Sample Exam

Math 31A

There will be eight questions, each worth 20 points.

1) Let . Using the definition of the derivative prove that

2. Given that

prove, using the definition of the derivative, that if f(x) = sin(x) then f'(x) = cos(x).

3. Find dy/dx given that

a) y = 4x² b) y = sin(x) + sin(2x²) * cos(3x³)

c) d) y = sec(tan(x+y))

4. Find a point where the curve y = x³ + 3 x² + 3x + 5 has a horizontal tangent.

5. A trough is 10 feet long and its ends have the shapes of isosceles triangles that are 3 ft across at the top and have a height of 1 foot. If the trough is filled with water at a rate of 12 ft³/min, how fast is the water level rising when the water is 9 inches deep?

6. Let C be the curve defined by x³ + xy + y³ = 3 and which goes through the point (1,1). What is the slope of the tangent line to C at (1,1)?

7. Give an example of a function of a function that is continuous on
I = [-1, 1] but not differentiable at at least two points in (-1,1).

8. A function f(x) and its first two derivatives are graphed below. Label which is which.