If we graph these functions on the same axes, as in Figure , we can use the graphs to understand the relationship between these two functions. First, we notice that is increasing over its entire domain, which means that the slopes of its tangent lines at all points are positive. Consequently, we expect for all values of in its domain. Furthermore, as increases, the slopes of the tangent lines to are decreasing and we expect to see a corresponding decrease in.
We also observe that is undefined and that , corresponding to a vertical tangent to at 0. The graphs of these functions are shown in Figure. Observe that is decreasing for. For these same values of. For values of is increasing and. Also, has a horizontal tangent at and. Use the following graph of to sketch a graph of. The solution is shown in the following graph. Observe that is increasing and on. Also, is decreasing and on and on. Also note that has horizontal tangents at -2 and 3, and and.
Sketch the graph of. On what interval is the graph of above the -axis? The graph of is positive where is increasing. First, we consider the relationship between differentiability and continuity. We will see that if a function is differentiable at a point, it must be continuous there; however, a function that is continuous at a point need not be differentiable at that point.
In fact, a function may be continuous at a point and fail to be differentiable at the point for one of several reasons. Let be a function and be in its domain. If is differentiable at , then is continuous at.
If is differentiable at , then exists and. We want to show that is continuous at by showing that. Therefore, since is defined and , we conclude that is continuous at.
We have just proven that differentiability implies continuity, but now we consider whether continuity implies differentiability. To determine an answer to this question, we examine the function. This function is continuous everywhere; however, is undefined. This observation leads us to believe that continuity does not imply differentiability. For ,. See Figure. Consider the function :. Thus does not exist. A quick look at the graph of clarifies the situation.
The function has a vertical tangent line at 0 Figure. The function also has a derivative that exhibits interesting behavior at 0. We see that. This limit does not exist, essentially because the slopes of the secant lines continuously change direction as they approach zero Figure. A toy company wants to design a track for a toy car that starts out along a parabolic curve and then converts to a straight line Figure.
The function that describes the track is to have the form , where and are in inches. For the car to move smoothly along the track, the function must be both continuous and differentiable at Find values of and that make both continuous and differentiable.
For the function to be continuous at. Thus, since. Equivalently, we have. Since is defined using different rules on the right and the left, we must evaluate this limit from the right and the left and then set them equal to each other:. This gives us. Thus and. Find values of and that make both continuous and differentiable at 3. Use Figure as a guide. The derivative of a function is itself a function, so we can find the derivative of a derivative.
For example, the derivative of a position function is the rate of change of position, or velocity. The derivative of velocity is the rate of change of velocity, which is acceleration. The new function obtained by differentiating the derivative is called the second derivative.
Furthermore, we can continue to take derivatives to obtain the third derivative, fourth derivative, and so on. Collectively, these are referred to as higher-order derivatives. The notation for the higher-order derivatives of can be expressed in any of the following forms:. It is interesting to note that the notation for may be viewed as an attempt to express more compactly.
For , find. First find. Next, find by taking the derivative of. Find for. We found in a previous checkpoint. Use Figure to find the derivative of. The position of a particle along a coordinate axis at time in seconds is given by in meters. We have just proven that differentiability implies continuity, but now we consider whether continuity implies differentiability.
This observation leads us to believe that continuity does not imply differentiability. Thus, since. The derivative of a function is itself a function, so we can find the derivative of a derivative. For example, the derivative of a position function is the rate of change of position, or velocity.
The derivative of velocity is the rate of change of velocity, which is acceleration. The new function obtained by differentiating the derivative is called the second derivative. Furthermore, we can continue to take derivatives to obtain the third derivative, fourth derivative, and so on.
Collectively, these are referred to as higher-order derivatives. Learning Objectives Define the derivative function of a given function. Graph a derivative function from the graph of a given function. State the connection between derivatives and continuity. Calculus, all content edition. Introduction to differential calculus. Newton, Leibniz, and Usain Bolt Opens a modal. Derivative as slope of curve Opens a modal. Derivative notation review Opens a modal.
Derivative as slope of curve. Derivative as slope of tangent line. Interpreting derivative challenge Opens a modal. Derivative as instantaneous rate of change. Tangent slope as instantaneous rate of change Opens a modal. Estimating derivatives with two consecutive secant lines Opens a modal. Approximating instantaneous rate of change with average rate of change Opens a modal. Secant lines. Slope of a line secant to a curve Opens a modal. Secant line with arbitrary difference Opens a modal.
Secant line with arbitrary point Opens a modal. Secant line with arbitrary difference with simplification Opens a modal. Secant line with arbitrary point with simplification Opens a modal. Secant lines: challenging problem 1 Opens a modal.
Secant lines: challenging problem 2 Opens a modal. Derivative as a limit. Formal definition of the derivative as a limit Opens a modal.
Derivative as a limit: numerical Opens a modal. Derivative as a limit: numerical. Formal definition of derivative. Formal and alternate form of the derivative Opens a modal. Worked example: Derivative as a limit Opens a modal. Worked example: Derivative from limit expression Opens a modal.
Using the formal definition of derivative. Limit expression for the derivative of a linear function Opens a modal. Limit expression for the derivative of cos x at a minimum point Opens a modal. Limit expression for the derivative of function graphical Opens a modal. Tangent lines and rates of change Opens a modal. Differentiability at a point: graphical Opens a modal.
Differentiability at a point: algebraic function is differentiable Opens a modal. Differentiability at a point: algebraic function isn't differentiable Opens a modal. Differentiability at a point old Opens a modal. Differentiability at a point: graphical. Differentiability at a point: algebraic. Derivative as a function. Connecting f and f' graphically Opens a modal. Visualizing derivatives. Review: Derivative basics. Learn No videos or articles available in this lesson. Derivatives basics challenge.
Basic differentiation rules. Basic derivative rules Part 1 Opens a modal. Basic derivative rules Part 2 Opens a modal. Basic derivative rules: find the error Opens a modal. Basic derivative rules: table Opens a modal. Basic differentiation review Opens a modal. Basic derivative rules: find the error. Basic derivative rules: table.
Power rule. Power rule Opens a modal. Justifying the power rule Opens a modal. Proof of power rule for positive integer powers Opens a modal. Proof of power rule for square root function Opens a modal.
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