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mathematics
calculus early transcendentals
Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions
Evaluate the following limits. Check your results by graphing. 3 1 lim X- 00
Evaluate the following limits or explain why they do not exist. Check your results by graphing. cos 0 lim_ (tan 0)°os cos e 0>T/2
What conditions must be met to ensure that a function has an absolute maximum value and an absolute minimum value on an interval?
a. Write an equation of the line that represents the linear approximation to the following functions at a.b. Graph the function and the linear approximation at a.c. Use the linear approximation to estimate the given quantity.f(x) = cos x; a = π/4; cos 0.8
Use analytical methods to find all local extrema of the function f(x) = xx, for x > 0. Verify your work using a graphing utility.
Determine the following indefinite integrals. Check your work by differentiation.∫(22x10 - 24 e12x) dx
Sketch a graph of a function f that is continuous on (-∞, ∞) and has the following properties.f'(x) < 0 and f"(x) > 0 on (-∞, 0); f'(x) < 0 and f"(x) < 0 on (0, ∞)
A rancher is building a horse pen on the corner of her property using 1000 ft of fencing. Because of the unusual shape of her property, the pen must be built in the shape of a trapezoid (see figure).a. Determine the lengths of the sides that maximize the area of the pen.b. Suppose there is already
a. Find the critical points of f on the given interval.b. Determine the absolute extreme values of f on the given interval.c. Use a graphing utility to confirm your conclusions.f(x) = (x - 2)1/2 on [2, 6]
Evaluate the following limits. Check your results by graphing. lim x/x х-
Evaluate the following limits or explain why they do not exist. Check your results by graphing. lim (1 + 4x)³/* 3/x х—0
a. Write an equation of the line that represents the linear approximation to the following functions at a.b. Graph the function and the linear approximation at a.c. Use the linear approximation to estimate the given quantity.f(x) = 1/(x + 1); a = 0; 1/1.1
Use analytical methods to find all local extrema of the function f(x) = x1/x, for x > 0. Verify your work using a graphing utility.
Determine the following indefinite integrals. Check your work by differentiation. dt
Sketch a graph of a function f that is continuous on (-∞, ∞) and has the following properties.f'(x) < 0 and f"(x) < 0 on (-∞, 0); f'(x) < 0 and f"(x) > 0 on (0, ∞)
Determine whether the following statements are true and give an explanation or counterexample.a. The function f (x) = √x has a local maximum on the interval [0, ∞].b. If a function has an absolute maximum on a closed interval, then the function must be continuous on that interval.c. A function
A right circular cone is inscribed inside a larger right circular cone with a volume of 150 cm3. The axes of the cones coincide and the vertex of the inner cone touches the center of the base of the outer cone. Find the ratio of the heights of the cones that maximizes the volume of the inner cone.
Evaluate the following limits. Check your results by graphing. lim | In x|* x→0*
Evaluate the following limits or explain why they do not exist. Check your results by graphing. .2x lim x x→0+
a. Write an equation of the line that represents the linear approximation to the following functions at a.b. Graph the function and the linear approximation at a.c. Use the linear approximation to estimate the given quantity.f(x) = tan x; a = 0; tan 3°
Consider the function f(x) = cos (ln x), for x > 0. Use analytical techniques and a graphing utility.a. Locate all local extrema on the interval (0, 4).b. Identify the inflection points on the interval (0, 4).c. Locate the three smallest zeros of f on the interval (0.1, ∞).d. Sketch the graph
Determine the following indefinite integrals. Check your work by differentiation. (49 – x²)-1/2 dx
Sketch a graph of a function f that is continuous on (-∞, ∞) and has the following properties.f'(x) < 0 and f"(x) > 0 on (-∞, 0); f'(x) > 0 and f"(x) > 0 on (0, ∞)
All rectangles with an area of 64 have a perimeter given by P(x) = 2x + 128/x, where x is the length of one side of the rectangle. Find the absolute minimum value of the perimeter function on the interval (0, ∞). What are the dimensions of the rectangle with minimum perimeter?
Suppose you own a tour bus and you book groups of 20 to 70 people for a day tour. The cost per person is $30 minus $0.25 for every ticket sold. If gas and other miscellaneous costs are $200, how many tickets should you sell to maximize your profit? Treat the number of tickets as a non negative real
Evaluate the following limits. Check your results by graphing.
Evaluate the following limits. lim (x – Vx? + 4x)
Estimate f(3.85) given that f(4) = 3 and f'(4) = 2.
