Practice Exercises. Chapter 3. Derivatives of Functions. Chapter 3 Practice Exercises 235. Find the derivatives of the functions in Exercises 1-40.

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1 Chapter Practice Eercises 5 Chapter Practice Eercises Derivatives of Functions Find the derivatives of the functions in Eercises y = y = y = 7 y = - s + p d p + 5. y = s + d s + d 6. y = s - 5ds4 - d - 7. y = su + sec u + d 8. t 9. s = 0. s = + t y = a- - csc u t - - u 4 b

2 6 Chapter : Differentiation. y = tan - sec. y = sin - sin ƒ() g() ƒ() g(). s = 4. s = cot a cos4 s - td t b 5. s = ssec t + tan td 5 6. s = csc 5 s - t + t d 7. r = u sin u 8. r = ucos u 9. r = sin u 0. r = sin Au + u + B. y = csc. y = sin. y = -> secsd 4. y = cscs + d 5. y = 5 cot 6. y = cot 5 7. y = sin s d 8. y = - sin s d 9. - s = a 4t - 0. s = t + b 5s5t - d. y = a. y = a + b + b. y = + B 4. y = sin u r = a cos u - b 6. r = a + sin u - cos u b 7. y = s + d + 8. y = 0s - 4d >4 s - 4d ->5 9. y = s5 + sin d > 40. y = s + cos d -> Implicit Differentiation In Eercises 4 48, find dy>d. 4. y + + y = 4. + y + y - 5 = y - y 4> = >5 + 0y 6>5 = y = 46. y = y + y = = + A - In Eercises 49 and 50, find dp>dq. 49. p + 4pq - q = 50. q = s5p + pd -> 0 5 Find the first derivatives of the following combinations at the given value of. a. 6ƒsd - gsd, = b. ƒsdg sd, = 0 c. ƒsd gsd +, = d. ƒsgsdd, = 0 e. gsƒsdd, = 0 f. s + ƒsdd >, = g. ƒs + gsdd, = Suppose that the function ƒ() and its first derivative have the following values at = 0 and =. ƒ() ƒ() >5 - > > -4 Find the first derivatives of the following combinations at the given value of. a. ƒsd, = b. ƒsd, = 0 c. ƒs d, = d. ƒs - 5 tan d, = 0 e. ƒsd + cos, = 0 f. 0 sin a p b ƒ sd, = 57. Find the value of dy>dt at t = 0 if y = sin and = t + p. 58. Find the value of ds>du at u = if s = t + 5t and t = su + ud >. 59. Find the value of dw>ds at s = 0 if w = sin A r - B and r = 8 sin ss + p>6d. 60. Find the value of dr>dt at t = 0 if r = su + 7d > and u t + u =. 6. If y + y = cos, find the value of d y>d at the point (0, ). 6. If > + y > = 4, find d y>d at the point (8, 8). In Eercises 5 and 5, find dr>ds. 5. r cos s + sin s = p 5. rs - r - s + s = - 5. Find d y>d by implicit differentiation: a. + y = b. y = a. By differentiating - y = implicitly, show that dy>d = >y. b. Then show that d y>d = ->y. Numerical Values of Derivatives 55. Suppose that functions ƒ() and g() and their first derivatives have the following values at = 0 and =. Derivative Definition In Eercises 6 and 64, find the derivative using the definition. 6. ƒstd = 64. gsd = + t a. Graph the function ƒsd = e, , 0. b. Is ƒ continuous at = 0? c. Is ƒ differentiable at = 0?

