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mathematics
calculus

Questions and Answers of Calculus

Find an equation of the tangent line at x = 3, assuming that f (3) = 5 and f ′(3) = 2?
What are the two ways of writing the difference quotient?
Describe the tangent line at an arbitrary point on the "curve" y = 2x + 8
Let f(x) = 1/x. Does f (- 2 +h) equal 1/-2 + h or 1/-2 + 1/h? compute the difference quotient at a = -2 with h = 0.5.
Let f (x) = 1/ ˆš x. Compute f €²(5) by showing that
27. F(x) = 2x2 + 10x, a = 3 29. F(t) = t - 2t2 , a = 3
Find a and h such thatis equal to the slope of the secant line between (3, f (3)) and (5, f (5)).
31. (x) = x3 + x, a = 0 33. f (x) = x −1, a = 8
35. F(x) = 1/x + 3, a = -2 37. F(x) = √ x + 4, a = 1
39. F(x) = 1/√x , a = 4 41. F(t) = √t2 + 1, a = 3
Which derivative is approximated by
f (x) = 1/x2 + 1, a = 0
Figure 13 displays data collected by the biologist Julian Huxley (1887-1975) on the average antler weight W of male red deer as a function of age t. Estimate the derivative at t = 4. For which values
Let f (x) = 4/1+ 2x(a) Plot f (x) over [−2, 2]. Then zoom in near x = 0 until the graph appears straight, and estimate the slope f′(0).(b) Use (a) to find an approximate equation to the tangent
Determine the intervals along the x-axis on which the derivative in Figure 15 is positive.
What do the following quantities represent in terms of the graph of f (x) = sin x?a. Sin 1.3 ˆ’ sin 0.9b.c. F'(0.9)
51. Each limit represents a derivative f (a). Find f (x) and a.53.
Apply the method of Example 6 to f (x) = sin x to determine F′(π/4) accurately to four decimal places.
For each graph in Figure 16, determine whether f€²(1) is larger or smaller than the slope of the secant line between x = 1 and x = 1 + h for h > 0. Explain.
Compute f'(a) in two ways, using Eq. (1) and Eq. (2). 3. f(x) = x2 + 9x, a = 0 5. (x) = 3x2 + 4x + 2, a = −1
Sketch the graph of f (x) = x5/2 on [0, 6].(a) Use the sketch to justify the inequalities for h > 0:(b) Use (a) to compute f €²(4) to four decimal places.(c) Use a graphing utility to plot f
Use a plot of f (x) = xx to estimate the value c such that f (c) = 0. Find c to sufficient accuracy so that
The vapor pressure of water at temperature T (in kelvins) is the atmospheric pressure P at which no net evaporation takes place. Use the following table to estimate P€²(T ) for T = 303,
The current (in amperes) at time t (in seconds) flowing in the circuit in Figure 19 is given by Kirchhoff's Law:i(t) = Cv €²(t) + R ˆ’1v(t)where v(t) is the voltage (in volts), C the
Find the slope of the secant line through (2, f (2)) and (2.5, f (2.5)). Is it larger or smaller than f ′(2)? Explain.
Explain how the symmetric difference quotient defined by Eq. (4) can be interpreted as the slope of a secant line.
Show that if f (x) is a quadratic polynomial, then the SDQ at x = a (for any h _= 0) is equal to f _ (a). Explain the graphical meaning of this result
Estimate f ′ (1) and f ′(2)
Use Eq. (1) to estimate Δf = f (3.02) − f (3). 1. f (x) = x2 3. f (x) = x−1
Estimate Δy using differentials [Eq. (3)]. 13. y = cos x, a = π/6, dx = 0.014 15. y = 10 − x2/2 + x2, a = 1, dx = 0.01
Estimate using the Linear Approximation and find the error using a calculator. 17. √26 - √25 19. 1/√101 - 1/10
Estimate g(1.2) − g(1) if g′(1) = 4.
Estimate f (4.03) for f (x) as in Figure 8.
Which is larger: √2.1 − √2 or √9.1 − √9? Explain using the Linear Approximation
Box office revenue at a multiplex cinema in Paris is R(p) = 3600p − 10p3 euros per showing when the ticket price is p euros. Calculate R(p) for p = 9 and use the Linear Approximation to estimate
Estimate f (2.1) if f (2) = 1 and f′(2) = 3.
A thin silver wire has length L = 18 cm when the temperature is T = 30oC. Estimate ΔL when T decreases to 25oC if the coefficient of thermal expansion is k = 1.9 × 10−5oC−1.
