Flashcards
Increasing and Decreasing functions
Cards in this deck (9)
- 1Where dy/dx is positive, the graph is.....Increasing
- 2Where dy/dx is negative, the graph is.....Decreasing
- 3If f'(x)>0 for all values, it's called an.....Increasing function
- 4If f'(x)<0 for all values, it's called a.....Decreasing function
- 5What does ∀x represent?"For all values for x"
- 6Show that 5x^5+3x^3+7x is an increasing function (explanation marks available)dy/dx = 25x^4+9x^2+7 , and x^4≥0 ∀x , and x^2≥0 ∀x, so 24x^4+9x^2≥0 ∀x.
Therefore 24x^4+9x^2+7>0 ∀x, and an increasing function has dy/dx>0, so 5x^5+3x^3+7x is an increasing function. - 7Find the x coordinates of the points of the curve y=3x^3+4/x where the gradient is equal to -35dy/dx: 9x^2-4x^-2=-35, 9x^2-4/root(x)+35=0
Multiply all by x^2 ; 9x^4-4+35x^2=0
Solve: 9(x^2)^2+35(x^2)-4=0 , x^2= 1/9 or -4
x^2=-4 can be ignored as it has no real solutions, so x^2=1/9 , x=1/3 or -1/3 - 8If we have the x-coordinates x=1/3 or -1/3, how would we find the y coordinates (original equation is y=3x^3+4/x)Plug in 1/3 and -1/3 as x into the y=3x^3+4/x , x=1/3 gives y=109/9 and x=-1/3 gives y=-109/9
Therefore our coordinates are (1/3,109/9) and (-1/3,-109/9) - 9Find the equations of the tangent lines to y=3x^3+4/x parallel to y+35x=2Rearrange to get y-2=-35x, therefore m=-35, find coordinates on curve where m=-35 (parallel lines have same gradient) (use method explained before this one). Now plug in values to y-y1=m(x-x1) to get the equations: y-109/9=-35(x-1/3) and y+109/9=-35(x+1/3)