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Editorial Reviews. About the Author. William Ma is a math consultant and former chair of the 5 Steps to a 5: AP Calculus AB 4th Edition, Kindle Edition. by . Download Now: anesi.info # PDF ~ 5 Steps to a 5 AP Calculus AB (5 Steps to a 5 on the. 5 STEPS TO A 5. ™. A Perfect Plan for the Perfect Score on the Advanced Placement Exams. AP Calculus AB/BC Questions to Know by Test Day Zachary.
Thus the minimum value of l is Differentiate V x. The function f is monotonic—strictly increasing or decreasing. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Write the given inequality in standard form with the polynomial on the left and zero on the right.
Example 1 The graph of a function f on [0. Method 1: Selected values of f are shown below. Differentiation Method 3: Differentiate with respect to y: Solve for dy. Differentiation Example 2 Example 1 could have been done by using implicit differentiation.
Since y 1 is strictly increasing. Differentiate each term implicitly with respect to x. You can continue to differentiate f as long as there is differentiability. But always select an answer to a multiple-choice question.
Dx3 y. Using the slope of the line segment joining 2. Using implicit differentiation. Differentiation 7.
Using a calculator. Part B Calculators are allowed. Find dy. Let f be a continuous and differentiable function. The graph of a function f on [1.
Applying the product rule. Calculator Let f be a continuous and differentiable function. Applying the quotient rule. Applying the power rule. Since 3e 5 is a constant. Thus f x has an inverse. Differentiate with respect to x: Thus the point 0. The graph of y 1 is strictly increasing. Since the degree of the polynomial in the denominator is greater than the degree of the polynomial in the numerator.
Therefore the slope is 0. Checking the three conditions of continuity: Differentiation Thus. Mean Value Theorem. Many questions on the AP Calculus AB exam involve working with graphs of a function and its derivatives.
Do not forget to change it back to Radians after you have finished using it in Degrees. See Figure 8. Mean Value Theorem If f is a function that satisfies the following conditions: To find c.
Find all values of c. Note f x is a polynomial and thus f x is continuous and differentiable everywhere. Using the Mean Value Theorem. Determine if the hypotheses of the Mean Value Theorem are satisfied on the interval [0. Since f x is defined for all real numbers. You might need this to find the volume of a solid whose cross sections are equilateral triangles. Since f x is a polynomial. Extreme Value Theorem If f is a continuous function on a closed interval [a.
Since f x is a x rational function. Since f x is not continuous on [0. Test for Increasing and Decreasing Functions. To find the critical numbers of f x.
Let f be a function defined at a number c. Figure 8. Find the intervals on which f is increasing or decreasing. Find the critical numbers of f. Set up a table. Determine intervals. Write a conclusion. You have to be careful in filling in the bubbles especially when you skip a question. Use the First Derivative Test. Find the relative extrema of f. Graphs of Functions and Derivatives 1. Find all critical numbers of f x. Find all critical numbers of f.
Apply the Second Derivative Test. Using the First Derivative Test Step 1: Using the First Derivative Test. See Figures 8. There are some textbooks that define a point of inflection as a point where the concavity changes and do not require the existence of a tangent at the point of inflection. The converse of the statement is not necessarily true.
There are two points of inflection: In that case. Graphs of Functions and Derivatives Note that if a point a. The graph of y 2. Using the [Zero] function. Graphs of Functions and Derivatives Step 4: Which of the following statements is true? Determine if the function has any symmetry. Using the numbers in Step 4. Set up a table using the intervals. Find any horizontal. Graphing without Calculators. Determine the domain and if possible the range of the function f x.
You can earn many more points from other problems. Sketch the graph.. Do not linger on a problem too long. If necessary. Set up a table: Possible points of inflection: Critical numbers: No asymptote 5. Range 0. Relative maximum: No symmetry 3. Here are some examples. Example 1 The graph of a function f is shown in Figure 8. The graph is concave downward on a. Which of the following is true for f on a. Since f is strictly increasing. Find where f has the points of inflection.
Find where f has its absolute extrema. Find the intervals where f is concave upward or downward. Sketch a possible graph of f. Find the values of x where f has change of concavity. Graphs of Functions and Derivatives 8. Given f is twice differentiable. Find the values of x where f has a relative minimum.
Given the function f in Figure 8. Show that the hypotheses of the Mean Value Theorem are satisfied on [0. The graph of f is shown in Figure 8.
