Functions  Logarithmic  Differentiation.
Applications  Test Yourself 1.
The questions on this page focus on:

Gradients  1. Show that the gradient of the curve y = log_{e} (x^{2} + 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 = x^{2} + log_{e} 2x can never have a stationary point.  
Tangents and normals  4. Find the equation of the tangent to y = 3 log_{e} 4x at x = 3.
Answer.y = x  3  3ln 12. 

5. Find the equation of the normal to y = log_{e} (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  5x^{2}) at the point where x = 0. Answer.y = 2 ln 2 

7.
The function y = log_{e}(x^{2}) is graphed above for x > 0.


8. Find the equation of the tangent to the curve y = x^{2} ln x at the point where x = e.
Answer.y = 3ex  2e^{2} 

9.  
10.  
Maximum and minimum questions  11. (i) Find the stationary point on the curve y = 4 ln (x^{2} + 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.
Answer.(ii) X = 2 (iii)D = 2. 

13. In the diagram,
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. 