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

Calculus - Differentiation - First derivative.
Gradients and equations of curves, tangents, etc - Test Yourself 2.


 

Remember that the first derivative is the gradient function of a function. So it is used to answer questions requiring information about the slope of a curve.

Gradients. 1. Find the coordinates of the point where the gradient of the tangent to f(x) = x2 - 4x is equal to 2.

Answer.At (0, 0).

 2. At what points on the curve
y = x3 - 4x2 + 2x are the tangents parallel to the line 2x + y - 3 = 0

Answer.At (2/3, -4/27) and at (2, -4).
  3. For what values of x do the graphs
y = x3 + x2 - x - 1 and y = x2 - 1 have the same gradient?

Answer.At x = -1, 0 and +1.

 4. For the graph of f(x) = x3 - x2 - 6x + 1, find the values of x for which f '(x) = -5.

Answer.x = -1/3 and x = 1.
5. Find the gradient of the parabola

y = 3x3 - 2x2 - 5x + 4 at the point where it crosses the y-axis.

Answer.Gradient = 123 at (0, 4).

 6. Find the gradients of the parabola
y = x2 - x - 6 at the points where it crosses the
x-axis.

Answer.Gradient = 5 at x = 3
Gradient = -3 at x = -2.
  7. Find the coordinates of the points on the parabola y = 2x2 +4x - 1 where the tangent:

(i) is parallel to the x axis.

(ii) makes an angle of 45° with the
x-axis.

(iii) is parallel to the line y = 2x.

Answer.(i) At (-1, -3).
(ii) At (-0.75, -2 7/8).
(iii) At (-0.5, -2.5)

 8. Find the points on the curve
y = x2 - 5x + 6 at which the tangent:

(i) makes an angle of 45° with the x-axis.

(ii) is parallel to the line
3x + y - 4 = 0.

(iii) is perpendicular to the line with equation 2y - x + 3 = 0.

Answer.(i) (3, 0).
(ii) at (1, 2).
(iii) at (1.5, 0.75)

  9. (i) Differentiate .

(ii) For x > 0, explain why all tangents have a positive gradient.

 10. If y = ax2 + bx has a maximum value at the point (2, 3), find the values of a and b.
Answer.a = -0.75 and b = 3.
Points of contact. 11. The point P where x = 2 lies on the curve given by the equation y = 2x3 - 5x2 + 3x -1:

(i) find the coordinates of the point Q where the tangent at P cuts the y axis;

(ii) find the coordinates of the point R where the normal at P cuts the x axis.

Answer.(i) Q is (0, -13).
(ii) R is (9, 0).		 12. A tangent to the curve y = 3x2 - 5x + 2 has a gradient of 4.

Find the coordinates of the point of contact.

Answer.At (1.5, 1.25).
  13. Given that y = -4x + L is a tangent to
y = x3 - 4x2 - 7x + 10 and x > 0, find the point with the value of L.
AnswerPOC is (3, -20)so I = -8.		 14. Find the value of a given that the curve

has a gradient of -1 when x = 6.

Answera = -4 or -2.
Equations of tangents & normals -
ordinary derivative.		 15. The tangent to the curve y = 3x3 - 8x2 at the point of contact A (2, -8) cuts the x-axis at B. The normal to the curve at the same point of contact cuts the y-axis at C.

(i) Find the equation of the tangent at A.

(ii) Find the equation of the normal at A.

(iii) Find the coordinates of B and C.

(iv) Find the length of the interval AB.

 16. A tangent is drawn to the parabola
y = x2 - 4x at the point P.

The tangent has a gradient of 6.

(i) Find the coordinates of P.

(ii) Find the equation of the tangent at P.

(iii) Find the gradient of the normal at P.
  17. Find the equations of the normals to the curve xy = 4 which are parallel to the line
4x - y = 2.
Answer.4x - y + 15 = 0 and
4x - y - 15 = 0.		 18. Find the equation of the normals to at x = 1/4.
Answer.32x + 24y - 11 = 0.
Chain rule. 19. Find the equation of the tangent to
y = (2x - 3)2 at the point where x = 3.
Answer.12x - y - 27 = 0.		 20. The graph of has a tangent T as shown in the diagram. The tangent makes an angle of 30° with the x axis as shown and its equation can be expressed as y = mx + x.

(i) Show that .

(ii) Find the x-coordinate of the point of contact P between the curve and the tangent.

Answer.x = 1.
Product rule. 21. Differentiate y = (2x - 3)(3x + 1)4 and determine the values of x for which the tangent is parallel to the x axis.

Answer.x = -1/3 and x = 17/15.

 22.
Quotient rule. 23. (i) Show that the derivative of is .

(ii) For which value(s) of x is the slope of this curve positive?

Answer.All x ≠ 0.		 24.
Increasing/ decreasing functions. 25. For what values of x is the curve

y = 4 + 36x - 3x2 - 2x3 decreasing?

Answer.Decreasing when x < -3 or when x > 2.		 26. For what values of x is the curve

y = x4 - 2x2 rising?

Answer.Rising for x > 0.
  27. For what values of x is the curve a monotonic decreasing function?
Answer.Monotonically decreasing
for x > -2.		 28.