Which of the following has the largest value: Sketch the graphs of the following functions indicating any relative extrema. Graphs of Functions and Derivatives Given the graph of f in Figure 8.
Find dx2 Take ln of both sides. Find critical numbers. Now we check the end points. Check for tangent line: And since the point 0. Check intervals. The correct answer is A. Step 6: Set up a table Table 8. Step 8: Graphs of Functions and Derivatives Table 8. Set up table as below: Step 7: Sketch the graph. Some textbooks define a point of inflection as a point where the concavity changes and do not require the existence of a tangent. Step 9: Using the functions of the calculator. A fundamental domain of y 1 is [0.
Note that the graph has a symmetry about the y -axis. Represent the given information and the unknowns by mathematical symbols.
Shadow Problem. Common Related Rate Problems. Read the problem and. If the equation contains more than one variable. Inverted Cone Water Tank Problem. Write an equation involving the rate of change to be determined. Two of the most common applications of derivatives involve solving related rate problems and applied maximum and minimum problems.
Common Related Rate Problems Example 1 When the area of a square is increasing twice as fast as its diagonals. Since Note: Solve the resulting equation for the desired rate of change. Example 2 Find the surface area of a sphere at the instant when the rate of increase of the volume of the sphere is nine times the rate of increase of the radius. Find the volume of the cone at the instant when the rate of increase of the volume is twelve times the rate of increase of the radius.
Example 3 The height of a right circular cone is always three times the radius. Surface area of a sphere: Differentiate each term of the equation with respect to time. Let z represent the diagonal of the square. Write the answer and indicate the units of measure. Substitute all known values and known rates of change into the resulting equation.
Define the variables. Applications of Derivatives Let r. Let V be the volume of water in the tank. Usually that is the correct one. See Using similar triangles. How fast is the water level rising when the water is 5 meters deep? See Figure 9. The height of the cone is 10 meters and the diameter of the base is 8 meters as shown in Figure 9.
Set up an equation: Building Light 6 ft ft Figure 9. Substitute known values. How fast is his shadow on the building becoming shorter when he is 40 feet from the building? Differentiate both sides of the equation with respect to t. Write an equation using similar triangles.
Applications of Derivatives Solution: Let x be the distance between the balloon and the ground. Find dt Step 4: Differentiate both sides with respect to t. Figure 9. Area and Volume Problems. Differentiate to obtain the first derivative and to find critical numbers.
Write an equation that is a function of the variable representing the quantity to be maximized or minimized. Distance Problem Find the shortest distance between the point A Let P x.
Determine what is given and what is to be found. Read the problem carefully and if appropriate. Distance Problem. If the equation involves other variables. Determine the appropriate interval for the equation i..
Applications of Derivatives 9. Write the answer s to the problem and. Draw a diagram. Check the function values at the end points of the interval. Using the distance formula. Apply the First Derivative Test. Special case: In distance problems. Using Synthetic Division. Find the dimensions of the rectangle so that its area is a maximum. Step 4 could have been done using a graphing calculator.
Apply the Second Derivative Test: Note that at the endpoints: The domain of V is [0. STEP 4. Let x be the length of a side of the square to be cut from each corner. Differentiate V x.
Applications of Derivatives 10 Step 6: Using the Second Derivative Test. Since it is the only relative 3 maximum on the interval. V is a relative maximum. In terms of the surface area. The radius of a sphere is increasing at a constant rate of 2 inches per minute. Apply Second Derivative Test. Write an equation. If h is the diameter of a circle and h is increasing at a constant rate of 0.
Write a solution. A foot ladder is leaning against a wall. How fast is the diameter increasing when the radius is 5 cm?
The first car is going due east at the rate of 40 mph and the second is going due south at the rate of 30 mph.
Find the volume of the balloon at the instant when the rate of increase of the surface area is eight times the rate of increase of the radius of the sphere. A water tank in the shape of an inverted cone has a height of 18 feet and a base radius of 12 feet.
Two cars leave an intersection at the same time. A spherical balloon is being inflated. How fast is her shadow lengthening? How fast is the distance between the two cars increasing when the first car is miles from the intersection? Wall 13 ft Ground Figure 9. Applications of Derivatives 4. If the perimeter of an isosceles triangle is 18 cm. Find the length of a side of the square being cut so that the box will have a maximum volume.
