Barrons AP Calculus - David Bock

as before. Formula (3) the previous page gives the general case where y = un and u is a differentiable function of x. Now suppos ...

SOLUTION: EXAMPLE 4 SOLUTION: EXAMPLE 5 If s(t) = (t^2 + 1)(1 − t)^2 , find s ′(t). SOLUTION: EXAMPLE 6 If f (t) = e^2 t sin 3t, ...

If x = cos^3 (1 − 3θ), find SOLUTION: EXAMPLE 11 If y = e(sin x) + 1, find SOLUTION: EXAMPLE 12 If y = (x + 1)ln^2 (x + 1), find ...

SOLUTION: EXAMPLE 17 Let y = 2u^3 − 4u^2 + 5u − 3 and u = x^2 − x. Find SOLUTION: EXAMPLE 18 If y = sin (ax + b), with a and b c ...

D. DIFFERENTIABILITY AND CONTINUITY If a function f has a derivative at x = c, then f is continuous at x = c. This statement is ...

E. ESTIMATING A DERIVATIVE E1. Numerically. EXAMPLE 22 The table shown gives the temperatures of a polar bear on a very cold arc ...

T ′(t) −3.05 −1.89 −1.16 −0.73 −0.47 −0.28 −0.17 −0.11 Note that the entries for T ′(t) also represent the approximate slopes of ...

FIGURE N3–4 E2. Graphically. If we have the graph of a function f (x), we can use it to graph f ′(x). We accomplish this by esti ...

FIGURE N3–6 From the graphs above we can make the following observations: (1) At the points where the slope of f (in Figure N3–5 ...

EXAMPLE 25 Find the equation of the tangent to the curve in Example 24 for SOLUTION: W h e n the slope of the tangent, equals −2 ...

G. IMPLICIT DIFFERENTIATION When a functional relationship between x and y is defined by an equation of the form F(x, y) = 0, we ...

where we substituted for from (1) in (2), then used the given equation to simplify in (3). EXAMPLE 31 Using implicit differentia ...

FIGURE N3–8 Simply put, the derivative of the inverse of a function at a point is the reciprocal of the derivative of the functi ...

EXAMPLE 35 Where is the tangent to the curve 4x^2 + 9y^2 = 36 vertical? SOLUTION: We differentiate the equation implicitly to ge ...

FIGURE N3–10 The Mean Value Theorem is one of the most useful laws when properly applied. EXAMPLE 36 You left home one morning a ...

BC ONLY Limits of the following forms are called indeterminate: To find the limit of an indeterminate form of the type we apply ...

EXAMPLE 40 (Example 13) is of type and thus equals as before. Note that is not the limit of an indeterminate form! EXAMPLE 41 is ...

(Note the easier solution Other indeterminate forms, such as 0^0 , 1∞ and ∞^0 , may be resolved by taking the natural logarithm ...

SOLUTION: is the derivative of f (x) = x^4 at the point where x = 2. Since f ′(x) = 4 x^3 the value of the given limit is f ′(2) ...

Practice Exercises Part A. Directions: Answer these questions without using your calculator. In each of Questions 1–20 a functio ...