?1. Graph the functionf(x) = x4 - 3x3 - 6x2 + 7x + 30in the viewing rectangle [-3, 5] by

(a) How is the number e defined? (b) Use a calculator to estimate the values of the limits \(\lim \limits_{h \rightarrow 0} \frac{2.7^{h}-1}{h}\) and \(\lim \limits_{h \rightarrow 0} \frac{2.8^{h}-1}{h}\) correct to two decimal places. What can you conclude about the value of e?

Find the linearization \(L(x)\) of the function at \(a\). \(f(x)=e^{3 x}, \quad a=0\) Equation Transcription: Text Transcription: L(x) a f(x) = e^3x, a = 0

Find the linearization \(L(x)\) of the function at \(a\). \(f(x)=\cos 2 x, \quad a=\pi / 6\) Equation Transcription: Text Transcription: L(x) a f(x) = cos 2x, a = pi/6

Find the linearization \(L(x)\) of the function at \(a\). \(f(x)=\sqrt[3]{x}, \quad a=8\) Equation Transcription: Text Transcription: L(x) a f(x) = 3 sqrt x, a = 8

Find the linear approximation of the function \(f(x)=\sqrt{1-x}\) at \(a = 0\) and use it to approximate the numbers \(\sqrt{0.9}\) and \(\sqrt{0.99}\). Illustrate by graphing \(f\) and the tangent line. Equation Transcription: Text Transcription: f(x) = sqrt 1-x a = 0 sqrt 0.9 sqrt 0.99 f

Find the linear approximation of the function \(g(x)=\sqrt[3]{1+x}\) at \(a=0\) and use it to approximate the numbers \(\sqrt[3]{0.95}\) and \(\sqrt[3]{1.1}\). Illustrate by graphing \(t\) and the tangent line. Equation Transcription: Text Transcription: g(x) = 3 sqrt 1+x a = 0 3 sqrt 0.95 3 sqrt 1.1 t

Verify the given linear approximation at \(a = 0\). Then determine the values of \(x\) for which the linear approximation is accurate to within \(0.1\). \(\tan ^{-1} x \approx x\) Equation Transcription: Text Transcription: a = 0 x 0.1 tan^-1 x approx x

3–34 Differentiate the function. F(t) = t3 + e3

3–34 Differentiate the function. W(v) = 1.8v-3

3–34 Differentiate the function. r(z) = z-5 - z1/2

3–34 Differentiate the function. f(x) = + x-3

Differentiate the function. \(V(t)=t^{-3 / 5}+t^{4}\)

3–34 Differentiate the function. s(t) =

3–34 Differentiate the function. r(t) =

3–34 Differentiate the function. y = 2x +

3–34 Differentiate the function. h(w) =

3–34 Differentiate the function. g(x) =

3–34 Differentiate the function. W(t) =

3–34 Differentiate the function. f(x) =

3–34 Differentiate the function. F(t) = (2t - 3)2

3–34 Differentiate the function. y = 3

3–34 Differentiate the function. S(R) = 4R2

3–34 Differentiate the function. f(x) =

Differentiate the function. \(y=\frac{\sqrt{x}+x}{x^{2}}\)

Differentiate the function. \(G(r)=\frac{3 r^{3 / 2}+r^{5 / 2}}{r}\)

3–34 Differentiate the function. G(t) = +

Differentiate the function. \(j(x)=x^{2.4}+e^{2.4}\)

3–34 Differentiate the function. k(r) =

3–34 Differentiate the function. F(z) =

Differentiate the function. \(G(q)=\left(1+q^{-1}\right)^{2}\)

3–34 Differentiate the function. D(t) =

3–34 Differentiate the function. f(v) =

3–34 Differentiate the function. P(w) =

3–34 Differentiate the function. y =

Find dy/dx and dy/dt. y = t

Find dy/dx and dy/dt. y =

Find an equation of the tangent line to the curve at the given point. y = 2 (1,3)

Find an equation of the tangent line to the curve at the given point. \(y=2 e^{x}+x, \quad(0,2)\) Text Transcription: y = 2e^x + x, (0,2)

Find an equation of the tangent line to the curve at the given point. y = x + (2,3)

Find an equation of the tangent line to the curve at the given point. y = (1,0)

Find equations of the tangent line and normal line to the curve at the given point. y = x4 + 2ex, (0, 2)

Find equations of the tangent line and normal line to the curve at the given point. y = x3/2, (1, 1)

Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen. y = 3x2 - x3, (1,2)

Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen. y = x - (1,0)

Find f’(x). Compare the graphs of f and f’ and use them to explain why your answer is reasonable. f(x) = x4 - 2

Find f’(x). Compare the graphs of f and f’ and use them to explain why your answer is reasonable. f(x) = x5 - 2

1. Graph the function f(x) = x4 - 3x3 - 6x2 + 7x + 30 in the viewing rectangle [-3, 5] by [-10, 50]. 2. Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of f’. (See Example 2.8.1.) 3. Calculate f’(x) and use this expression to graph f’. Compare with your sketch in part (b).

1. Graph the function g(x) = ex - 3x2 in the viewing rectangle [-1, 4] by [-8, 8]. 2. (b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of g’. (See Example 2.8.1.) 3. (c) Calculate g,(x) and use this expression to graph t9. Compare with your sketch in part (b).

