If our function is the position of $$x\text{,}$$ then the first derivative is the rate of change or the velocity of $$f(x)\text{. The graph of a di erentiable function does not have any sharp corners. 4.5.1 Explain how the sign of the first derivative affects the shape of a function's graph. So at x = 0, the second derivative of f(x) is 12, so we know that the graph of f(x) is concave down at x = 0. ection of a line has slope the reciprocal of the slope of the original line. Type in a function and see its slope below (as calculated by the program). Graphically, this means that the derivative is the slope of the graph of that function. 4. Steps . Identify and classify the x-coordinate of each critical value . More INFO about graphs of functions. Math 132Derivatives and GraphsStewart x3.3 Increasing and decreasing functions. 3 Example #1. The derivative function Rules of differentiation Equations Of tangents . 7 v240 Y1x3J PKzuZt daN YSVopf9txw Ia MrSes L5L zC M.C f WAnl 4l D Frli kgjh Jt Asi Hr1eZs5emr3v Eeed m.m l EMpavdOeb Sw vi wtch3 GI3nXf ZiBn3iqtMeT BC2a 1l ac CuSl0uxs 5. k Worksheet by Kuta Software LLC Now that we have the concept of limits, we can make this more precise. = . On the same coordinate plane, sketch a graph of , the derivative of . Congratulations, you've just learned some calculus. a. Notation for derivatives: original function derivative . For #7-10, find all points of inflection of the function. The graph of , the derivative of , is shown. Justify your answer. Typically, you will also see was a factor in the integrand as well. Notes and Solutions . Write the functions in part a in terms of f and g. (For example, if h(x) = 2x2 you can write h in terms of f as h(x) = 2f(x).) If y= f(x) is a function of x, then we also use the . Applying this principle, we nd that the 17th derivative of the sine function is equal to the 1st derivative, so d17 dx17 sin(x) = d dx sin(x) = cos(x) The derivatives of cos(x) have the same behavior, repeating every cycle of 4. In one example we saw that tells us how steep . We said that the derivative of a quadratic function at a point is the slope of the tangent line to the graph of that function at that point. Sketching Graphs of Derivative Functions Previously, we have seen that if f(x) is a polynomial of degree n, then its derivative is one degree lower (i.e., n (One exception to this is the case where f(x) is a constant function and so has degree n = 0.) Connecting , ', and ''. 3. The interactive function graphs are computed in the browser and displayed within a canvas element (HTML5). From this graph determine the intervals in which the function increases and decreases. As the last problem shows, it is often useful to simplify between taking the first and second derivatives. Lectures #9 and #10. Match the derivatives in the table with the points a,b,c,d,e (Assume that the axes have equal scales Graph Matching Background review: estimating derivatives, one point at a time: The derivative of a function at a point represents the slope (or rate of change) of a function at Make a guess and check your answer by clicking the red question mark buttons Make a guess and check your answer by . max. 4 Example #1 Points to note: (1) the f(x) has a minimum at x=2 and the derivative has an x-intercept at x=2 (2) the f(x) decreases on (-,2) and Evaluate fa and fb . Record these in the microscope row as horizontal line segments (or 0sif the derivative does not exist). Since 3x is a first degree polynomial, we know that it will always have the same slope, and therefor the same derivative. This corresponds to a graph that does not have any concavity, such as the line above. Locate all intervals on which the original function's graph is increasing and describe the characteristics of the derivative over those same intervals . Example 2. When reading a derivative graph(fx() c): x-intercepts represent x-values where horizontal tangents occur on original function AND In our last lecture, we talked about the derivative of a quadratic function. , sketch a possi ble graph of f on the same axes. 3. Connecting f, f', and f'' graphically. It is important for student's conceptual understanding of the relationship between a function and its derivatives that students don't simply rely on their knowledge of derivative rules to be able to match a function graph with its derivative graph. Solve 3x 2 - 8x + 4 = 0. solutions are: x = 2 and x = 2/3, see table of sign below that also shows interval of increase/decrease and maximum and minimum points. ; 4.5.3 Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function's graph. The graph of a function, drawn in black, and a tangent line to that function, drawn in red. For an example of nding and using the second derivative of a function, take f(x) = 3x3 6x2 + 2x 1 as above. We take the derivative of f (x) to obtain f' (x) = 2x. When you think you have a good representation of f (x), click the "Show results!" button below the applet. 