[link] .
The type of limit we compute in order to find the slope of the line tangent to a function at a point occurs in many applications across many disciplines. These applications include velocity and acceleration in physics, marginal profit functions in business, and growth rates in biology. This limit occurs so frequently that we give this value a special name: the
derivative . The process of finding a derivative is called
differentiation .
Definition
Let
be a function defined in an open interval containing
The derivative of the function
at
denoted by
is defined by
provided this limit exists.
Alternatively, we may also define the derivative of
at
as
Now that we can evaluate a derivative, we can use it in velocity applications. Recall that if
is the position of an object moving along a coordinate axis, the
average velocity of the object over a time interval
if
or
if
is given by the difference quotient
Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ...
Step 2: Find each score's deviation from the mean. ...
Step 3: Square each deviation from the mean. ...
Step 4: Find the sum of squares. ...
Step 5: Divide the sum of squares by n – 1 or N.
The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.
Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400
a. what is the probability of getting more than 12,000 hits?
b. what is the probability of getting fewer than 9,000 hits?
Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400.
a. What is the probability of getting more than 12,000 hits