Find the indefinite integral and check the result by differentiation. (Use C for the constant of integration.) ∫(7(sec(θ))2 − 2 sin(θ))dθ
Find the equation of y, given the derivative and the indicated point on the curve. y = dy/dx = 2(x−1)
Tutorial Exercise Use a(t) = −9.8 m/sec2 as the acceleration due to gravity. (Neglect air resistance. ) With what initial velocity must an object be thrown upward (from a height of 3 meters) to reach a maximum height of 470 meters?
Consider a particle moving along the x-axis where x(t) is the position of the particle at time t, x′(t) is its velocity, and x′′(t) is its acceleration. x(t) = t3 − 6t2 + 9t − 3, 0 ≤ t ≤ 10 (a) Find the velocity and acceleration of the particle. x′(t) = x′′(t) = (b) Find the open t-intervals on which the particle is moving to the right. (Enter your answer using interval notation.) (c) Find the velocity of the particle when the acceleration is 0.
Consider a particle moving along the x-axis where x(t) is the position of the particle at time t, x′(t) is its velocity, and x′′(t) is its acceleration. A particle, initially at rest, moves along the x-axis such that its acceleration at time t > 0 is given by a(t) = 2cos(t). At the time t = 0, its position is x = 7. (a) Find the velocity and position functions for the particle. v(t) = f(t) = (b) Find the values of t for which the particle is at rest. (Use k as an arbitrary non-negative integer. ) t =
Tutorial Exercise Find the sum. Use the summation capabilities of a graphing utility to verify your result. ∑i = 1 6 (9i + 8)
Find the sum. Use the summation capabilities of a graphing utility to verify your result. (Round your answer to four decimal places. ) ∑k = 2 6 1 k2 + 4
Use sigma notation to write the sum. [(9 n)6 − 9 n](9 n)+⋯+[(9n n)6 − 9n n](9 n) ∑i = 1
Tutorial Exercise Use the properties of summation and the Summation Formulas Theorem to evaluate the sum. Use the summation capabilities of a graphing utility to verify your result. ∑i = 1 24 9i
Tutorial Exercise Use the summation capabilities of a graphing utility to evaluate the sum. Then use the properties of summation and the Summation Formulas Theorem from the chapter to verify the sum. ∑i = 1 20 (i2 + 6)
Use the summation capabilities of a graphing utility to evaluate the sum. Then use the properties of summation and the Summation Formulas Theorem to verify the sum. ∑i = 1 23 (i3 − 2i)
Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval. (Round your answers to four decimal places. ) f(x) = cos(x), [0, π/2], 4 rectangles < Area <
Tutorial Exercise Use the summation formulas to rewrite the expression without the summation notation. Use the result to find the sums for n = 10, 100, 1000, and 10, 000 . ∑i = 1 n 6i+4 n2
Tutorial Exercise Find a formula for the sum of n terms. Use the formula to find the limit as n→∞. limn→∞ ∑i = 1 n 32i n2
Find a formula for the sum of n terms. Use the formula to find the limit as n→∞. limn→∞ ∑i = 1 n (3i n)(2 n)
Tutorial Exercise Find a formula for the sum of n terms. Use the formula to find the limit as n→∞. limn→∞ ∑i = 1 n(1 + 6i n)3(2 n)
Tutorial Exercise Use the limit process to find the area of the region between the graph of the function and the x-axis over the given interval. Sketch the region. y = 25 − x2, [−5, 5]
The Midpoint Rule is shown below. Area ≅∑i = 1 n f(xi + xi−1 2)Δx Use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of the function over the x-axis over the given interval. f(x) = sin(x), [0, π2]
Use the graph to estimate the slope of the tangent line to y = xn at the point (1, 1). Verify your answer analytically. (a) y = x−1/5 y′(1) = (b) y = x−5 y′(1) =
Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. f(x) = 7 x2, (1, 7) f′(1) =
Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. f(θ) = 6 sin(θ) − θ, (0, 0) f′(0) =
Tutorial Exercise Find k such that the line is tangent to the graph of the function. function line f(x) = k x y = 5x + 20
A silver dollar is dropped from the top of a building that is 1393 feet tall. Use the position function below for free-falling objects. s(t) = −16t2 + v0t + s0 (a) Determine the position and velocity functions for the coin. s(t) = v(t) = (b) Determine the average velocity on the interval [2, 3]. ft/s (c) Find the instantaneous velocities when t = 2 seconds and t = 3 seconds. v(2) = ft/s v(3) = ft/s (d) Find the time required for the coin to reach the ground level. (Round your answer to three decimal places. ) t = s (e) Find the velocity of the coin at impact. (Round your answer to three decimal places.) ft/s
A projectile is shot upward from the surface of Earth with an initial velocity of 119 meters per second. Use the position function below for free-falling objects. What is its velocity after 2 seconds? After 11 seconds? (Round your answers to one decimal place.) s(t) = −4.9t2 + v0t + s0 2 s m/s 11 s m/s
To estimate the height of a building, a stone is dropped from the top of the building into a pool of water at ground level. How high is the building if the splash is seen 6.3 seconds after the stone is dropped? Use the position function for freefalling objects given below. (Round your answer to one decimal place.) s(t) = −4.9t2 + v0t + s0 m
