Year11 MATH 3-2-3 Applications of the Derivative warm-up workbook

Unit1

Focus: Tangents, normals, and stationary points.

1. Find the gradient of the tangent to $y = x^2 + 2x$ at the point $(1, 3)$.
2. Find the equation of the tangent to $y = x^2$ at $x = 2$.
3. Determine the coordinates of the stationary point for $y = x^2 – 4x + 5$.
4. Determine the nature of the stationary point for $y = x^2$ (Max, Min, or Inflexion).
5. If the displacement of a particle is $s(t) = t^2 + 4t$, find the velocity at $t = 2$.
6. Find the acceleration if velocity is $v(t) = 3t^2 – 5$.
7. At what value of $x$ is the tangent to $y = x^2 – 8x$ horizontal?
8. Find the equation of the normal to $y = 2x^2$ at $x = 1$.
9. A curve has $\frac{dy}{dx} = 2x – 4$. Find the intervals where the function is increasing.
10. Find the average rate of change of $f(x) = x^2$ between $x = 1$ and $x = 3$.

Answers:

1. 4 | 2. $y = 4x – 4$ | 3. $(2, 1)$ | 4. Minimum | 5. 8 units/s | 6. $6t$ | 7. $x = 4$ | 8. $y = -\frac{1}{4}x + \frac{9}{4}$ | 9. $x > 2$ | 10. 4.