Derivative
The derivative measures the _____________________________.
gradient of the tangent line
Write the equation of the tangent line of f(x)=2x3+x
at the point (1,3).
y=7x-4.
A stationary point is a point where ______________________________________________________________________.
If the function is increasing, then the derivative is ____________________. If the function is decreasing, then the derivative is ____________________________.
positive; negative
A rectangular piece of card measures 24 cm by 9 cm. Equal squares of length x cm are cut from each corner of the card. What is left is then folded into an open box of length l cm and width w cm.
Write expressions, in terms of x, for the length and the width.
l=24-2x
w=9-2x
The Leibniz notation for the derivative is
dy/dx
Write the equation of the normal line of f(x)=2x3+x
at the point (1,3).
y=-1/7 x +22/7
Outline the steps to find a maximum or minimum of a function.
1. Find the derivative of the function.
2. Find the stationary points.
3. Make a number line where you plot the x-values of the stationary points.
4. Use test values to determine if the function is increasing or decreasing over that interval.
5. If the function goes from inc to dec, the stationary point is a max. If the function goes from dec to inc, the stationary point is a min.
True or False: A function is can be increasing or decreasing at a stationary point.
False, the derivative is 0 at a stationary point.
A rectangular piece of card measures 24 cm by 9 cm. Equal squares of length x cm are cut from each corner of the card. What is left is then folded into an open box of length l cm and width w cm.
Find an equation for the volume of the box.
V=4x3-66x2+216x
Compute the derivative of f(x)=2x4+3x2-x
f'(x)=8x3+6x-1
The normal line at x1 and the tangent line at x1 are _____________.
Perpendicular.
Find all of the local maxima and minima of the function f(x)=(x+1)2-2
Local minimum at (-1,-2)
The endpoints of the intervals of increasing or decreasing are always the _________ ____________ of the function.
A rectangular piece of card measures 24 cm by 9 cm. Equal squares of length x cm are cut from each corner of the card. What is left is then folded into an open box of length l cm and width w cm.
What is the maximum volume of the box?
200 cm3
Find the slope of the tangent of line of f(x)=x3-2x at x=-1.
f'(-1)=1
The slope of the normal line is the ____________________________ of the slope of the tangent line.
negative reciprocal
Find the local extrema of h = 3 + 14t − 5t2
Local maximum of 12.8 at 1.4
Find the interval(s) on which the function
f(x) = (1/3)x3 + 2.5x2 – 14x + 25 increases.
(-infinity, -7) U (2, infinity)
A closed box has a square base of side x and height h. Write down an expression for volume, V, of the box. Write down an expression for the total surface area, A, of the box.
V=x2h
A= 2x2+4xh
Find dy/dx of f(x)= 3/x2 at x=2.
f'(2)=-3/4
The equation of the normal line on f(x) at x1 is
y-4=(1/2)x-1
Find the equation of the tangent line.
y=2x
Find the stationary points of f(x)=2x3−3x2+5
(0,5) and (1,4)
Find the interval(s) where the following function is increasing.
y=13x3+2x2−5x−6
(-infinity, -5) U (1, infinity)
A closed box has a square base of side x and height h. If the volume of the box is 1000 cm3, find the minimum surface area the box can obtain.
600 cm2