Point of Inflection Show that the cubic polynomial has

Relative Extrema Graph the fourth-degree polynomial for various values of the constant (a) Determine the values of for which has exactly one relative minimum. (b) Determine the values of for which has exactly one relative maximum. (c) Determine the values of for which has exactly two relative minima. (d) Show that the graph of cannot have exactly two relative extrema.

Relative Extrema (a) Graph the fourth-degree polynomial for 0, 1, 2, and 3. For what values of the constant does have a relative minimum or relative maximum? (b) Show that has a relative maximum for all values of the constant (c) Determine analytically the values of for which has a relative minimum. (d) Let be a relative extremum of Show that lies on the graph of Verify this result graphically by graphing together with the seven curves from part (a).

Relative Minimum Let Determine all values of the constant such that has a relative minimum, but no relative maximum.

Points of Inflection (a) Let be a quadratic polynomial. How many points of inflection does the graph of have? (b) Let be a cubic polynomial. How many points of inflection does the graph of have? (c) Suppose the function satisfies the equation where and are positive constants. Show that the graph of has a point of inflection at the point where (This equation is called the logistic differential equation.)

Extended Mean Value Theorem Prove the following Extended Mean Value Theorem. If and are continuous on the closed interval and if exists in the open interval then there exists a number in such that

Illumination The amount of illumination of a surface is proportional to the intensity of the light source, inversely proportional to the square of the distance from the light source, and proportional to where is the angle at which the light strikes the surface. A rectangular room measures 10 feet by 24 feet, with a 10-foot ceiling (see figure). Determine the height at which the light should be placed to allow the corners of the floor to receive as much light as possible.

Minimum Distance Consider a room in the shape of a cube, 4 meters on each side. A bug at point wants to walk to point at the opposite corner, as shown in the figure. Use calculus to determine the shortest path. Explain how you can solve this problem without calculus. (Hint: Consider the two walls as one wall.) Figure for 7 Figure for 8

Areas of Triangles The line joining and crosses the two parallel lines, as shown in the figure. The point is units from How far from should the point be positioned so that the sum of the areas of the two shaded triangles is a minimum? So that the sum is a maximum?

Mean Value Theorem Determine the values and such that the function satisfies the hypotheses of the Mean Value Theorem on the interval

Mean Value Theorem Determine the values and such that the function satisfies the hypotheses of the Mean Value Theorem on the interval fx

Proof Let and be functions that are continuous on and differentiable on . Prove that if and for all in then

Proof (a) Prove that (b) Prove that (c) Let be a real number. Prove that if then

Tangent Lines Find the point on the graph of (see figure) where the tangent line has the greatest slope, and the point where the tangent line has the least slope.

Stopping Distance The police department must determine the speed limit on a bridge such that the flow rate of cars is maximum per unit time. The greater the speed limit, the farther apart the cars must be in order to keep a safe stopping distance. Experimental data on the stopping distances (in meters) for various speeds (in kilometers per hour) are shown in the table. (a) Convert the speeds in the table to speeds in meters per second. Use the regression capabilities of a graphing utility to find a model of the form for the data. (b) Consider two consecutive vehicles of average length 5.5 meters, traveling at a safe speed on the bridge. Let be the difference between the times (in seconds) when the front bumpers of the vehicles pass a given point on the bridge. Verify that this difference in times is given by (c) Use a graphing utility to graph the function and estimate the speed that minimizes the time between vehicles. (d) Use calculus to determine the speed that minimizes What is the minimum value of Convert the required speed to kilometers per hour. (e) Find the optimal distance between vehicles for the posted speed limit determined in part (d).

Darbouxs Theorem Prove Darbouxs Theorem: Let be differentiable on the closed interval such that and If lies between and , then there exists in such that

Maximum Area The figures show a rectangle, a circle, and a semicircle inscribed in a triangle bounded by the coordinate axes and the first-quadrant portion of the line with intercepts and Find the dimensions of each inscribed figure such that its area is maximum. State whether calculus was helpful in finding the required dimensions. Explain your reasoning.

Point of Inflection Show that the cubic polynomial has exactly one point of inflection where and Use this formula to find the point of inflection of

Minimum Length A legal-sized sheet of paper (8.5 inches by 14 inches) is folded so that corner touches the opposite 14-inch edge at (see figure). Note: (a) Show that (b) What is the domain of (c) Determine the -value that minimizes (d) Determine the minimum length

Quadratic Approximation The polynomial is the quadratic approximation of the function at when and (a) Find the quadratic approximation of at (b) Use a graphing utility to graph and in the same viewing window.