Question: Consider the function f(x) = x3/2
Consider the function f(x) = x3/2 with the solution point (4, 8).
(a) Use a graphing utility to graph f. Use the zoom feature to obtain successive magnifications of the graph in the neighborhood of the point (4, 8). After zooming in a few times, the graph should appear nearly linear. Use the trace feature to determine the coordinates of a point near (4, 8). Find an equation of the secant line S(x) through the two points.
(b) Find the equation of the line T(x) = f′(4)(x − 4) + f(4) tangent to the graph of f passing through the given point. Why are the linear functions S and T nearly the same?
(c) Use a graphing utility to graph f and T on the same set of coordinate axes. Note that T is a good approximation of f when x is close to 4. What happens to the accuracy of the approximation as you move farther away from the point of tangency?
(d) Demonstrate the conclusion in part (c) by completing the table.
> Find the derivative of the trigonometric function.
> Find the derivative of the trigonometric function.
> Find the derivative of the trigonometric function.
> Find the derivative of the trigonometric function.
> y = csc (1 – 2x)2 Find the derivative of the trigonometric function.
> Complete the table.
> Find the derivative of the trigonometric function.
> Find the derivative of the trigonometric function.
> Find the derivative of the trigonometric function.
> Find the derivative of the trigonometric function. y = cos 4x
> Find the derivative of the function. g(x) – (2 + (x2 + 1)4)3
> Find the derivative of the function. f(x) = ((x2 + 3)5 + x)2
> Find the derivative of the function.
> Find all differentiable functions f : R→R such that for all real numbers x and all positive integers n
> Prove that d/dx [cos x] = -Sin x.
> Where are the functions f1(x) = ∣sin x∣ and f2(x) = sin ∣x∣ differentiable?
> Find a and b such that f is differentiable everywhere.
> Find a and b such that f is differentiable everywhere.
> Find the equation(s) of the tangent line(s) to the graph of the parabola y = x2 through the given point not on the graph. (a) (0, a) (b) (a, 0) Are there any restrictions on the constant a?
> Find the equation(s) of the tangent line(s) to the graph of the curve y = x3 − 9x through the point (1, −9) not on the graph.
> Let (a, b) be an arbitrary point on the graph of y = 1/x, x > 0. Prove that the area of the triangle formed by the tangent line through (a, b) and the coordinate axes is 2.
> Find an equation of the parabola y = ax2 + bx + c that passes through (0, 1) and is tangent to the line y = x − 1 at (1, 0).
> Find the derivative of the function.
> The annual inventory cost C for a manufacturer is where Q is the order size when the inventory is replenished. Find the change in annual cost when Q is increased from 350 to 351 and compare this with the instantaneous rate of change when Q = 350.
> Verify that the average velocity over the time interval [t0 − ∆t, t0 + ∆t] is the same as the instantaneous velocity at t = t0 for the position function
> A car is driven 15,000 miles a year and gets x miles per gallon. Assume that the average fuel cost is $3.48 per gallon. Find the annual cost of fuel C as a function of x and use this function to complete the table. Who would benefit more from a one-mile-
> The stopping distance of an automobile, on dry, level pavement, traveling at a speed v (in kilometers per hour) is the distance R (in meters) the car travels during the reaction time of the driver plus the distance B (in meters) the car travels after the
> The area of a square with sides of length s is given by A = s2. Find the rate of change of the area with respect to s when s = 6 meters.
> The volume of a cube with sides of length s is given by V = s3. Find the rate of change of the volume with respect to s when s = 6 centimeters.
> The graph of the velocity function (see figure) represents the velocity in miles per hour during a 10-minute trip to work. Make a sketch of the corresponding position function.
> The graph of the position function (see figure) represents the distance in miles that a person drives during a 10-minute trip to work. Make a sketch of the corresponding velocity function.
> Use the position function s(t) = −4.9t 2 + v0 t + s0 for free-falling objects. A rock is dropped from the edge of a cliff that is 214 meters above water. (a) Determine the position and velocity functions for the rock. (b) Determine the average velocit
> Use the position function s(t) = −4.9t 2 + v0 t + s0 for free-falling objects. A projectile is shot upward from the surface of Earth with an initial velocity of 120 meters per second. What is its veloc ity after 5 seconds? After 10 seconds?
> Find the derivative of the function.
> Complete the table.
> Use the position function s(t) = −16t 2 + v0 t + s0 for free-falling objects A ball is thrown straight down from the top of a 220-foot building with an initial velocity of −22 feet per second. What is its velocity after 3 seconds? What is its velocity a
> Use the position function s(t) = −16t 2 + v0 t + s0 for free-falling objects. A silver dollar is dropped from the top of a building that is 1362 feet tall. (a) Determine the position and velocity functions for the coin. (b) Determine the average veloc
> Find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval.
> Find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval.
