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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Show that the subtangent to y = x3 at x = a is equal to 1/3a.
Formulate and prove a generalization of Exercise 88 for y = xn.Data From Exercise 88Show that the subtangent to y = x3 at x = a is equal to 1/3a.
Two small arches have the shape of parabolas. The first is given by ƒ x) = 1 − x2 for −1 ≤ x ≤ 1 and the second by g(x) = 4 − (x − 4)2 for 2 ≤ x ≤ 6. A board is placed on top of
A vase is formed by rotating y = x2 around the y-axis. If we drop in a marble, it will either touch the bottom point of the vase or be suspended above the bottom by touching the sides (Figure 21).
Let n be a whole number. Calculate the derivative of ƒ(x) = x−n by showing that f(x+h)-f(x) h -1 x(x + h)" (x + h)" - xn h
Verify the Power Rule for the exponent 1, where n is a positive integer, using the following trick: Rewrite the difference quotient for y = x1 at x = b in terms of u = (b +h)¹/n and a bl/n =
If and exist but are not equal, then ƒ is not differentiable at c, and the graph of ƒ has a corner at c. Prove that f is continuous at c. lim h→0- f(c+h)-f(c) h
For which values of c does the equation x2 + 4 = cx have a unique solution?
In Exercises 7–14, use the Power Rule to compute the derivative. d dt 8=11
Is 20 equal to 10 or 20? lim x-10
With t in hours, at t = 0 Dale entered Highway 1. At t = 2 he was 126 miles down the highway, on the side of the road with a flat tire. At t = 3 he was still on the side of the road, waiting for road
What is the graphical interpretation of the IVT?
The method for evaluating limits discussed in this section is sometimes called simplify and plug in. Explain how it actually relies on the property of continuity.
What is the graphical interpretation of instantaneous velocity at a specific time t = t0?
Show that the following statement is false by drawing a graph that provides a counterexample:If ƒ is continuous and has a root in [a, b], then ƒ (a) and ƒ (b) have opposite signs.
Determine as in Figure 10. lim_ f(x) for f x-0.5
The position of a particle at time t is s(t) = t3 + t. Compute the average velocity over the time interval [1, 4] and estimate the instantaneous velocity at t = 1
Adapt the argument in Example 3 to prove rigorously that lim x-1 = for all c = 0.
The following statement is incorrect: “If ƒ has a horizontal asymptote y = L at ∞, then the graph of ƒ approaches the line y = L as x gets greater and greater, but never touches it.” In
Can Corollary 2 be applied to ƒ(x) = x−1 on [−1, 1]? Does ƒ have any roots?
On Wednesday at noon the weather was fair in Boston with a barometric pressure of 1018 mb. At the same time, a low-pressure storm system was passing by Buffalo, where the pressure was 996 mb. At noon
Let P(n) be the perimeter of an n-gon inscribed in a unit circle (Figure 7).(a) Explain, intuitively, why P(n) approaches 2π as n→ ∞.(b) Show that P(n) = 2n sin (π).(c) Combine (a) and (b) to
[x] refers to the least integer function. It is defined by x = n, where n is the unique integer such that n − 1 < x ≤ n. In each case, provide the graph of ƒ , indicate the points of
[x] refers to the least integer function. It is defined by x = n, where n is the unique integer such that n − 1 < x ≤ n. In each case, provide the graph of ƒ , indicate the points of
Using a diagram of the unit circle and the Pythagorean Theorem, show thatConclude that sin2 θ ≤ 2(1 − cos θ) ≤ θ2 and use this to give an alternative proof that the limit in Exercise 51
Which of the lines in Figure 12 are tangent to the curve? A B C D
Are the following statements true or false? If false, state the correct version.(a) ƒg denotes the function whose value at x is ƒ(g(x)).(b) ƒ/g denotes the function whose value at x is
Identify the outside and inside functions for each of these composite functions.(a) y = √4x + 9x2 (b) y = tan(x2 + 1)(c) y = sec5 x (d) y = (1 + x12)4
In Exercises 1–6, use the Product Rule to calculate the derivative.ƒ(x) = x3(2x2 + 1)
Which units might be used for each rate of change?(a) Pressure (in atmospheres) in a water tank with respect to depth(b) The rate of a chemical reaction (change in concentration with respect to time
For each headline, rephrase as a statement about first and second derivatives and sketch a possible graph.• “Stocks Go Higher, Though the Pace of Their Gains Slows”• “Recent Rains Slow
Fill in a table of the following type:ƒ(u) = u3/2, g(x) = x4 + 1 f(g(x)) f'(u) f(g(x)) g'(x) (fog)'(x)
Which differentiation rule is used to show d dx sin y = cos y dy -? dx
Let ƒ(x) = 5x2. Show that ƒ(3 + h) = 5h2 + 30h + 45. Then show thatand compute ƒ(3) by taking the limit as h → 0. f(3+h)-f(3) h 5h + 30
Determine the sign (+ or −) that yields the correct formula for the following: (a) (b) (c) d -(sin x + cos x) = ± sin x ± cos x dx d dx d dx sec x = sec x tan x cotx = ±csc² x
Show that if you differentiate both sides of x2 + 2y3 = 6, the result is 2x + 6y2 dy/dx = 0. Then solve for dy/dx and evaluate it at the point (2, 1).
