Application Of Integration | Stanford University

University:
Stanford University
Course:
Introduction to Mathematical Thinking
Academic year:

2023
Views:

277

Pages:

3
Author:

MAlphen

Application of Integration (Equation of Curves) A curve is such that 𝑑 2𝑦 𝑑𝑥2 = 𝑎𝑥 − 2, where 𝑎 is a constant. The curve has a minimum gradient 1 at 𝑥 = . 3 a) Show that 𝑎 = 6. The tangent to the curve at (1, 4) is 𝑦 = 2𝑥 + 2. b) Find the equation of the curve. Ans: Application of Integration – Area Under the Graph In the diagram, the curve 𝑥 = (𝑦 − 1)2 + 4 and the line 𝑦 + 𝑥 = 7 intersects at 𝐴 and 𝐵. (i) Find the coordinates of 𝐴 and of 𝐵. (ii) Calculate the area of the shaded region. Ans: Application of Differentiation and Integration (Kinematics) A particle moves in a straight line so that its acceleration, 𝑎 ms-2, is given by 𝑎 = 2𝑡 − 13,where 𝑡 is the time in seconds after passing a fixed point 𝑂. The particle first comes to instantaneous rest at 𝑡 = 5𝑠. Find, (i) the velocity when the particle passes through 𝑂. (ii) the total distance travelled by the particle when it next comes to rest. (iii) the minimum velocity of the particle. Ans: