Recall that the graph of a function is symmetric with

Find the value of that maximizes the angle shown in the figure. What is the approximate measure of this angle?

Recall that the graph of a function is symmetric with respect to the origin if whenever is a point on the graph, is also a point on the graph. The graph of the function is symmetric with respect to the point if, whenever is a point on the graph, is also a point on the graph, as shown in the figure. (a) Sketch the graph of on the interval Write a short paragraph explaining how the symmetry of the graph with respect to the point allows you to conclude that (b) Sketch the graph of on the interval Use the symmetry of the graph with respect to the point to evaluate the integral (c) Sketch the graph of on the interval Use the symmetry of the graph to evaluate the integral (d) Evaluate the integral

(a) Use a graphing utility to graph on the interval (b) Use the graph to estimate (c) Use the definition of derivative to prove your answer to part (b).

Let (a) Determine the domain of the function (b) Find two values of satisfying (c) Find two values of satisfying (d) What is the range of the function (e) Calculate and use calculus to find the maximum value of on the interval (f) Use a graphing utility to graph in the viewing window and estimate if it exists. (g) Determine analytically, if it exists.

Graph the exponential function for 1.2, and 2.0. Which of these curves intersects the line Determine all positive numbers for which the curve intersects the line

(a) Let be a point on the unit circlein the first quadrant (see figure). Show that is equal to twicethe area of the shaded circular sector(b) Let be a point on the unit hyperbolain the first quadrant (see figure). Show that isequal to twice the area of the shaded region Begin byshowing that the area of the shaded region is given bythe formulax1O 1PA(1, 0)ytAt 12cosh t sinh t

cosh t1x2
1 dx.AOPAOP.x2
y2  1 tPcosh t, sinh tx1O 1Pt A(1, 0)yAOP.

Consider the three regions and determined by the graph of as shown in the figure. (a) Calculate the areas of regions and (b) Use your answers in part (a) to evaluate the integral (c) Use your answers in part (a) to evaluate the integral (d) Use your answers in part (a) to evaluate the integral

Let be the tangent line to the graph of the function at the point Show that the distance between and is always equal to 1.

Let be the tangent line to the graph of the function at the point Show that the distance between and is always equal to 1.

Use integration by substitution to find the area under the curve between and

Use integration by substitution to find the area under the curve between and

(a) Use a graphing utility to compare the graph of the function with the graphs of each of the given functions. (i) (ii) (iii) (b) Identify the pattern of successive polynomials in part (a) and extend the pattern one more term and compare the graph of the resulting polynomial function with the graph of (c) What do you think this pattern implies?

A $120,000 home mortgage for 35 years at has a monthly payment of $985.93. Part of the monthly payment goes for the interest charge on the unpaid balance and the remainder of the payment is used to reduce the principal. The amount that goes for interest is and the amount that goes toward reduction of the principal is In these formulas, is the amount of the mortgage, is the interest rate, is the monthly payment, and is the time in years. (a) Use a graphing utility to graph each function in the same viewing window. (The viewing window should show all 35 years of mortgage payments.) (b) In the early years of the mortgage, the larger part of the monthly payment goes for what purpose? Approximate the time when the monthly payment is evenly divided between interest and principal reduction. (c) Use the graphs in part (a) to make a conjecture about the relationship between the slopes of the tangent lines to the two curves for a specified value of Give an analytical argument to verify your conjecture. Find and (d) Repeat parts (a) and (b) for a repayment period of 20 years What can you conclude?