Marginal revenue A bus will hold 60 people. The number x

Problem 1PE Derivatives of Functions Find the derivatives of the functions

Problem 2AAE If the identity sin (x + a) = sin x cos a + cos x sin ais differentiated with respect to x, is the resulting equation also an identity? Does this principle apply to the equation x2 - 2x - 8 = 0? Explain.

Problem 150PE Related Rates 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 = r? to find how fast (radians per second) the spool is turning when the layer of radius 1.2 ft is being unwound.

Problem 2PE Derivatives of Functions Find the derivatives of the functions

Problem 1QGY What is the derivative of a function ƒ? How is its domain related to the domain of ƒ? Give examples.

Problem 5QGY How are derivatives and one-sided derivatives related?

Problem 2QGY What role does the derivative play in defining slopes, tangents, and rates of change?

Problem 5PE Derivatives of Functions Find the derivatives of the functions

a. Find values for the constants \(a, b\), and \(c\) that will make \(f(x)=\cos x \quad \text { and } \quad g(x)=a+b x+c x^{2}\) satisfy the conditions \(f(0)=g(0), \quad f^{\prime}(0)=g^{\prime}(0), \quad \text { and } \quad f^{\prime \prime}(0)=g^{\prime \prime}(0)\). b. Find values for \(b\) and \(c\) that will make \(f(x)=\sin (x+a) \quad \text { and } \quad g(x)=b \sin x+c \cos x\) satisfy the conditions \(f(0)=g(0) \quad \text { and } \quad f^{\prime}(0)=g^{\prime}(0)\) c. For the determined values of \(a,b\) and \(c\), what happens for the third and fourth derivatives of \(f\) and \(g\) in each of parts (a) and (b)? Equation Transcription: Text Transcription: a,b c f(x)=cos x and g(x)=a+bx+cx2 f(0)=g(0), f'(0)=g'(0), and f"(0)=g"(0) b c f(x)=sin (x+a) and g(x)=b sin x+ c cos x f(0)=g(0) and f'(0)=g'(0) a,b c f g

Problem 12PE Derivatives of Functions Find the derivatives of the functions

Problem 12QGY What is the derivative of the exponential function ex? How does the domain of the derivative compare with the domain of the function?

Problem 15PE Derivatives of Functions Find the derivatives of the functions

Problem 15QGY How can derivatives arise in economics?

Problem 13AAE Velocity of a particle A particle of constant mass m moves along the x-axis. Its velocity and position x satisfy the equation where k, y0 , and x0 are constants. Show that whenever

Problem 16AAE Does the function have a derivative at x = 0? Explain.

Problem 16QGY Give examples of still other applications of derivatives

Problem 14PE Derivatives of Functions Find the derivatives of the functions

Problem 16PE Derivatives of Functions Find the derivatives of the functions

Problem 17PE Derivatives of Functions Find the derivatives of the functions

Problem 18PE Derivatives of Functions Find the derivatives of the functions

Problem 17QGY What do the limits have to do with the derivatives of the sine and cosine functions? What are the derivatives of these functions?

Problem 19PE Derivatives of Functions Find the derivatives of the functions

Problem 19QGY At what points are the six basic trigonometric functions continuous? How do you know?

Problem 22QGY What is implicit differentiation? When do you need it? Give examples.

Problem 30PE Derivatives of Functions Find the derivatives of the functions

Problem 30QGY How do related rates problems arise? Give examples.

Problem 24QGY What is the derivative of the exponential function and ? What is the geometric significance of the limit of as ? What is the limit when a is the number e?

