MATH 1142 Section 1.2 & 1.3 Worksheet
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Slope of a Curve at a Point
We want to extend the notion of of a slope of a line to the slope of more general curves. However, this is somewhat
problematic. Why?
One way to think of extending the notion of the slope of a line is to look for the slope of the line tangent to the
curve at a particular point.
Slope of a Curve
The slope of a curve at a point Pis defined to be the slope of the tangent line
to the curve at P.
What does this mean?
The slope of a curve at a point Pmeasures the instantaneous rate of change measures the rate of change of the
curve as it passes through P.
Will the slope of a curve be the same for any point on the curve? Why or why not?
What does this mean about the slope of a curve?
For most functions, there is a formula for finding the slope of a curve, f(x), this formula is called the called the
derivative (or sometimes the slope formula) and is denoted f0(x).
Recall that we already know the slope of a line g(x) = mx +bis this means that the derivative of the line
is g0(x) = . Calculating this derivative can be more complicated when dealing with more complicated curves
since we are trying to find the slope of a line for which we only know one point.
One way to approximate the derivative is by calculating the slope of the secant line as opposed to the tangent line
(since we will know two points). Calculating the slope of the secant line is the same as finding the average rate of
change of the function between two points.
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