[SOLVED] If x2 + xy + y3 = 1, find the

> The graph of f is given. State, with reasons, the numbers at which f is not differentiable.

> Find f'(x). Compare the graphs of f and f' and use them to explain why your answer is reasonable. f (x) = 3x5 - 20x3 + 50x

> The unemployment rate U (t) varies with time. The table (from the Bureau of Labor Statistics) gives the percentage of unemployed in the US labor force from 1998 to 2007. (a). What is the meaning of U' (t)? What are its units? (b). Construct a table of

> (a). Use the definition of derivative to calculate f'. (b). Check to see that your answer is reasonable by comparing the graphs of f and f'. f (t) = t2 - √t

> (a). Sketch, by hand, the graph of the function f (x) = ex, paying particular attention to how the graph crosses the y-axis. What fact allows you to do this? (b). What types of functions are f (x) = ex and g (x) = xe? Compare the differentiation formulas

> (a). Use the definition of derivative to calculate f'. (b). Check to see that your answer is reasonable by comparing the graphs of f and f'. f (x) = x + 1/x

> Differentiate the function. f (x) = 186.5

> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. f (x) = x4

> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. f (x) = x +√ x

> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. f (x) = x2 - 2x3

> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. f (x) = 1.5x2 - x + 3.7

> Use a linear approximation (or differentials) to estimate the given number. e-0.015

> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. f (t) = 5t - 9t2

> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. f (x) = mx + b

> Use the given graph to estimate the value of each derivative. Then sketch the graph of f'. (a). f'(0) (b). f'(1) (c). f'(2) (d). (3) (e). f'(4) (f). f'(5) (g). f'(6) (h). f'(7)

> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

> Let f (x) = x3. (a). Estimate the values of f'(0), f'(1/2), f'(1), f'(2), and f'(3) by using a graphing device to zoom in on the graph of f. (b). Use symmetry to deduce the values of f'(-1/2), f'(-1), f'(-2), and f'(-3). (c). Use the values from parts (a

> Let f (x) = x2. (a). Estimate the values of f'(0), f'(1/2), f'(2), and by using a graphing device to zoom in on the graph of f. (b). Use symmetry to deduce the values of f'(-1/2), f'(-1), and f'(-2). (c). Use the results from parts (a) and (b) to guess a

> Make a careful sketch of the graph of f and below it sketch the graph of f' in the same manner as in Exercises 4–11. Can you guess a formula for f' (x) from its graph? f (x) = ln x

> Make a careful sketch of the graph of f and below it sketch the graph of f' in the same manner as in Exercises 4–11. Can you guess a formula for f' (x) from its graph? f (x) = ex

> If y = f (u) and u = g (x), where f and g are twice differentiable functions, show that

> (a). Write |x| = √x2 and use the Chain Rule to show that d/dx |x| = x/|x| (b). If f (x) = |sin x|, find f'(x) and sketch the graphs of f and f'. Where is f not differentiable? (c). If g (x) = sin |x|, find g'(x) and sketch the graphs of g and g'. Where i

> The graph of the derivative f' of a continuous function f is shown. (a). On what intervals f is increasing? Decreasing? (b). At what values of x does f have a local maximum? Local minimum? (c). On what intervals is f concave upward? Concave downward? (

> Use the Chain Rule to show that if is measured in degrees, then (This gives one reason for the convention that radian measure is always used when dealing with trigonometric functions in calculus: The differentiation formulas would not be as simple if we

> Differentiate the function. A (x) = 12/x5

> (a). If is a positive integer, prove that d/dx (sinnx cos mx) = n sinn-1x cos (n + 1) x (b). Find a formula for the derivative of y = cosnx cos nx that is similar to the one in part (a).

> (a). Use a CAS to differentiate the function and to simplify the result. (b). Where does the graph of f have horizontal tangents? (c). Graph f and f' on the same screen. Are the graphs consistent with your answer to part (b)?

> Show that the curve y = 6x3 + 5x - 3 has no tangent line with slope 4.

> For what values of does the graph of f (x) = x3 + 3x2 + x + 3 have a horizontal tangent?

> A curve C is defined by the parametric equations x = t2, y = t3 – 3t. (a). Show that C has two tangents at the point (3, 0) and find their equations. (b). Find the points on C where the tangent is horizontal or vertical. (c). Illustrate parts (a) and (b)

> The equation of motion of a particle is s = t3 – 3t, where s is in meters and is in seconds. Find (a). the velocity and acceleration as functions of t, (b). the acceleration after 2 s, and (c). the acceleration when the velocity is 0.

> Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work.

> Coffee is being poured into the mug shown in the figure at a constant rate (measured in volume per unit time). Sketch a rough graph of the depth of the coffee in the mug as a function of time. Account for the shape of the graph in terms of concavity. Wha

> Find an equation of the tangent line to the curve at the point corresponding to the given value of the parameter.

> Find equations of the tangent line and normal line to the curve at the given point. y = (1 + 2x)2 (1, 9)

> The flash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the flash is set off. The following data describe the charge Q remaining on the capacitor (measured in microcoulombs, ÂµC) at time (measured i

> Air is being pumped into a spherical weather balloon. At any time t, the volume of the balloon is V (t) and its radius is r (t). (a). What do the derivatives dV/dr and dV/dt represent? (b). Express dV/dt in terms of dr/dt.

> A particle moves along a straight line with displacement s (t) velocity v (t), and acceleration a (t). Show that a (t) = v (t) = dv/ds Explain the difference between the meanings of the derivatives dv/dt and dv/ds.

> Under certain circumstances a rumor spreads according to the Equation P (t) = 1/1 + ae-kt where P (t) is the proportion of the population that knows the rumor at time t and a and k are positive constants. [In Section 7.5 we will see that this is a reason

> The motion of a spring that is subject to a frictional force or a damping force (such as a shock absorber in a car) is often modeled by the product of an exponential function and a sine or cosine function. Suppose the equation of motion of a point on suc

> In Example 4 in Section 1.3 we arrived at a model for the length of daylight (in hours) in Philadelphia on the tth day of the year: Use this model to compare how the number of hours of daylight is increasing in Philadelphia on March 21 and May 21.

> (a). Find the average rate of change of the area of a circle with respect to its radius r as r changes from (i). 2 to 3 (ii). 2 to 2.5 (iii). 2 to 2.1 (b). Find the instantaneous rate of change when r = 2. (c). Show that the rate of change of the area

> In the study of ecosystems, predator-prey models are often used to study the interaction between species. Consider populations of tundra wolves, given by W (t), and caribou, given by C (t), in northern Canada. The interaction has been modeled by the equa

> In a fish farm, a population of fish is introduced into a pond and harvested regularly. A model for the rate of change of the fish population is given by the equation where r0 is the birth rate of the fish, Pc is the maximum population that the pond can

> Refer to the law of laminar flow given in Example 7. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference 3000 dynes/cm2, and viscosity η = 0.027. (a). Find the velocity of the blood along the centerline r = 0, at radius r = 0.00

> Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f' below it.

> Match the graph of each function in (a)â(d) with the graph of its derivative in IâIV. Give reasons for your choices.

> (a). The volume of a growing spherical cell is V = 4/3πr3, where the radius is measured in micrometers (1 µm = 10-6 m). Find the average rate of change of with respect to when changes from (i). 5 to 8 µm (ii). 5 to 6 µm (iii). 5 to 5.1 µm (b). Find the i

> A particle moves according to a law of motion s = f (t), t > 0, where is measured in seconds and in feet. (a). Find the velocity at time t. (b). What is the velocity after 3 s? (c). When is the particle at rest? (d). When is the particle moving in the po

> If xy + ey = e, find the value of y" at the point where x = 0.

> (a). Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate dV/dx when x = 3 mm and explain its meaning. (b)

> Let f (x) = 2x – tan x, -π/2 < x < π/2. On what interval is f concave downward?

> The graph of a function is shown in the figure. Make a rough sketch of an antiderivative F, given that F (0) = 1.

> The graph of a function f is shown. Which graph is an antiderivative of f and why?

> Shown is the graph of the population function P (t) for yeast cells in a laboratory culture. Use the method of Example 1 to graph the derivative P' (t). What does the graph of P' tell us about the yeast population?

> Find the derivative of the function. y = r/r2 + 1

> Suppose f'(x) = xe-x2. (a). On what interval is f increasing? On what interval is f decreasing? (b). Does f have a maximum value? Minimum value?

> Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f' below it.

> (a). A company makes computer chips from square wafers of silicon. It wants to keep the side length of a wafer very close to 15 mm and it wants to know how the area A (x) of a wafer changes when the side length x changes. Find A'(15) and explain its mean

> The graph of the derivative f' of a continuous function f is shown. (a). On what intervals f is increasing? Decreasing? (b). At what values of x does f have a local maximum? Local minimum? (c). On what intervals is f concave upward? Concave downward? (

> Let K (t) be a measure of the knowledge you gain by studying for a test for t hours. Which do you think is larger, K (8) – K (7) or k (3) – K (2)? Is the graph of K concave upward or concave downward? Why?