3.3 Increasing & Decreasing Functions and the 1st Derivative Test Definitions: A function f is increasing on an interval if, for any two numbers x1 and x2 in the interval, x1 < x2 implies that f(x1) < f(x 2). Imagine a point moving along the original graph, and the tangent to the graph at that point. Using a straight edge, draw tangent lines to the graph of the function at specified points on the curve. The graph flattens out for x=0, where the derivative or rate of change of g(x) becomes zero. (b)sinx cosx is the derivative and cosx is an antiderivative. You just take the derivative of that function and plug the x coordinate of the given point into the derivative. Example 2 . It plots your function in blue, and plots the slope of the function on the graph below in red (by calculating the difference between each point in the original function . We will see how to determine the im- portant features of a graph y = f(x) from the derivatives f0(x) and f00(x), sum- marizing our Method on the last page. Write any intervals in which the . The graph of the new function is easy to describe: just take every point in the graph of f(x), and move it up a distance of d. That . The . Have fun with derivatives! We know that carries important information about the original function . Find the derivative of ( )y f x mx = = + b. Lecture 9: Partial derivatives If f(x,y) is a function of two variables, then x f(x,y) is dened as the derivative of the function g(x) = f(x,y), where y is considered a constant. Denition 3.1 Given two functions f : D ! Find the derivative of ( )y f x mx = = + b. Suppose that you are given the derivative of a function, such as cjx = f' (x) = 2x - 1 dx and you want to visualize what the original function's graph looks like. For problems 1 & 2 the graph of a function is given. 2. 2. (What qualitative feature or shape does . FINDING THE DERIVATIVE FUNCTION FROM A GRAPH Procedure: The graph of a function is drawn below for you. Choose the one alternative that best completes the statement or answers the question. Below are three pairs of graphs. Repeat question 3a 3c using the alternative definition of the derivative. Often this involves nding the maximum or minimum value of some function: the minimum time to make a certain journey, the minimum cost for doing a task, the maximum power that can be generated by a device . 3. 1. 5 Unit #3 : Dierentiability, Computing Derivatives, Trig Review Goals: Determine when a function is dierentiable at a point Relate the derivative graph to the the graph of an original function You can begin by sketching tangent lines at a few random points, and determining whether the slope. Determine the intervals on which the function increases and decreases. Identify the abs. time and acceleration vs Basic Graphs Worksheets Exponential and second derivative graphs functions can avoid common functions Graph of derivative 15 y x3 2 2 1 x 2 Free Response Question Let f be a function defined on the closed interval 5dxd5 with f 1 3 y x3 2 2 1 x 2 Free Response Question Let f be a function defined on the closed . . In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. function fx on the interval ab, use the following process. Use the following table to find and classify the critical points for the original function fx( ). (Graphs of Derivative) (Graphs of Function) 13. Sketch given the graph 5.8 Sketching Graphs of Derivatives Calculus The graph of a function is shown. inverse is the domain of the original function. Select a simple function w(x) that appears in the integral. since g(h(x))=sinh(x)& derivative sinis cos because h(x)=x2& its derivative is 2x Therefore In each of these cases we pretend that the inner function is a single variable and derive it as such 2.Another way to view it f (x)=e sin(x**2) Create temp variables u=sin v, v=x2, then f (u)=euwith computational graph: 19 df dx = df dg dg . Calculate the slope of each of the tangent lines drawn. (c) x5 5x4 is the derivative and x6=6 is an antiderivative. (e)3e2x 6e2x is the derivative and (3=2)e2x is an . 2. The slope of a line tangent to the graph at a is - lim This is the derivative of the function. F by g f(x):=g(f(x)). Derivatives are a fundamental tool of calculus.For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the . 2. On the same coordinate plane, sketch a graph of , the derivative of . Think about this one graphically, too. As the point moves along the graph, the slope of the tangent changes. The partial derivative with respect to y is dened similarly. Transformations of the Functions. Thus, the slope of the line tangent to the . Derivative Plotter. (1 point) The graph shown is the graph of the SLOPE of the tangent line of the original function. Chapter 9 - GRAPHS and the DERIVATIVE 199 Procedure 9.1 Graphing y= f[x] with the First Derivative 1. Practice: Visualizing derivatives. x() 2 xc2 Notice when reading the graph of the derivative, the y-values represent the slope of the graph of the original function at the same x-value. Basic Elementary Functions. 2.Find derivatives and antiderivatives of the following. 1. Search: Graphing Derivatives Worksheet With Answers. Write it in the form dw = . 