2.5.1 Concepts
- Write the definition of each of the following concepts:
- \(\displaystyle \lim_{x\to a}f(x) =L\);
- \(f\) is continuous at \(a\);
- \(f\) is differentiable at \(a\);
- \(f\) is continuous on an interval \((c,d)\); \(\vphantom{\displaystyle \lim_{x\to a}f(x) =L}\)
- \(f\) is continuous on an interval \([c,d]\);
- \(f\) is differentiable on an interval \((c,d)\).
- State the three types of discontinuities of a function at a given point \(x\). Draw a picture for each type.
- Draw a picture of the possible cases where \(f\) fails to be differentiable at \(a\in D_f\).
2.5.2 Exercises
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For each function \(f\), (i) use the definition of derivativeto find the derivative of \(f\) at the given \(a\); (ii) use a rule of differentiation to find \(f'(a)\); (iii) use your answer in (i) or (ii) to find the slope-intercept equation of the line tangent to the graph at the point with \(x\)-coordinate \(a\).
(a) \(\displaystyle f(x)=2 x^5 – x^3\), \(a=-1\); (b) \(\displaystyle f(x)=\frac{x}{x^2-9}\), \(a=-2\); (c) \(\displaystyle f(x)=\sqrt[6]{x}\), \(a=1\).
- Find the points on the graph of \(f(x)=x^4-8 x^2+16\) whose tangent line is horizontal.
- Reading comprehension.
Definition. Let \(f\) be a function defined on the open interval \((a,b)\). A number \(c\in (a,b)\) is a critical number of \(f\) if \(f'(c)=0\) or \(f'(c)\) is undefined.
(a) Find the critical numbers of the function \(f(x)=x^3-8 x^2+16 x-18\).
(b) Assume that \(f\) is a function such that \(f'(x) = 2\sin^2 2x -1\). Find its critical numbers.
- For what values of \(a\) and \(b\) is the line \(y=2x-3\) tangent to the parabola \(y=a x^2+b x -2\) at \(x=-2\)?
- Find the \(x\)-coordinates of the points on the graph of \(y=3 x^2+2 x+4\), whose tangent lines pass through \(A(-1,-4)\).
- Use the definition of derivative to find the derivative of \(\displaystyle f(x)=\sqrt[5]{x^2}\). Hint: \(\displaystyle \sqrt[5]{x^2} =\left(\sqrt[5]{x}\right)^2 \).
- Let \(n\in \mathbb{N}\). Use the definition of derivative to show that
- if \(f(x)=x^n\), then \(f'(x)=nx^{n-1}\), for all \(x\in \mathbb{R}\).
- if \(f(x)=x^{-n}\), then \(f'(x)=-nx^{-n-1}\), for all \(x\in \mathbb{R}\), \(x\neq 0\).
- if \(f(x)=\sqrt[n]{x}\), then \(\displaystyle f'(x)=\frac{1}{n}\frac{1}{\displaystyle \sqrt[n]{x^{n-1}}}\), for all \(x\in D_f\), \(x\neq 0\).
Show all details.
- if \(f(x)=x^n\), then \(f'(x)=nx^{n-1}\), for all \(x\in \mathbb{R}\).
- Use the definition of derivative to show that \(f(x)=|x-2|\) is not differentiable at \(x=2\).
- Show that \(\displaystyle f(x)=\frac{|x-2|}{x-2}\) is not continuous at \(x=2\). Sketch its graph.
- Show that the area of the triangle in the first quadrant formed by the tangent line to the graph of \(\displaystyle y=\frac{1}{x}\), \(x>0\), and the coordinate axes is independent of the point of tangency, and find the constant value for the area.