大学高等数学测试题

Quiz

Chinese Name: Class: Date: 2017-2 A. Translation

B. Please solve the problems in English.

1. Find the equation of the tangent to the curve

which is perpendicular to the line .

2. Find the equation of the normal to the curve

which is parallel to the line .

3. Differentiate each of these functions f(x).

Give your answers f’(x) in a similar form, without negative or fractional indices(负指数或分数指数).

(a) (b) (c)

(d) (e) (f)

4. Given the function , find

(a) f’(-2) (b) the values of a such that f’(a)=56.

5. Find the coordinates of the stationary points on the graphs of the function , and find whether theses points are maxima or minima.

6. Differentiate the following with respect to x.

(a) (b)

(c) (d)

Additional paper(附加题):

7. The sum of two real numbers x and y is 12. Find the maximum value of their product xy.

8. The volume of a cylinder(圆柱体) is given by the formula(公式)

. Find the greatest and least values of V if r+h=6.

9. Use differentiation to find the coordinates of the stationary points on the curve and determine whether each stationary point is a maximum or a minimum point.

Find the set of values of x for which y increases as x increases.

10. Air is being lost from a spherical balloon(圆形的气球)at a constant rate of 0.6m3s-1. Find the rate at which the radius is decreasing at the instant(在那个瞬间) when the radius is 2.5m.