calc at university
Try these freshmnan calc tests on for size:
2300H, test 2, smith, Fall 2000 Name:
(25)I. (i) Give the limit definition of the derivative f’(a).
(ii) Give the neighborhood definition: limx-->a f(x) = L if and only if:
(iii) Give the explicit definition (using epsilon, delta, M, N as appropriate):
limx-->+inf f(x) = L (where L is a real number) if and only if:
(iv) Give the explicit definition: limx-->a+ f(x) = -inf, if and only if:
(v) State the intermediate value theorem:
(20)II. True or False?
(i) If f is continuous on (a,b), f is globally bounded there.
(ii) If g is continuous on (-inf,+inf), g is locally bounded at each point.
(iii) f(x) = 1/x is globally bounded on [1,2].
(iv) g(x) = cos(1/x) is globally bounded on (0,1).
(v) If both one sided limits limx-->a+ f(x) and limx-->a- f(x) exist, then limx-->a f(x) also exists.
(30)III. Find the limits, or if they do not exist, say so. A limit is considered to “exist” if it equals either a finite real number or +inf or -inf.
(i) If [x] = the greatest integer not greater than x, limx-->0- [x] = ?
(ii) If g(x) = xcos(1/x), then limx-->0 g(x) = ?
(iii) limx-->(-5)(x^2-25)/(x+5)= ?
(iv) limx-->2+ (x-3)/(x^2-4) = ?
(v) limx-->+inf (cos(1+x^3))/x = ?
(vi) limx-->0 (x^2+x)/(4x-sin(x)) = ?
(10)IV. Prove there is a real number x such that x^3 + x - 9 = 0. If you use any big theorems, explain why they apply to this situation.
(15)V. Either: prove the “power rule” that if f is differentiable at a, then so is f^n for every natural number n,
Or: prove that sin’(x) = cos(x), (assuming the basic trig limits and the trig addition formulas).
EXTRA: Either prove (5) a differentiable function is continuous,
or (10) a locally bounded function on [0,1], is globally bounded.
2310H Test 2 Fall 2004, Smith NAME:
no calculators, good luck! (use the backs)
1. (a) Define "Lipschitz continuity" for a function f on an interval I.
(b) State a criterion for recognizing Lipschitz continuity in the case of a differentiable function f on an interval I.
(c) Determine which of the following functions is or is not Lipschitz continuous, and explain briefly why in each case.
(i) The function is f(x) = x1/3, on the interval (0, ).
(ii) The function is G(x) = indefinite integral of [t], on the interval [0,10], (where [t] = "the greatest integer not greater than t", i.e. [t] = 0 for t in [0,1), [t] = 1 for t in [1,2), [t] = 2 for t in [2,3), etc...[t] = 9 for t in [9,10), [10] = 10.)
(iii) The function is h(x) = x + cos(x) on the interval (- , ).
2. (i) State the "fundamental theorem of calculus", i.e. state the key properties of the indefinite integral function G(x) = indefinite integral of f from a to x, associated to an integrable function f on a closed bounded interval [a,b]. You may assume f is continuous everywhere on [a,b] if you wish.
(ii) Explain carefully why the definite integral of f from a to b, of a continuous function f, equals H(b)-H(a), whenever H is any "antiderivative" of f, i.e. whenever H'(x) = f(x) for all x in [a,b]. Justify the use of any theorems to which you appeal by verifying their hypotheses.
(iii) Is there a differentiable function G(x) with G'(x) = cos(1/[1+x^4])?
If so, give one, if not say why not.
3. Let S be the solid obtained by revolving the graph of y = e^x around the x-axis between x=0 and x=3. Define the moving volume function V(x) = that part of the volume of S lying between 0 and x. (draw a picture.)
(i) What is dV/dx = ?
(ii) Write an integral for the volume of S, and compute that volume.
4. Consider a pyramid of height H, with base a square of side B. Define a moving volume function V(x) = that part of the volume of the pyramid lying between the top of the pyramid, and a plane which is parallel to the base and at a distance x from the top.
(i) Find the derivative dV/dx. [Hint: Use similarity.]
(ii) Find the volume V(H).
(iii) Make a conjecture about the volume of a pyramid of height H with base of any planar shape whatsoever, and base area B.
EXTRA: Either: Prove the FTC. from part 2(i), you may draw pictures and assume your f is monotone and continuous if you like.
Or: ask and answer your own question.