Problem 34PE Derivatives of Functions Find the derivatives of the functions

Problem 35PE Derivatives of Functions Find the derivatives of the functions

Problem 33PE Derivatives of Functions Find the derivatives of the functions

Problem 36PE Derivatives of Functions Find the derivatives of the functions

Problem 38PE Derivatives of Functions Find the derivatives of the functions

Problem 37PE Derivatives of Functions Find the derivatives of the functions

Problem 40PE Derivatives of Functions Find the derivatives of the functions

Problem 43PE Derivatives of Functions Find the derivatives of the functions

Problem 39PE Derivatives of Functions Find the derivatives of the functions

Problem 42PE Derivatives of Functions Find the derivatives of the functions

Problem 41PE Derivatives of Functions Find the derivatives of the functions

Problem 44PE Derivatives of Functions Find the derivatives of the functions

Problem 46PE Derivatives of Functions Find the derivatives of the functions

Problem 47PE Derivatives of Functions Find the derivatives of the functions

Problem 45PE Derivatives of Functions Find the derivatives of the functions

Problem 56PE Derivatives of Functions Find the derivatives of the functions

Problem 48PE Derivatives of Functions Find the derivatives of the functions

Problem 57PE Derivatives of Functions Find the derivatives of the functions

Problem 55PE Derivatives of Functions Find the derivatives of the functions

Problem 77PE Implicit Differentiation Find dy/dx by implicit differentiation.

Problem 76PE Implicit Differentiation Find dy/dx by implicit differentiation.

Problem 78PE Implicit Differentiation Find dy/dx by implicit differentiation.

Problem 82PE Implicit Differentiation Find dr/ds.

Problem 85PE Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1. Find the first derivatives of the following combinations at the given value of x.

Problem 86PE Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1. Find the first derivatives of the following combinations at the given value of x.

Problem 87PE Find the value of

Problem 83PE Implicit Differentiation Find by implicit differentiation:

Problem 84PE Implicit Differentiation a. By differentiating implicitly, show that b. Then show that

Problem 88PE Find the value of

Problem 89PE Find the value of

Problem 91PE If find the value of at the point (0, 1).

Problem 90PE Find the value of at if and .

Problem 94PE Applying the Derivative Definition Find the derivative using the definition.

Problem 93PE Applying the Derivative Definition Find the derivative using the definition.

Problem 92PE If at the point (8, 8).

Problem 97PE Applying the Derivative Definition a. Graph the function b. Is ƒ continuous at x = 1? c. Is ƒ differentiable at x = 1? Give reasons for your answers.

Problem 96PE Applying the Derivative Definition a. Graph the function b. Is ƒ continuous at x = 0? c. Is ƒ differentiable at x = 0? Give reasons for your answers.

Problem 95PE Applying the Derivative Definition a. Graph the function b. Is ƒ continuous at x = 0? c. Is ƒ differentiable at x = 0? Give reasons for your answers.

Problem 127PE Trigonometric Limits Find the limits

Problem 99PE Slopes, Tangents, and Normals Tangents with specified slope Are there any points on the curve where the slope is –3/2? If so, find them.

Problem 107PE Slopes, Tangents, and Normals Tangent parabola The parabola is to be tangent to the line y = x.Find C.

Problem 128PE Trigonometric Limits Find the limits

Problem 136PE Logarithmic Differentiation Use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.

Problem 129PE Trigonometric Limits Find the limits

Problem 135PE Logarithmic Differentiation Use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.

Problem 137PE Logarithmic Differentiation Use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.

Problem 138PE Logarithmic Differentiation Use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.

Problem 133PE Trigonometric Limits Show how to extend the functions to be continuous at the origin.

Problem 134PE Trigonometric Limits Show how to extend the functions to be continuous at the origin.

Problem 140PE Logarithmic Differentiation Use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.

Problem 143PE Related Rates Circle’s changing area The radius of a circle is changing at the rate of At what rate is the circle’s area changing when r = 10 m?

Problem 145PE Related Rates Resistors connected in parallel If two resistors of R1 and R2 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 If R1 is decreasing at the rate of 1 ohm sec and R2 is increasing at the rate of 0.5 ohm sec, at what rate is R changing when R1 = 75 ohms and R2 = 50 ohms ?

Problem 139PE Logarithmic Differentiation Use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.

Problem 141PE Related Rates 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 a. How is dS/dt related to dr/dt if h is constant? b. How is dS/dt related to dr/dt if r is constant? c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant? d. How is dS/dt related to dr/dt if S is constant?

Problem 144PE Related Rates Cube’s changing edges The volume of a cube is increasing at the rate of 1200 cm3/min at the instant its edges are 20 cm long. At what rate are the lengths of the edges changing at that instant?

Problem 146PE Related Rates 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 If R is increasing at 3 ohms/sec and X is decreasing at 2 ohms/sec, at what rate is Z changing when and changing when R = 10 ohms and X = 20 ohms?

