Unit 4-5 Test Review Practice Problems

Front

What is the linear approximation of a function f(x) at a point x = a?

Back

The linear approximation of a function f(x) at a point x = a is given by the formula: y = f(a) + f'(a)(x - a).

Front

How do you find the acceleration from a position function x(t)?

Back

To find the acceleration from a position function x(t), take the second derivative of x(t) with respect to time t: a(t) = x''(t).

Front

What does it mean if f'(c) = 0 and f''(c) > 0 at a point c?

Back

If f'(c) = 0 and f''(c) > 0, then there is a local minimum at x = c.

Front

How can you determine if a particle is moving left based on its velocity function?

Back

A particle is moving left when its velocity function v(t) is negative, i.e., v(t) < 0.

Front

What is a relative minimum in the context of a function's graph?

Back

A relative minimum is a point on the graph of a function where the function value is lower than the values of the function at nearby points.

Front

What is the significance of the first derivative f'(x)?

Back

The first derivative f'(x) represents the rate of change of the function f(x) and indicates the slope of the tangent line at any point x.

Front

What does the second derivative f''(x) tell us about a function?

Back

The second derivative f''(x) indicates the concavity of the function: if f''(x) > 0, the function is concave up; if f''(x) < 0, the function is concave down.

Front

How do you find the critical points of a function?

Back

Critical points occur where the first derivative f'(x) is zero or undefined.

Front

What is the relationship between the velocity and position functions?

Back

The velocity function v(t) is the first derivative of the position function x(t) with respect to time: v(t) = x'(t).

Front

What is the formula for linear approximation at x = 1 for f(x) = x^4 + 2x^3 + x^2?

Back

The linear approximation at x = 1 is y = 12x - 8.