Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.


x ƒ(x) g(x) ƒ'(x) g'(x)
0 1 1 -3 1/2
1 3 5 1/2 -4


Find the first derivatives of the following combinations at the given value of x.


c. ƒ(x) , x = 1
g(x) + 1

Computer Explorations

Use a CAS to perform the following steps in Exercises 55–62.

b. Using implicit differentiation, find a formula for the derivative dy/dx and evaluate it at the given point P.

2y² + (xy)¹/³ = x² + 2, P(1,1)

The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’s

b. area?

Theory and Examples

In Exercises 51–54,

c. For what values of x, if any, is f' positive? Zero? Negative?

y = −x²

The folium of Descartes (See Figure 3.27)

c. Find the coordinates of the point A in Figure 3.29 where the folium has a vertical tangent line.

Area The area A of a triangle with sides of lengths a and b enclosing an angle of measure θ is

A = (1/2) ab sinθ.

b. How is dA/dt related to dθ/dt and da/dt if only b is constant?

Differentiability and Continuity on an Interval

Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

c. neither continuous nor differentiable?

Give reasons for your answers.