In Exercises 1–8, find the rate of change.Area of a square with respect to its side s when s = 5
Find an equation of the tangent line at the point indicated.y = sin x, x = π/4
Two trains travel from New Orleans to Memphis in 4 h. The first train travels at a constant velocity of 90 mph, but the velocity of the second train varies. What was the second train’s average
Fill in a table of the following type:ƒ(u) = u3, g(x) = 3x + 5 f(g(x)) f'(u) f(g(x)) g'(x) (fog)'(x)
Sketch a graph of position as a function of time for an object that is slowing down and has positive acceleration.
One of (a)–(c) is incorrect. Find and correct the mistake. (a) (b) (c) d dy d dx d dx -sin(y²) = 2y cos(y²) - sin(x²) = 2x cos(x²) sin(y²) = 2y cos(y²)
Determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits are ∞ or −∞, and describe the corresponding vertical asymptote. lim |x|¹/x X X-
Determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits are ∞ or −∞, and describe the corresponding vertical asymptote. lim X-0* x - sin |x| x3
Determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits are ∞ or −∞, and describe the corresponding vertical asymptote. x³ + x-2 lim x+1+ x²
Verify each limit using the limit definition. For example, in Exercise 13, show that |3x − 12| can be made as small as desired by taking x close to 4. lim 3x = 12 x-4
Verify each limit using the limit definition. For example, in Exercise 13, show that |3x − 12| can be made as small as desired by taking x close to 4. lim 3 = 3 x-5
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. lim h→0 3-1 h
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. lim sinh cos h→0 1 h
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. tan x - x lim x-0 sin x - x
Determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits are ∞ or −∞, and describe the corresponding vertical asymptote. sin. X lim x→0+ |x|
Determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits are ∞ or −∞, and describe the corresponding vertical asymptote. lim X-4 x + 1 X-4
Determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits are ∞ or −∞, and describe the corresponding vertical asymptote. lim x2= 4x² + 7 x³ +
Determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits are ∞ or −∞, and describe the corresponding vertical asymptote. x² lim x+3+ x² -
Determine the infinite one-and two-sided limits in Figure 14. -1 y 3 5 x
Give an example where exists but neither lim(f(x) + g(x)) X-0
Give an example where exists but neither lim(f(x) · g(x)) X-0
Assume that L(a) = lim at - 1 x-0 X exists for all a > 0. Assume also that lim at = 1. x-0
Give an example where exists but neither lim f(x) X-0 g(x)
Refer to the function g whose graph appears in Figure 16.At which point c does g have a removable discontinuity? How should g(c) be redefined to make g continuous at x = c?
Sketch the graph of ƒ. At each point of discontinuity, state whether ƒ is left-or right continuous. f(x): = x² 2-x for x ≤ 1 for x>1
Sketch the graph of ƒ. At each point of discontinuity, state whether ƒ is left- or right continuous. x + 1 for x < 1 f(x)=1 for x ≥ 1
Sketch the graph of ƒ. At each point of discontinuity, state whether ƒ is left- or right continuous. f(x) = x²-3x+2 |x - 21 0 x 2 x = 2
Sketch the graph of ƒ. At each point of discontinuity, state whether ƒ is left- or right continuous. ³+1 f(x) = -x + 1 -x² + 10x for -∞< x≤0 for 0 < x < 2 15 for x ≥ 2
H is the Heaviside function, defined byIn each case, sketch the graph of ƒ, indicate whether or not ƒ is continuous, and—if ƒ is not continuous—identify the points of discontinuity. H(x) 6 0
Define ƒ(x) = x sin 1/x + 2 for x ≠ 0. Plot ƒ. How should ƒ(0) be defined so that ƒ is continuous at x = 0?
