2.1-2.4 Derivative Concepts Flashcards
Describe how to estimate a derivative if you only have a table of values.
Find the slope of the two points closest to the time in question. Make sure to show the difference quotient.
Why can’t a function be differentiable at a corner (sharp turn)
You could draw many different tangent lines on the graph.
Why can’t a function be differentiable at a vertical tangent?
A vertical tangent would have an undefined (infinite) slope.
Why can’t a function be differentiable at a discontinuity?
How could draw a tangent line? If there is a hole in the graph, you won’t have a point to be tangent to. If there is a jump, you’d have more than one tangent line.
List 3 things that cause differentiability problems
Discontinuities (holes, jumps, VA), sharp turns (corners or cusps), vertical tangents.
What are some synonyms for a derivative?
slope of the graph, slope of the tangent line, instantaneous rate of change
How do you write the equation of a tangent line?
Every line needs a point and a slope. So, y equals y-value plus slope time x minus x-value. f(a) + f’(a) (x-a). “a” is almost always given. f(a) is often supplied as well or you plug “a” into an equation or look at a graph of f. The slope will be found by finding the derivative at a. This could be by finding the slope of a graph, by using the limit definition of a derivative, or just by plugging it into a provided equation of the derivative.
Why would we want to write the equation of a tangent line?
Curves are hard to work with but straight lines aren’t. If you zoom in enough, the function will behave exactly like the tangent line.
How do you find where a function has horizontal tangent lines?
A horizontal tangent is when the slope is 0. So set the derivative = 0 and solve. Often this will involve using derivative rules to find the derivative first.
How can you make sure that a piecewise function is differentiable?
Set the different pieces equal to each other at the x-value of the overlap AND set the derivative of each piece equal at the overlap. If there are any constants, solve using this system of two equations.
What two things do the equation of a tangent line and a function have in common?
They share the point of tangency (x,y) and they have the same slope at the x value (m of the line = f-prime of the x-value)
