1. (5 marks). Suppose that f and g are differentiable functions and let h(x) = f(x)9(x) Note: this question requires content covered in Section 3.6. (a) (3 marks). Show that h' = f⁹−¹ (ƒ'g+ ƒg' ln ƒ) (b) (1 mark). If ƒ(x)=b (where b is a strictly positive constant), show that the formula in part (a) simplifies into a formula we have seen earlier in this course. (c) (1 mark). If g(x) = n (where n is a constant), show that the formula in part (a)simplifies into a familiar formula we have seen earlier in this course. 2. (5 marks). When you are at a height x above the surface of the earth, your weight can be found by the following formula: W(x)=m g\left(1+\frac{x}{R}\right)^{-2} where m is the mass of your body, g is the acceleration due to gravity, and R is the radius of the earth.

For example, W(0) is your weight when you are on the surface of the earth and W(R)is your weight when you are at the height R above the surface of the earth (yes, you are far away from the earth's atmosphere).

In this question, we will approximate W(0) and W(R).

Note: we have posted a video on Linear Approximations that will help with this question. (a) (1 mark). Find Lo(x), the linear approximation of W(x) at x = 0.

(b) (1.5 marks). Use L₁(x) to approximate W(0) and W(R).Which approximation is more accurate? Why?

(c) (1 mark). Find LR(x), the linear approximation of W (x) at x = R.

(d) (1.5 marks). Use LÂ(x) to approximate W(0) and W(R).Which approximation is more accurate? Why? 3. (5 marks). Determine whether each of the following statements are true or false.

If true, provide justification (calculations that justify your assertion and 1-2 sentences explaining your answer).

If false, provide a counterexample (an example of a function that satisfies the hypotheses of the statement but not the conclusion, and 1-2 sentences explaining your answer). (a) (1 mark).d/dx− (f(x)g(x)) = f'(x)g'(x)

(b) (2 marks). Every quadratic function (i.e. every function of the form f(x) =ax² +bx+c with a, b and c constants and a ‡ 0) has exactly one tangent line that is horizontal.

(c) (2 marks). Every cubic function (i.e. function of the form f(x) = ax³+bx²+cx+d with a, b, c and d constants, and a ‡ 0) has at least one tangent line that is horizontal. 4. (5 marks). An ordinary differential equation (ODE) is an equation that contains an unknown function y(x) and some of it's derivatives y'(x), y"(x), etc.

Many of you will take a course in your second year that will show you how to solve ODES. Solving an ODE means finding a function y(x) that satisfies the equation.

In this question, we've done the solving for you! You will be showing that a given function is a solution to an ODE.

Consider the following ordinary differential equation y^{\prime \prime}(x)+y^{\prime}(x)-2 y(x)=-4 x^{2} (a) (2 marks). Show that y(x) = 2x² + 2x + 3 is a solution to the ODE given above.(b) (3 marks). For what values of r is y(x) = erª +2x²+2x+3 also a solution to the ODE given above? 5. (5 marks). Let f(x)=\frac{1}{x} (a) (1 mark). Use the limit definition of derivative to find ƒ'(x).

(b) (1 mark). Verify your answer to (a) using the Power Rule.

(c) (3 marks). Use the Power Rule to find ƒ"(x), ƒ""(x) and ƒ(¹)(x).Find a pattern in the derivatives and make a conjecture for a formula for ƒ(¹)(x),the nth derivative of f(x).