Problem 147PE Related Rates Speed of moving particle The coordinates of a particle moving in the metric xy-plane are differentiable functions of time t with dx/dt = 10 m/sec and dy/dt = 5 m/sec. How fast is the particle moving away from the origin as it passes through the point (3,–4)

Problem 149PE Related Rates Draining a tank Water drains from the conical tank shown in the accompanying figure at the rate of 5 ft3/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?

Problem 1AAE An equation like sin2 ? + cos2?= 1 is called an identity because it holds for all values of An equation like sin ? = 0.5is not an identity because it holds only for selected values of not all. If you differentiate both sides of a trigonometric identity in ? with respect to ? the resulting new equation will also be an identity. Differentiate the following to show that the resulting equations hold for all ?

Problem 160PE Differential Estimates of Change Finding height To find the height of a lamppost (see accompanying figure), you stand a 6 ft pole 20 ft from the lamp and measure the length a of its shadow, finding it to be 15 ft, give or take an inch. Calculate the height of the lamppost using the value a = 15 and estimate the possible error in the result.

Problem 3PE Derivatives of Functions Find the derivatives of the functions

Problem 3QGY How can you sometimes graph the derivative of a function when all you have is a table of the function’s values?

Problem 4PE Derivatives of Functions Find the derivatives of the functions

Problem 4AAE Solutions to differential equations a. Show that y = sin x, y = cos x, and y = a cos x + b sin x (a and b constants) all satisfy the equation b. How would you modify the functions in part (a) to satisfy the equation Generalize this result.

Problem 4QGY What does it mean for a function to be differentiable on an open interval? On a closed interval?

Problem 5AAE An osculating circle Find the values of h, k, and a that make the circle (x – h)2 + ( y – k)2 = a2 tangent to the parabola at the point (1, 2) and that also make the second derivatives d2y/dx2have the same value on both curves there. Circles like this one that are tangent to a curve and have the same second derivative as the curve at the point of tangency are called osculating circles (from the Latin osculari, meaning “to kiss”). We encounter them again in Chapter 13.

Problem 6AAE Marginal revenue A bus will hold 60 people. The number x of people per trip who use the bus is related to the fare charged ( p dollars) by the law Write an expression for the total revenue r (x) per trip received by the bus company. What number of people per trip will make the marginal revenue dr/dx equal to zero? What is the corresponding fare? (This fare is the one that maximizes the revenue, so the bus company should probably rethink its fare policy.)

Problem 6PE Derivatives of Functions Find the derivatives of the functions

Problem 6QGY Describe geometrically when a function typically does not have a derivative at a point.

Problem 7AAE Industrial production a. Economists often use the expression “rate of growth” in relative rather than absolute terms. For example, let u = ƒ(t) be the number of people in the labor force at time t in a given industry. (We treat this function as though it were differentiable even though it is an integer-valued step function.) Let y = g(t) be the average production per person in the labor force at time t. The total production is then y = uv. If the labor force is growing at the rate of 4% per year (dv/dt = 0.04v) and the production per worker is growing at the rate of 5% per year find the rate of growth of the total production, y. b. Suppose that the labor force in part (a) is decreasing at the rate of 2% per year while the production per person is increasing at the rate of 3% per year. Is the total production increasing, or is it decreasing, and at what rate?

Problem 7PE Derivatives of Functions Find the derivatives of the functions

Problem 7QGY How is a function’s differentiability at a point related to its continuity there, if at all?

Problem 8AAE Designing a gondola The designer of a 30-ft-diameter spherical hot air balloon wants to suspend the gondola 8 ft below the bottom of the balloon with cables tangent to the surface of the balloon, as shown. Two of the cables are shown running from the top edges of the gondola to their points of tangency, (–12,9 ) and (12, –9) How wide should the gondola be?

Problem 8PE Derivatives of Functions Find the derivatives of the functions

Problem 8QGY What rules do you know for calculating derivatives? Give some examples.

Problem 9AAE Pisa by parachute On August 5, 1988, Mike McCarthy of London jumped from the top of the Tower of Pisa. He then opened his parachute in what he said was a world record low-level parachute jump of 179 ft. Make a rough sketch to show the shape of the graph of his speed during the jump. (Source: Boston Globe, Aug. 6, 1988.)

