2729 investigate the motion of a projectile shotfrom a cannon. The fixed parameters are

Find the tangent line approximation for 1 + x near x = 0.

What is the tangent line approximation to ex near x = 0?

Find the tangent line approximation to 1/x near x = 1.

Find the local linearization of f(x) = x2 near x = 1.

What is the local linearization of ex2 near x = 1?

Show that 1 x/2 is the tangent line approximation to 1/ 1 + x near x = 0.

Show that ex 1 x near x = 0

Local linearization gives values too small for the function x2 and too large for the function x. Draw pictures to explain why

Using a graph like Figure 3.41, estimate to one decimal place the magnitude of the error in approximating sin x by x for 1 x 1. Is the approximation an over- or an underestimate?

For x near 0, local linearization gives e x 1 + x. Using a graph, decide if the approximation is an overor underestimate, and estimate to one decimal place the magnitude of the error for 1 x 1.

(a) Find the best linear approximation, L(x), to f(x) = ex near x = 0. (b) What is the sign of the error, E(x) = f(x) L(x) for x near 0? (c) Find the true value of the function at x = 1. What is the error? (Give decimal answers.) Illustrate with a graph. (d) Before doing any calculations, explain which you expect to be larger, E(0.1) or E(1), and why. (e) Find E(0.1).

(a) Find the tangent line approximation to cos x at x = /4. (b) Use a graph to explain how you know whether the tangent line approximation is an under- or overestimate for 0 x /2. (c) To one decimal place, estimate the error in the approximation for 0 x /2.

(a) Graph f(x) = x3 3x2 + 3x + 1. (b) Find and add to your sketch the local linearization to f(x) at x = 2. (c) Mark on your sketch the true value of f(1.5), the tangent line approximation to f(1.5) and the error in the approximation.

(a) Show that 1+kx is the local linearization of(1+x) k near x = 0. (b) Someone claims that the square root of 1.1 is about 1.05. Without using a calculator, do you think that this estimate is about right? (c) Is the actual number above or below 1.05?

. Figure 3.43 shows f(x) and its local linearization at x = a. What is the value of a? Of f(a)? Is the approximation an under- or overestimate? Use the linearization to approximate the value of f(1.2).

In Problems 1617, the equation has a solution near x = 0. By replacing the left side of the equation by its linearization, find an approximate value for the solution. e x + x = 2

In Problems 1617, the equation has a solution near x = 0. By replacing the left side of the equation by its linearization, find an approximate value for the solution. x + ln(1 + x)=0.2

(a) Given that f(7) = 13 and f (7) = 0.38, estimate f(7.1). (b) Suppose also f(x) < 0 for all x. Does this make your answer to part (a) an under- or overestimate?

(a) Explain why the following equation has a solution near 0: et = 0.02t + 1.098. (b) Replace et by its linearization near 0. Solve the new equation to get an approximate solution to the original equation.

The speed of sound in dry air is f(T ) = 331.3 1 + T 273.15 meters/second where T is the temperature in degrees Celsius. Find a linear function that approximates the speed of sound for temperatures near 0C.

Air pressure at sea level is 30 inches of mercury. At an altitude of h feet above sea level, the air pressure, P, in inches of mercury, is given by P = 30e3.23105h (a) Sketch a graph of P against h. (b) Find the equation of the tangent line at h = 0. (c) A rule of thumb used by travelers is that air pressure drops about 1 inch for every 1000-foot increase in height above sea level. Write a formula for the air pressure given by this rule of thumb. (d) What is the relation between your answers to parts (b) and (c)? Explain why the rule of thumb works. (e) Are the predictions made by the rule of thumb too large or too small? Why?

On October 7, 2010, the Wall Street Journal8 reported that Android cell phone users had increased to 10.9 million by the end of August 2010 from 866,000 a year earlier. During the same period, iPhone users increased to 13.5 million, up from 7.8 million a year earlier. Let A(t) be the number of Android users, in millions, at time t in years since the end of August 2009. Let P(t) be the number of iPhone users in millions. (a) Estimate A (0). Give units. (b) Estimate P (0). Give units. (c) Using the tangent line approximation, when are the numbers of Android and iPhone users predicted to be the same? (d) What assumptions did you make in part (c)?

Writing g for the acceleration due to gravity, the period, T , of a pendulum of length l is given by T = 2 l g . (a) Show that if the length of the pendulum changes by l, the change in the period, T , is given by T T 2l l. (b) If the length of the pendulum increases by 2%, by what percent does the period change?

Suppose now the length of the pendulum in Problem 23 remains constant, but that the acceleration due to gravity changes. (a) Use the method of the preceding problem to relate T approximately to g, the change in g. (b) If g increases by 1%, find the percent change in T .

Suppose f has a continuous positive second derivative for all x. Which is larger, f(1+x) or f(1)+f (1)x? Explain.

Suppose f (x) is a differentiable decreasing function for all x. In each of the following pairs, which number is the larger? Give a reason for your answer. (a) f (5) and f (6) (b) f(5) and 0 (c) f(5 + x) and f(5) + f (5)x

Problems 2729 investigate the motion of a projectile shot from a cannon. The fixed parameters are the acceleration of gravity, g = 9.8 m/sec2, and the muzzle velocity, v0 = 500 m/sec, at which the projectile leaves the cannon. The angle , in degrees, between the muzzle of the cannon and the ground can vary. The range of the projectile is f() = v2 0 g sin 90 wq = 25510 sin 90 meters. (a) Find the range with = 20. (b) Find a linear function of that approximates the range for angles near 20. (c) Find the range and its approximation from part (b) for 21.