3 Chapter Practice Eercises a. Graph the function b. Is ƒ continuous at = 0? c. Is ƒ differentiable at = 0? 67. a. Graph the function b. Is ƒ continuous at =? c. Is ƒ differentiable at =? 68. For what value or values of the constant m, if any, is a. continuous at = 0? b. differentiable at = 0?, ƒsd = e tan, 0 p>4. ƒsd = e, 0 -, 6. sin, 0 ƒsd = e m, 7 0 Slopes, Tangents, and Normals 69. Tangents with specified slope Are there any points on the curve y = s>d + >s - 4d where the slope is ->? If so, find them. 70. Tangents with specified slope Are there any points on the curve y = - >sd where the slope is? If so, find them. 7. Horizontal tangents Find the points on the curve y = where the tangent is parallel to the - ais. 7. Tangent intercepts Find the - and y-intercepts of the line that is tangent to the curve y = at the point s -, -8d. 7. Tangents perpendicular or parallel to lines Find the points on the curve y = where the tangent is a. perpendicular to the line y = - s>4d. b. parallel to the line y = Intersecting tangents Show that the tangents to the curve y = sp sin d> at = p and = -p intersect at right angles. 75. Normals parallel to a line Find the points on the curve y = tan, -p> 6 6 p>, where the normal is parallel to the line y = ->. Sketch the curve and normals together, labeling each with its equation. 76. Tangent and normal lines Find equations for the tangent and normal to the curve y = + cos at the point sp>, d. Sketch the curve, tangent, and normal together, labeling each with its equation. 77. Tangent parabola The parabola y = + C is to be tangent to the line y =. Find C. 78. Slope of tangent Show that the tangent to the curve y = at any point sa, a d meets the curve again at a point where the slope is four times the slope at sa, a d. 79. Tangent curve For what value of c is the curve y = c>s + d tangent to the line through the points s0, d and s5, -d? 80. Normal to a circle Show that the normal line at any point of the circle + y = a passes through the origin. Tangents and Normals to Implicitly Defined Curves In Eercises 8 86, find equations for the lines that are tangent and normal to the curve at the given point y = 9, s, d 8. + y =, s, d 8. y + - 5y =, s, d 84. s y - d = + 4, s6, d y = 6, s4, d 86. > + y > = 7, s, 4d 87. Find the slope of the curve y + y = + y at the points (, ) and s, -d. 88. The graph shown suggests that the curve y = sin s - sin d might have horizontal tangents at the -ais. Does it? Give reasons for your answer. Tangents to Parametrized Curves In Eercises 89 and 90, find an equation for the line in the y-plane that is tangent to the curve at the point corresponding to the given value of t. Also, find the value of d y>d at this point. 89. = s>d tan t, y = s>d sec t, t = p> 90. = + >t, y = - >t, t = Analyzing Graphs y 0 y sin ( sin ) Each of the figures in Eercises 9 and 9 shows two graphs, the graph of a function y = ƒsd together with the graph of its derivative ƒ sd. Which graph is which? How do you know?

4 8 Chapter : Differentiation 9. y 9. A 0 B y 4 0 A B (0, 700) Number of foes Number of rabbits Initial no. rabbits 000 Initial no. foes Time (days) (a) 9. Use the following information to graph the function y = ƒsd for - 6. i. The graph of ƒ is made of line segments joined end to end. ii. The graph starts at the point s -, d. iii. The derivative of ƒ, where defined, agrees with the step function shown here. 94. Repeat Eercise 9, supposing that the graph starts at s -, 0d instead of s -, d. Eercises 95 and 96 are about the graphs in Figure.5 (right-hand column). The graphs in part (a) show the numbers of rabbits and foes in a small arctic population. They are plotted as functions of time for 00 days. The number of rabbits increases at first, as the rabbits reproduce. But the foes prey on rabbits and, as the number of foes increases, the rabbit population levels off and then drops. Figure.5b shows the graph of the derivative of the rabbit population. We made it by plotting slopes. 95. a. What is the value of the derivative of the rabbit population in Figure.5 when the number of rabbits is largest? Smallest? b. What is the size of the rabbit population in Figure.5 when its derivative is largest? Smallest (negative value)? 96. In what units should the slopes of the rabbit and fo population curves be measured? Trigonometric Limits y y f'() sin - tan :0 - : sin r tan r (0, 40) Time (days) Derivative of the rabbit population (b) FIGURE.5 r:0 4 tan u + tan u + u:sp>d - tan u cot u u:0 + 5 cot u - 7 cot u - 8 sin cos :0 Show how to etend the functions in Eercises 05 and 06 to be continuous at the origin. tan stan d 05. gsd = 06. ƒsd = tan Related Rates Rabbits and foes in an arctic predator-prey food chain. sin ssin ud u:0 u - cos u u:0 u tan stan d sin ssin d 07. Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = pr + prh. a. How is ds>dt related to dr>dt if h is constant? b. How is ds>dt related to dh>dt if r is constant?

