Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)
Understanding the sign of the first derivative is a fundamental concept in calculus, specifically when analyzing the behavior of functions. It serves as a key tool in determining whether a function is increasing or decreasing over a given interval. The first derivative provides valuable insights into the rate of change of a function at any given point, shedding light on its overall behavior. By interpreting the sign of the first derivative, mathematicians and scientists can make informed predictions about the direction of a function's growth or decline, making it a crucial aspect of mathematical analysis.
Questions
- Is #f(x)=(12x^2-16x-12)/(x+2)# increasing or decreasing at #x=5#?
- Is #f(x)=(-2x^3+9x^2-5x+6)/(x-2)# increasing or decreasing at #x=0#?
- Is #f(x)=1/x-1/x^3+1/x^5# increasing or decreasing at #x=-1#?
- Is #f(x)=cos^2x+sin2x# increasing or decreasing at #x=pi/6#?
- Is #f(x)=(x-2)(2x-3)(2x-1)# increasing or decreasing at #x=-2#?
- Is #f(x)=(x^3+3x^2-4x-9)/(x+1)# increasing or decreasing at #x=0#?
- Is #f(x)=x-e^x/sinx# increasing or decreasing at #x=pi/3#?
- Is #f(x)=(3x^3-2x^2-2x+5)/(x+2)# increasing or decreasing at #x=3#?
- Is #f(x)=x^2-3lnx# increasing or decreasing at #x=1#?
- Is #f(x)=(x-2)e^x # increasing or decreasing at #x=-2 #?
- Is #f(x)=(x-2)(x+5)(x+2)# increasing or decreasing at #x=-3#?
- Is #f(x)=(2x+4)^2-6x-7 # increasing or decreasing at #x=-2 #?
- Is #f(x)=tanx# increasing or decreasing at #x=0#?
- Is #f(x)=(-2x^3+4x^2-x-2)/(x+3)# increasing or decreasing at #x=-2#?
- Is #f(x)=(x^2+2x-6)/(2x+1)# increasing or decreasing at #x=0#?
- Is #f(x)=(-7x^3-x^2-2x+2)/(x^2+3x)# increasing or decreasing at #x=1#?
- How do you find the intervals of increasing and decreasing using the first derivative given #y=x(x^2-3)#?
- Is #f(x)= cot(3x+(5pi)/8) # increasing or decreasing at #x=pi/4 #?
- Is #f(x)=x^2-x# increasing or decreasing at #x=1#?
- Is #f(x)=(x^2+2x-2)/(2x-4)# increasing or decreasing at #x=0#?