Section 3
Practise questions
1. Differentiate the following functions.
a) \(y = 75x - 4x^{2}\)
b) \(y = 4000x - 33x^{2}\)
c) \(y = -x^{3} - 30x^{2} + 360x - 500\)
d) \(y = 35 + 5x - 2x^{2} + x^{3}\)
e) \(y = 1.5x + 10 + \frac{46}{x}\)
f) \(y = x^{2} - 21x + 500 + \frac{200}{x}\)
g) \(y = (4x^{2} - 5\)(3x^{4})
h) \(y = x^{2} - 10x\)
i) \(y = x^{0.5}\)
j) \(y = 10x^{0.4}\)
k) \(y = (5x + 8)^{3}\)
l) \(y = \frac{(5x - 1)(3x + 2)^{3}(3x^{2} + 2x + 5)^{5}}{(x^{2} + 2x + 2)^{2}}\)
a) \({y}' = 75 - 8x\)
b) \({y}' = 4000 - 66x\)
c) \({y}' = -3x^{2} - 60x + 360\)
d) \({y}' = 5 - 4x + 3x^{2}\)
e) \({y}' = 1.5 - \frac{46}{x^{2}}\)
f) \({y}' = 2x - 21 - \frac{200}{x^{2}}\)
g) \({y}' = (4x^{2} - 5) \cdot \frac{d}{dx}(3x^{4}) + 3x^{4} \cdot \frac{d}{dx}(4x^{2} - 5)\)
\(\qquad = 12x^{3}(4x^{2} - 5) + 24x^{5}\)
h) \({y}' = 2x - 10\)
i) \({y}' = 0.5x^{0.5 - 1}\)
\(\qquad = 0.5x^{0.5 - 1}\)
\(\qquad = \frac{0.5}{x^{0.5}}\)
j) \({y}' = 10(0.4)x^{0.4 - 1}\)
\(\qquad = 4x^{-0.6}\)
\(\qquad = \frac{4}{x^{0.6}}\)
k) \({y}' = 3(5x + 8)^{3 - 1} \cdot \frac{d}{dx}(5x + 8)\)
\(\qquad = 15(5x + 8)^{2}\)
l) \({y}' = \qquad\) In economics, we do not need to deal with such complicated equations.
2. For each of the following functions, find the first-order derivative and evaluate the first-order derivative at \(x = 5\).
a) \(f(x) = x^{3} - 10x^{2} + 40x\)
b) \(f(x) = 10 + 2x\)
c) \(f(x) = 10x - x^{2}\)
d) \(f(x) = \frac{x}{1 - x}\)
e) \(f(x) = 23 + 10x + 2x^{2}\)
a) \({f}'(5) = 15\)
b) \({f}'(5) = 2\)
c) \({f}'(5) = 0\)
d) \({f}'(5) = \frac{1}{16}\)
e) \({f}'(5) = 30\)
3. Find the slope of the following functions at \(x = 0\), \(x = 1\),\(x = 2\),\(x = -1\), and \(x = -2\).
a) \(f(x) = x^{2}\)
b) \(f(x) = x^{3}\)
c) \(f(x) = 4x - x^{2}\)
a) \(f(0) = 0\), \(f(1) = 2\), \(f(2) = 4\), \(f(-1) = -2\), \(f(-2) = -4\)
a) \(f(0) = 0\), \(f(1) = 3\), \(f(2) = 12\), \(f(-1) = 3\), \(f(-2) = 12\)
a) \(f(4) = 0\), \(f(1) = 2\), \(f(2) = 0\), \(f(-1) = 6\), \(f(-2) = 8\)
UWO Economics Math Resources by Mohammed Iftekher Hossain is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.