Questions

1. Question 1

   Find by definition, the derivatives w.r.t. ‘x’ of the following functions defined as:

   (i)

   $2x^2 + 1$

   (ii)

   $2 - \sqrt{x}$

   (iii)

   $\dfrac{1}{\sqrt{x}}$

   (iv)

   $x(x - 3)$

   SolutionTheory
2. Question 2

   Find $\dfrac{dy}{dx}$ from first principle and find gradient of the curve at the given point:

   (i)

   $\sqrt{x + 2}$ at $x = 6$

   (ii)

   $\dfrac{1}{\sqrt{x + a}}$ at $x = a$

   SolutionTheory
3. Question 3

   (i)

   Find the derivative of $x^{\frac{2}{3}}$ at $x = 8$ from the first principle.

   (ii)

   Find the derivative of $x^2 + 2x + 3$ by definition.

   SolutionTheory
4. Question 4

   Find from first principle, the derivatives of the following expressions w.r.t. their respective independent variables:

   (i)

   $(3x - 2)^{-2}$

   (ii)

   $(2t + 3)^5$

   (iii)

   $(aw + b)^7$

   SolutionTheory
5. Question 5

   Find the gradient and equation of the tangent line to $y = 3x^2 - 4x + 1$ at $x = 2$.

   SolutionTheory
6. Question 6

   For the function $f(x) = 2x^3 + x$, calculate the equation of the tangent line at $x = -1$.

   SolutionTheory
7. Question 7

   Find the coordinates of the point of tangency and the equation of the tangent line for $f(x) = x^3 - 2x + 1$ at $x = 1$.

   SolutionTheory
8. Question 8

   Find the gradient of the curve $f(x) = 3x^2 + 2x$ at $x = 1$.

   SolutionTheory
9. Question 9

   Find the gradient and an equation of tangent line to the graph of $f(x) = \sqrt{x}$ at $x = 9$.

   SolutionTheory
10. Question 10

    The position of a car after $t$ hours is given by: $s(t) = 2t^3 - 3t^2 + t$ (in kilometres)

    (i)

    Find the average velocity over the interval $[1, 4]$

    (ii)

    Find the instantaneous velocity at $t = 2$

    SolutionTheory
11. Question 11

    A stone is thrown upwards and its height after $t$ seconds is given by: $s(t) = -16t^2 + 32t + 10$ (in feet). Find the instantaneous velocity at $t = 1$.

    SolutionTheory
12. Question 12

    The outdoor temperature (in °C) over time is modeled by: $T(t) = -t^2 + 12t + 10$, where $t$ is the time in hours. Find the instantaneous rate of change at $t = 2$.

    SolutionTheory