2. Problems range in difficulty from average to challenging. . The top graph is the original function, f (x), and the bottom graph is the derivative, f' (x). BCcampus Open Publishing - Open Textbooks Adapted and Created by BC Faculty If f(x) is a function, then remember that we de ne f0(x) = lim h!0 (a) For 5 x 5 1). Unit #3 : Dierentiability, Computing Derivatives, Trig Review Goals: Determine when a function is dierentiable at a point Relate the derivative graph to the the graph of an original function 2) f is continuous on [-10, 10] because f is differentiable (as explained in #1). Sketch a graph of the function whose derivative satisfies the properties given in the table below. We also use the short hand notation . 2. Graph's Intercepts. 4. Then see if you can figure out the derivative yourself. Below is the graph of the derivative of a function. The graph of f', the derivative of f, consists of two semicircles and two line segments, as shown above. derivatives for classes of functions, including polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric. First, we consider where the graph is rising %and falling &. Likewise, at x = 1, the second derivative of f(x) is f00(1) = 18 112 = 1812 = 6; so the graph of f(x) is . Think about this one graphically, too. Plus each one comes with an answer key Beyond Calculus is a free online video book for AP Calculus AB y x3 2 2 1 x 2 Free Response Question Let f be a function defined on the closed interval 5dxd5 with f 1 3 You will split each absolute value equation into two separate equations, then solve them to find your two solutions Solutions B Answers . Compute f0[x] and nd all values of xwhere f0[x]=0(or f0[x] does not exist). This reveals the true graph of f (x), drawn in red. NMSI suggests using this activity before teaching derivative rules to students. Locate any x-intercepts of the derivative graph, and describe the characteristics of the original function at those same values of x. To the right is a graph of the function y fx= , a function whose domain is the interval [A,G]. f (x) = 511 (5 . In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Evaluate fx at all points found in Step 1. dx 3. Applications of the Derivative 6.1 tion Optimiza Many important applied problems involve nding the best way to accomplish some task. (a) ex ex is both the derivative and an antiderivative. The graph of , the derivative of , is shown. Lecture 9: Partial derivatives If f(x,y) is a function of two variables, then x f(x,y) is dened as the derivative of the function g(x) = f(x,y), where y is considered a constant. We can extend this notion of derivative to many other functions. ANSWERS 1) f is differentiable on [-10, 10] because f' is continuous on this interval. Answer (1 of 11): The first thing to remember geometrically is the derivative is the slope of the line tangent to the graph. Find dw dx by dierentiating. Suppose d 2 R is some number that is greater than 0, and you are asked to graph the function f(x)+d.  The partial derivative with respect to y is dened similarly. Example 2 . Te Collee oar: 1 . To retrieve these formulas we rewrite the de nition of the hyperbolic function as a degree two polynomial in ex; then we solve for ex and invert the exponential. Unit10- GraphsofAntiderivatives;SubstitutionIntegrals 21 Steps in the Method Of Substitution 1. To the right is a graph of the function y fx= , a function whose domain is the interval [A,G]. Make . to the original result of the sine function. Then f0(x) = 9x2 12x + 2, and f00(x) = 18x 12. Every function can have at most one y-intercept. 2. Solution. We can also verify this by looking at the graph, noticing that . 1) The graph of y = f(x) in the accompanying figure is made of line segments joined end to end. How can you change the graph of f to obtain the graphs of the rst three functions? 7. y xxe 8. f xtan 1 9. f 1 x x 3 x 4 10. y x3 2 2 1 x 2 Free Response Question Let f be a function defined on the closed interval 5dxd5 with f 1 3. This illustrates a general principle: . A function, f, is decreasing over an interval if the graph, y = f(x), falls from left to right; in other words, if y decreases as x increases. This is a linear function, so its graph is its own tangent line! Compare these derivatives to the graph above. Worksheet for Week 3: Graphs of f(x) and f0(x) In this worksheet you'll practice getting information about a derivative from the graph of a function, and vice versa. Drag the blue points up and down so that together they follow the shape of the graph of f (x). This is the currently selected item. The gure below shows that the formula agrees with the fact that the graph of f 1 is the re ection across the 45 line y= xof the graph of f. Such a . We learn how to sketch a function y = f(x) given the graph of its derivative y = f'(x) and how to interpret, or read, the graph of a function's derivative fu. On the same coordinate plane, sketch a possible graph of . The slope of the tangent line equals the derivative of the function at the marked point. ; 4.5.4 Explain the concavity test for a function over an open interval. For each function to be graphed, the calculator creates a JavaScript function, which is then evaluated in small steps in order to draw the graph. Free derivative calculator - differentiate functions with all the steps. x, the rst derivative f0(x) > 0, so the function f(x) is always increasing. Most of the trip is on rural interstate highway at the 65 mph speed limit. 