Each of the following graphs contain a curve, f \left( x \right), along with one of its tangents, g \left( x \right).
State the coordinates of the point at which g \left( x \right) is a tangent to the curve f \left( x \right).
State the gradient of the tangent.
Hence determine the equation of the line y = g \left( x \right).
Each of the following graphs contain a curve, f \left( x \right), along with one of its tangents, g \left( x \right).
State the coordinates of the point at which g \left( x \right) is a tangent to the curve f \left( x \right).
State the gradient of the tangent.
Hence determine the equation of the line y = g \left( x \right).
State the x-coordinate of the point on the curve at which we could draw a tangent that has the same gradient as g \left( x \right).
Consider the graph of the function f \left( x \right):
Sketch the graph of the function \\ g \left( x \right) = 2 x + 3 on the same number plane.
Is g \left( x \right) a tangent to f \left( x \right)? Explain your answer.
Consider the graph of the function f \left( x \right):
Sketch the graph of the function \\ g \left( x \right) = 2 x-1 on the same number plane.
Is g \left( x \right) a tangent to f \left( x \right)? Explain your answer.
Consider the graph of the function f \left( x \right):
Sketch the graph of the function \\ g \left( x \right) = 3 x + 3 on the same number plane.
Is g \left( x \right) a tangent to f \left( x \right)? Explain your answer.
Consider the graph of the function f \left( x \right):
Sketch the graph of the function \\ g \left( x \right) = 4 x + 7 on the same number plane.
Is g \left( x \right) a tangent to f \left( x \right)? Explain your answer.
Consider the curve given by the function f \left( x \right) = x^{2} - 1.
Find the gradient of the tangent to the curve at the point \left(1, 0\right).
Graph the curve and the tangent at the point \left(1, 0\right) on a number plane.
Consider the curve given by the function f \left( x \right) = x^{2} - 4 x + 2.
Find the gradient of the tangent to the curve at the point \left(3, -1\right).
State the coordinates of the vertex of the parabola f \left( x \right) = x^{2} - 4 x + 2.
Graph the curve and the tangent at the point \left(3, -1\right) on a number plane.
Consider the curve given by the function f \left( x \right) = x^{3} + 5 x.
Find the gradient of the tangent at the point \left(2, 18\right).
Consider the parabola f \left( x \right) = x^{2} + 3 x - 10.
Find the x-intercepts.
Find the gradient of the tangent at the positive x-intercept.
Consider the function y = x^{2} - 3 x + 4. State the x-coordinate of the point on the curve where the tangent makes an angle of 45 \degree with the x-axis.
Given that y = \dfrac{1}{3} x^{3} + \dfrac{1}{6} x^{6} + 2 x:
Find \dfrac{dy}{dx}.
Evaluate \dfrac{dy}{dx} when x = - 7.
Consider the tangent to the curve f \left( x \right) = x^{2} at the point \left( - 1 , 1\right).
Find the gradient of the function f \left( x \right) = x^{2} at this point.
Hence, find the equation of the tangent to the curve f \left( x \right) = x^{2} at the point \left( - 1 , 1\right).
Consider the tangent to the curve f \left( x \right) = x^{3} at the point \left( - 1 , -1\right).
Find the gradient of the function f \left( x \right) = x^{3} at this point.
Hence, find the equation of the tangent to the curve f \left( x \right) = x^{3} at the point \left( - 1 , -1\right).
Consider the tangent to the curve f \left( x \right) = - x^{2} at the point \left( - 3 , - 9 \right).
Find the gradient of the function f \left( x \right) = - x^{2} at this point.
Hence, find the equation of the tangent to the curve f \left( x \right) = - x^{2} at the point \left( - 3 , - 9 \right).
Consider the tangent to the curve f \left( x \right) = - x^{3} at the point \left(2, - 8 \right).
Find the gradient of the function f \left( x \right) = - x^{3} at this point.
Hence, find the equation of the tangent to the curve f \left( x \right) = - x^{3} at the point \left(2, - 8 \right).
Consider the tangent to the curve f \left( x \right) = x^{2} at the point \left( - 2 , 4\right).
Find the gradient of the function f \left( x \right) = x^{2} at this point.
Hence, find the equation of the tangent to the curve f \left( x \right) = x^{2} at the point \left( - 2 , 4\right)
Consider the tangent to the curve f \left( x \right) = 6 \sqrt{x} at the point \left(4, 12\right).
Find the gradient of the function f \left( x \right) = 6 \sqrt{x} at this point.
Hence, find the equation of the tangent to the curve f \left( x \right) = 6 \sqrt{x} at the point \left(4, 12\right).
Consider the tangent to the curve f \left( x \right) = 5 \sqrt{x} at the point \left(\dfrac{1}{9}, \dfrac{5}{3}\right).
Find the gradient of the function f \left( x \right) = 5 \sqrt{x} at this point.
Hence, find the equation of the tangent to the curve f \left( x \right) = 5 \sqrt{x} at the point \left(\dfrac{1}{9}, \dfrac{5}{3}\right).
Find the equation of the tangent of the following curves at the given point:
Consider the tangent to the curve f \left( x \right) = 3 x^{3} at x = 2.
Describe what must be done to find the equation of the tangent to the curve f(x) at \\ x=2.
Hence, find the equation of the tangent to the curve f \left( x \right) = 3 x^{3} at x = 2.
Consider the tangent to the curve f \left( x \right) = \dfrac{2}{x^{3}} at x = - 2.
Find the gradient of the function f \left( x \right) = \dfrac{2}{x^{3}} at x = - 2.
