MATH 1A FINAL-

Problem 1- (i) Give the precise definition of the definite integral using Riemann sums.

(ii) Write an expression for the definite integral ₁∫⁵ e^x/e^x+3 dx, giving your answer as a limit and using right hand endpoints as sample points.

(iii) Calculate the integral ₁∫⁵ e^x/e^x+3 dx using any methods you like.

Problem 2- (i) State carefully and in full the Fundamental Theorem of Calculus.

(ii) Suppose that a particle moves along the x-axis with velocity sin(πt/2), where t is time in seconds. What is the net change in position between time t = 0 seconds and time t = 1 seconds?

Problem 3- (i) Let f be a real-valued function, and let a and L be real numbers. What does it mean to say that lim_x→a^+ f(x) = L?

(ii) State the domain of the function f(x) = 8√(x - 15).

(iii) Prove, using the definition you gave in part (i), that

lim_x→15^+8√(x - 15) = 0.

Problem 4- (i) Let y = y(t) be the number of bacterial cells growing in a culture, where t is time measured in days. Suppose the rate of growth of y(t) is proportional to y(t). That is, suppose that y'(t) = Cy(t) where C is constant. Prove carefully that y(t) = y(0)e^Ct.

(ii) You are given that at time t = 0 there are 3 cells in the culture. After 6 days there are 27 cells in the culture. How many days will it take for there to be 81 cells?

Problem 5- (i) A bowl is made by rotating the region bounded by the curve y = x³, the x-axis, and the line x = 2 about the y-axis. Find the volume of the bowl.

(ii) A football is made in the shape of y = sin x, rotated about the x-axis between x = 0 and x = π. Find the volume of the football. (You may use the fact that cos(2θ) = 1 - 2 sin²θ without proof.)

Problem 6- A ladder that is 13 meters long is leaning against a vertical wall and standing on horizontal ground. The bottom of the ladder slips. Assume that the top of the ladder stays in contact with the wall, and the bottom of the ladder stays in contact with the ground. Calculate the speed that the bottom of the ladder is moving away from the wall when the top of the ladder is 5 meters above the ground and moving downwards at 3 meters per second.

Problem 7- (i) Suppose that yx - sin(x⁴) = 5^x. Find dy/dx. You may leave your answer in terms of both x and y.

(ii) Suppose y = x^5x. Find dy/dx.

Problem 8- Calculate the following limits:

(i) lim_x→∞ x/e_2x

(ii) lim_x→0(1-cos(3x)/x²)

(iii) lim_x→∞(3x²-7x+3/5x²-14)

If you use any special rules to calculate your answers, you should state the rule each time you use it.

Problem 9- Consider the graph of y = x³ - 3x².

(i) Give the coordinates of the x-intercepts.

(ii) Give the coordinates of all critical points, and indicate for each if it is a local maximum or local minimum.

(iii) Give the coordinates of any points of inflection.

(iv) The graph passes through the point (-1, -4). Describe the concavity of the graph at this point.

Problem 10- Let f and g be real-valued functions. Suppose that lim_x→a f(x) = L and lim_x→a g(x) = K. Prove that lim_x→a(f(x) + g(x)) = L + K.