...These are not difficult subjects, but they are tricky problems. To prepare for a test our professor gave us some practice problems to make sure we understand the concepts. I got most of them, but here are a few I can't figure out.
45) Consider the curve defined by -8x^2 +5xy+y^3 =-149. Write and equation for the line tangent to f at the point (4, -1) and using this line approximate f(4.2).
a straight-forward implicit derivative ... what did you get for dy/dx ?
- a) -0.4
- b) 0.4
- c) -0.373
- d) 0.373
- e) -0.6
CONCEPT: linearization, (and implicit differentiation).
39) If f(x) = x3 +x and g(x) is the inverse function of f(x), then g'(1) =
- a) -0.5
- b) 0.003
- c) 0.077
- d) 0.25
- e) 0.417
- THIS ONE I FIGURED OUT ON WOLFRAM ALPHA, BUT CANNOT DO ON MY TI-89 CALCULATOR
if f and g are inverses, then f[g(x)] = x ... take the derivative of this equation w/r to x and determine a relationship between the derivatives of the inverse functions
37) An object traveling in a straight line has position s(t) at time t. If the initial position is s(0) = 3 and the velocity of the object is v(t) = (1+2t2 )(1/3) , what is the position of the object at time t=2?
fundamental theorem ...
- THE VELOCITY FUNCTION IS EXACTLY HOW THEY HAVE IT. WITHOUT AN EXTRA t IT IS HARD TO INTEGRATE, BUT I THINK I'M SUPPOSE TO BE ABLE TO FIGURE IT OUT CONCEPTUALLY.
18) Let g be a twice-differentiable function with g'(x) > 0 and g''(x) > 0 for all real numbers x, such that g(4) = 12 and g(5) = 18. Of the following, which is a possible value for g(6)?
g'(x) > 0 and g''(x) > 0 ... what does this tell you about the rate of change of g ?
16) The function f is differentiable and has values as shown in the table below. Both f and f' are strictly increasing on the interval 0<=x<=5. Which of the following could be the value of f'(3)?
same concept as the previous question
x = 2.5 | 2.8 | 3.0| 3.1
f(x)= 31.25 | 39.2 | 45| 48.05