Reference no: EM131076787
MATH 1A MIDTERM 1-
Problem 1-
(i) State the Squeeze Theorem.
(ii) Prove the Squeeze Theorem.
(iii) Using a carefully justified application of the Squeeze Theorem, find limx→0x7 sin(1/x).
Problem 2- Evaluate:
limx→0(√(1 + x) - √(1 - x)/x).
You should show your reasoning carefully, however you may use any of the limit laws without explanation or proof.
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 limx→af(x) = L?
(ii) Prove carefully, using the definition you gave in part (i), that limx→1(x3 + x2 + x - 5) = -2.
Problem 4-
(i) State carefully the Intermediate Value Theorem.
(ii) Prove that there is a root of the equation: 2x3 = 3x in the interval (1, 2).
Problem 5- The figure below shows the graph of y = f(x) when
For each of the following statements, indicate if it is true or false.
(i) limx→4 f(x) = 3
(ii) limx→5^+ f(x) = ∞
(iii) limx→5 f(x) = ∞
(iv) limx→-3 f(x) exists
(v) limx→- π/2^+ f(x) = ∞
(vi) limx→∞ f(x) = ∞
(vii) The graph y = f(x) has a horizontal asymptote at y = 0.
(viii) The graph y = f(x) has two horizontal asymptotes.
(ix) The graph y = f(x) has two vertical asymptotes.
(x) f(x) is continuous at x = 0.
(xi) f(x) is continuous at x = 1.
(xii) f(x) is continuous on the interval [1, 2].
Problem 6- Using the limit definition of derivative, show that if f(x) = x2, then f'(x) = 2x.
Problem 7- What is log48?
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