If the speed of the plane is mph. If water is being drained from the conical. Find the rate of change of the angle of elevation of the camera 5 sec after the rocket went up.
A trough is 10 meters long and 4 meters wide.
How fast is the water level rising when the water is 2 meters high? A camera m away is recording the rocket. Two water containers are being used. Find the maximum area that the man can obtain. What is the shortest distance between the 1 point 2. The other container is a right circular cylinder with a radius of 6 feet and a height of 8 feet. An open box is to be made using a piece of cardboard 8 cm by 15 cm by cutting a square from each corner and folding the sides up.
A man with meters of fence plans to enclose a rectangular piece of land using a river on one side and a fence on the other three sides. The two sides of the trough are equilateral triangles. Find a number in the interval 0. The wall of a building has a parallel fence that is 6 feet high and 8 feet from the wall. Evaluate lim If the hypotenuse passes through the point 0. Fence 6 ft How fast is the water level rising in the cylindrical container? Wall A right triangle is in the first quadrant with a vertex at the origin and the other two vertices on the x.
Find where the function f: What is the length of the shortest ladder that passes over the fence and leans on the wall? Using similar triangles. Light 20 ft 5 ft y x Figure 9.
Calculator Find the shortest distance between the point 1. Applying the Pythagorean Theorem. Substitute all known values into the equation: Using the Pythagorean Theorem. Applications of Derivatives Step 2: By the Pythagorean Theorem. Second Derivative Test: Check endpoints.
Since it is the only relative extremum.
Let x be the number and reciprocal. Using the quadratic formula. First Derivative Test: The dx 2 9 critical numbers are and 6. V is 3 the absolute maximum. Set Step 4: Average Cost: Z has a minimum. Distance Formula: Substitute known values: Differentiate with respect to t. River Substitute known values into the equation: Let h be the height of the trough and 4 be a side of one of the two equilateral triangles. The dt plane is gaining altitude at The angle dt of elevation is changing at 0.
Let x be the distance of the foot of the ladder from the higher wall. Pythagorean Theorem: Let y be the height of the point where the ladder touches the higher wall. Thus the minimum value of l is The graph of the average cost function is shown in Figure 9. Verify the result with the First Derivative Test. Apply the First Derivative Test: Note that: The 2. The slope of the hypotenuse: Distance formula: Calculator See Figure 9. Solve for y: Since f has only one relative extremum.
Thus the shortest distance is approximately 0. Verify the result with the First Deriative Test. Tangent Lines. You will also learn to apply derivatives to solve rectilinear motion problems. See Figure See 2 Figure Write an equation of each vertical tangent. Find dx. Find points of tangency. Find the x -coordinate of points of tangency. Write equations for vertical tangents: See Figures Example 6 Using your calculator. Using the [Intersection] function of the calculator for y 2 and y 3.
Find the x -coordinate of the points of tangency. Using the [Solve] function of the calculator. More Applications of Derivatives Example 5 Using your calculator. At these points. Special Cases: Find m normal.
Write equation of normal line. Find m tangent. If yes. Find the point on the graph. Tangent Line Approximation. Equation of normal: Find other points of intersection. More Applications of Derivatives Step 3: Write equation of normal.
Estimating the nth Root of a Number. Use the tangent line to find the approximate values of f 1. Using tangent line approximation.
Differentiate f x: The point 3. The slope of a function at any point x. Instantaneous Velocity and Acceleration. Vertical Motion. Find a the position. Instantaneous Velocity: Instantaneous speed: The graph of a t indicates that: Example 3 The graph of the velocity function is shown in Figure Vertical Motion Example From a foot tower.
Describe the motion of the particle.
More Applications of Derivatives Step 2: Then it reverses direction moving to the right again and speeds up indefinitely. Published in: Full Name Comment goes here. Are you sure you want to Yes No. Be the first to like this. No Downloads. Views Total views. Actions Shares. Embeds 0 No embeds. No notes for slide.
AP Calculus AB 2. Book Details Author: William Ma Pages: McGraw-Hill Education Brand: Actions Shares. Embeds 0 No embeds. No notes for slide. Book details Author: William Ma Pages: McGraw-Hill Education Language: English ISBN Description this book Please continue to the next page5 Steps to a 5: If you want to download this book, click link in the last page 5.
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