4. since g(h(x))=sinh(x)& derivative sinis cos because h(x)=x2& its derivative is 2x Therefore In each of these cases we pretend that the inner function is a single variable and derive it as such 2.Another way to view it f (x)=e sin(x**2) Create temp variables u=sin v, v=x2, then f (u)=euwith computational graph: 19 df dx = df dg dg . Solve the problem. }$$ The second derivative is acceleration or how fast velocity changes.. Graphically, the first derivative gives the slope of the graph at a point. We know that the derivative means the rate of change of the function. The . The y-coordinate of the point where a function's graph intersects the y-axis is called the y-intercept of the graph. Use the original definition of the derivative to find the derivative of each function at the indicated point. On the other hand, it is the height of the graph of the derivative f0 above 2.

1. Search: Graphing Derivatives Worksheet With Answers. Warning: For a di erentiable function f(x), any place where it has a local . A function is decreasing on an interval I if, for any pair of points, < in 1, f(xl) > For example, the graph of y = f(x) = x3 decreases throughout its domain. 3. Use first and second derivative theorems to graph function f defined by. The second thing to remember is start with positive, negative, or zero. Find all critical points of fx in ab, . If the slope of f (x) is negative, then the graph of f' (x) will be below the x-axis. Afterwards, we just plug the x coordinate of (2,4) into f' (x). The graph has a vertical tangent line at and horizontal tangent lines at and What are all values of : x, at which : f: is continuous but not differentiable? It is called partial derivative of f with respect to x. d) at 2- 34 c. 4. at a = 2 KO b) at a =2. On the same coordinate plane, sketch a possible graph of . Differential calculus.

(This slope is also called the derivative of f.) For each interval, enter all letters whose corresponding state- ments are true for that interval. 4. F,wecan dene the composite function g f : D ! Exponential and Log Functions Worksheet Exponential Functions and Inverse of a Function 1 Review Exercises We have 10 unit tests which cover the major topics of this course, followed by a full-length AP Calculus AB practice exam Advanced Placement Calculus (also known as AP Calculus, AP Calc, or simply AB / BC) is a set of two distinct Advanced . Use the graph of f below to answer each question. It is called partial derivative of f with respect to x. Section 4-5 : The Shape of a Graph, Part I. We also use the short hand notation . Chapter 3 The Derivative Name_____ MULTIPLE CHOICE. Notes and Solutions . ; 4.5.2 State the first derivative test for critical points. symmetric around zero) and the odd order derivatives are odd functions (antisymmetric around zero). f (x) = x 3 - 4x 2 + 4x. = . An easy to follow tutorial on function derivatives and their computation using the definition of a derivative along with examples. (d)5x4 20x3 is the derivative and x5 is an antiderivative. (a)Go to the website http://www.shodor.org/interactivate/activities/Derivate/ (b)Enter the function y = x2x 2. When x>0, g(x) increases quadratically with the increase in x . ii. a. useful function, denoted by f0(x), is called the derivative function of f. De nition: Let f(x) be a function of x, the derivative function of f at xis given by: f0(x) = lim h!0 f(x+ h) f(x) h If the limit exists, f is said to be di erentiable at x, otherwise f is non-di erentiable at x. a . 15.2.Relating graph of function to graph of derivative We give a series of examples with the graph of a function on the . The slope of the tangent line, the derivative, is the slope of the line: ' ( ) = f x m. Rule: The derivative of a linear function is its slope . The accompanying figure shows the graph of the derivative of a function f. The domain of f is the closed interval > 3,3@. The derivative at a point is the slope of the tangent to the graph of y = f ( x) at that point. Let us draw the graph of a function f(x) on an xy-plane. (Note: this is a formula for the of the original derivative function). At the end, you'll match some graphs of functions to graphs of their derivatives. This is how the graphs of Gaussian derivative functions look like, from order 0 up to order 7 (note the marked increase in amplitude for higher order of differentiation): 53 4.1 . It also does not have any points with verticle slope. 1. A function f is decreasing on an interval if, Definitions of y-intercept. Absolute maximum and minimum values at endpoints and where f0(x) = 0. x y Figure 11.

