Math Archive: Questions from 2024-05-23

Find dy/dx by implicit differentiation. x4 + y9 = 28

Find dy/dx by implicit differentiation. xy = x7y + 77 dy/dx =

Find dy/dx by implicit differentiation. cot⁡(y) = x − 5y dy/dx =

Find dy/dx by implicit differentiation. ln xy + 5x = 26 dy dx =

Find dy/dx by implicit differentiation and evaluate the derivative at the given point. (5x + 5y)3 = 125x3 + 125y3, (−1, 1)

Find dy/dx by implicit differentiation and evaluate the derivative at the given point. tan⁡(4x + y) = 4x, (0, 0) dy dx = At (0, 0): y′ =

Find the slope of the tangent line to the graph at the given point. x3 + y3 − 6xy = 0, (4/3, 8/3) Folium of Descartes:

Find an equation of the tangent line to the graph at the given point. (x + 2)2 + (y − 3)2 = 37, (−1, −3) y = Circle

Use logarithmic differentiation to find dy/dx. y = x2−1 x2+1, x > 1 dy dx =

Use logarithmic differentiation to find dy/dx. y = (x+1)(x−9) (x−1)(x+9), x > 9

Use the graph of f to find the following. (a) The largest open interval on which f is increasing. (−3, 1) (1, 3) (3, 4) (4, 5) (5, 7) (b) The largest open interval on which f is decreasing. (−3, 1) (1, 3) (3, 4) (4, 5) (5, 7)

Use the graph to estimate the open intervals on which the function is increasing or decreasing. Then find the open intervals analytically. (Select all that apply.) y = x3 16 − 3x Increasing: (−∞, −4) (−4, 4) (4, ∞) none of these Decreasing: (−∞, −4) (−4, 4) (4, ∞) none of these

Identify the open intervals on which the function is increasing or decreasing. g(x) = x2 − 4x − 192

Identify the open intervals on which the function is increasing or decreasing. (Select all that apply. ) f(x) = sin⁡x+9, 0 < x < 2π Increasing: (3π/2, 2π) (π/2, 3π/2) (0, π/2) (−∞, 0) (0, ∞) Decreasing: (0, ∞) (π/2, 3π/2) (0, π/2) (−∞, 0) (3π/2, 2π)

Identify the open intervals on which the function is increasing or decreasing. (Select all that apply. ) f(x) = x + 2cos⁡(x), 0 < x < 2π Increasing: (0, π6) (π6, 5π6) (5π6, 2π) none of these Decreasing: (0, π6) (π6, 5π6) (5π6, 2π) none of these

Identify the open intervals on which the function is increasing or decreasing. (Select all that apply. ) f(x) = x − 2cos⁡(x), 0 < x < 2π Increasing: (0, 7π6) (7π6, 11π6) (11π6, 2π) none of these Decreasing: (0, 7π6) (7π6, 11π6) (11π6, 2π) none of these

Consider the function below. f(x) = −3x2 + 6x + 2 Exercise (a) Find the critical numbers of f. Exercise (b) Find the open intervals on which the function is increasing or decreasing. Exercise (c) Apply the First Derivative Test to identify the relative extrema.

Consider the following function. f(x) = (x+2)2(x−1) (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) x = (b) Find the open intervals on which the function is increasing or decreasing. (Select all that apply. ) Increasing: (−∞, −2) (−2, 0) (0, ∞) (−∞, ∞) Decreasing: (−∞, −2) (−2, 0) (0, ∞) (−∞, ∞) (c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.) relative maximum (x, y) = ( ) relative minimum (x, y) = ( ) (d) Use a graphing utility to confirm your results.

Consider the following function. f(x) = x1/7 + 5 (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) x = (b) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation. If an answer does not exist, enter DNE. ) increasing decreasing (c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.) relative maximum (x, y) = ( ) relative minimum (x, y) = ()

Consider the following function. f(x) = x1/3 + 9 (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) x = (b) Find the open intervals on which the function is increasing or decreasing. Increasing: (−∞, 0) (0, ∞) (−∞, ∞) (9, ∞) (−∞, 9) none of the above Decreasing: (−∞, 0) (0, ∞) (−∞, ∞) (9, ∞) (−∞, 9) none of the above (c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.) relative maximum (x, y) = ( ) relative minimum (x, y) = ( ) (d) Use a graphing utility to confirm your results.

Consider the following function. f(x) = 2 − |x−9| (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) x = (b) Find the open intervals on which the function is increasing or decreasing. Increasing: (−∞, 9) (−∞, −9) (9, ∞) (−9, ∞) Decreasing: (−∞, 9) (−∞, −9) (9, ∞) (−9, ∞) (c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.) relative maximum (x, y) = ( ) relative minimum (x, y) = ( ) (d) Use a graphing utility to confirm your results.

Consider the following function. f(x) = x2 x2−9 (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) x = (b) Find the open intervals on which the function is increasing or decreasing. (Select all that apply. ) Increasing: (−∞, −3) (−3, 0) (0, 3) (3, ∞) none of these Decreasing: (−∞, −3) (−3, 0) (0, 3) (3, ∞) none of these (c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.) relative maximum (x, y) = ( ) relative minimum (x, y) = ( ) (d) Use a graphing utility to confirm your results.

Consider the following function. f(x) = {16 − x2, x ≤ 0 −5x, x > 0 (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) x = (b) Find the open intervals on which the function is increasing or decreasing. Increasing: (−∞, 0) (−∞, 16) (0, ∞) (16, ∞) Decreasing: (−∞, 0) (−∞, 16) (0, ∞) (16, ∞) (c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.) relative maximum (x, y) = ( ) relative minimum (x, y) = ( ) (d) Use a graphing utility to confirm your results.

Find the critical numbers of f (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results. f(x) = x5 5 − ln⁡x Critical numbers: (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) x = Increasing: (Select all that apply. ) none of these (0, 1) (1, ∞) (−∞, 0) (0, ∞) Decreasing: (Select all that apply. ) (−∞, 0) (0, 1) (0, ∞) (1, ∞) none of these Relative extrema: (If an answer does not exist, enter DNE. ) relative maximum (x, y) = ( ) relative minimum (x, y) = ( )

Consider the function on the interval (0, 2π). f(x) = x/2 + cos⁡(x) (a) Find the open intervals on which the function is increasing or decreasing. (Select all that apply.) Increasing: (0, π6) (π6, 5π6) (5π6, 2π) none of these Decreasing: (0, π6) (π6, 5π6) (5π6, 2π) none of these (b) Apply the First Derivative Test to identify the relative extrema. relative maximum (x, y) = relative minimum (x, y) = (c) Use a graphing utility to confirm your results.

The function s(t) describes the motion of a particle along a line. s(t) = 18t − t2 (a) Find the velocity function v(t) of the particle at any time t ≥ 0. v(t) = (b) Identify the time interval in which the particle is moving in a positive direction. (0, ∞) (0, 9) (0, 18) (18, ∞) (9, ∞) None (c) Identify the time interval in which the particle is moving in a negative direction. (0, ∞) (0, 9) (0, 18) (18, ∞) (9, ∞) None (d) Identify the time at which the particle changes direction. t =