Lecture 11 Flashcards
existence of derivatives
when do derivatives exist?
for a smooth graph with no sharp corners, cusps, or vertical tangents
what is the formula to find the derivative for the power rule?
what do the variables represent?
x^(p)–>px^(p-1)
p is a number
what is the formula to find the derivative of exponentials?
a^(x)–> a^(x) ln(a)
what does it mean if two variables are inversely proportional to one another in terms of increasing and decreasing?
the increase in one causes a decrease in the other
what is the general formula to find the chain rule?
f ‘ [g(x)] g ‘ (x)
how can you tell if a function is an underestimate?
if a function is concave up, then the tangent line will lie below the curve. Hence the tangent line of approximation will be an underestimate
how can you tell if a function is an overestimate?
if a function is concave down, then the tangent line will lie above the curve. Hence the tangent line of approximation will be an overestimate
what does it mean if the first derivative is positive?
the function is increasing
how can you tell if a derivative is negative on a graph?
the slope is going from uphill to downhill
how can you tell if a derivative is positive on a graph?
the slope is going from downhill to uphill
if you have two negative slopes, how can you tell which one is smaller?
the stepper negative slope is the smaller slope