Straight lines and gradients：練習問題

The cost C (in USD) of producing x units of a product is given by the function \(C(x) = 5x + 500\). If a company produces \(120\) units, what is the total cost?

The population P of a town t years after 2020 can be modeled by the function \(P(t) = 25000 \cdot (1.03)^t\). In what year will the population first exceed \(30000\)?

The height h (in metres) of a ball thrown vertically upwards is modelled by the function \(h(t) = -4.9t^2 + 19.6t + 1.5\), where t is the time in seconds after it is thrown. Find the maximum height reached by the ball.

Given the function $$f(x) = 2x^2 - 3x + 1$$, find the value of $$f(-1)$$ .

Given the functions $$f(x) = 3x - 1$$ and $$g(x) = x^2 + 2$$, determine the expression for $$(g \circ f)(x)$$ .

Find the vertical asymptote of the rational function given by the equation \(f(x) = \frac{3x - 1}{2x + 5}\).

The relationship between variables \(x\) and \(y\) is modelled by the equation \(y = ax^k\). If a scatter plot of \(\ln y\) against \(\ln x\) results in a straight line passing through the points \((1.5, 2.7)\) and \((4.5, 9.3)\), find the value of the exponent \(k\).

In the diagram showing the intersection of f(x) = 3^x and g(x) = 10 - x^2, there are two intersection points. Use technology to find the coordinates of the intersection point located in the second quadrant, correct to three significant figures.

A pharmaceutical company models the concentration of a specific drug in a patient's bloodstream using the function
$$C(t) = \frac{5t}{t^2 + 10}, \quad t \ge 0$$
where \(C\) is the concentration in milligrams per litre (mg/L) and \(t\) is the time in hours after the drug is administered.

a) Calculate the concentration of the drug after 2 hours, giving your answer correct to three significant figures.

b) The minimum concentration required for the drug to be detectable is \(0.8\) mg/L. Determine the time, \(t > 0\), when the concentration first reaches \(0.8\) mg/L. Give your answer correct to three significant figures.

c) Use technology to find the maximum concentration reached and the time, \(t_{max}\), when this maximum concentration occurs.

d) The drug is considered effective when the concentration is between \(0.5\) mg/L and \(1.5\) mg/L. Calculate the total time interval (in hours) during which the drug is effective. Give your answer correct to three significant figures.

Consider the logarithmic function f(x) = \lo\(g_{2}(x + 3) + 1\), for x > -3.

a) State the equation of the vertical asymptote of the graph of f.

b) Find the x-intercept of the graph of f.

c) Find an expression for the inverse function, \(f^{-1}(x)\).

d) The graph of g is obtained by translating the graph of f by the vector \(\begin{pmatrix} -1 \\ 2 \end{pmatrix}\). Find the equation of g(x) in the form g(x) = \lo\(g_{2}(x + a) + b\).