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Chapter 2: Q16E (page 77)

Make a careful sketch of the graph of \(f\) and below it sketch the graph \(f'\) in the same manner as in Exercise 4-11. Can you guess a formula for \(f'\left( x \right)\) from its graph?

16. \(f\left( x \right) = \sin x\)

Short Answer

Expert verified

The formula for \(f'\left( x \right)\) is \(f'\left( x \right) = \cos x\).

Step by step solution

01

Sketch the graph of \(f\)and  \(f'\)

Sketch the graph of \(f\left( x \right) = \sin x\) as shown below:

Consider some points \(A,B,C,D,E,F\) and \(G\) to plot on the graph of the given function \(f\left( x \right)\). We can estimate the graph of derivatives from these points.

To determine the value of the derivative at any value of \(x\), we can draw the tangent at the point \(\left( {x,f\left( x \right)} \right)\) and estimate the slope.

The tangent \(A,C,E,G\) is horizontal. Therefore the derivative is 0.

The slope of the tangent at \(B\) appears to be \( - \pi \), the value of \(f'\left( B \right) = - \pi \).

The graph of \(f\) at the points \(D\) crosses the \(x\)-axis (where \(y = 0\)). The value of \(f'\left( D \right) = 0\).

The function \(f\) is decreasing at this point \(F\). The value of \(f'\left( F \right) = - \pi \).

Use the above points to sketch the graph of \(f'\left( x \right)\) as shown below:

02

Guess the formula for \(f'\left( x \right)\) from its graph

The graph of \(f'\left( x \right)\) is similar to the graph of the cosine function. We might assume the formula is \(f'\left( x \right) = \cos x\).

Thus, the formula for \(f'\left( x \right)\) is \(f'\left( x \right) = \cos x\).

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Most popular questions from this chapter

The point \(P\left( {{\bf{0}}.{\bf{5}},{\bf{0}}} \right)\) lies on the curve \(y = {\bf{cos}}\pi x\).

(a) If Q is the point \(\left( {x,{\bf{cos}}\pi x} \right)\), find the slope of the secant line PQ (correct to six decimal places) for the following values of x:

(i) 0 (ii) 0.4 (iii) 0.49 (iv) 0.499 (v) 1 (vi) 0.6 (vii) 0.51 (viii) 0.501

(b) Using the results of part (a), guess the value of the slope of tangent line to the curve at \(P\left( {{\bf{0}}.{\bf{5}},{\bf{0}}} \right)\).

(c) Using the slope from part (b), find an equation of the tangent line to the curve at \(P\left( {{\bf{0}}.{\bf{5}},{\bf{0}}} \right)\).

(d) Sketch the curve, two of the secant lines, and the tangent line.

A roast turkey is taken from an oven when its temperature has reached \({\bf{185}}\;^\circ {\bf{F}}\) and is placed on a table in a room where the temperature \({\bf{75}}\;^\circ {\bf{F}}\). The graph shows how the temperature of the turkey decreases and eventually approaches room temperature. By measuring the slope of the tangent, estimate the rate of change of the temperature after an hour.

For the function h whose graph is given, state the value of each quantity if it exists. If it does not exist, explain why.

(a)\(\mathop {{\bf{lim}}}\limits_{x \to - {{\bf{3}}^ - }} h\left( x \right)\)

(b)\(\mathop {{\bf{lim}}}\limits_{x \to - {{\bf{3}}^ + }} h\left( x \right)\)

(c) \(\mathop {{\bf{lim}}}\limits_{x \to - {\bf{3}}} h\left( x \right)\)

(d) \(h\left( { - {\bf{3}}} \right)\)

(e) \(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{0}}^ - }} h\left( x \right)\)

(f) \(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{0}}^ + }} h\left( x \right)\)

(g) \(\mathop {{\bf{lim}}}\limits_{x \to {\bf{0}}} h\left( x \right)\)

(h) \(h\left( {\bf{0}} \right)\)

(i) \(\mathop {{\bf{lim}}}\limits_{x \to {\bf{2}}} h\left( x \right)\)

(j) \(h\left( {\bf{2}} \right)\)

(k) \(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{5}}^ + }} h\left( x \right)\)

(l) \(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{5}}^ - }} h\left( x \right)\)

The deck of a bridge is suspended 275 feet above a river. If a pebble falls of the side of the bridge, the height, in feet of the pebble above the water surface after t seconds is given by\(y = {\bf{275}} - {\bf{16}}{t^{\bf{2}}}\)

(a) Find the average velocity of the pebble for the time period beginning when\(t = {\bf{4}}\)and lasting

(i) 0.1 seconds (ii) 0.05 seconds (iii) 0.01 seconds

(b) Estimate the instaneous velocity of pebble after 4 seconds

Use thegiven graph of f to state the value of each quantity, if it exists. If it does not exists, explain why.

(a)\(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{2}}^ - }} f\left( x \right)\)

(b)\(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{2}}^ + }} f\left( x \right)\)

(c) \(\mathop {{\bf{lim}}}\limits_{x \to {\bf{2}}} f\left( x \right)\)

(d) \(f\left( {\bf{2}} \right)\)

(e) \(\mathop {{\bf{lim}}}\limits_{x \to {\bf{4}}} f\left( x \right)\)

(f) \(f\left( {\bf{4}} \right)\)
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