The graph of f is shown. State the signs of f′ and f′′ on the interval (0, 2). f′? ∇0 f′′? ∇0
Determine the open intervals on which the graph is concave upward or concave downward. y = 2x2 − x − 5 Concave upward: (−∞, ∞) (−∞, 4) (4, ∞) (0, ∞) none Concave downward: (−∞, ∞) (−∞, 4) (4, ∞) (0, ∞) none
Determine the open intervals on which the graph is concave upward or concave downward. g(x) = 9x2 − x3 Concave upward: (−∞, −3) (−∞, 3) (−3, ∞) (3, ∞) none of these Concave downward: (−∞, −3) (−∞, 3) (−3, ∞) (3, ∞) none of these
Determine the open intervals on which the graph is concave upward or concave downward. f(x) = 13 x2+27 Concave upward: (−∞, −3) (−3, 3) (−∞, −3)∪(3, ∞) (−∞, ∞) none of these Concave downward: (−∞, −3) (−3, 3) (−∞, −3)∪(3, ∞) (−∞, ∞) none of these
Determine the open intervals on which the graph is concave upward or concave downward. g(x) = x2+1 9−x2 Concave upward: (−∞, −3) (−3, 3) (3, ∞) (−∞, ∞) none of these Concave downward: (−∞, −3)∪(3, ∞) (−∞, −3) (−3, 3) (−∞, ∞) none of these
Find the points of inflection of the graph of the function. (If an answer does not exist, enter DNE. ) f(x) = 15 x4 + 4x3 (x, y) = ( ) smaller x-value (x, y) = ( ) larger x-value ) larger x-value Describe the concavity. Concave upward: (−∞, −10)∪(0, ∞) (−∞, 0)∪(10, ∞) (−10, 0) (0, 10) none of these Concave downward: (−∞, −10)∪(0, ∞) (−∞, 0)∪(10, ∞) (−10, 0) (0, 10) none of these
Find the point of inflection of the graph of the function. (If an answer does not exist, enter DNE.) (x, y) = f(x) = x3 − 6x2 + 13x Describe the concavity. Concave upward: (−∞, 2) (−∞, −2) (2, ∞) (−2, ∞) none of these Concave downward: (−∞, 2) (−∞, −2) (2, ∞) (−2, ∞) none of these
Tutorial Exercise Find the points of inflection and discuss the concavity of the graph of the function. f(x) = x+8 x
Find the point of inflection of the graph of the function. (If an answer does not exist, enter DNE.) f(x) = x + 3cosx, [0, 2π] (x, y) = ( ) (smallest x-value) (x, y) = ( ) (largest x-value) Describe the concavity. (Select all that apply. ) Concave upward: (0, π/2) (π/2, 3π/2) (3π/2, 2π) (0, 2π) none of these Concave downward: (0, π/2) (π/2, 3π/2) (3π/2, 2π) (0, 2π) none of these
Find all relative extrema. Use the Second-Derivative Test where applicable. (If an answer does not exist, enter DNE. ) f(x) = (3x − 8)2 relative maximum (x, y) = ( ) relative minimum (x, y) = ( )
Find all relative extrema. Use the Second Derivative Test where applicable. (If an answer does not exist, enter DNE. ) f(x) = x3 − 6x2 + 4 relative maximum (x, y) = ( ) relative minimum (x, y) = ( )
Find all relative extrema. Use the Second Derivative Test where applicable. (If an answer does not exist, enter DNE. ) f(x) = x8/9 − 6 relative minimum (x, y) = ( ) relative maximum (x, y) = ( )
Find a, b, c, and d such that the cubic f(x) = ax3 + bx2 + cx + d satisfies the given conditions. Relative maximum: (3, 19) Relative minimum: (5, 17) Inflection point: (4, 18) a = b = c = d =
Find all relative extrema. Use the Second Derivative Test where applicable. (If an answer does not exist, enter DNE. ) f(x) = xe −7x relative minimum (x, y) = ( ) relative maximum (x, y) = ( )
Tutorial Exercise A manufacturer has determined that the total cost C of operating a factory is C = 2x2 + 60x + 200, where x is the number of units produced. At what level of production will the average cost per unit be minimized? (The average cost per unit is C/x.)
Use the graph to determine the limit. (If an answer does not exist, enter DNE.) (a) limx→c+ f(x) = (b) limx→c− f(x) = (c) limx→c f(x) = Is the function continuous at x = −3 ? Yes No
Use the graph to determine the limit. (If an answer does not exist, enter DNE.) (a) limx→c+ f(x) = (b) limx→c− f(x) = (c) limx→c f(x) = Is the function continuous at x = 5 ? Yes No
Tutorial Exercise Find the limit (if it exists). limx→6 f(x), where f(x) = {x2 − 5x + 4 if x < 6 −x2 + 5x + 16 if x ≥ 6
Discuss the continuity of the function. f(x) = 1 2 [[x]] + x - 2 f is discontinuous at every rational number. f is continuous for all real x. f is discontinuous at x = 0. f is discontinuous at every integer.
Consider the following. f(x) = 5/x Find the x-value at which f is not continuous. Is the discontinuity removable? (Enter NONE in any unused answer blanks.) x =
Tutorial Exercise Find the x-values at which f is not continuous. Which of the discontinuities are removable? (Enter your answers from smallest to largest. Enter NONE in any unused answer blanks. ) f(x) = x x2−4
Consider the following. f(x) = |x−2| x−2 Find the x-value at which f is not continuous. Is the discontinuity removable? (Enter NONE in any unused answer blanks. ) x =
Consider the following. f(x) = {ln(x + 1), x ≥ 0 5 − x2, x < 0 Find the x-value at which f is not continuous. Is the discontinuity removable? (Enter NONE in any unused answer blanks.)
Find the constant a such that the function is continuous on the entire real line. g(x) = {3sinx x if x < 0 a − 7x if x ≥ 0 a =
Describe the interval on which the function below is continuous. f(x) = x x2 + 3x + 9 (−∞, ∞) [3, 9) (−∞, 3) [3, ∞) (3, 9)
Describe the intervals on which the function below is continuous. (Select all that apply.) f(x) = sec(πx 4) (−6, 6) (−6, 2) (−2, 6) (−2, 2) (−6, −2) (2, 6)