Determine the following indefinite integrals. Check your work by differentiation. 1 xVx? – 25
Sketch a graph of a function f that is continuous on (-∞, ∞) and has the following properties.f'(x) > 0, f"(x) > 0
Suppose a tour guide has a bus that holds a maximum of 100 people. Assume his profit (in dollars) for taking n people on a city tour is P(n) = n(50 - 0.5n) - 100. (Although P is defined only for positive integers, treat it as a continuous function.)a. How many people should the guide take on a tour
A right circular cylinder is placed inside a cone of radius R and height H so that the base of the cylinder lies on the base of the cone. a. Find the dimensions of the cylinder with maximum volume. Specifically, show that the volume of the maximum-volume cylinder is 4/9 the volume of the
The following third- and fourth degree polynomials have a property that makes them relatively easy to graph. Make a complete graph and describe the property.f(x) = x3 - 3x2 - 144x - 140
Evaluate the following limits. Check your results by graphing. lim (sin x)tan x х- T/2
Evaluate the following limits. lim (tan 0 – sec 0) 0→T/2¯
A sales analyst determines that the revenue from sales of fruit smoothies is given by R(x) = -60x2 + 300x, where x is the price in dollars charged per item, for 0 ≤ x ≤ 5.a. Find the critical points of the revenue function.b. Determine the absolute maximum value of the revenue function and the
Find the dimensions and area of the rectangle of maximum area that can be inscribed in the following figures.a. A right triangle with a given hypotenuse length L b. An equilateral triangle with a given side length Lc. A right triangle with a given area A d. An arbitrary triangle with a
The following third- and fourth degree polynomials have a property that makes them relatively easy to graph. Make a complete graph and describe the property.f(x) = x3 - 147x + 286
Evaluate the following limits. Check your results by graphing. lim (1 + x)cot.x r→0+ x-
Evaluate the following limits. lim (x – Vx² + 1)
Estimate f(5.1) given that f(5) = 10 and f'(5) = -2.
Determine the following indefinite integrals. Check your work by differentiation. dz 16z? + 25
Verify that the following functions satisfy the conditions of Theorem 4.5 on their domains. Then find the location and value of the absolute extremum guaranteed by the theorem.f(x) = x√3 - x
A stone is launched vertically upward from a cliff 192 feet above the ground at a speed of 64 ft/s. Its height above the ground t seconds after the launch is given by s = -16t2 + 64t + 192, for 0 ≤ t ≤ 6. When does the stone reach its maximum height?
Sketch the graph of a function continuous on the given interval that satisfies the following conditions.f is continuous on the interval [-4, 4]; f'(x) = 0 for x = -2, 0, and 3; f has an absolute minimum at x = 3; f has a local minimum at x = -2; f has a local maximum at x = 0; f has an absolute
Can the graph of a polynomial have vertical or horizontal asymptotes? Explain.
Explain why Rolle’s Theorem cannot be applied to the function f(x) = |x| on the interval [-a, a], for any a > 0.
Suppose the objective function is Q = x2y and you know that x + y = 10. Write the objective function first in terms of x and then in terms of y.
Why are special methods, such as l’Hôpital’s Rule, needed to evaluate indeterminate forms (as opposed to substitution)?
Suppose you find the linear approximation to a differentiable function at a local maximum of that function. Describe the graph of the linear approximation.
Describe the set of antiderivatives of f(x) = 0.
Explain how to apply the First Derivative Test.
Explain how the iteration formula for Newton’s method works.
What are local maximum and minimum values of a function?
Consider the graph of a function f on the interval [-3, 3].a. Give the approximate coordinates of the local maxima and minima of f.b. Give the approximate coordinates of the absolute maximum and minimum of f (if they exist).c. Give the approximate coordinates of the inflection point(s) of f.d. Give
Explain why it is useful to know about symmetry in a function.
Draw the graph of a function for which the conclusion of Rolle’s Theorem does not hold.
If the objective function involves more than one independent variable, how are the extra variables eliminated?
Explain with examples what is meant by the indeterminate form 0/0.
Determine whether the following statements are true and give an explanation or counterexample.a. The linear approximation to f(x) = x2 at x = 0 is L(x) = 0.b. Linear approximation at x = 0 provides a good approximation to f(x) = |x|.c. If f(x) = mx + b, then the linear approximation to f at any
Determine the following indefinite integrals. Check your work by differentiation. dx .2 xVx? - 100
Verify that the following functions satisfy the conditions of Theorem 4.5 on their domains. Then find the location and value of the absolute extremum guaranteed by the theorem.A(r) = 24/r + 2πpr2, r > 0
Sketch the graph of a smooth function f and label a point P(a, (f(a)) on the curve. Draw the line that represents the linear approximation to f at P.
Fill in the blanks with either of the words the derivative or an antiderivative: If F'(x) = f(x), then f is ________ of F and F is ________ of f.
Explain how the first derivative of a function determines where the function is increasing and decreasing.
Give a geometric explanation of Newton’s method.
What does it mean for a function to have an absolute extreme value at a point c of an interval [a, b]?
