11.04 Tangents and normals
Find the gradient of the tangent to f \left( x \right) = x^{3} + 5 x at the point \left(2, 18\right).
Consider the function y = \dfrac{1}{3} x^{3} + \dfrac{1}{6} x^{6} + 2 x.
Find \dfrac{dy}{dx}.
Evaluate \dfrac{dy}{dx} when x = - 7.
Find the gradient of f \left( x \right) = \dfrac{11}{x} + \dfrac{10}{x^{2}} at the point \left(4, \dfrac{27}{8}\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 = 4 x^{2} - 5 x + 2.
Find \dfrac{dy}{dx}.
Find the value of x at which the tangent to the parabola is parallel to the x-axis.
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 the point.
Hence, find the equation of the tangent to the curve at the point.
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 the point.
Hence, find the equation of the tangent to the curve at the point.
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 the point.
Hence, find the equation of the tangent to the curve at the point.
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 of intersection between the tangent line and the curve.
Hence, find the equation of the tangent to the curve 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 line and the curve.
Find the equation of the tangent to the curve at x = 1.
Consider the tangent to the curve f \left( x \right) = 3 x^{3} at x = 2.
Describe the steps required to find the equation of the tangent line at x = 2.
Hence, find the equation of the tangent to the curve.
Consider the function f \left( x \right) = \dfrac{9 x + 4}{3 x}.
Find the y-coordinate of the point on the curve at x = - 1.
Hence, find the equation of the tangent to the curve at x = - 1.
Consider the function g \left( x \right) = \dfrac{8 x^{7} - 6 x^{6} + 4 x^{5} + 7}{2 x^{2}}.
Find the derivative of the function.
Find the equation in of the tangent to the curve at x = 1.
Find the equation of the tangent to f \left( x \right) = x^{2} + x at the point \left(2, 6\right).
Find the equation of the tangent to f \left( x \right) = \left( 3 x - 1\right) \left( 2 x - 5\right) at the point \left(2, - 5 \right).
Consider the function f \left( x \right) = x^{3} - 6 x^{2}. Find 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 coordinates of the points on the curve where the gradient is 5.
Consider the function f \left( x \right) = 2 x^{2} - 216 \sqrt{x}. Find the coordinates of the point on the curve where the gradient is 0.
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.
At point M, the equation of the tangent to the curve y = x^{2} is given by y = 4 x - 4. Find the coordinates of M.
At point M, the equation of the tangent to the curve y = x^{3} is given by y = 12 x - 16. Find the coordinates of M.
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 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).
Find the coordinates of the vertex of the parabola f \left( x \right).
Sketch the graph of the curve and the tangent at the point \left(3, -1\right).
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).
Sketch the graph of the curve and the tangent at the point \left(1, 0\right).
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 at each of the x-coordinates found in part (a).
The curve f \left( x \right) = k \sqrt{x} - 5 x has a gradient of 0 at x = 16. Find k.
In the figure, the straight line y = \dfrac{1}{10} x + b is tangent to the graph of f \left( x \right) = 6 \sqrt{x} at \\ x = a.
Find a.
Find b.
5 x + y + 2 = 0 is the tangent line 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.
State the gradient of the tangent to the curve at x = 9.
Find the value of b.
Find the value of c.
The curve y = a x^{3} + b x^{2} + 2 x - 17 has a gradient of 58 at the point \left(2, 31\right).
Use the given information to write two equations expressing b in terms of a.
Hence, find the value of a and b.
The graph of y = a x^{3} + b x^{2} + c x + d has the following properties:
- It intersects the y-axis at y = - 28, where the tangent is parallel to the x-axis.
- It intersects the x-axis at \left(2, 0\right), where it has a gradient of 36.
Use the information given in the first point above to find the value of c and d.
Use the information given in the second point above to write two equations expressing b in terms of a.
Hence, find the value of a and b.
From an external point \left(3, 2\right), two tangents L_{1} and L_{2} are drawn to the curve y = x^{2} - 6.
Let the gradient of a tangent line to the curve be m. Find the two possible values of m that correspond to the gradients of L_{1} and L_{2}.
If L_{1} is the tangent with the larger gradient, find the equation of L_{1}.
Find the equation of L_{2}.
Consider the function y = 10 x^{5} - 2 x^{4} + 8 x^{3} - 6 x^{2} + 4 x - 164.
Find \dfrac{dy}{dx}.
Find the gradient of the tangent to the curve at the point (3, 2278).
Find the gradient of the normal to the curve at the point (3, 2278).
The gradient of the curve f \left( x \right) at x = 3 is 2.
Find the gradient of the tangent line at x = 3.
Find the gradient of the normal line at x = 3.
Consider the curve f \left( x \right) = x^{2} + 8 x + 15 at the point \left(4, 63\right).
