State and prove the intermediate value theorem prove that

Question

Practice Final-

Problem 1- Give the precise meaning of the following statements.

(i) "lim_x→a f(x) = L"

(ii) "lim_x→a^+ f(x) = L"

(iii) "lim_x→+∞ f(x) = L"

(iv) "lim_x→+∞ f(x) = -∞"

(v) "lim_x→a^- f(x) = -∞"

Problem 2- Prove the following statements using the limit definitions.

(i) "lim_x→0(1/x²+1) = 1"

(ii) "lim_x→1(x²-4x+5/x+4) = 2/5"

(iii) "lim_x→+∞(e^x/e^x+x) = 1"

Problem 3- (i) State and prove the Squeeze Theorem.

(ii) Use the Squeeze Theorem to compute - lim_x→0x²(sin x)⁴(cos x)³. Justify your answer carefully.

Problem 4- Let f and g be functions, and suppose lim_x→a f(x) = L and lim_x→a g(x) = K. Prove that lim_x→a(f(x) + g(x)) = L + K.

Problem 5- Evaluate

lim_x→0((1/√(1 + x)) - (1/1 + x))²

You should show you're reasoning carefully; however you may use any of the limit laws without explanation or proof.

Problem 6- Indicate "true" if the statement is always true; indicate "false" if there exists a counterexample.

(i) "If lim_x→a f(x) = L, then lim_x→a^+ f(x) = L."

(ii) "If lim_x→a^+ f(x) = L, then lim_x→a f(x) = L."

(iii) "If lim_x→∞ f(x) = 0, then lim_x→∞ f(x)e^x = 0."

(iv) "If lim_x→a(f(x))² = 1, then lim_x→a f(x) = 1."

Problem 7- (i) Give the precise meaning of the statement "f is continuous at x = a".

(ii) Using the definition in (i), show that f(x) = x is continuous at x = 1.

Problem 8- (i) State and prove the Intermediate Value Theorem.

(ii) Prove that e^x sin x = 40 has a solution in (0, ∞).

Problem 9- (i) Give the precise meaning of the statement "f is differentiable at x = a".

(ii) Using the definition in (i), show that f(x) = x is differentiable at x = 1.

Problem 10- (i) State Rolle's Theorem.

(ii) State the Mean Value Theorem.

(iii) Prove the Mean Value Theorem using Rolle's Theorem.

Problem 11- In each of the following cases, evaluate dy/dx.

(i) y = 2x/x²+1

(ii) y = arctan((sin x)²)

(iii) y² + 3xy + x² = e^xcos x

(iv) y = x^{x^x}

Problem 12- Alexander Coward's youtube channel has 21 subscribers at time t = 0, and the number of subscribers grows exponentially with respect to time. At time t = 4, he has 103 subscribers. After how long will Alexander have 106 subscribers?

Problem 13- Which point on the graph of y = x² is closest to the point (5, -1)?

Problem 14- The interior of a bowl is a "conic frustum", where the top surface is a disk of radius 2 and the bottom surface is a disk of radius 1 and the height of the cup is 3. A liquid is being poured into the bowl at a constant rate of 4. How fast is the height of the water increasing when the bowl is full?

Problem 15- Showing your work carefully, evaluate the limit

lim_x→0((1 + sin x)² - (cos x)²/x²).

Problem 16- (i) Give the precise definition of the definite integral using Riemann sums.

(ii) What's the difference between a definite integral and an indefinite integral?

(iii) Using the definition in (i), compute ₀∫²x² dx.

Problem 17- (i) State the Fundamental Theorem of Calculus.

(ii) Let f : R → R be a differentiable function. Prove that if g is an anti-derivative of f', then there exists a constant C such that f(x) = g(x) + C for all x.

(iii) Are all continuous functions differentiable?

(iv) Do all continuous functions have anti-derivatives?

Problem 18- Compute an anti-derivative of the following functions.

(i) f(x) = 8x³ + 3x²

(ii) f(x) = (5√x + 1)²

(iii) f(x) = x√(1 + x²)

(iv) f(x) = tan(arcsin(x))

(v) f(x) = x³/√(x²+1)

Problem 19- (i) Find the volume of the solid obtained by rotating the region {(x, y): 0 ≤ x ≤ e^y, 1 ≤ y ≤ 2} about the y-axis.

(ii) Find the volume of the solid obtained by rotating about the y-axis the region between y = √x and y = x².

Problem 20- Simplify log_{log_3 9}(log₄2).