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Unit #4 - Inverse Trig, Interpreting Derivatives, Newton’S Method

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Unit #4 - Inverse Trig, Interpreting Derivatives, Newton’S Method Unit #4 - Inverse Trig, Interpreting Derivatives, Newton's Method Some problems and solutions selected or adapted from Hughes-Hallett Calculus. Computing Inverse Trig Derivatives 1. Starting with the inverse property that 3. Starting with the inverse property that sin(arcsin(x)) = x, find the derivative of arcsin(x). tan(arctan(x)) = x, find the derivative of arctan(x). You will need to use the trig identity You will need to use the trig identity 2 2 sin2(x) + cos2(x) = 1. sin (x)+cos (x) = 1, or its related form, dividing each term by cos2(x), 2. Starting with the inverse property that tan2(x) + 1 = sec2(x) cos(arccos(x)) = x, find the derivative of arccos(x). You will need to use the trig identity sin2(x) + cos2(x) = 1. Interpreting Derivatives 4. The graph of y = x3 − 9x2 − 16x + 1 has a slope of 5 At what rate was the population changing on January at two points. Find the coordinates of the points. 1st, 2010, in units of people/year? 5. Determine coefficients a and b such that p(x) = x2 + 10. The value of an automobile can be approximated by ax + b satisfies p(1) = 3 and p0(1) = 1. the function V (t) = 25(0:85)t; 6. A ball is thrown up in the air, and its height over time is given by where t is in years from the date of purchase, and V (t) f(t) = −4:9t2 + 25t + 3 is its value, in thousands of dollars. where t is in seconds and f(t) is in meters. (a) Evaluate and interpret V (4). (b) Find an expression for V 0(t). (a) What is the average velocity of the ball during the (c) Evaluate and interpret V 0(4). first two seconds? Include units in your answer. (b) Find the instantaneous velocity of the ball at t = 2. 11. The quantity, q of a certain skateboard sold depends on the selling price, p, in dollars, with q = f(p). For (c) Compute the acceleration of the ball at t = 2. this particular model of skateboard, f(140) = 15; 000 (d) What is the highest height reached by the ball? and f 0(140) = −100. (e) How long is the ball in the air? (a) What do f(140) = 15; 000 and f 0(140) = −100 tell 7. The height of a sand dune (in centimeters) is repre- you about the sales of this skateboard model? sented by f(t) = 800 − 5t2 cm, where t is measured in (b) The total revenue, R, earned by the sale of this years since 1995. Find the values f(8) and f 0(8), in- skateboard is given by cluding units, and determine what each means in terms of the sand dune. R = (price per unit)(quantity sold) = p · q dR 8. With a yearly inflation rate of 5%, prices are given by Find the rate of change of revenue, when dp t p = 140. This is sometimes written as P (t) = P0(1:05) dR where P0 is the price in dollars when t = 0 and t is dp p=140 time in years. Suppose P0 = 1. How fast (in cents per year) are prices rising when t = 10? dR (c) From the sign of , decide whether the 9. With t in years since January 1st, 1990, the population dp p=140 P of a small US town has been given by company would actually increase revenues by in- creasing the price of this skateboard from $140 to P = 35; 000(0:98)t $141. 1 12. The theory of relativity predicts that an object whose (b) Let h(v) be the car's gas consumption in liters per mass is m0 when it is at rest will appear heavier when hour. In other words, h(v) tells you how many moving at speeds near the speed of light. When the liters of gas the car uses in one hour if it is go- object is moving at speed v, its mass m is given by ing at velocity v. What is the relationship between 0 m0 h(v) and f(v)? Find h(80) and h (80). m = p ; where c is the speed of light 1 − (v2=c2) (c) How would you explain the practical meaning of the values of these function and their derivatives dm (a) Find . to a driver who knows no calculus? dv dm (b) In terms of physics, what does tell you? 15. If P (x) is a polynomial, and it