The atmospheric pressure P at altitude h = 20 km is P = 5.5 kilopascals. Estimate P at altitude h = 20.5 km assuming that dP/dh = −0.87
Newton's Law of Gravitation shows that if a person weighs w pounds on the surface of the earth, then his or her weight at distance x from the center of the earth iswhere R = 3960 miles is the radius
A one tossed vertically into the air with Initial velocity v cm/s reaches a maximum height of h = v2/ 1960 cm. (a) Estimate Δh if v = 700 cm/s and Δv = 1 cm/s. (b) Estimate Δh if v = 1000 cm/s and
Use the following fact derived from Newton's Laws: An object released at an angle Î¸ with initial velocity v ft/s travels a horizontal distanceA player located 18.1 ft from the basket
The radius of a spherical ball is measured at r = 25 cm. Estimate the maximum error in the volume and surface area if r is accurate to within 0.5 cm.
The volume (in liters) and pressure P (in atmospheres) of a certain gas satisfy PV = 24. A measurement yields V = 4 with a possible error of ±0.3 L. Compute P and estimate the maximum error in this
Find the linearization at x = a. 45. f(x) = x4, a = 1 47. f(θ) = sin2 θ, a = π/4
Estimate √16.2 using the linearization L(x) of f (x) = √x at a = 16. Plot f (x) and L(x) on the same set of axes and determine whether the estimate is too large or too small.
Approximate using linearization and use a calculator to compute the percentage error 59. 1/√17 61. 1/(10.03)2
Show that the Linear Approximation to f (x) = √x at x = 9 yields the estimate √9 + h − 3 ≈ 1/6 h. Set K = 0.01 and show that |f′′(x)| ≤ K for x ≥ 9. Then verify numerically that the
The cube root of 27 is 3. How much larger is the cube root of 27.2? Estimate using the Linear Approximation.
Compute dy/dx at the point P = (2, 1) on the curve y3 + 3xy = 7 and show that the linearization at P is L(x) = −1/3x + 5/3. Use L(x) to estimate the y-coordinate of the point on the curve where x =
Apply the method of Exercise 71 to P = (−1, 2) on y4 + 7xy = 2 to estimate the solution of y4 − 7.7y = 2 near y = 2.
Let Δf = f (5 + h) − f (5), where f (x) = x2. Verify directly that E = |Δf − f′(5)h| satisfies (5) with K = 2.
Use Eq. (1) to estimate Δf. Use a calculator to compute both the error and the percentage error. 9. f (x) = √1 + x, a = 3, Δx = 0.2 11. f (x) = 1/1 + x2, a = 3, Δx = 0.5
The following questions refer to Figure 15.(a) How many critical points does f (x) have on [0, 8]?(b) What is the maximum value of f (x) on [0, 8]?(c) What are the local maximum values of f (x)?(d)
Choose the correct conclusion. 2. If f (x) is not continuous on [0, 1], then (a) f (x) has no extreme values on [0, 1]. (b) f (x) might not have any extreme values on [0, 1]. 3. If f (x) is
Plot f (x) = 2 √x − x on [0, 4] and determine the maximum value graphically. Then verify your answer using calculus.
Approximate the critical points of g(x) = x cos x on I = [0, 2π] and estimate the minimum value of g(x) on I.
Find the min and max of the function on the given interval by comparing values at the critical points and endpoints. 25. y = 2x2 + 4x + 5, [−2, 2] 27. y = 6t − t2, [0, 5]
Find all critical points of the function. 3. f(x) = x2 − 2x + 4 5. f(x) = x3 - 9/2 x2 − 54x + 2
Find the critical points and the extreme values on [0, 4].53. y = |x ˆ’ 2| 55. y = |x2 + 4x ˆ’ 12|
Verify Rolle's Theorem for the given interval. 57. f(x) = x + x−1, [1/2, 2] 59. f(x) = x2/8x − 15, [3, 5]
Prove that f (x) = x5 + 2x3 + 4x − 12 has precisely one real root.
Prove that f (x) = x4 + 5x3 + 4x has no root c satisfying c > 0.
The position of a mass oscillating at the end of a spring is s(t) = A sin ωt, where A is the amplitude and ω is the angular frequency. Show that the speed |v(t)| is at a maximum when the
In 1919, physicist Alfred Betz argued that the maximum efficiency of a wind turbine is around 59%. If wind enters a turbine with speed v1 and exits with speed v2, then the power extracted is the
The response of a circuit or other oscillatory system to an input of frequency Ï‰ ("omega") is described by the functionBoth Ï‰0 (the natural frequency of the system) and D (the
Find the maximum of y = xa − xb on [0, 1] where 0 < a < b. In particular, find the maximum of y = x5 − x10 on [0, 1].