Find f' \left( x \right).
Find the gradient of the tangent to the curve at the point.
Find the equation of the tangent to the curve at the point.
Find the gradient of the normal to the curve at the point.
Find the equation of the normal to the curve at the point.
Consider the curve f \left( x \right) = 4 x + \dfrac{64}{x} at the point \left(4, 32\right).
Find the equation of the tangent to the curve at the point.
Find the equation of the normal to the curve at the point.
- 4 x + y + 1 = 0 is the normal to the curve y = x^{2} + b x + c at the point \left( - 8 , - 32 \right).
Find the derivative \dfrac{d y}{d x} of y = x^{2} + b x + c.
State the gradient of the normal to the curve at x = - 8.
Find the value of b.
Find the value of c.
Find the angle of inclination of a line to the x-axis, whose gradient is 4. Write your answer correct to the nearest degree.
Consider the function y = x^{2} - 3 x + 4. Find the x-coordinate of the point on the curve where the tangent makes an angle of 45 \degree with the x-axis.
The given line has an angle of inclination of \theta with the positive x-axis:
Find the gradient of the line.
Hence, find \theta. Give your answer in degrees to two decimal places.
Consider the graph of the line y = \dfrac{1}{2} x + 1 given and the triangle whose vertices are the origin, the x-intercept and the y-intercept. \theta is the acute angle in the triangle that the line makes with the x-axis.
State the gradient of the line.
Find the value of \tan \theta.
A line with equation y = 6 x + 2 has an angle of inclination of \theta with the positive x-axis. Find the value of \theta to the nearest degree.
If a line has a gradient of - 4 and an angle of inclination of \theta with the positive x-axis, which of the following is true of the value of \theta?
-90\degree < \theta < 0\degree
0\degree < \theta < 90\degree
90\degree < \theta < 180\degree
180\degree < \theta < 270\degree
The given line passing through the origin makes an angle of \theta with the positive x-axis. The point on the line has coordinates of \left(5.2, 5.72\right).
Find \theta. Give your answer in degrees, correct to four decimal places.
A line with gradient m has an angle of inclination of 65 \degree with the positive x-axis, such that \tan 65 \degree = m. Without solving for m, find the value of \theta such that \tan \theta = - \dfrac{1}{m}.
A line with equation 5 x + 6 y - 2 = 0 has an angle of inclination of \theta with the positive x-axis. Find the value of \theta, correct to the nearest degree.
A line is drawn on the Cartesian plane at an angle of inclination \theta with the positive x-axis. Given that \sin \theta = \dfrac{12}{13}, find the gradient m of the line.
A line passing through the points \left(2, - 3 \right) and \left(x, 9\right) makes an angle of 120 \degree with the positive x-axis. Find the exact value of x.
Consider the curve y = x^{3} + 2 x^{2} + 4 x - 21 at the point \left(8, 651 \right).
Find \dfrac{dy}{dx}.
Find the gradient of the tangent to the curve at the point.
Find the gradient of the normal to the curve at the point.
Find the angle of the inclination, \alpha, of the tangent to the curve at the point, correct to two decimal places.
Find the angle of the inclination, \beta, of the normal to the curve at the point, correct to two decimal places.
Two curves C_{1}: y = 3 x^{2} and C_{2}: y = \dfrac{24}{x} intersect at point P. Tangents are drawn to both curves at point P such that the tangent to C_{1} makes an angle of \theta degrees with the positive x-axis, and the tangent to C_{2} makes an angle of \alpha degrees with the positive x-axis.
Find the x-coordinate of point P.
Find exact value of \theta.
Find exact value of \alpha.
Find the angle between the two tangents at point P, correct to one decimal place.
The steepness of a ski run is defined by the angle of inclination of the run with the horizontal ground.
By how many degrees is Run A steeper than Run B? Round your answer to one decimal place.
Consider the curve f \left( x \right) = 36 - x^{2} at the point P\left(3, 27\right).
Find f' \left( x \right).
Find the gradient of the tangent to the curve at the point.
Find the equation of the tangent to the curve at the point.
Find the gradient of the normal to the curve at the point.
Find the equation of the normal to the curve at the point.
The tangent line at \left(3, 27\right) meets the x-axis at point A. Find the x-coordinate of A.
The normal line at \left(3, 27\right) meets the x-axis at point B. Find the x-coordinate of B.
Find the length of AB.
Find the exact area of \triangle ABP.
The normal to the curve y = x \left(x - 5\right)^{2} at the point A \left(6, 6\right) cuts the x-axis at B.
Find the gradient function.
Find the gradient of the tangent at A.
Find the gradient of the normal at A.
Hence, find the x-coordinate of point B.
Find the area of triangle whose vertices are the origin, point A and point B.