Find the y-coordinate of the point on the curve where x=-2.
Hence, find the equation of the tangent to the curve f \left( x \right) = \dfrac{2}{x^{3}} at x = - 2.
Consider the function f \left( x \right) = 3 x^{2}.
Find the gradient of the function at x = 2
Find the y-coordinate of the point on the function at x=2.
Hence, find the equation of the tangent to the curve f \left( x \right) = 3 x^{2} at x = 2.
Consider the tangent to the curve f \left( x \right) = - 2 x^{2} + 8 x + 2 at x = 1.
Find the y-coordinate of the point of intersection between the tangent and the curve.
Hence, determine the equation of the tangent to the curve f \left( x \right) = - 2 x^{2} + 8 x + 2 at \\ x = 1.
Find the equation of the tangent to the curve f \left( x \right) = 0.3 x^{3} - 5 x^{2} - x + 4 at x = 1.
Find the equation of the tangent to the curve f \left( x \right) = \dfrac{9 x + 4}{3 x} at x = - 1.
Find the equation of the tangent to the curve g \left( x \right) = \dfrac{8 x^{7} - 6 x^{6} + 4 x^{5} + 7}{2 x^{2}} at x = 1.
From an external point \left(3, 2\right), two tangents are drawn to the curve y = x^{2} - 6.
Find the gradient of both tangents.
Find the equation of both tangents.
Find the equation of the tangent to the parabola y = 2 x^{2} + 8 x - 5 at the point where the gradient is 0.
Consider the curve y = x^{3} - x^{2} and the line 4 x - y = 11.
Find the x-coordinates of the points on the curve at which the tangents are perpendicular to the line 4 x - y = 11.
Find the equation of the tangent to the curve y = x^{3} - x^{2} at each of these \\ x-coordinates.
Consider the function f \left( x \right) = x^{3} + 2 x - 1.
Find f' \left( x \right).
Complete the table using three iterations of Newton's method, starting with x = 0.5. Round each answer to four decimal places.
Hence, state the approximate solution to f \left( x \right) = 0. Round your answer to four decimal places.
| n | x_n | x_{n + 1} |
|---|---|---|
| 0 | 0.5 | |
| 1 | ||
| 2 |
For each of the following functions:
Find f' \left( x \right).
Complete the table using three iterations of Newton's method, starting with x = 1. Round each answer to four decimal places.
Hence, state the approximate solution to f \left( x \right) = 0. Round your answer to four decimal places.
| n | x_n | x_{n + 1} |
|---|---|---|
| 0 | 1 | |
| 1 | ||
| 2 |
For each of the following:
Find the x-coordinate of point M.
Find the y-coordinate of point M.
At point M\left(x, y\right), the equation of the tangent to the curve y = x^{2} is given by y = 4 x - 4.
At point M\left(x, y\right), the equation of the tangent to the curve y = x^{3} is given by \\ y = 12 x - 16.
Consider the function f \left( x \right) = \dfrac{4 x^{3}}{3} + \dfrac{5 x^{2}}{2} - 3 x + 7. Find the x-coordinates of the points on the curve whose tangent is parallel to the line y = 3 x + 7.
5 x + y + 2 = 0 is the tangent to the curve y = x^{2} + b x + c at the point \left(9, - 47 \right).
Find the derivative \dfrac{d y}{d x} of y = x^{2} + b x + c.
Find the gradient of the tangent to the curve at x = 9.
Solve for the value of b.
Solve for the value of c.
Consider the function f \left( x \right) = x^{2} + 5 x.
Find the x-coordinate of the point at which f \left( x \right) has a gradient of 13.
Hence, state the coordinates of the point on the curve where the gradient is 13.
Consider the function f \left( x \right) = x^{3} - 6 x^{2}.
Find the x-coordinates of the points at which f \left( x \right) has a gradient of 495.
Hence, state the coordinates of the points on the curve where the gradient is 495.
Consider the function f \left( x \right) = x^{3} + 3 x^{2} - 19 x + 2.
Find the x-coordinates of the points at which f \left( x \right) has a gradient of 5.
Hence, state the coordinates of the points on the curve where the gradient is 5.
Consider the function f \left( x \right) = x^{3} + 6 x^{2} - 14 x - 2.
Find the x-coordinates of the points on the curve where the gradient is the same as that of g \left( x \right) = 22 x - 4.
Hence, state the coordinates of the points on the curve where the gradient is the same as that of g \left( x \right) = 22 x - 4.
The curve f \left( x \right) = k \sqrt{x} - 5 x has a gradient of 0 at x = 16.Find the value of k.
Consider the function y = 4 x^{2} - 5 x + 2.
Find \dfrac{dy}{dx}.
Hence, find the value of x at which the tangent to the parabola is parallel to the x-axis.
Consider the function f \left( x \right) = 5 x^{2} + \dfrac{4}{x} - 1. The tangent to the curve at the point \left(2, 21\right) makes an angle of \theta with the x-axis. Find \theta, correct to the nearest degree.
The curve y = a x^{3} + b x^{2} + 2 x - 17 has a gradient of 58 at the point \left(2, 31\right). Find the values of a and b.
The graph of y = a x^{3} + b x^{2} + c x + d intersects the x-axis at \left(2, 0\right), where it has a gradient of 36. It also intersects the y-axis at y = - 28, where the tangent is parallel to the x-axis.
Find the values of a, b, c and d.
In the following graph, the line y = \dfrac{x}{10} + b is a tangent to the graph of f \left( x \right) = 6 \sqrt{x} at x = a.
Find the value of a.
Find the value of b.