Differentiation and integration
Connect a function with its gradient and use antiderivatives to evaluate signed area.
What you will learn
- Locate stationary points.
- Evaluate a definite integral with limits.
Differentiate to study change
For a power term axⁿ, the derivative is anxⁿ⁻¹. Differentiate a sum term by term. A stationary point has derivative zero; finding the point also requires substituting x into the original function.
A second derivative can classify a stationary point when it is non-zero: positive indicates a local minimum and negative a local maximum. If it is zero, this test alone is inconclusive.
Integration reverses differentiation
For n≠−1, an antiderivative of axⁿ is axⁿ⁺¹/(n+1). Add a constant for an indefinite integral. For a definite integral, calculate F(upper)−F(lower); the arbitrary constant cancels.
A definite integral gives signed area. Regions below the x-axis contribute negatively. If the question asks for total geometric area, split at roots and account for the signs.
Worked example
Evaluate the integral of 2x+3 from x=1 to x=4.
Show the worked solution
- An antiderivative is F(x)=x²+3x.
- F(4)=16+12=28 and F(1)=1+3=4.
- Subtract lower from upper: 28−4.
Answer 24
Common mistakes
- Do not substitute limits into the derivative instead of the antiderivative.
- A stationary x-coordinate is not a full coordinate pair.
What can you explain now?
Find the stationary point of y=x²−6x+2.
Compare with the explanation
(3,−7), a minimum
The derivative 2x−6 is zero at x=3. Substitution gives −7, and the second derivative is 2>0.
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