1.4 Review

1.4 Review Problems

1.4.1 Concepts

1. 1. State the two central concepts of calculus;
   2. state the two additional concepts that are essential for a rigorous development of calculus.
2. 1. State the the two geometric problems that are related with the central concepts of calculus.
   2. Write a brief paragraph explaining the approximation process used to solve the geometric problems in (a).
3. 1. Write the definition of circle.
   2. Write the formula to find the area of a circular sector of radius \(r\) that subtends an angle \(\theta\) (in radians).
4. Write a brief paragraph explaining why it is intuitively plausible that for each secant line \(\mathcal{L}\) to the parabola \(\mathcal{P}:\, y = a x^2 + bx + c\), \(a \neq 0\), there should be a point on \(\mathcal{P}\) whose tangent line is parallel to \(\mathcal{L}\).

 

5. State its domain and range, and basic properties of the function. (a) \(y = \arccos x\); (b) \(y = \arcsin x\); (c) \(y = \arctan x\).
6. Write down the (a) addition; (b) difference identities for sine and cosine.

1.4.2 Exercises

1. Find and simplify the difference quotient \(\displaystyle \frac{f(x)-f(a)}{x-a}\) for each function.

   (a) \(\displaystyle f(x) = 2 x^2-3 x^4\);   (b) \(\displaystyle f(x) = \frac{x^3}{2x+3} \);  (c) \(\displaystyle f(x) = \frac{x}{\sqrt{4 x^2 + 1}}\);   (d) \(\displaystyle f(x) = \sqrt[3]{x^2} + \frac{x}{\sqrt[4]{x}}\).

 

2. A diameter of a circle has endpoints \(A(2,-3)\) and \(B(4,5)\).
   1. Find an equation for the circle;
   2. find the slope-intercept equation of the line tangent to the circle at \(A\).

    

3. Find the points on the circle \(x^2+y^2=4\) whose tangent lines pass through the point \(A(6,2)\).

 

4. For each function, (i) sketch its graph; (ii) sketch a rectangular approximation with the indicated number \(n\) of rectangles; follow the pattern of the examples in 1.3.4; (iii) find the exact value of the sum of the areas of the rectangles.
    

   (a) \(f(x)=2\cos x + 2\), \(\displaystyle x \in \left [0, \pi \right ]\); \(n =6\).

   (b) \(\displaystyle f(x)=\arcsin x +\frac{\pi}{2}\), \(\displaystyle x\in [-1,1] \vphantom{\left [ \frac{1}{3},\frac{4}{3}\right ]}\); \(n=4\).

   (c) \(f(x)=2\ln x \), \(x\in \left [\,\sqrt{e},4\sqrt{e}\,\right ]\); \(n=3\).

   (d) \(f(x)=e^x \), \(\displaystyle x\in \left [ \frac{1}{3},\frac{4}{3}\right ]\); \(n=3\).

 

5. Given the points \(A(2,3)\) and \(B(-2,1)\). Show that \((x-2)(x+2)+(y-3)(y-1)=0\) is an equation for the circle that has \(A\) and \(B\) as the endpoints of a diameter.
    
6. Given \(f(x)=\sqrt{4-x^2}\), \(x \in [-2,2]\), find the are under its graph (and above the \(x\)-axis) over the interval (a) \(\left [-1, \sqrt{3}\right ]\); (b) \(\left [-\sqrt{2} , -1\right ]\).

 

7. Given the function \(f(x)=2x- 2|x-1|+8\), \(x\in [-3,2]\), sketch and find the area under its graph.

 

8. Given \(f(x)=4-|x-2|\),

   (a) sketch its graph and find the area under its graph and over the interval \([-2,4]\);

   (b) let \(\mathcal{R}\) be the rectangle inscribed in the region enclosed by the graph of \(f\) and the \(x\)-axis, with base on the interval \([-1,5]\) on the \(x\)-axis, and the other two vertices on the graph of \(f\), sketch \(\mathcal{R}\) and find its area;

   (c) let \(\mathcal{R}\) be the region enclosed by the graphs of \(f\) and \(\displaystyle g(x)= \frac{1}{3}x+\frac{2}{3}\), sketch \(\mathcal{R}\) and find its area.

 

9. Given the parabola \(y= x^2 – 4 x + 6\), and the point \(A(0,-10)\), find the points on the parabola whose tangent lines pass through \(A\), and find the slope-intercept equations of the corresponding tangent lines.

 

10. Let \(f(x)=\sqrt{r^2-x^2}\), \(x \in [-r,r]\), and let \(A\) be the area enclosed by the graph of \(f\) and the \(x\)-axis over \([0,b]\) with \(0 < x_1 < r\). Show that the following formulas are valid:
     

    (a) \(\displaystyle A =\frac{1}{2}x_1\sqrt{r^2-x_1^2} + \frac{1}{2}r^2 \arctan \frac{x_1}{\displaystyle \sqrt{r^2-b^2}}\);

    (b) \(\displaystyle A =\frac{1}{2}x_1\sqrt{r^2-x_1^2} + \frac{1}{2}r^2 \arcsin \frac{x_1}{r}\);

    (c) \(\displaystyle A =\frac{1}{2}x_1\sqrt{r^2-x_1^2} + \frac{1}{2}r^2 \left( \frac{\pi}{2} – \arccos \frac{x_1}{r}\right)\).

 

11. Given the parabola \(\mathcal{P}:\) \(y=f(x)= -(x-2) (x+3)\),

    (a) find the \(x\)- and \(y\)-intercepts;

    (b) complete the square to find the vertex of the parabola and sketch its graph;

    (c) Let \(\mathcal{L}_S\) be the secant line passing through the points \(A\) and \(B\) with \(x\)-coordinates, respectively, \(x=-2\) and \(x=2\), find the point on the parabola whose tangent line is parallel \(\mathcal{L}_S\).

    (d) Find the area of the region enclosed by the parabola \(\mathcal{P}\) and the secant line \(\mathcal{L}_S\);

    (e) find the area of the region enclosed by the parabola and the \(x\)-axis.

 

12. Recall the following definition.

    Definition. Let \(A\) and \(B\) be two distinct points in \(\mathbb{R}^2\). The perpendicular bisector of the line segment \(AB\) is the set of point \(\mathcal{S}\) equidistant from \(A\) and \(B\).

    Given \(\displaystyle A\left (-\frac{3}{4},\frac{1}{2}\right )\) and \(\displaystyle B\left (\frac{5}{2}, -\frac{1}{4}\right )\), find an equation for the perpendicular bisector of the line segment \(AB\). Do this in two different ways:

    (a) using the distance formula and the definition of perpendicular bisector;

    (b) using the midpoint of \(AB\) and equations of lines.

 

13. Show that \(\displaystyle \frac{\cos a(x+h)-\cos ax}{h} = \cos ax \frac{\cos ah-1}{h} – \frac{a\sin ah}{ah}\sin ax , \;\; 0\neq a \in \mathbb{R}\).

 

14. Show that \(\displaystyle \frac{\tan x-\tan a}{x-a} =\frac{\sin (x-a)}{x-a}\sec x \sec a\).