Find the equation of the tangent line T to the graph of f at the given point. Use this linear approximation to complete the table. (Round your answers to four decimal places.) f(x) = x2, (3, 9) T(x) =
Find the equation of the tangent line T to the graph of f at the given point. Use this linear approximation to complete the table. (Round your answer to four decimal places.) f(x) = x, (3, 3)
Use the information to evaluate and compare Δy and dy. (Round your answers to three decimal places.) y = x3 x = 3 Δx = dx = 0.1 Δy = dy =
Find the differential dy of the given function. (Use " dx " for dx. ) dy = y = 115 cos(5πx − 1 2)
Use differentials and the graph of f to approximate the following. (Round your answers to two decimal (a) f(3.9) ≈ (b) f(4.04) ≈
Use differentials and the graph of f to approximate the following. (Round your answers to two decimal places.) (a) f(1.7) ≈ (b) f(2.07) ≈
The side of a square floor tile is measured to be 24 inches, with a possible error of 1/32 inch. Use differentials to approximate the possible error and the relative error in computing the area of the square.
The measurement of the radius of the end of a log is found to be 8 inches, with a possible error of 1/8 inch. Use differentials to approximate the possible propagated error in computing the area of the end of the log. in2
The measurement of a side of a square is found to be 11 centimeters, with a possible error of 0.05 centimeter. (a) Approximate the percent error in computing the area of the square. % (b) Estimate the maximum allowable percent error in measuring the side if the error in computing the area cannot exceed 2.5%. %
The total stopping distance T of a vehicle is shown below, where T is in feet and x is the speed in miles per hour. T = 2.5x + 0.5x2 Approximate the change and percent change in total stopping distance as speed changes from x = 27 to x = 31 miles per hour.
Use differentials to approximate the value of the expression. Compare your answer with that of a calculator. (Round your answers to four decimal places.) 3 25 using differentials using a calculator
Use the function f and the given real number a to find (f−1)′(a). (Hint: See Example 5. ) f(x) = x3 − 2, a = 123
Verify that f has an inverse. Then use the function f and the given real number a to find (f−1)′(a). (Hint: See Example 1 . If an answer does not exist, enter DNE. ) Function Real Number f(x) = x+3 x−5, x > 5 a = 3(f−1)′(3) =
Consider the following. function point f(x) = arcsec 5x (2/5, π/4) (a) Find an equation of the tangent line to the graph of f at the given point. (b) Use a graphing utility to graph the function and its tangent line at the point.
Find an equation of the tangent line to the graph of the equation at the given point. x2 + xarctan(y) = y−1, (−π/4, 1)
Find an equation of the tangent line to the graph of the function at the given point. y = arcsec 4x y =
Find an equation of the tangent line to the graph of the equation at the given point. arctan(x + y) = y2 + π/4, (1, 0)
And airplane flies at an altitude of 5 miles toward a point directly over an observer. Consider θ and x as shown in the figure below. Part (a) Write θ as a function of x. Part (b) The speed of the plane is 327 miles per hour. Find dθ/dt when x = 7 and x = 6.
Use a graphing utility to graph the function and visually estimate the limits. h(x) = −x2 + 7x (a) limx→7 h(x) = (b) limx→−1 h(x) =
Consider the following function and its graph. Use the graph to determine the limit visually (if it exists). (If an answer does not exist, enter DNE. ) (a) limx→0 g(x) (b) limx→−3 g(x) Write a simpler function that agrees with the given function at all but one point. g2(x) =
Find the limit (if it exists). (If an answer does not exist, enter DNE. ) limx→7 7−x x2−49
Determine the limit of the trigonometric function (if it exists). (If an answer does not exist, enter DNE.) limθ→0 cos(4 θ)tan(4 θ) θ
Use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. (You may round your answer to three decimal places. ) limx→0 x+6 − 6 x
Use the position function s(t) = −4.9t2 + 650, which gives the height (in meters) of an object that has fallen for t seconds from a height of 650 meters. The velocity at time t = a seconds is given by the following. limt→a s(a)−s(t) a−t Find the velocity of the object when t = 4.
Consider the following limit. limx→0 9sin(x) x Complete the table below. (Round your answers to five decimal places.) x -0.1 -0.01 -0.001 0.001 0.01 0.1 f(x) Use the table to estimate the limit. Use a graphing utility to graph the function to confirm your result. (Round your answer to five decimal places.)
Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. (Round your answers to four decimal places. If an answer does not exist, enter DNE.) limx→2 lnx − ln2 x−2 ≈ x 1.9 1.99 1.999 2.001 2.01 2.1 f(x)
Consider the following. limx→1 x−2 x2+6x−16 Create a table of values for the function. x 0.9 0.99 0.999 1.001 1.01 1.1 f(x) (Round your answers to four decimal places.) Use the table to estimate the limit. Use a graphing utility to graph the function to confirm your result. (Round your answer to four decimal places.)
Consider the following. limx→-3 |x+3| x+3 Use the graph to find the limit (if it exists). (If an answer does not exist, enter DNE.)
Consider the following. limx→-2 sin(πx) Use the graph to find the limit (if it exists). (If an answer does not exist, enter DNE.)
Use the graph of the function f to decide whether the value of the given quantity exists. If it does, find it. If it does not, enter DNE. (a) f(1) (b) limx→1 f(x) (c) f(4) (d) limx→4 f(x)
Sketch the graph of f. Then identify the values of c for which limx→ cf(x) exists. f(x) = {x2, x ≤ 2 5.6 − 0.8x, 2 < x < 7 5, x ≥ 7
Sketch a graph of a function f that satisfies the given values. f(0) is undefined. limx→0 f(x) = 4 f(4) = 6 limx→4 f(x) = 2
For a long-distance phone call, a hotel charges $9.25 for the first minute and $0.64 for each additional minute or fraction thereof. A formula for the cost is given below, where t is the time in minutes. (Note: [[x]] = greatest integer n such that n ≤ x. For example [[3.2]] = 3 and [[−1.6]] = −2. C(t) = 9.25 − 0.64[[−(t−1)]] (a) Use a graphing utility to graph the cost function for 0 < t ≤ 5. (b) Use the graph to complete the table below and observe the behavior of the function as t approaches 3.5 . Use the graph and the table to find the following limit. limt→3.5 C(t) (c) Use the graph to complete the table below and observe the behavior of the function as t approaches 3. Does the limit of C(t) as t approaches 3 exist? The limit does not exist. The limit exists.
Find the limit L. Then find the largest δ > 0 such that |f(x)−L| < 0.01 whenever 0 < |x−c| < δ. limx→2 (11x + 2)
Find the limit L. Then use the ε−δ definition to prove that the limit is L. limx→−4 (x2 + 4x) L =
Find the value of the derivative (if it exists) at the indicated extremum. (If an answer does not exist, enter DNE.) f(x) = x2 x2+4 f'(0) =
Find the value of the derivative (if it exists) at the indicated extremum. (-4/3 4 6/3) f(x) = -3x x+2
Approximate the critical numbers of the function shown in the graph. Determine whether the function has a relative maximum, a relative minimum, an absolute maximum, an absolute minimum, or none of these at each critical number on the interval shown. (Enter your answers as a comma-separated list. ) Approximate the critical numbers. x = List the critical numbers at which each phenomenon occurs. (If an answer does not exist, enter DNE.) relative maxima x = relative minima x = absolute maxima x = absolute minima x =
Find the critical numbers of the function. (Enter your answers as a comma-separated list.) t = g(t) = t 5−t, t < 11 3
Find the absolute extrema of the function on the closed interval. f(x) = 5x2+3, [−1, 2] minimum (x, y) = ( ) maximum (x, y) = ( )
Find the absolute extrema of the function on the closed interval. f(x) = x3 −32 x2, [−5, 2] minimum (x, y) = ( ) maximum (x, y) = ( )
Find the absolute extrema of the function on the closed interval. y = 2 − |t−2|, [−9, 3] minimum (t, y) = ( ) maximum (t, y) = ( )
Locate the absolute extrema of the function on the closed interval. g(x) = x2 − 6x, [0, 8] minimum (x, y) = ( ) maximum (x, y) = ( )
Locate the absolute extrema of the function on the closed interval. h(s) = 1 s−5, [1, 2] minimum (s, h) = ( ) maximum (s, h) = ( )
Locate the absolute extrema of the function on the closed interval. f(x) = cos(πx), [0, 1 3] minimum (x, y) = ( ) maximum (x, y) = ( )
Locate the absolute extrema of the function on the closed interval. y = 8 cosx, [0, 2π]
Locate the absolute extrema of the function (if any exist) over each interval. (If an answer does not exist, enter DNE. ) f(x) = 2x − 6 (a) [−3, 4] minimum (x, y) = ( ) maximum (x, y) = ( ) (b) [−3, 4) minimum (x, y) = ( ) maximum (x, y) = ( ) (c) (−3, 4] minimum (x, y) = ( ) maximum (x, y) = ( ) (d) (−3, 4) minimum (x, y) = ( ) maximum (x, y) = ( )