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Key Concepts

Key Concepts

4.1 Related Rates

  • To solve a related rates problem, first draw a picture that illustrates the relationship between the two or more related quantities that are changing with respect to time.
  • In terms of the quantities, state the information given and the rate to be found.
  • Find an equation relating the quantities.
  • Use differentiation, applying the chain rule as necessary, to find an equation that relates the rates.
  • Be sure not to substitute a variable quantity for one of the variables until after finding an equation relating the rates.

4.2 Linear Approximations and Differentials

  • A differentiable function y=f(x)y=f(x) can be approximated at aa by the linear function
    L(x)=f(a)+f′(a)(x−a).L(x)=f(a)+f′(a)(x−a).
  • For a function y=f(x),y=f(x), if xx changes from aa to a+dx,a+dx, then
    dy=f′(x)dxdy=f′(x)dx

     is an approximation for the change in y.y. The actual change in yy is
    Δy=f(a+dx)−f(a).Δy=f(a+dx)−f(a).
  • A measurement error dxdx can lead to an error in a calculated quantity f(x).f(x). The error in the calculated quantity is known as the propagated error. The propagated error can be estimated by
    dy≈f′(x)dx.dy≈f′(x)dx.
  • To estimate the relative error of a particular quantity q,q, we estimate Δqq.Δqq.

4.3 Maxima and Minima

  • A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum.
  • If a function has a local extremum, the point at which it occurs must be a critical point. However, a function need not have a local extremum at a critical point.
  • A continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. Each extremum occurs at a critical point or an endpoint.

4.4 The Mean Value Theorem

  • If ff is continuous over [a,b][a,b] and differentiable over (a,b)(a,b) and f(a)=f(b),f(a)=f(b), then there exists a point c∈(a,b)c∈(a,b) such that f′(c)=0.f′(c)=0. This is Rolle’s theorem.
  • If ff is continuous over [a,b][a,b] and differentiable over (a,b),(a,b), then there exists a point c∈(a,b)c∈(a,b) such that
    f′(c)=f(b)−f(a)b−a.f′(c)=f(b)−f(a)b−a.

     This is the Mean Value Theorem.
  • If f′(x)=0f′(x)=0 over an interval I,I, then ff is constant over I.I.
  • If two differentiable functions ff and gg satisfy f′(x)=g′(x)f′(x)=g′(x) over I,I, then f(x)=g(x)+Cf(x)=g(x)+C for some constant C.C.
  • If f′(x)>0f′(x)>0 over an interval I,I, then ff is increasing over I.I. If f′(x)<0f′(x)<0 over I,I, then ff is decreasing over I.I.

4.5 Derivatives and the Shape of a Graph

  • If cc is a critical point of ff and f′(x)>0f′(x)>0 for x<cx<c and f′(x)<0f′(x)<0 for x>c,x>c, then ff has a local maximum at c.c.
  • If cc is a critical point of ff and f′(x)<0f′(x)<0 for x<cx<c and f′(x)>0f′(x)>0 for x>c,x>c, then ff has a local minimum at c.c.
  • If f″(x)>0f″(x)>0 over an interval I,I, then ff is concave up over I.I.
  • If f″(x)<0f″(x)<0 over an interval I,I, then ff is concave down over I.I.
  • If f′(c)=0f′(c)=0 and f″(c)>0,f″(c)>0, then ff has a local minimum at c.c.
  • If f′(c)=0f′(c)=0 and f″(c)<0,f″(c)<0, then ff has a local maximum at c.c.
  • If f′(c)=0f′(c)=0 and f″(c)=0,f″(c)=0, then evaluate f′(x)f′(x) at a test point xx to the left of cc and a test point xx to the right of c,c, to determine whether ff has a local extremum at c.c.

4.6 Limits at Infinity and Asymptotes

  • The limit of f(x)f(x) is LL as x→∞x→∞ (or as x→−∞)x→−∞) if the values f(x)f(x) become arbitrarily close to LL as xx becomes sufficiently large.
  • The limit of f(x)f(x) is ∞∞ as x→∞x→∞ if f(x)f(x) becomes arbitrarily large as xx becomes sufficiently large. The limit of f(x)f(x) is −∞−∞ as x→∞x→∞ if f(x)<0f(x)<0 and |f(x)||f(x)| becomes arbitrarily large as xx becomes sufficiently large. We can define the limit of f(x)f(x) as xx approaches −∞−∞ similarly.
  • For a polynomial function p(x)=anxn+an−1xn−1+…+a1x+a0,p(x)=anxn+an−1xn−1+…+a1x+a0, where an≠0,an≠0, the end behavior is determined by the leading term anxn.anxn. If n≠0,n≠0, p(x)p(x) approaches ∞∞ or −∞−∞ at each end.
  • For a rational function f(x)=p(x)q(x),f(x)=p(x)q(x), the end behavior is determined by the relationship between the degree of pp and the degree of q.q. If the degree of pp is less than the degree of q,q, the line y=0y=0 is a horizontal asymptote for f.f. If the degree of pp is equal to the degree of q,q, then the line y=anbny=anbn is a horizontal asymptote, where anan and bnbn are the leading coefficients of pp and q,q, respectively. If the degree of pp is greater than the degree of q,q, then ff approaches ∞∞ or −∞−∞ at each end.

4.7 Applied Optimization Problems

  • To solve an optimization problem, begin by drawing a picture and introducing variables.
  • Find an equation relating the variables.
  • Find a function of one variable to describe the quantity that is to be minimized or maximized.
  • Look for critical points to locate local extrema.

4.8 L’Hôpital’s Rule

  • L’Hôpital’s rule can be used to evaluate the limit of a quotient when the indeterminate form 0000 or ∞/∞∞/∞ arises.
  • L’Hôpital’s rule can also be applied to other indeterminate forms if they can be rewritten in terms of a limit involving a quotient that has the indeterminate form 0000 or ∞/∞.∞/∞.
  • The exponential function exex grows faster than any power function xp,xp, p>0.p>0.
  • The logarithmic function lnxlnx grows more slowly than any power function xp,xp, p>0.p>0.

4.9 Newton’s Method

  • Newton’s method approximates roots of f(x)=0f(x)=0 by starting with an initial approximation x0,x0, then uses tangent lines to the graph of ff to create a sequence of approximations x1,x2,x3,….x1,x2,x3,….
  • Typically, Newton’s method is an efficient method for finding a particular root. In certain cases, Newton’s method fails to work because the list of numbers x0,x1,x2,…x0,x1,x2,… does not approach a finite value or it approaches a value other than the root sought.
  • Any process in which a list of numbers x0,x1,x2,…x0,x1,x2,… is generated by defining an initial number x0x0 and defining the subsequent numbers by the equation xn=F(xn−1)xn=F(xn−1) for some function FF is an iterative process. Newton’s method is an example of an iterative process, where the function F(x)=x−[f(x)f′(x)]F(x)=x−[f(x)f′(x)] for a given function f.f.

4.10 Antiderivatives

  • If FF is an antiderivative of f,f, then every antiderivative of ff is of the form F(x)+CF(x)+C for some constant C.C.
  • Solving the initial-value problem
    dydx=f(x),y(x0)=y0dydx=f(x),y(x0)=y0

     requires us first to find the set of antiderivatives of ff and then to look for the particular antiderivative that also satisfies the initial condition.