(b) Use the tangent line equation you found in (a) to approximate g 3 0. What is the error in your linear approximation?Ģ(a) For g(x) = sec x, find an equation of the linear function that best fits at x (c) Find f(3) by using the function f(x). (b) Use the tangent line equation you found in (a) to approximate f(3).
Point on the graph of f, there would be different tangent line approximation.įor x < 0, is the linear approximation values smaller or larger than the values of f(x)? Why?įor x > 0, is the linear approximation values smaller or larger than the values of f(x)? Why?ġ(a) For f(x) = 0 4, find an equation of the linear function that best fits f at 3.This linear approximation of f(x) = 1 + sin x depends on the point of tangency. Values of y given by this linear approximation with the values of f(x) near x = 0. Graph f(x) and the tangent line in your calculator.
Find the equation of the tangent line at (0, 1).Ĭ. Find the derivative of f(x) = 1 + sin x.ī. Then use a table toĬompare the y-values of the linear function with those of f(x) on an open interval containingĪ. In other words, as xc, the limit of y is f(c).įind the tangent line approximation of f(x) = 1 + sin x at point (0, 1). Moreover, by restricting the values of x to be sufficientlyĬlose to c, the values of y can be used as approximations (to any desired accuracy) of the Because c is aĬonstant, y is a linear function of x. The equation for the tangent line atĪnd is called the tangent line approximation or linear approximation of f at c. To begin, consider a function f that is differentiable at c. What is the volume of the cylinderįind the points of inflection and discuss the concavity of the graph of the function: What is the maximum possible volume that could be generated?Ī cylinder is inscribed in a sphere with radius 3 inches. What are the twoĪ rectangle with perimeter of 36 inches is revolved about one of its sides to form a cylinder. What is volume of the cone of the greatest possible volume?
Find the dimensions of the circleĪnd square that produce a minimum total area.Ī tank with a rectangular base and rectangular sides is open at the top. The combined perimeter of a circle and a square is 16. Of the triangle and square that produce a minimum total area. The combined perimeter of an equilateral triangle and a square is 10. How much of the wire shouldīe used for the square and how much should be used for the circle to enclose the maximum If its width is x, express its length and area inįour feet of wire is to be used to form a square and a circle. Express the area in terms of x,Īnd find the value of x that gives the greatest area.Ī rectangle has a perimeter of 80 cm. Using a wall as one side and 120 m of fencing for Give all decimalįind two positive numbers such that the sum of the first and twice the second is 100 andĪ gardener wants to make a rectangular enclosure Write a function for each problem, and justify your answers using Calculus. Section 3 Worksheet – Optimization Problems The sum of the first number squared and the second number is 54 and the product is a.Hypotheses are satisfied – AP College Boardįind two positive numbers that satisfy the given requirement: To justify the location of a relative extremum of a function, a student could invoke the First or Second Derivative Test accompanied by evidence that the When instructed to “justify an answer”, students are expected to provide an explanation of the mathematical basis for their results or conclusions. Meters of framing materials, what must the dimensions of the window be to let in the most light? A window is being built and at the bottom is a rectangle and at the top is a semicircle.Of the field of the largest area that can be enclosed with $900 of fencing. The material for the fenceĬost $2 per foot for the two ends and $3 per foot for the side parallel to the river. A rectangular field is going to be fenced off along the bank of a river.What dimensions will give the greatest volume? An open box is to be made out of a 10’’ x 12’’ piece of cardboard by cutting squares out of theĬorners.AP Calculus AB Chapter 3 Applications of DifferentiationĤ Antiderivatives and Indefinite Integration
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