Problem 9PE Derivatives of Functions Find the derivatives of the functions

Problem 9QGY Explain how the three formulas enable us to differentiate any polynomial.

Problem 10AAE Motion of a particle The position at time t ? 0of a particle moving along a coordinate line is a. What is the particle’s starting position (t = 0) ? b. What are the points farthest to the left and right of the origin reached by the particle? c. Find the particle’s velocity and acceleration at the points in part (b). d. When does the particle first reach the origin? What are its velocity, speed, and acceleration then?

Problem 10PE Derivatives of Functions Find the derivatives of the functions

Problem 10QGY What formula do we need, in addition to the three listed in Question 9, to differentiate rational functions?

Problem 11AAE Shooting a paper clip On Earth, you can easily shoot a paper clip 64 ft straight up into the air with a rubber band. In t sec after firing, the paper clip is above your hand. a. How long does it take the paper clip to reach its maximum height? With what velocity does it leave your hand? b. On the moon, the same acceleration will send the paper clip to a height of ft in t sec. About how long will it take the paper clip to reach its maximum height, and how high will it go?

Problem 11PE Derivatives of Functions Find the derivatives of the functions

Problem 11QGY What is a second derivative? A third derivative? How many derivatives do the functions you know have? Give examples.

Problem 12AAE Velocities of two particles At time t sec, the positions of two particles on a coordinate line are s1 = 3t 3 - 12t 2 + 18t + 5 and s2 = -t 3 + 9t 2 - 12t m. When do the particles have the same velocities?

Problem 13PE Derivatives of Functions Find the derivatives of the functions

Problem 13QGY What is the relationship between a function’s average and instantaneous rates of change? Give an example.

Problem 14AAE Average and instantaneous velocity a. Show that if the position x of a moving point is given by a quadratic function of t, x = At2 + Bt + Cthen the average velocity over any time interval [t1, t2] is equal to the instantaneous velocity at the midpoint of the time interval. b. What is the geometric significance of the result in part (a)?

Problem 14QGY How do derivatives arise in the study of motion? What can you learn about a body’s motion along a line by examining the derivatives of the body’s position function? Give examples.

Problem 15AAE Find all values of the constants m and b for which the function is a. continuous at b. differentiable at

Problem 17AAE a. For what values of a and b will be differentiable for all values of x? b. Discuss the geometry of the resulting graph of ƒ.

Problem 18QGY Once you know the derivatives of sin x and cos x, how can you find the derivatives of tan x, cot x, sec x, and csc x? What are the derivatives of these functions?

Problem 18AAE a. For what values of a and b will be differentiable for all values of x? b. Discuss the geometry of the resulting graph of g.

Problem 19AAE Odd differentiable functions Is there anything special about the derivative of an odd differentiable function of x? Give reasons for your answer.

Problem 20PE Derivatives of Functions Find the derivatives of the functions

Problem 20QGY What is the rule for calculating the derivative of a composite of two differentiable functions? How is such a derivative evaluated? Give examples.

Problem 21PE Derivatives of Functions Find the derivatives of the functions

Problem 21QGY If u is a differentiable function of x, how do you find if n is an integer? If n is a real number? Give examples.

Problem 22AAE Use the result of Exercise 21 to show that the following functions are differentiable at x = 0 Reference: Exercise 21 Suppose that the functions ƒ and g are defined throughout an open interval containing the point x0 that ƒ is differentiable at is differentiable at x = 0.

Problem 22PE Derivatives of Functions Find the derivatives of the functions

Problem 21AAE Suppose that the functions ƒ and g are defined throughout an open interval containing the point x0 that ƒ is differentiable at that and that g is continuous at x0. Show that the product ƒg is differentiable at x0. This process shows, for example, that although is not differentiable at x = 0, the product is differentiable at x = 0.

Problem 23AAE Is the derivative of Give reasons for your answers.

Problem 23PE Derivatives of Functions Find the derivatives of the functions

Problem 23QGY What is the derivative of the natural logarithm function ln x? How does the domain of the derivative compare with the domain of the function?