can be factored so that dv P (x) = (x − a)2Q(x), where Q(x) is also a polynomial, 13. A museum has decided to sell one of its paintings and we call x = a a double zero of the polynomial P (x). to invest the proceeds. If the picture is sold between the years 2000 and 2020 and the money from the sale (a) If x = a is a double zero (i.e. P (x) can be written is invested in a bank account earning 5% interest per as (x − a)2Q(x)), show that both P (a) = 0 and year, compounded annually, then the balance in the P 0(a) = 0. year 2020 depends on the time t when the painting is sold. Let P (t) be the price the painting will sell for if (b) Show that if P (x) is a polynomial, and both P (a) = 0 sold in year t, and let B(t) be the balance from the sale 0 and P (a) = 0, then we can factor P (x) into the 2 (after interest) in 2020, again depending on the year of form (x − a) Q(x). sale, t. 16. If g(x) is a polynomial, it is said to have a zero of mul- If t = 0 in the year 2000, so 0 ≤ t ≤ 20, then tiplicity m at x = a if it can be factored so that B(t) = P (t)(1:05)20−t g(x) = (x − a)mh(x) (a) Explain why B(t) is given by this formula. (b) Show that the formula for B(t) is equivalent to where h(x) is a polynomial such that h(a) 6= 0. P (t) Explain why having a polynomial having a zero of mul- B(t) = (1:05)20 (1:05)t tiplicity m at x = a must then satisfy g(a) = 0; g0(a) = 0;:::; and g(m)(a) = 0. 0 (c) Find B (10), given that P (10) = 150; 000 and (Note: g(m) indicates the m-th derivative of g(x).) P 0(10) = 5000. (d) Decide whether, if the museum still has the paint- 17. (a) Find the eighth derivative of f(x) = x7 + 5x5 − ing in 2010 (at t = 10), it should continue to hold 4x3 + 6x − 7. Look for patterns as you go... on to the painting, or sell it immediately if its goal is to maximize its financial future. (b) Find the seventh derivative of f(x). 14. Let f(v) be the gas consumption (in liters/km) of a car 18. (a) Use the formula for the area of a circle of radius r, going at velocity v (in km/hr). In other words, f(v) dA A = πr2, to find . tells you how many liters of gas the car uses to go one dr kilometer, if it is travelling at velocity v. You are told (b) The result from part (a) should look familiar. that dA 0 What does represent geometrically? f(80) = 0:05 and f (80) = 0:0005 dr (a) Let g(v) be the distance the same car goes on one (c) Use the difference quotient definition of the deriva- liter of gas at velocity v. What is the relationship tive to explain the observation you made in part between f(v) and g(v)? Find g(80) and g0(80). (b). Linear Approximations and Tangent Lines 19. Find the equation of the tangent line to the graph of f (b) Sketch the curve and the tangent line on the same at (1,1), where f is given by f(x) = 2x3 − 2x2 + 1. axes, and decide whether using the tangent line to approximate f(x) = x3 would produce over- or 20. (a) Find the equation of the tangent line to f(x) = x3 under-estimates of f(x) near x = 2. at x = 2. 2 21. Find the equation of the line tangent to the graph of f 30. Find the quadratic polynomial g(x) = ax2 + bx + c at (3; 57), where f is given by f(x) = 4x3 − 7x2 + 12. which best fits the function f(x) = ex at x = 0, in the sense that 22. Given a power function of the form f(x) = axn, with f 0(3) = 16 and f 0(6) = 128, find n and a. g(0) = f(0); g0(0) = f 0(0); and g00(0) = f 00(0) 23. Find the equation of the line tangent to the graph of f at (2; 1), where f is given by f(x) = 2x3 − 5x2 + 5. 8 31. Consider the graphs of y = sin(x) (regular sine graph), 24. Find all values of x where the tangent lines to y = x −x 9 and y = ke (exponential decay, but scaled vertically and y = x are parallel. by k). 25. Consider the function f(x) = 9 − ex. If k ≥ 1, the two graphs will intersect. What is the smallest value of k for which two graphs will be tan- (a) Find the slope of the graph of f(x) at the point gent at that intersection point? where the graph crosses the x-axis.
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