3 Composite Functions Apart from addition, subtraction, multiplication and division to get new functions, there is another useful way to obtain new functions from old called composition . 3. Type in any function derivative to get the solution, steps and graph more. Place the values of : ;, : ;, and : ; in increasing order for each point on the graph of . Chapter 9 - GRAPHS and the DERIVATIVE 197 Exercise Set 9.2 Make a qualitative rough sketch of a graph of the distance traveled as a function of time on the following hypothetical trip: You travel a total of 100 miles in 2 hours. While graphing, singularities (e. g. poles) are detected and treated specially. For example: y = sinhx = ex e x 2,e2x 2yex 1 = 0 ,ex = y p y2 + 1 and since the exponential must be positive we select the positive sign . 1. i. You can continue to move points and see how the accuracy changes. 2. Example 4 Find f0(x) and f00(x) if f(x) = x x1. If you nd more than one way of writing these functions in terms of f and g, show that they are equivalent. Find the derivative of ( ) f x =135. GRAPHS OF FUNCTIONS AND DERIVATIVES 5 x y Figure 10. On the one hand, it is the slope of the line tangent to the graph of the original function f above 2. Also, for all x, the second derivative is 0. This is a linear function, so its graph is its own tangent line! (smallest function value) from the evaluations in Steps 2 & 3. From a graph of a function, sketch its derivative 2. Steps . (largest function value) and the abs. Solution. Find the derivative of ( ) f x =135. 22 Derivative of inverse function 22.1 Statement Any time we have a . The nth derivative of Write any intervals in which the . From a graph of a derivative, graph an original function. If you have a graph, you can estimate the derivative one point at a time by drawing the tangent line at that point, then calculating the slope of that tangent line (remember, slope is rise over run). increasing x decreases g(x) and hence g'(x) < 0 for x<0. Since the derivative represents velocity, imagine that before t= 1 you are driving toward one direction with a velocity of, say, 30 miles per hour. Use the following table to find and classify the critical points for the original function fx( ). a. The graphical relationship between a function & its derivative (part 1) The graphical relationship between a function & its derivative (part 2) Matching functions & their derivatives graphically. Assignment I Example Find f '(x) of the function. Solution to Example 2. step 1: f ' (x) = 3x 2 - 8x + 4. Graph Of Derivative To Original Function What do you notice about each pair? Learning Objectives. Describe the general shape of the derivative graph. Absolute maximum and minimum values at endpoints and where f0(x) does not exist. CH.2.4, 2.5. Compare the graphs of a function and its derivative below. the zeroth order) derivative functions are even functions (i.e. E and g : E ! min. Graph the derivative of f. x y 3. Graphing Using First and Second Derivatives GRAPHING OF FUNCTIONS USING FIRST AND SECOND DERIVATIVES The following problems illustrate detailed graphing of functions of one variable using the first and second derivatives. 2.4 The Derivative Function. So say we have f (x) = x^2 and we want to evaluate the derivative at point (2, 4). You do not know any actual points on the graph of the function, but the derivative tells you that at any particular point (xo , Yo), the local slope of the function graph is 2xo - 1 . We have seen how to create, or derive, a new function from a function , summarized in the paragraph containing equation 2.1.1.

x y Figure 12. (Note: this is a formula for the of the original derivative function). Transformations "after" the original function Suppose you know what the graph of a function f(x) looks like. The slope of the tangent line, the derivative, is the slope of the line: ' ( ) = f x m. Rule: The derivative of a linear function is its slope . The graph of y = x3 x on [0,1.5]. That tells you how the derivative is changing.