> Find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. f(t) = t 2 − 7, [3, 3.1]
> f(t) = 3t + 5, [1, 2] Find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval.
> Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
> Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f(x) = 0, then f′(x) is undefined.
> Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f(x) = −g(x) + b, then f′(x) = −g′(x).
> Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If y = π2 , then dy/dx = 2π.
> Find the derivative of the function.
> Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If y = x a+2 + bx, then dy/dx = (a + 2)x a+1 + b.
> Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f′(x) = g′(x), then f(x) = g(x).
> Repeat Exercise 83 for the function f(x) = x3 , where T(x) is the line tangent to the graph at the point (1, 1). Explain why the accuracy of the linear approximation decreases more rapidly than in Exercise 83.
> Find an equation of the tangent line to the graph of the function f through the point (x0, y0) not on the graph. To find the point of tangency (x, y) on the graph of f, solve the equation (x0, y0) = (5, 0)
> Find an equation of the tangent line to the graph of the function f through the point (x0, y0) not on the graph. To find the point of tangency (x, y) on the graph of f, solve the equation f(x) = √x (x0,y0) = ( - 4, 0)
> Show that the graph of the function f (x) = x5 + 3x3 + 5x
> Show that the graph of the function f (x) = 3x + sin x + 2 does not have a horizontal tangent line.
> Show that the graphs of the two equations have tangent lines that are perpendicular to each other at their point of intersection.
> Sketch the graphs of y = x2 and y = −x2 + 6x − 5, and sketch the two lines that are tangent to both graphs. Find equations of these lines.
> Find the derivative of the function.
> Use the graph of f to answer each question. To print an enlarged copy of the graph, go to MathGraphs.com. (a) Between which two consecutive points is the average rate of change of the function greatest? (b) Is the average rate of change of the functio
> Sketch the graph of a function f such that f′ > 0 for all x and the rate of change of the function is decreasing.
> The graphs of a function f and its derivative f′ are shown on the same set of coordinate axes. Label the graphs as f or f′ and write a short paragraph stating the criteria you used in making your selection. To print an
> The graphs of a function f and its derivative f′ are shown on the same set of coordinate axes. Label the graphs as f or f′ and write a short paragraph stating the criteria you used in making your selection. To print an
> g(x) = 3 f(x) − 1 The relationship between f and g is given. Explain the relationship between f′ and g′.
> The relationship between f and g is given. Explain the relationship between f′ and g′. g(x) = −5 f(x)
> The relationship between f and g is given. Explain the relationship between f′ and g′. g(x) = 2 f(x)
> g(x) = f(x) + 6 The relationship between f and g is given. Explain the relationship between f′ and g′.
> Find k such that the line is tangent to the graph of the function. Function: f(x) = k√x Line: y = x + 4
> Find k such that the line is tangent to the graph of the function. Function: / Line: /
> Find the derivative of the function.
> Function: f(x) = kx2 Line: = −2x + 3 Find k such that the line is tangent to the graph of the function.
> Function: f(x) = k − x2 Line: y = −6x + 1 Find k such that the line is tangent to the graph of the function.
> y = √3x + 2 cos x, 0 ≤ x < 2π Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.
> y = x + sin x, 0 ≤ x < 2π Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.
> y = x2 + 9 Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.
> Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.
> y = x3 + x Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.
> y = x4 − 2x2 + 3 Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.
> (a) find an equation of the tangent line to the graph of the function at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the tangent feature of a graphing utility to confirm your results.
> (a) find an equation of the tangent line to the graph of the function at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the tangent feature of a graphing utility to confirm your results.
> Find the derivative of the function.
> (a) find an equation of the tangent line to the graph of the function at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the tangent feature of a graphing utility to confirm your results.
> (a) find an equation of the tangent line to the graph of the function at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the tangent feature of a graphing utility to confirm your results.
> Find the derivative of the function.
> Find the derivative of the function.
> Find the derivative of the function.
> f(x) = 6√x + 5 cos x Find the derivative of the function.
> f(t) = t 2/3 − t 1/3 + 4 Find the derivative of the function.
> f(x) = √x − 6√ 3 x Find the derivative of the function.
> y = x2 (2x2 − 3x) Find the derivative of the function.
> y = x(x2 + 1) Find the derivative of the function.
> Find the derivative of the function.
> Find the derivative of the function.
> Find the derivative of the function.
> Find the derivative of the function.
> Find the derivative of the function.
> Find the derivative of the function.
> Find the derivative of the function.
> f(x) = x3 − 2x + 3x−3 Find the derivative of the function.
> f(x) = x2 + 5 − 3x−2 Find the derivative of the function
> Function: g(t) = −2 cos t + 5 Point: (π, 7) 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.
> 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. Function: f(θ) = 4 sin θ − θ. Point: (0, 0 )
> f(x) = x(2x - 5)3 Find the derivative of the function.