Plot the function using a graphing utility and find its critical points and extreme values on [ˆ’5, 5].
(a) Use implicit differentiation to find the critical points on the curve 27x2 = (x2 + y2)3. (b) Plot the curve and the horizontal tangent lines on the same set of axes.
Sketch the graph of a continuous function on (0, 4) having a local minimum but no absolute minimum.
Sketch the graph of a function f(x) on [0, 4] with a discontinuity such that f (x) has an absolute minimum but no absolute maximum.
Show that the extreme values of f (x) = a sin x + b cos x are ±√a2 + b2.
Show that if the quadratic polynomial f (x) = x2 + rx + s takes on both positive and negative values, then its minimum value occurs at the midpoint between the two roots.
A cubic polynomial may have a local min and max, or it may have neither. Find conditions on the coefficients a and b ofthat ensure that f has neither a local min nor a local max.
Prove that if f is continuous and f (a) and f (b) are local minima where a < b, then there exists a value c between a and b such that f (c) is a local maximum. Show that continuity is a necessary
For which value of m is the following statement correct? If f (2) = 3 and f (4) = 9, and f (x) is differentiable, then f has a tangent line of slope m.
Determine the intervals on which f€²(x) is positive and negative, assuming that Figure 13 is the graph of f (x).
State whether f (2) and f (4) are local minima or local maxima, assuming that Figure 13 is the graph of f€²(x).
Sketch the graph of a function f (x) whose derivative f′(x) has the given description 15. f′(x) > 0 for x > 3 and f′(x) < 0 for x < 3 17. f′(x) is negative on (1, 3) and positive everywhere
Find all critical points of f and use the First Derivative Test to determine whether they are local minima or maxima. 19. f(x) = 4 + 6x − x2 21. f(x) = x2/x + 1
Find a point c satisfying the conclusion of the MVT for the given function and interval. 1. y = x−1, [2, 8] 3. y = cos x − sin x, [0, 2π]
Find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or
Can a function that takes on only negative values have a positive derivative? If so, sketch an example.
For f (x) with derivative as in Figure 12:(a) Is f (c) a local minimum or maximum?(b) Is f (x) a decreasing function?
Find the minimum value of f (x) = xx for x > 0.
Show that f (x) = x3 − 2x2 + 2x is an increasing function.
Let h(x) = x(x2 − 1)/x2 + 1 and suppose that f′(x) = h(x). Plot h(x) and use the plot to describe the local extrema and the increasing/decreasing behavior of f(x). Sketch a plausible graph for f
Determine where f (x) = (1000 − x)2 + x2 is decreasing. Use this to decide which is larger: 8002 + 2002 or 6002 + 4002.
Which values of c satisfy the conclusion of the MVT on the interval [a, b] if f (x) is a linear function?
Suppose that f (0) = 2 and f′(x) ≤ 3 for x > 0. Apply the MVT to the interval [0, 4] to prove that f (4) ≤ 14. Prove more generally that f (x) ≤ 2 + 3x for all x > 0.
Show that if f (2) = 5 and f′(x) ≥ 10 for x > 2, then f (x) ≥ 10x − 15 for all x > 2.
Prove that if f (0) = g(0) and f′(x) ≤ g′(x) for x ≥ 0, then f (x) ≤ g(x) for all x ≥ 0.
Use Exercise 59 and the inequality sin x ≤ x for x ≥ 0 (established in Theorem 3 of Section 2.6) to prove the following assertions for all x ≥ 0 (each assertion follows from the previous
Assume that f″ exists and f″(x) = 0 for all x. Prove that f (x) = mx + b, where m = f′(0) and b = f (0).
Suppose that f (x) satisfies the following equation (an example of a differential equation): f″(x) = −f (x) (a) Show that f (x)2 + f′(x)2 = f (0)2 + f′(0)2 for all x. (b) Verify that sin x
Use Exercise 66 to prove: f (x) = sin x is the unique solution of Eq. (1) such that f (0) = 0 and f′(0) = 1; and g(x) = cos x is the unique solution such that g(0) = 1 and g′(0) = 0. This result
Let f(x) = x5 + x2. The secant line between x = 0 and x = 1 has slope 2 (check this), so by the MVT, f′(c) = 2 for some c ∈ (0, 1). Plot f (x) and the secant line on the same axes. Then plot y =
Which of the following sequences converge to zero?(a)(b) 2n (c)
Which of the following sequences is defined recursively? (a) an = √4 + n (b) bn = √4 + bn−1
Theorem 5 says that every convergent sequence is bounded. Determine if the following statements are true or false and if false, give a counterexample. (a) If {an} is bounded, then it converges. (b)
Match each sequence with its general term:

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