Questions:1, 3, 7, 15. Thank you
SECTION 1./ Bites of Change in the Natural soul Social Sciences marginal cost and marginal profit in economics; rate of heat flow in geology rate of Improvement of performance in psychology, rate of spread of a rumor in sociology these are all special eases of a single mathematical concept, the derivative. This is an illustration of the fact that part of the power of mathematics lies in its achstraciness, A single abstract mathematical concept (such as the derivative ) can have dif ferear interpretations in each of the sciences. When we develop the properties of the mathematical concept once and for all, we can then turn wound and apply these results to all of the sciences. This is much more efficient than developing properties of special con- cepts in each separate science. The French mathematician Joseph Fourier (1768-1830) put it succinctly: "Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them." 3.7 EXERCISES 1-4 A particle moves according to a law of motion s - f(), 7. The height (in meters) of a projectile shot vertically upward 0, where ? is measured in sexists and's in fort. from a point 2 m above ground level with an initial velocity (a) Find the velocity at time 1, of 24.5 m/sish - 2 + 24 51 - 4.9r after I seconds, () What is the velacity after I second? (a) Find the velocity after 2 s and after 4 s. (c) When is the park le at resa? (b) When does the projectile reach its maximum height ? in When is the particle moving in the positive direction? (c) What is the maximum height? je) Find the total distance traveled during the first 6 seconds (d) When does it hit the ground? 1 Draw a diagram like Figure 2 to illustrate the motion of the (e) With what velocity does it hit the ground? particle. (p) Find the acceleration of time / and after I second 8. If a ball is thrown vertically upward with a velocity of () Graph the position, velocity, and acceleration functions 80 tv/s, then its height after I seconds is s - 801 - 161 (a) What is the maximum height reached by the ball? () When is the particle spending up? When is it slowing down? (b) What is the velocity of the ball when it is 90 fi above the ground on its way up? On its way down? (1. An -!' - 8+241 2. /(1) -- F+ 9 * 9. If a rock is thrown vertically upwand from the surface of Mars with velocity 15 m/s, its height after f seconds is 3. /t) - sin(=1/2) 4. f(1) - re h = 151 -1.861. (a) What is the velocity of the rock after 2 s? 5. Graphs of the velocity functions of two particles are shown. (b) What is the velocity of the rock when its height is 25 m on its way up? On its way down? where ! is measured in seconds. When is each particle spending up? When is it slowing down? Explain. 10. A particle moves with position function (@) re (b) F. 5my -41-2013 + 201 120 (a) At what time does the particle have a velocity of 20 m/s? (b) At what time is the acceleration ()? What is the signifi- cance of this value of ? 11. (a) A company makes computer chips from square wafers 6. Graphs of the position functions of two particles are shown. of silicon, Is wants to keep the side length of a wafer very where ( is measured in seconds. When is each particle close to 15 min and it wants to know how the area A(x) of speeding up? When is is slowing down? Explain a wafer changes when the side length x changes. Find (b) 4. A'(15) and explain its meaning in this situation, (b) Show that the rate of change of the area of a square with respect to its side length is half its perimeter. Try to explain geometrically why this is true by drawing a square whose side length x is increased by an amount Ax. How can you approximate the resulting change in area AA if Ax is small