5 Chapter Practice Eercises 9 c. How is ds>dt related to dr>dt and dh>dt if neither r nor h is constant? d. How is dr>dt related to dh>dt if S is constant? 08. Right circular cone The lateral surface area S of a right circular cone is related to the base radius r and height h by the equation S = prr + h. a. How is ds>dt related to dr>dt if h is constant? b. How is ds>dt related to dh>dt if r is constant? c. How is ds>dt related to dr>dt and dh>dt if neither r nor h is constant? 09. Circle s changing area The radius of a circle is changing at the rate of ->p m>sec. At what rate is the circle s area changing when r = 0 m? 0. Cube s changing edges The volume of a cube is increasing at the rate of 00 cm >min at the instant its edges are 0 cm long. At what rate are the lengths of the edges changing at that instant?. Resistors connected in parallel If two resistors of R and R ohms are connected in parallel in an electric circuit to make an R-ohm resistor, the value of R can be found from the equation Eit rate: 5 ft /min 6. Rotating spool As television cable is pulled from a large spool to be strung from the telephone poles along a street, it unwinds from the spool in layers of constant radius (see accompanying figure). If the truck pulling the cable moves at a steady 6 ft> sec (a touch over 4 mph), use the equation s = ru to find how fast (radians per second) the spool is turning when the layer of radius. ft is being unwound. r 4' h 0' R = R + R. R R R.' If R is decreasing at the rate of ohm> sec and R is increasing at the rate of 0.5 ohm> sec, at what rate is R changing when R = 75 ohms and R = 50 ohms?. Impedance in a series circuit The impedance Z (ohms) in a series circuit is related to the resistance R (ohms) and reactance X (ohms) by the equation Z = R + X. If R is increasing at ohms> sec and X is decreasing at ohms> sec, at what rate is Z changing when R = 0 ohms and X = 0 ohms?. Speed of moving particle The coordinates of a particle moving in the metric y-plane are differentiable functions of time t with d>dt = 0 m/sec and dy>dt = 5 m/sec. How fast is the particle moving away from the origin as it passes through the point s, -4d? 4. Motion of a particle A particle moves along the curve y = > in the first quadrant in such a way that its distance from the origin increases at the rate of units per second. Find d>dt when =. 5. Draining a tank Water drains from the conical tank shown in the accompanying figure at the rate of 5 ft >min. a. What is the relation between the variables h and r in the figure? b. How fast is the water level dropping when h = 6 ft? 7. Moving searchlight beam The figure shows a boat km offshore, sweeping the shore with a searchlight. The light turns at a constant rate, du>dt = -0.6 rad/sec. a. How fast is the light moving along the shore when it reaches point A? b. How many revolutions per minute is 0.6 rad> sec? km 8. Points moving on coordinate aes Points A and B move along the - and y-aes, respectively, in such a way that the distance r (meters) along the perpendicular from the origin to the line AB remains constant. How fast is OA changing, and is it increasing, or decreasing, when OB = r and B is moving toward O at the rate of 0.r m> sec? A

6 40 Chapter : Differentiation Linearization 9. Find the linearizations of a. tan at = -p>4 b. sec at = -p>4. Graph the curves and linearizations together. 0. We can obtain a useful linear approimation of the function ƒsd = >s + tan d at = 0 by combining the approimations to get L - and tan L + + tan L -. Show that this result is the standard linear approimation of >s + tan d at = 0.. Find the linearization of ƒsd = + + sin at = 0.. Find the linearization of ƒsd = >s - d at = 0. Differential Estimates of Change. Surface area of a cone Write a formula that estimates the change that occurs in the lateral surface area of a right circular cone when the height changes from h 0 to h 0 + dh and the radius does not change. 4. Controlling error a. How accurately should you measure the edge of a cube to be reasonably sure of calculating the cube s surface area with an error of no more than %? b. Suppose that the edge is measured with the accuracy required in part (a). About how accurately can the cube s volume be calculated from the edge measurement? To find out, estimate the percentage error in the volume calculation that might result from using the edge measurement. 5. Compounding error The circumference of the equator of a sphere is measured as 0 cm with a possible error of 0.4 cm. This measurement is then used to calculate the radius. The radius is then used to calculate the surface area and volume of the sphere. Estimate the percentage errors in the calculated values of a. the radius. b. the surface area. c. the volume. 6. Finding height To find the height of a lamppost (see accompanying figure), you stand a 6 ft pole 0 ft from the lamp and measure the length a of its shadow, finding it to be 5 ft, give or take an inch. Calculate the height of the lamppost using the value a = 5 and estimate the possible error in the result. h h r V r h S rr h (Lateral surface area) 0 ft 6 ft a