Problem 24AAE Suppose that a function ƒ satisfies the following conditions for all real values of x and y: Show that the derivative exists at every value of x and that

Problem 24PE Derivatives of Functions Find the derivatives of the functions

Problem 25AAE The generalized product rule Use mathematical induction to prove that if is a finite product of differentiable functions, then is differentiable on their common domain and

Problem 25PE Derivatives of Functions Find the derivatives of the functions

Problem 25QGY What is the derivative of Are there any restrictions on a?

Problem 26AAE Leibniz’s rule for higher-order derivatives of products Leibniz’s rule for higher-order derivatives of products of differentiable functions says that The equations in parts (a) and (b) are special cases of the equation in part (c). Derive the equation in part (c) by mathematical induction, using

Problem 26PE Derivatives of Functions Find the derivatives of the functions

Problem 27AAE The period of a clock pendulum The period T of a clock pendulum (time for one full swing and back) is given by the formula where T is measured in seconds, and L, the length of the pendulum, is measured in feet. Find approximately a. the length of a clock pendulum whose period is T = 1 sec b. the change dT in T if the pendulum in part (a) is lengthened 0.01 ft. c. the amount the clock gains or loses in a day as a result of the period’s changing by the amount dT found in part (b).

Problem 26QGY What is logarithmic differentiation? Give an example.

Problem 28AAE The melting ice cube Assume that an ice cube retains its cubical shape as it melts. If we call its edge length s, its volume is V = s3 and its surface area is 6s.2We assume that V and s are differentiable functions of time t. We assume also that the cube’s volume decreases at a rate that is proportional to its surface area. (This latter assumption seems reasonable enough when we think that the melting takes place at the surface: Changing the amount of surface changes the amount of ice exposed to melt.) In mathematical terms, The minus sign indicates that the volume is decreasing. We assume that the proportionality factor k is constant. (It probably depends on many things, such as the relative humidity of the surrounding air, the air temperature, and the incidence or absence of sunlight, to name only a few.) Assume a particular set of conditions in which the cube lost 1/ 4 of its volume during the first hour, and that the volume is V0 when t = 0. How long will it take the ice cube to melt?

Problem 28PE Derivatives of Functions Find the derivatives of the functions

Problem 27PE Derivatives of Functions Find the derivatives of the functions

Problem 27QGY How can you write any real power of x as a power of e? Are there any restrictions on x? How does this lead to the Power Rule for differentiating arbitrary real powers?

Problem 28QGY What is one way of expressing the special number e as a limit? What is an approximate numerical value of e correct to 7 decimal places?

Problem 29PE Derivatives of Functions Find the derivatives of the functions

Problem 29QGY What are the derivatives of the inverse trigonometric functions? How do the domains of the derivatives compare with the domains of the functions?

Problem 31QGY Outline a strategy for solving related rates problems. Illustrate with an example.

Problem 31PE Derivatives of Functions Find the derivatives of the functions

Problem 32PE Derivatives of Functions Find the derivatives of the functions

Problem 32QGY What is the linearization L(x) of a function ƒ(x) at a point x = a?What is required of ƒ at a for the linearization to exist? How are linearizations used? Give examples.

Problem 33QGY If x moves from a to a nearby value a + dx how do you estimate the corresponding change in the value of a differentiable function ƒ(x)? How do you estimate the relative change? The percentage change? Give an example.

Problem 49PE Derivatives of Functions Find the derivatives of the functions

Problem 50PE Derivatives of Functions Find the derivatives of the functions

Problem 51PE Derivatives of Functions Find the derivatives of the functions

Problem 53PE Derivatives of Functions Find the derivatives of the functions

Problem 54PE Derivatives of Functions Find the derivatives of the functions

Problem 52PE Derivatives of Functions Find the derivatives of the functions

Problem 58PE Derivatives of Functions Find the derivatives of the functions

Problem 59PE Derivatives of Functions Find the derivatives of the functions

Problem 61PE Derivatives of Functions Find the derivatives of the functions

Problem 62PE Derivatives of Functions Find the derivatives of the functions

Problem 60PE Derivatives of Functions Find the derivatives of the functions

Problem 63PE Derivatives of Functions Find the derivatives of the functions

Problem 64PE Derivatives of Functions Find the derivatives of the functions

Problem 65PE Implicit Differentiation Find dy/dx by implicit differentiation.

