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Calculus and its Applications

2023–24

Assignment 1

for "Calculus and its Applications" by

11:00am on Monday, February 5, 2024.

Each question will be marked out of 5, for an overall maximum of 10 marks.

5. Perfect (or nearly so)

4. Good grade A quality. (Correct except for relatively minor points)

3. Grade B quality (the majority of the work is correct)

2. Bare passing quality. (Some substantial progress, but no more than half complete)

1. Very little progress made.

Missing submissions or meaningless attempts will be awarded 0 marks.

1. Consider the function f(x) = x/(x+3).

a) Find the domain and the range of the function.

b) Calculate the derivative f′ of f from first principles, i.e., using only the definition and none of the rules of differentiation.

c) Use the expression for f′ from item a) to calculate the second derivative f′′ of f from first principles.

d) Verify your expressions for f′ and f′′ from items a) and b) by applying the rules of differentiation to f.

2. a) With justification provide a counter-example to the following statement.

Assume limx→a f(x) + g(x) exists then both limx→a f(x), limx→a g(x) exist and limx→a f(x) + g(x) = f(a) + g(a).

b) Explain why the Limit Laws do not apply in this case.

c) Given the function f(x) = e−4x sin(8x), show that it satisfies the equation

f"(x) + 8f′ (x) + 80f(x) = 0.