Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Functions - Logarithmic - Differentiation.
Applications - Test Yourself 1.


 

The questions on this page focus on:

  1. determining gradients;
  2. equations of tangents and normals;
  3. maximum and minimum questions;

 

Gradients 1. Show that the gradient of the curve y = loge (x2 + 1) is always positive.
  2. Find the exact coordinates of the point on the curve
where the gradient equals zero.
Answer.At (1/√e, 1/(2e)).
  3. Prove that the curve y = x2 + loge 2x can never have a stationary point.
Tangents and normals 4. Find the equation of the tangent to y = 3 loge 4x at x = 3.
Answer.y = x - 3 - 3ln 12.
  5. Find the equation of the normal to y = loge (3x - 2)
at the point (1, 0) in general form.
Answer.x + 3y - 1 = 0.
  6. Determine the equation of the tangent to the curve
y = 2ln (2 - 5x2) at the point where x = 0.
Answer.y = 2 ln 2
  7.

The function y = loge(x2) is graphed above for x > 0.

(i) Show that the tangent to the curve at the point P (e, 2) passes through the origin O.

(ii) Find the equation of the normal to the curve at P and then find the point Q where that normal meets the y-axis.

(iii) Show that the area of the triangle OPQ is u2.
  8. Find the equation of the tangent to the curve y = x2 ln x at the point where x = e.
Answer.y = 3ex - 2e2
  9.
  10.
Maximum and minimum questions 11. (i) Find the stationary point on the curve y = 4 ln (x2 + 1).

(ii) For what values of x is the concavity of this curve always positive?

Answer.(i) SP at (0, 0)
(ii) Positive concavity: -1 < x < 1.
  12. (i) Draw the graphs of y = ln(x - 1) and y = x on the same set of axes.

(ii) If D represents the distance between the two curves, write down an expression for D.

(iii) If X represents the value on the x-axis where this distance between the two curves is the shortest, find the value for X.

(iii) Find the minimum distance between the two curves.

Answer.(ii) X = 2
(iii)D = 2.
 

13. In the diagram,
  • P is a point on the curve y = x2 - 4x + 7
  • Q is a point on the curve y = ln (x + 2)
  • P and Q have the same x coordinates.

Find the exact minimum distance for PQ between the two curves.

Answer.Min PQ = 1.60 (2 dec places).
  14. For what value of x is the value for the ratio
a maximum? Answer.x = e.
 

15.