Exercises

Find by differentiation the slopes of the tangents to the following curves at the points indicated. Verify each result by drawing the curve and its tangent.

1. $y = x^2 - 4$, where $x = 2$. (Ans. $4$.)

2. $y = 6 − 3x^2$ where $x=1$. (Ans. $-6$.)

3. $y = x^3$, where $x = -1$. (Ans. $-3$.)

4. $y = \frac{2}{x}$, where $x = -1$. (Ans. $-\frac{1}{2}$.)

5. $y = x - x^2$, where $x = 0$. (Ans. $1$.)

6. $y = \frac{1}{x - 1}$, where $x=3$. (Ans. $-\frac{1}{4}$.)
7. $y = \frac{1}{2} x^2$, where $x = 4$. (Ans. $4$.)

8. $y = x^2 - 2x + 3$, where $x=1$. (Ans. 0.)

9. $y = 9 − x^2$, where $x = -3$. (Ans. $6$.)

10. Find the slope of the tangent to the curve $y = 2x^3 - 6x + 5$, (a) at the point where $x=1$; (b) at the point where $x = 0$.

    (Ans. (a) 0; (b) $-6$.)

11. (a) Find the slopes of the tangents to the two curves $y = 3x^2 - 1$ and $y = 2x^2 + 3$ at their points of intersection. (b) At what angle do they intersect?

    (Ans. (a) $\pm 12$, $\pm 8$; (b) $\arctan \frac{4}{97}$.)

    Here's how to use SAGE to verify these:

    [fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\sage]
    
    sage: solve(3*x^2 - 1 == 2*x^2 + 3,x)
    [x == -2, x == 2]
    sage: g(x) = diff(3*x^2 - 1,x)
    sage: h(x) = diff(2*x^2 + 3,x)
    sage: g(2); g(-2)
    12
    -12
    sage: h(2); h(-2)
    8
    -8
    sage: atan(12)-atan(8)
    atan(12) - atan(8)
    sage: atan(12.0)-atan(8.0)
    0.0412137626583202
    sage: RR(atan(4/97))
    0.0412137626583202
    

12. The curves on a railway track are often made parabolic in form. Suppose that a track has the form of the parabola $y=x^2$ (see Figure 4.2 in §4.9), the directions $OX$ and $OY$ being east and north respectively, and the unit of measurement $1$ mile. If the train is going east when passing through $O$, in what direction will it be going

    (a)

    when $\frac{1}{2}$ mi. east of $OY$? (Ans. Northeast.)

    (b)

    when $\frac{1}{2}$ mi. west of $OY$? (Ans. Southeast.)

    (c)

    when $\frac{\sqrt{3}}{2}$ mi. east of $OY$? (Ans. N. $30^o$E.)

    (d)

    when $\frac{1}{12}$ mi. north of $OX$? (Ans. E. $30^o$S., or E. $30^o$N.)

13. A street-car track has the form of the cubic $y = x^3$. Assume the same directions and unit as in the last example. If a car is going west when passing through $O$, in what direction will it be going

    (a)

    when $\frac{1}{\sqrt{3}}$ mi. east of $OY$? (Ans. Southwest.)

    (b)

    when $\frac{1}{\sqrt{3}}$ mi. west of $OY$? (Ans. Southwest.)

    (c)

    when $\frac{1}{2}$ mi. north of $OX$? (Ans. S. $27^o$ $43'$ W.)

    (d)

    when $2$ mi. south of $OX$?

    (e)

    when equidistant from $OX$ and $OY$?