Problems 2729 investigate the motion of a projectile shot from a cannon. The fixed parameters are the acceleration of gravity, g = 9.8 m/sec2, and the muzzle velocity, v0 = 500 m/sec, at which the projectile leaves the cannon. The angle , in degrees, between the muzzle of the cannon and the ground can vary. The time that the projectile stays in the air is t() = 2v0 g sin 180 = 102 sin 180 seconds. (a) Find the time in the air for = 20. (b) Find a linear function of that approximates the time in the air for angles near 20. (c) Find the time in air and its approximation from part (b) for 21.

Problems 2729 investigate the motion of a projectile shot from a cannon. The fixed parameters are the acceleration of gravity, g = 9.8 m/sec2, and the muzzle velocity, v0 = 500 m/sec, at which the projectile leaves the cannon. The angle , in degrees, between the muzzle of the cannon and the ground can vary. At its highest point the projectile reaches a peak altitude given by h() = v2 0 2g sin2 180 = 12755 sin2 180 meters. (a) Find the peak altitude for = 20. (b) Find a linear function of that approximates the peak altitude for angles near 20. (c) Find the peak altitude and its approximation from part (b) for 21

In Problems 3032, find the local linearization of f(x) near 0 and use this to approximate the value of a. f(x) = (1 + x) r , a = (1.2)3/5

In Problems 3032, find the local linearization of f(x) near 0 and use this to approximate the value of a. f(x) = ekx, a = e0.3

In Problems 3032, find the local linearization of f(x) near 0 and use this to approximate the value of a. f(x) = b2 + x, a = 26

In Problems 3337, find a formula for the error E(x) in the tangent line approximation to the function near x = a. Using a table of values for E(x)/(xa) near x = a, find a value of k such that E(x)/(x a) k(x a). Check that, approximately, k = f(a)/2 and that E(x) (f(a)/2)(x a) 2. f(x) = x4, a = 1

In Problems 3337, find a formula for the error E(x) in the tangent line approximation to the function near x = a. Using a table of values for E(x)/(xa) near x = a, find a value of k such that E(x)/(x a) k(x a). Check that, approximately, k = f(a)/2 and that E(x) (f(a)/2)(x a) 2. f(x) = cos x, a = 0

In Problems 3337, find a formula for the error E(x) in the tangent line approximation to the function near x = a. Using a table of values for E(x)/(xa) near x = a, find a value of k such that E(x)/(x a) k(x a). Check that, approximately, k = f(a)/2 and that E(x) (f(a)/2)(x a) 2. f(x) = ex, a = 0

In Problems 3337, find a formula for the error E(x) in the tangent line approximation to the function near x = a. Using a table of values for E(x)/(xa) near x = a, find a value of k such that E(x)/(x a) k(x a). Check that, approximately, k = f(a)/2 and that E(x) (f(a)/2)(x a) 2. f(x) = x, a = 1

In Problems 3337, find a formula for the error E(x) in the tangent line approximation to the function near x = a. Using a table of values for E(x)/(xa) near x = a, find a value of k such that E(x)/(x a) k(x a). Check that, approximately, k = f(a)/2 and that E(x) (f(a)/2)(x a) 2. f(x) = ln x, a = 1

Multiply the local linearization of ex near x = 0 by itself to obtain an approximation for e2x. Compare this with the actual local linearization of e2x. Explain why these two approximations are consistent, and discuss which one is more accurate.

(a) Show that 1 x is the local linearization of 1 1 + x near x = 0. (b) From your answer to part (a), show that near x = 0, 1 1 + x2 1 x2 . (c) Without differentiating, what do you think the derivative of 1 1 + x2 is at x = 0?

From the local linearizations of ex and sin x near x = 0, write down the local linearization of the function ex sin x. From this result, write down the derivative of ex sin x at x = 0. Using this technique, write down the derivative of ex sin x/(1 + x) at x = 0.

Use local linearization to derive the product rule, [f(x)g(x)] = f (x)g(x) + f(x)g (x). [Hint: Use the definition of the derivative and the local linearizations f(x+h) f(x)+f (x)h and g(x+h) g(x) + g (x)h.]

Derive the chain rule using local linearization. [Hint: In other words, differentiate f(g(x)), using g(x + h) g(x) + g (x)h and f(z + k) f(z) + f (z)k.]

Consider a function f and a point a. Suppose there is a number L such that the linear function g g(x) = f(a) + L(x a) is a good approximation to f. By good approximation, we mean that lim xa EL(x) x a = 0, where EL(x) is the approximation error defined by f(x) = g(x) + EL(x) = f(a) + L(x a) + EL(x). Show that f is differentiable at x = a and that f (a) = L. Thus the tangent line approximation is the only good linear approximation.

Consider the graph of f(x) = x2 near x = 1. Find an interval around x = 1 with the property that throughout any smaller interval, the graph of f(x) = x2 never differs from its local linearization at x = 1 by more than 0.1|x 1|.

In Problems 4546, explain what is wrong with the statement. To approximate f(x) = ex, we can always use the linear approximation f(x) = ex x + 1.

In Problems 4546, explain what is wrong with the statement. The linear approximation for F(x) = x3 + 1 near x = 0 is an underestimate for the function F for all x, x = 0.

In Problems 4749, give an example of: Two different functions that have the same linear approximation near x = 0.

In Problems 4749, give an example of: A non-polynomial function that has the tangent line approximation f(x) 1 near x = 0.

In Problems 4749, give an example of: A function that does not have a linear approximation at x = 1.

Let f be a differentiable function and let L be the linear function L(x) = f(a) + k(x a) for some constant a. Decide whether the following statements are true or false for all constants k. Explain your answer. (a) L is the local linearization for f near x = a, (b) If lim xa (f(x) L(x)) = 0, then L is the local linearization for f near x = a.