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

Calculus - Differentiation - Applications of Calculus.
Gradients and equations of curves, tangents, concavity, etc - Test Yourself 1.


 

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

The questions on this page focus on:
1. Gradients.
2. Points of contact between curves and tangents.
3. Obtaining equations of tangents and normals using various structures of the functions.
4. Increasing or decreasing functions.
5. Extended applications.

 

Gradients.   1.Find the co-ordinates of the point(s) on the curve y = 2x3 - 6x at which the gradient equals zero?
Answer.at (1, -4) and at (-1, 4).		 2. At what point(s) on the curve does the gradient equal 3?
Answer.At (7, -179/3) and (-1, 5/3).
    3. At what point on the curve
y = 2x2(x - 1)4 is the gradient equal to zero?

Answer.at x = 0, 1 and 1/3.

 4. At what point on the curve does the gradient equal -1?

Answer.at (7/4, 1/2).
    5. The graph of
y = x3 + ax2 + bx - 10 cuts the
x-axis at x = 2 and it has a gradient of zero at x = -1.

Find the values of a and b.

Answer.a = 1 and b = -1.

 6. Evaluate f '(1) if f(x) = 5x2(4x2 - 1)5.

Answer.f '(1) = 18,630.
    7. Given that f(x) = x2 + x, find the coordinates of the points for which

f"(x) = f(x).

Answer.Points are (-2, 5) and (1, 3).		 8. Given the equation f(x) = x2 - 3x - 3, find the value of α for which f(α) = f'(α).
Answer.a = 0 or a = 5.
    9. Evaluate f '(0) if .

Answer.f '(0) = 2.

 10. Prove that the tangents at the points
(2, -3) and (1, -3) on the curve
y = x2 - 3x -1 are perpendicular to one another.
    11. The gradient of the curve
y = ax2 + bx equals -8 at the point
(2, 1).

Find the values of a and b.

Answer.a = -17/4 and b = 9.

 12. For the curve f(x) = ax2 + bx + c where a, b and c are constants, it is given that, when x = 1, y = a and
f '(x) = 1.

Find the relationship between a and c.
Answer.a = (1 + c)/2.
Points of contact.   13. The gradient of a tangent to the curve y = x2 - 5x is -3.

Find the coordinates of the point of contact of the tangent to the curve.

Answer.The point is (1, -4).		 14. The gradient of a tangent to the curve y = x2 + 7x - 3 is 9.

Find the coordinates of the point of contact of the tangent to the curve.

Answer.At (1, 5).
    15. A tangent to the curve
y = x2 - 2x - 8 has gradient 8. Its equation is y = ax + b.

Find the values of a and b.

Answer.a = 8 and b = -33.		 16. The tangent to the curve
y = 2x2 + 3x - 1 makes an angle of 135° with the positive direction of the x-axis.

What is the equation of this tangent?

Answer.x + y + 3 = 0.
    17. Find the value of b given that

y = 2x - 5 is a tangent to the curve

y = x2 + bx + 4

at the point where x = 3.

Answer. b = -4.		 18. At what point on is the tangent parallel to 3x - y + 2 = 0?
Answer.At (6, 4).
Equations of tangents & normals. Ordinary derivative. 19. Find the equation of the normal to the curve
y = 3x - 2x2 at x = 1.

Answer.y = x.

 20. Find the equation of the tangent to the curve at the point
where x = 1.

Answer.4x - y - 1 = 0.
    21. The curve y = ax2 - 2x - 14 has a gradient of 10 when x = 2. Find the value of a.

Answer.a = 3

 22. Find the equation of the tangent to
y = 3x4 + 1 at the point (1, 4).

Answer.x + 12y + 47 = 0
    23. (i) Find the equation of the tangent to y = x3 - 9x2 + 20x - 8 at the point (1, 4).

(ii) For what values of x are the tangents to the above cubic curve parallel to the line y = -4x + 3?

Answer.(i)y = 5x - 1
(ii)x = 4 and for x = 2.		 24. Find the equations of the two tangents to the curve y = 3x2 - 6x at the points where it crosses the x axis.
Answer.y = -6x and y = 6x - 12.
  Chain rule 25. Find the equation of the tangent to y = (2x - 3)2 at the point where x = 3.

Answer.y = 12x - 25.

 26. Find the equation of the normal to the curve f(x) = (2x - 1)2 at x = 2.

Answer.x + 12y - 110 = 0
  Product rule. 27. There are three points on the curve y = x2 (3x - 2)3 where the gradient equals zero.

Find the x values of these points and write down the equations of the normals at those points.
Answer.x = 0, x = 4/15 and x = 2/3.

 28. Evaluate f'(1) if f(x) = 5x2(4x2 - 1)5
Answer.f'(1) = 230×34.
    29. Find the equation of the normal to where
x = 1.

Answer.x + 3y - 4 = 0.

 30. (i) Show that the equation of the tangent to y = x(x-1)4 at x = 2 is
y = 9x - 16.

(ii) If the tangent cuts the x-axis at A and the y-axis at B (with O as the origin), determine the area of the triangle OAB.

Answer.14.2 u2.
Quotient rule. 31. Find the equation of the tangent to at the point
x = 0.

Answer.2x + 3y - 3 = 0.

 32. Find the equation of the normal to the curve at x = -4.
Answer.Eqn is 8x - 2y + 35 = 0.
Increasing/ decreasing functions.   33. For what values of x is the function f(x) = 6 - 3x - x2 increasing?

Answer.4x - y - 1 = 0.

 34. For what values of x is the function

f(x) = x3 + x2 - 5x - 6 decreasing?

Answer.4x - y - 1 = 0.
    35. Show that is a monotonically decreasing function for all values of x (x ≠1.5). 36. For what values of x is the function decreasing?
(Answer to 3 significant figures).

Answer.0 < x < 0.488.
Extended applications.   37. Consider the curve y = 8 - 2x2.

(i) Sketch the curve showing all intercepts with the coordinate axes.

(ii) Let P and Q be the x-intercepts of the curve. Find the equations of the normals at P and Q.

(iii) Let the point of intersection of the normals at P and Q be R.
Show that the coordinates of R are (0, -0.25).

(iv) Hence find the area enclosed by the normal at P, the normal at Q and the x-axis.

Answer.(ii) At p: x - 8y - 2 = 0 and at Q: x + 8y + 2 = 0.
(iv) Area = 0.5 u2
    38. Consider the line y = x + 3 and the parabola y = 5x - x2.

(i) Find the coordinates of the points of intersection of the parabola and the line. Let these points be P and Q.

(ii) Find the equations of the tangents to the parabola at P and Q.

(iii) Let A and B be the points where the tangents at P and Q meet the y axis respectively. Show that the length of AB is 8 units.

Answer.(i) P is (3, 6) and Q is (1, 4).
(ii)At P: x + y - 9 = 0 and
at Q: 3x - y + 1 = 0.
   

39. The cubic y = ax3 bx2 + cx + d has a point of inflexion at x = p.

Show that .