> A particle is moving along a horizontal straight line. The graph of its position function (the distance to the right of a fixed point as a function of time) is shown. (a). When is the particle moving toward the right and when is it moving toward the le

> The table gives population densities for ring-necked pheasants (in number of pheasants per acre) on Pelee Island, Ontario. (a). Describe how the rate of change of population varies. (b). Estimate the inflection points of the graph. What is the signific

> A graph of a population of yeast cells in a new laboratory culture as a function of time is shown. (a). Describe how the rate of population increase varies. (b). When is this rate highest? (c). On what intervals is the population function concave upwar

> The graphs of a function f and its derivative f' are shown. Which is bigger, f' (-1) or f"(1)?

> Use the given graph of f to estimate the intervals on which the derivative f' is increasing or decreasing.

> The graph of the derivative f' of a function f is shown. (a). On what intervals is f increasing? Decreasing? (b). At what values of x does f have a local maximum? Local minimum? (c). If it is known that f (0) = 0, sketch a possible graph of f.

> (a). Find y' by implicit differentiation. (b). Solve the equation explicitly for and differentiate to get y' in terms of x. (c). Check that your solutions to parts (a) and (b) are consistent by substituting the expression for into your solution for part

> If a ball is thrown vertically upward with a velocity of 80 ft/s, then its height after seconds is s = 80t – 16t2. (a). What is the maximum height reached by the ball? (b). What is the velocity of the ball when it is 96 ft above the ground on its way up?

> The turkey in Example 1 is removed from the oven when its temperature reaches 1850F and is placed on a table in a room where the temperature is 750F. After 10 minutes the temperature of the turkey is and after 20 minutes it is 1600F. Use a linear approxi

> Evaluate the limit and justify each step by indicating the appropriate Limit Law(s).   lim (r + 1)'(t + 3)5 1--1

> Let F (x) = x2 – 1/|x – 1|. (a) Find (i). limx→1+ F (x) (ii). limx→1- F (x) (b). Does limx→1 F (x) exist? (c). Sketch the graph of F.

> Find the limit, if it exists. If the limit does not exist, explain why. 2x + 12 lim I-6 |x + 6||

> Prove that limx→0+ √x esinπ/x = 0.

> If 2x < g (x) < x4 – x2 + 2 for all x, evaluate limx→1 g (x).

> If 4x – 9 < f (x) < x2 – 4x + 7 for x > 0, find limx→4 f (x).

> Use the Squeeze Theorem to show that limx→0 √x3 + x2 sin π/x = 0 Illustrate by graphing the functions f, g and h (in the notation of the Squeeze Theorem) on the same screen.

> (a). Estimate the value of limx→0 x/ 1 + 3x – 1 by graphing the function f (x) = x/ (√1 + 3x – 1). (b). Make a table of values of f (x) for x close to 0 and guess the value of the limit. (c). Use the Limit Laws to prove that your guess is correct.

> Evaluate the limit, if it exists. lim x → -4 √x2 + 9 – 5/ x + 4

> Evaluate the limit, if it exists. lim t → 0 (1/t√1 + t – 1/t)

> The point P (1, 0) lies on the curve y = sin (10π/x). (a). If Q is the point (x, sin (10π/x)), find the slope of the secant line PQ (correct to four decimal places) for x = 2, 1.5, 1.4, 1.3, 1.2, 1.1, 0.5, 0.6, 0.7, 0.8, and 0.9. Do the slopes appear to

> Evaluate the limit, if it exists. lim t → 0 (1/t – 1/t2 + t)

> Evaluate the limit, if it exists. lim x → 16 4 - √x / 16x – x2

> Evaluate the limit, if it exists. lim x → -1 x2 + 2x + 1/ x4 - 1

> Explain what it means to say that limx→1- = f (x) = 3 and limx→1+ = f (x) = 7 In this situation is it possible that limx→1 = f (x) = 5 exists? Explain.

> Evaluate the limit, if it exists. lim h → 0 √1 + h - 1/h

> Evaluate the limit, if it exists. lim x → 2 x + 2 / x3 + 8

> Evaluate the limit, if it exists. lim h → 0 (2 + h)3 – 8 / h

Question: If x2 + xy + y3 = 1, find the