Homework, Page 158 1. Find the ROC of the area of a square with respect to the length of its side s when s = 3 and s = 5. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 158 5. Calculate the ROC dV/dr, where V is the volume of a cylinder whose height is equal to its radius. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 158 9. Refer to Figure 10. (a) Estimate the average velocity over [0.5, 1]. (b) Is the average velocity greater over [1, 2] or [2, 3]? (c) At what time is velocity maximum? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 158 13. A stone is tossed vertically upward with an initial velocity of 25 ft/s from the top of a 30-ft high building. a. What is the height of the stone after 0.25 s? b. Find the velocity of the stone after 1 s. c. When does the stone hit the ground? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 158 17. The Earth exerts a gravitational force of F (r) = (2.99 x 1016)/r 2 N on an object of mass 75 kg where r is the distance from the center of the Earth. Assuming the radius of the Earth is 6.77 x 106 m, calculate the ROC of the force with respect to distance at the Earth’s surface. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 158 21. By Faraday’s law, if a conducting wire of length l meters moves at a velocity of v m/s perpendicular to a magnetic field of strength B (in teslas), a voltage of size V = –Blv is induced in the wire. Assume that B = 2 and l = 0.5. (a) Find the rate of change dV/dv. (b) Find the ROC of V with respect to t, if v = 4t + 9 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 158 25. Ethan finds that with h hours of tutoring, he is able to answer correctly S(h) percent of the problems on a math exam. What is the meaning of the derivative S′(h)? Which would you expect to be larger, S′(3) or S′(30)? Explain The derivative S′(h) would be the rate of change in the percentage correct after h hours of tutoring. I would expect S′(3) to be greater than S′(30) since the percentage increase would first be rapid and then slow down as the percent correct approaches 100%. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 158 29. According to a formula used for determining drug dosage, a person’s body surface area (BSA) (in m2) is given by the formula , h is height in cm and w is weight in kg. Calculate the ROC of BSA with respect to weight for a person of height h = 180. What is the ROC for w = 70 and w = 80? Does BSA increase more rapidly with respect to weight for higher or lower weights? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 158 33. A ball is tossed up vertically from ground level and returns to Earth 4 s later. What was the initial velocity of the ball and how high did it go? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 158 37. Show that for an object rising and falling according to Galileo’s formula, the average velocity over any time interval [t1, t2] is equal to the average of the instantaneous velocities at t1 and t2. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 158 37. Continued. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 158 41. Let F (s) = 1.1s + 0.03s2 be the stopping distance. Calculate F (65) and estimate the increase in stopping distance if speed is increased from 65 to 66 mph. Compare your estimate to the actual. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 158 45. The demand for a commodity generally decreases as the price increases. Suppose that demand for oil is D(p) =900/p barrels, where p is the price per barrel in dollars. Find the demand when p = $40. Estimate the decrease in demand if p rises to $41 and the increase if p falls to $39. The demand per person at $40 per barrel is 22.5 barrels/year. If the price increases to $41, the demand will fall by 0.5625 barrels/year and if the price decreases to $39, the demand will increase by 0.5625 barrels per year. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Chapter 3: Differentiation Section 3.5: Higher Derivatives Jon Rogawski Calculus, ET First Edition Chapter 3: Differentiation Section 3.5: Higher Derivatives Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

As shown in the graph and table below, the population of Sweden increased at a decreasing rate from 1993 to 1997. P′ (t) is positive, as the slope is positive, but P″(t) is negative as the rate of increase is decreasing. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Higher derivatives are derivatives of derivatives Higher derivatives are derivatives of derivatives. The second derivative is the derivative of the first derivative, the third derivative is the derivative of the second derivative, and so forth. Symbolically, we write the derivatives as f ′, f ″, f ′″, f (4), …,f (n), … In Leibniz notation, we write: Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Table 2 tabulates the six derivatives of f (x) = x5. In general, an nth order function with all positive, whole number exponents, will have n +1 derivatives, the last of which will be zero. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 165 Calculate the second and third derivatives. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 165 Calculate the second and third derivatives. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 165 Calculate the second and third derivatives. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 165 Calculate the second and third derivatives. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 165 Calculate the second and third derivatives. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 165 Calculate the second and third derivatives. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Figure 2 shows the graphs of height versus time and velocity versus time for a ball tossed vertically into the air with an initial velocity of 40 ft/s. Notice that the velocity plots as a negative linear function. This indicates a constant negative acceleration due to gravity. For most calculations, this acceleration may be estimated at 32 ft/s or 9.8 m/s. In general, given position as a function of time, s (t), velocity is the first derivative of position and acceleration is the second derivative or v (t) = s′ (t) and a (t) = v′ (t) = s″ (t). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

The larger the magnitude of the second derivative, the more quickly the slope of the tangent changes. If the second derivative is zero, then the function is linear and the slope is unchanging. The sign of the second derivative tells us if the slope is increasing (positive) or decreasing (negative). In general, if the graph of the function lies above the graph of the tangent, the second derivative is positive, and if the graph of the function lies below the graph of the tangent, the second derivative is negative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 165 Find a general formula for f (n) (x). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 165 Find a general formula for f (n) (x). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

of two tangent lines in red. On the right are the graphs of the first Figure 4, on the left, shows the graph of a function in blue and graphs of two tangent lines in red. On the right are the graphs of the first and second derivatives, in red and green, respectively. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

WS 3.5.pdf Rogawski Calculus Copyright © 2008 W. H. Freeman and Company