Apply polynomial derivatives

Which value gives the tangent gradient at x = 2?

Move through all five inputs. Check how the function value, derivative and tangent intercept differ

At input a = 0, which statement follows?

Choose the correct equation

Substitute into the derivative to find a rate. To find an input for a required rate, solve an equation involving the derivative. Use the original rule to recover a point height

1. 1A rate at an input Differentiate the rule, then substitute the input into the derivative. Use the original rule for a point height \(\displaystyle m=f\prime(a),\quad y_0=f(a)\)
2. 2An input for a rate Set the derivative equal to the required rate, solve exactly, then filter against the original domain \(\displaystyle f\prime(x)=m\)
3. 3Stationary inputs Set the derivative equal to zero. Repeated roots give one distinct input. Zero gradient alone does not tell you whether the curve turns

Given

\(\displaystyle f(x)=x^3-2x+4\)

Find the instantaneous rate at x = −1

Given

\(\displaystyle f(x)=x^2+x\)

Find the input where the gradient equals 5

The gradient condition has candidates −1 and 2. The domain includes inputs greater than −1 and up to and including 3. Which set is admissible?

The derivative is the square of the difference between the input and 1 on the whole real line. Which claim is justified at input 1?

Construct a tangent from a slope and a point

Find the point from the original polynomial and the slope from its derivative. Both are needed. The vertical intercept is the point height minus slope times input

1. 1Tangent through the curve point Use the derivative slope together with the original function value. Rearrange only after forming the point-slope equation \(\displaystyle y-f(a)=f\prime(a)(x-a)\)

Given

\(\displaystyle f(x)=x^2+1\)

Find the slope of the tangent at x = 2

Given

\(\displaystyle f(x)=x^2+1\)

Find the tangent vertical intercept at x = 2

Keep the rate units and interval

If V is measured in litres and t in minutes, V′(t) is measured in litres per minute. It describes local change at that time, not an exact change over every later minute. Use a context model only on its stated time interval

1. 1Average and instantaneous rates An average uses two endpoints; an instantaneous rate uses the derivative at one input. Their units agree but their values need not \(\displaystyle \text{average}=\frac{f(b)-f(a)}{b-a},\quad\text{instantaneous at }a=f\prime(a)\)

Given

\(\displaystyle f(x)=x^2+1\)

Find the average rate from x = 1 to x = 3