Problem 66PE Implicit Differentiation Find dy/dx by implicit differentiation.

Problem 67PE Implicit Differentiation Find dy/dx by implicit differentiation.

Problem 68PE Implicit Differentiation Find dy/dx by implicit differentiation.

Problem 70PE Implicit Differentiation Find dy/dx by implicit differentiation.

Problem 71PE Implicit Differentiation Find dy/dx by implicit differentiation.

Problem 69PE Implicit Differentiation Find dy/dx by implicit differentiation.

Problem 73PE Implicit Differentiation Find dy/dx by implicit differentiation.

Problem 72PE Implicit Differentiation Find dy/dx by implicit differentiation.

Problem 75PE Implicit Differentiation Find dy/dx by implicit differentiation.

Problem 74PE Implicit Differentiation Find dy/dx by implicit differentiation.

Problem 79PE Implicit Differentiation Find dp/dq.

Problem 80PE Implicit Differentiation Find dp/dq.

Problem 81PE Implicit Differentiation Find dr/ds.

Problem 100PE Slopes, Tangents, and Normals Tangents with specified slope Are there any points on the curve where the slope is 2? If so, find them.

Problem 98PE Applying the Derivative Definition For what value or values of the constant m, if any, is a. continuous at x = 0? b. differentiable at x = 0? Give reasons for your answers.

Problem 101PE Slopes, Tangents, and Normals Horizontal tangents Find the points on the curve y = where the tangent is parallel to the x-axis

Problem 103PE Slopes, Tangents, and Normals Tangents perpendicular or parallel to lines Find the points on the curve where the tangent is a. perpendicular to the line b. parallel to the line

Problem 102PE Slopes, Tangents, and Normals Tangent intercepts Find the x- and y-intercepts of the line that is tangent to the curve at the point ( -2, -8)

Problem 104PE Slopes, Tangents, and Normals Intersecting tangents Show that the tangents to the curve at and intersect at right angles.

Problem 105PE Slopes, Tangents, and Normals Normals parallel to a line Find the points on the curve where the normal is parallel to the line y = -x/2.Sketch the curve and normals together, labeling each with its equation.

Problem 106PE Slopes, Tangents, and Normals Tangent and normal lines Find equations for the tangent and normal to the curve y = 1 + cos x at the point Sketch the curve, tangent, and normal together, labeling each with its equation.

Problem 108PE Slopes, Tangents, and Normals Slope of tangent Show that the tangent to the curve

Problem 110PE Slopes, Tangents, and Normals Normal to a circle Show that the normal line at any point of the circle passes through the origin.

Problem 113PE Slopes, Tangents, and Normals Find equations for the lines that are tangent and normal to the curve at the given point.

Problem 111PE Slopes, Tangents, and Normals Find equations for the lines that are tangent and normal to the curve at the given point.

Problem 109PE Slopes, Tangents, and Normals Tangent curve For what value of c is the curve tangent to the line through the points (0, 3) and (5, -2) ?

Problem 114PE Slopes, Tangents, and Normals Find equations for the lines that are tangent and normal to the curve at the given point.

Problem 112PE Slopes, Tangents, and Normals Find equations for the lines that are tangent and normal to the curve at the given point.

Problem 116PE Slopes, Tangents, and Normals Find equations for the lines that are tangent and normal to the curve at the given point.

Problem 117PE Slopes, Tangents, and Normals Find the slope of the curve at the points (1, 1) and (1,–1)

Problem 119PE Analyzing Graphs Each of the figures in shows two graphs, the graph of a function y = ƒ(x) together with the graph of its derivative f '(x). Which graph is which? How do you know?

Problem 115PE Slopes, Tangents, and Normals Find equations for the lines that are tangent and normal to the curve at the given point.

Problem 118PE Slopes, Tangents, and Normals The graph shown suggests that the curve y = sin (x - sin x) might have horizontal tangents at the x-axis. Does it? Give reasons for your answer.

Problem 120PE Analyzing Graphs Each of the figures in shows two graphs, the graph of a function y = ƒ(x) together with the graph of its derivative f '(x). Which graph is which? How do you know?