Determine whether the following statements are true and give an explanation or counterexample.a. If f'(c) = 0, then f has a local maximum or minimum at c.b. If f'(c) = 0, then f has an inflection point at c.c. F(x) = x2 + 10 and G(x) = x2 - 100 are antiderivatives of the same function.d. Between
Why is it important to determine the domain of f before graphing f?
Explain Rolle’s Theorem with a sketch.
a. Determine an equation of the tangent line and normal line at the given point (x0, y0) on the following curves.b. Graph the tangent and normal lines on the given graph.(x2 + y2)2 =25/3 (x2 - y2);(x0, y0) = (2, -1)(lemniscate of Bernoulli) (2, – 1)
a. Determine an equation of the tangent line and normal line at the given point (x0, y0) on the following curves.b. Graph the tangent and normal lines on the given graph.(x2 + y2 - 2x)2 = 2(x2 + y)2;(x0, y0) = (2, 2)(limaçon of Pascal) УА (2, 2) 2 х
a. Determine an equation of the tangent line and normal line at the given point (x0, y0) on the following curves.b. Graph the tangent and normal lines on the given graph.x4 = 2x2 + 2y2;(x0, y0) = (2, 2)(kampyle of Eudoxus) yA (2, 2) х
a. Determine an equation of the tangent line and normal line at the given point (x0, y0) on the following curves.b. Graph the tangent and normal lines on the given graph.3x3 + 7y3 = 10y;(x0, y0) = (1, 1) УА (1, 1) + х 1
Fill in the blanks: The goal of an optimization problem is to find the maximum or minimum value of the _______ function subject to the _______.
The figure in Exercise 69 shows the particle traveling away from the sensor, which may have influenced your solution (we expect you used the inverse sine function). Suppose instead that the particle approaches the sensor (see figure). How would this change the solution? Explain the differences in
Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure they actually
Find dy/dx, where√3x7 + y2 = sin2 y + 100xy.
Suppose f is differentiable for all real numbers with f(0) = -3, f(1) = 3, f'(0) = 3, and f'(1) = 5. Let g(x) = sin (πf(x)). Evaluate the following expressions.a. g'(0)b. g'(1)
Use the properties of logarithms to simplify the following functions before computing f'(x). 2x f(x) = In (x² + 1)*
Calculate the derivative of the following functions(a) Using the fact that bx = ex ln b.(b) By using logarithmic differentiation. Verify that both answers are the same.y = g(x)h(x)
Calculate the derivative of the following functions(a) Using the fact that bx = ex ln b.(b) By using logarithmic differentiation. Verify that both answers are the same.y = 3x
Determine equations of the lines tangent to the graph of y = x√5 - x2 at the points (1, 2) and (-2, -2). Graph the function and the tangent lines.
Find d2y/dx2 for the following functions.y = x cos x2
Normal lines A normal line at a point P on a curve passes through P and is perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point. Illustrate your work by graphing the curve with the
Determine whether the following statements are true and give an explanation or counterexample.a. The derivative of log2 9 = 1/(9 ln 2).b. ln (x + 1) + ln (x - 1) = ln (x2 - 1), for all x.c. The exponential function 2x + 1 can be written in base e as e2 ln (x + 1).d. d/dx (√2x) = x√2x - 1.e.
Use the General Power Rule where appropriate to find the derivative of the following functions.r = e2θ
Calculate y'(x) for the following relations. ey еь 1 + sin x
Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find f-1.f(x) = 3x + 4; (16, 4)
Use the General Power Rule where appropriate to find the derivative of the following functions.g(y) = ey • ye
Find dy / dxy = x5/4
Second derivatives Find d2y / dx2.sin x + x2y = 10
A hot-air balloon is 150 ft above the ground when a motorcycle (traveling in a straight line on a horizontal road) passes directly beneath it going 40 mi/hr (58.67 ft/s). If the balloon rises vertically at a rate of 10 ft/s, what is the rate of change of the distance between the motorcycle and the
An observer stands 300 ft from the launch site of a hot-air balloon. The balloon is launched vertically and maintains a constant upward velocity of 20 ft/s. What is the rate of change of the angle of elevation of the balloon when it is 400 ft from the ground? The angle of elevation is the angle θ
Evaluate and simplify the following derivatives.f'(1) when f(x) = tan-1 (4x2)
Find an equation of the line tangent to the graph of f at the given point.f(x) = sec-1 (ex); (ln 2, π/3)
If two opposite sides of a rectangle increase in length, how must the other two opposite sides change if the area of the rectangle is to remain constant?
Evaluate and simplify the following derivatives. d (2 х3 + тх? + 7х + 1 dx \3
Evaluate and simplify the following derivatives.d/dθ (4 tan (θ2 + 3θ + 2))
Use the General Power Rule where appropriate to find the derivative of the following functions.f(x) = 2x
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