If f'(x) = 6x^2 - 4x + 1 and f(1) = 5, what is the constant of integration C?
Answer and explanation
Correct answer: A. C = 4
- Step 1: Anti-differentiate f'(x): f(x) = 6(x^3/3) - 4(x^2/2) + x + C = 2x^3 - 2x^2 + x + C. Step 2. Use boundary condition f(1) = 5: 2(1)^3 - 2(1)^2 + (1) + C = 5 => 2 - 2 + 1 + C = 5 => 1 + C = 5 => C = 4. Answer: C = 4. Why not the others: f(1) includes the contribution of the polynomial terms, so C is not simply 5.
Practise more Calculus Methods, Derivatives, Kinematics and Coordinate Geometry.
Free questions with an explanation for every one, and full timed National Certificate of Educational Achievement mock tests.
More Calculus Methods, Derivatives, Kinematics and Coordinate Geometry questions
- What is the derivative of the function f(x) = 4x^3 - 5x^2 + 7x - 9?
- What is the gradient of the curve y = x^2 - 6x + 8 at the point where x = 5?
- At what x-coordinate does the curve f(x) = x^3 - 12x have a local minimum?
- What is the equation of the tangent line to the curve y = x^2 + 3x at the point (1, 4)?
- The displacement of a toy rocket in metres is given by s(t) = 30t - 5t^2, where t is in seconds. What is its velocity at t = 2 seconds?
- What is the acceleration of an object whose displacement is given by s(t) = 2t^3 - 9t^2 + 12t at t = 3 seconds?