Let \(f(x)=[\cos x], -\pi \leqslant x \leqslant \pi\) (a) Sketch the graph of \(f\). (b) Evaluate each limit, if it exists. \(\text { (i) } \lim _{x \rightarrow 0} f(x)\) \(\text { (ii) } \lim _{x \rightarrow(\pi / 2)^{-}} f(x\) \(\text { (iii) } \lim _{x \rightarrow(\pi / 2)^{+}} f(x\) \(\text { (iv) } \lim _{x \rightarrow \pi / 2}f(x)\) (c) For what values of \(a\) does \(\lim _{x \rightarrow a} f(x) \text { exist? }\) exist? Equation Transcription: , ? ? Text Transcription: f(x)=[cos x],-pi leqslant x leqslant pi f (i)lim over x rightarrow 0 f(x) (ii)lim over x rightarrow (pi/2)^- f(x) (iii)lim over x rightarrow (pi/2)^+ f(x) (iv)lim over x rightarrow pi/2 f(x) a Lim_x rightarrow a f(x)

Problem 122PE Analyzing Graphs Repeat Exercise 121, supposing that the graph starts at ( -1, 0) instead of ( -1, 2) . Reference: Exercise 121 Use the following information to graph the function y = ƒ(x) for –1? x ? 6 i) The graph of ƒ is made of line segments joined end to end. ii) The graph starts at the point ( -1, 2) . iii) The derivative of ƒ, where defined, agrees with the step function shown here.

Problem 125PE Trigonometric Limits Find the limits

Problem 122PE Analyzing Graphs Repeat Exercise 121, supposing that the graph starts at ( -1, 0) instead of ( -1, 2) . Reference: Exercise 121 Use the following information to graph the function? y ?= ƒ(?x) ?for –1? ?x ?? 6 i) ?The graph of ƒ is made of line segments joined end to end. ii) ?The graph starts at the point ( -1, 2) . iii) ?The derivative of ƒ, where defined, agrees with the step function shown here.

Problem 126PE Trigonometric Limits Find the limits

Problem 124PE Analyzing Graphs The graphs in part (a) show the numbers of rabbits and foxes in a small arctic population. They are plotted as functions of time for 200 days. The number of rabbits increases at first, as the rabbits reproduce. But the foxes prey on rabbits and, as the number of foxes increases, the rabbit population levels off and then drops. Part (b) shows the graph of the derivative of the rabbit population, made by plotting slopes. In what units should the slopes of the rabbit and fox population curves be measured?

Problem 130PE Trigonometric Limits Find the limits

Problem 131PE Trigonometric Limits Find the limits

Problem 132PE Trigonometric Limits Find the limits

Problem 142PE Related Rates 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 a. How is dS/dt related to dr/dt if h is constant? b. How is dS/dt related to dr/dt if r is constant? c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant?

Problem 148PE Related Rates Motion of a particle A particle moves along the curve in the first quadrant in such away that its distance from the origin increases at the rate of 11 units per second. Find dx/dt when x = 3.

Problem 151PE Related Rates Moving searchlight beam The figure shows a boat 1 km offshore, sweeping the shore with a searchlight. The light turns at a constant rate, d?/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?

Problem 153PE Find the linearizations of a. Graph the curves and linearizations together.

Problem 152PE Related Rates Points moving on coordinate axes Points A and B move along the x- and y-axes, 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 = 2r and B is moving toward O at therate of 0.3r m/sec?

Problem 155PE Find the linearization of

Problem 154PE We can obtain a useful linear approximation of the function by combining the approximations to get Show that this result is the standard linear approximation of 1/(1 + tan x) at x = 0.

Problem 156PE Find the linearization of at x = 0.

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 \(h_{0}+d h\) and the radius does not change. Equation Transcription: Text Transcription: h_{0} h_{0}+dh V=frac{1}{3} pi r^{2} h S=pi r sqrt{r^{2}+h^{2}}

Problem 159PE Differential Estimates of Change Compounding error The circumference of the equator of a sphere is measured as 10 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.

Problem 158PE 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 2%? 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.

Problem 20AAE Even differentiable functions Is there anything special about the derivative of an even differentiable function of x? Give reasons for your answer.