H is the Heaviside function, defined byAssume that ƒ is defined and continuous for all x. Under what condition on ƒ are we assured that the function g, defined by g(x) = H(x − a) ƒ(x), is
Find the value of the constant (a, b, or c) that makes the function continuous. f(x) = [x² - c for x < 5 4x + 2c for x ≥ 5
Find the value of the constant (a, b, or c) that makes the function continuous. f(x) = 2x+9x¹ -4x + c for x ≤ 3 for x > 3
Find the value of the constant (a, b, or c) that makes the function continuous. (x-1 for x < -1 f(x) = ax + b for −1≤x≤ // for x> //
DefineFind a value of c such that g is(a) Left-continuous (b) Right-continuousIn each case, sketch the graph of g. x + 3 for x < -1 g(x) = {cx x+2 for 1 ≤ x ≤2 for x> 2
Define Answer the following questions, using a plot if necessary.(a) Can g(1) be defined so that g is continuous at t = 1? If yes, how?(b) Can g(−1) be defined so that g is continuous at t =
Each of the following statements is false. For each statement, sketch the graph of a function that provides a counterexample. (a) If lim f(x) exists, then f is continuous at x = a. x→a (b) If f has
Draw the graph of a function on [0, 5] with the given properties.ƒ is not continuous at x = 1, but lim f(x) and lim f(x) exist and are equal. X→ 1+ x-1-
Draw the graph of a function on [0, 5] with the given properties.ƒ has a removable discontinuity at x = 1, a jump discontinuity at x = 2, and lim f(x) = -00, x-3- lim f(x) = 2 X→3+
Draw the graph of a function on [0, 5] with the given properties.ƒ is left-continuous but not continuous at x = 2, and right-continuous but not continuous at x = 3.
Evaluate using substitution. lim (2x³ - 4) X-1
Draw the graph of a function on [0, 5] with the given properties.ƒ is right- but not left-continuous at x = 1, left- but not right-continuous at x = 2, and neither left- nor right continuous at x =
Evaluate using substitution. lim(5x-12x-2) X-2
Evaluate using substitution. x + 2 lim x+3x² + 2x
Evaluate using substitution. lim sin X-X - A
Evaluate using substitution. lim tan(3x)
Evaluate using substitution. 1 lim X→7 COS X X
Evaluate using substitution. limx-5/2 x→4
Evaluate using substitution. lim √x³ + 4x x3 X→2
Evaluate using substitution. lim (1-8x³)³/2
Evaluate using substitution. 7x+2 2/3 lim x-2 4-X
Evaluate using substitution. lim 10²-2x x→3
Evaluate using substitution. lim 3sin.x π
Suppose that ƒ and g are discontinuous at x = c. Does it follow that ƒ + g is discontinuous at x = c? If not, give a counterexample. Does this contradict Theorem 1(i)? THEOREM 1 Basic Laws of
Prove that ƒ(x) = |x| is continuous for all x. To prove continuity at x = 0, consider the one-sided limits.
Use the result of Exercise 88 to prove that if g is continuous, then ƒ(x) = |g(x)| is also continuous.Data From Exercise 88Prove that ƒ(x) = |x| is continuous for all x. To prove continuity at x =
Which of the following quantities would be represented by continuous functions of time and which would have one or more discontinuities?(a) Velocity of an airplane during a flight(b) Temperature in a
In 2017, the federal income tax T on income of x dollars (up to $91,900) was determined by the formulaSketch the graph of T. Does T have any discontinuities? Explain why, if T had a jump
Explain why exists but does not exist. What is lim sin X→∞0 X
Show that the following function is continuous only at x = 0: f(x) = -x for x rational for x irrational
Show that ƒ is a discontinuous function for all x, where ƒ(x) is defined as follows:Show that ƒ2 is continuous for all x. f(x) = 1 -1 for x rational for x irrational
If ƒ has a removable discontinuity at x = c, then it is possible to redefine ƒ(c) so that ƒ is continuous at x = c. Can this be done in more than one way?
Give an example of functions ƒ and g such that ƒ (g(x)) is continuous but g has at least one discontinuity.
Let Prove rigorously that does not exist. Show that for any L, there always exists some x such that but no matter how small δ is taken. f(x) = X |xx|

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