Derivatives
Basics
Basics
This function grows faster and faster, and its rate of change at any point is triple the square of the input. Name its derivative if the function is x3
What is 3x2
For the Parabold f(x) = x2, this is the slope of the tangent line when x = 2
What is 4
f(x)=x2−6x+5
Does the stationary point represent a maximum or a minimum?
What is a Minimum?
An expression for h in terms of r of a box with a square base and a volume of 108 cm3.
What is 108/r2?
Differentiate the function
f(x) = 3x3 - 5x2 + 4x - 7
What is
f′(x)=9x2−10x+4
A function refuses to change no matter what you do to x
What is a constant function?
This is the equation of the tangent line to f(x) = x2 + 4x at x = 0
What is y = 4x
C(x)=0.5x2−10x+80
Find the stationary point of the function.
What is (10,30)?
A formula giving the volume of a can in terms of r and h.
What is πr2h?
Find the gradient of the curve
y = x2 + 2x
at x=1
What is 4
The function f(x) = x-1 is popular with fractions. Its derivative flips the sign and increases the exponent by one. Identify it.
What is - 1/x2
At x = 1, the line perpendicular to the tangent of f(x) = x2 has this slope.
What is -½
f(x)=x3−3x2−9x+2
Find all stationary points of the function -> State whether each point is a minimum or a maximum
What is (-1,7) is a local maximum and (3,-25) is a local minimum
A formula for the surface area of a can with a volume of 1L only in terms of r.
What is A = 2πr2 + 2000/r?
Find the equation of the tangent line to
y = x3
at x=2
what is y=12x−16
The derivative of f(x) = √x
What is 1/2√x ?
This is the equation of the line orthogonal to the curve f(x) = x2 at the point where x = 1
What is y = - 1/2x + 3/2?
The height of a ball is modeled by
h(t) = -4.9t2 + 14t + 1
Find the time and height at which the ball reaches its maximum.
What is 11m at 1.43 seconds
dA/dr of a box with a square base and volume of 108cm3 only in terms of r.
What is 4r - 432/r2?
Find the equation of the normal line to
y = x2 - 4x
at x=1
What is y+3=1/2(x−1)
The function that is squared after being built from 2x2 + 3x.
What is 2(2x2+ 3x)(4x+3)
This term refers to the unique point on a curve where the tangent line is horizontal, meaning the derivative equals zero.
What is a critical point?
The profit from selling x products is given by
P(x) = -2x2 + 40x - 120
How many products should be sold to achieve maximum profit, and what is that profit?
What is $80 occurs when 10 products are sold?
The value of r that minimizes the surface area of a can with a volume of 1L in centimeters.
What is 10.8cm?
The area of a rectangle is
A=x(20−x)
Find the value of x that maximizes the area.
What is x=10