Find the first and second derivatives of the function. f(x) = 0.001x5 - 0.02x3

Find the first and second derivatives of the function. G(r) =

Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of f, f’, and f’. f(x) = 2x - 5x3/4

Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of \(f, f^{\prime}\), and \(f^{\prime \prime}\). \(f(x)=e^x-x^3\)

The equation of motion of a particle is s = t3 - 3t, where s is in meters and t is in seconds. Find (a) the velocity and acceleration as functions of t, (b) the acceleration after 2 s, and (c) the acceleration when the velocity is 0.

The equation of motion of a particle is s = t4 - 2t3 + t2 - t, where s is in meters and t is in seconds. (a) Find the velocity and acceleration as functions of t. (b) Find the acceleration after 1 s. (c) Graph the position, velocity, and acceleration functions on the same screen.

Biologists have proposed a cubic polynomial to model the length L of Alaskan rockfish at age A: L = 0.0155A3 - 0.372A2 + 3.95A + 1.21 where L is measured in inches and A in years. Calculate and interpret your answer.

The number of tree species S in a given area A in a forest reserve has been modeled by the power function S(A) = 0.882A0.842 where A is measured in square meters. Find S’(100)and interpret your answer.

Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the pressure P of the gas is inversely proportional to the volume V of the gas. 1. Suppose that the pressure of a sample of air that occupies 0.106 m3 at 25°C is 50 kPa. Write V as a function of P. 2. Calculate dV/dP when P ? 50 kPa. What is the meaning of the derivative? What are its units?

Car tires need to be inflated properly because overinflation or underinflation can cause premature tread wear. The data in the table show tire life L (in thousands of miles) for a certain type of tire at various pressures P (in lb/in2). 1. Use a calculator or computer to model tire life with a quadratic function of the pressure. 2. Use the model to estimate dL/dP when P = 30 and when P = 40. What is the meaning of the derivative? What are the units? What is the significance of the signs of the derivatives?

Find the points on the curve y = x3 + 3x2 - 9x + 10 where the tangent is horizontal.

For what value of x does the graph of \(f(x)=e^{x}-2 x\) have a horizontal tangent?

Show that the curve y = 2ex + 3x + 5x3 has no tangent line with slope 2.

Find an equation of the line that is both tangent to the curve y = x4 + 1 and parallel to the line 32x - y = 15.

Find equations for two lines that are both tangent to the curve y = x3 - 3x2 - 3x - 3 and parallel to the line 3x - y = 15.

At what point on the curve y = 1 + 2ex - 3x is the tangent line parallel to the line 3x - y = 5? Illustrate by graphing the curve and both lines.

Find an equation of the normal line to the curve y = that is parallel to the line 2x + y = 1.

Where does the normal line to the parabola y = x2 - 1 at the point (-1, 0) intersect the parabola a second time? Illustrate with a sketch.

Draw a diagram to show that there are two tangent lines to the parabola y = x2 that pass through the point (0, -4). Find the coordinates of the points where these tangent lines intersect the parabola.

1. Find equations of both lines through the point (2, -3) that are tangent to the parabola y = x2 + x. 2. Show that there is no line through the point (2, 7) that is tangent to the parabola. Then draw a diagram to see why.

Use the definition of a derivative to show that if f(x)=1/x, then \(f^{\prime}(x)=-1 / x^2\). (This proves the Power Rule for the case n=-1.

Find the nth derivative of each function by calculating the first few derivatives and observing the pattern that occurs. (a) \(f(x)=x^n\) (b) f(x)=1/x

Find a second-degree polynomial P such that P(2) = 5, P’(2) = 3, and P”(2) = 2.

The equation \(y^{\prime \prime}+y^{\prime}-2 y=x^2\) is called a differential equation because it involves an unknown function y and its derivatives \(y^{\prime}\) and \(y^{\prime \prime}\). Find constants A, B, and C such that the function \(y=A x^2+B x+C\) satisfies this equation. (Differential equations will be studied in detail in Chapter 9.)

Find a cubic function y = ax3 + bx2 + cx + d whose graph has horizontal tangents at the points (-2, 6) and (2, 0).

Find a parabola with equation \(y=a x^{2}+b x+c\) that has slope 4 at x = 1, slope -8 at x = -1, and passes through the point (2, 15).

Let Is f differentiable at 1? Sketch the graphs of f and f’.

At what numbers is the following function t differentiable? Give a formula for g’ and sketch the graphs of g and g’.

(a) For what values of x is the function \(f(x)=\left|x^2-9\right|\) differentiable? Find a formula for \(f^{\prime}\). (b) Sketch the graphs of f and \(f^{\prime}\).

Where is the function h(x) = | x - 1 | + | x - 2 | differentiable? Give a formula for h’ and sketch the graphs of h and h’.

Find the parabola with equation y = ax2 + bx whose tangent line at (1, 1) has equation y = 3x - 2.

Suppose the curve y = x4 + ax3 + bx2 + cx 1d has a tangent line when x = 0 with equation y = 2x + 1 and a tangent line when x = 1 with equation y = 2 - 3x. Find the values of a, b, c, and d.

For what values of a and b is the line 2x+y=b tangent to the parabola \(y=a x^2\) when x=2 ?

Find the value of c such that the line y = + 6 is tangent to the curve y = c .

What is the value of c such that the line y = 2x + 3 is tangent to the parabola \(y=c x^{2}\)?

The graph of any quadratic function f(x) = ax2 + bx + c is a parabola. Prove that the average of the slopes of the tangent lines to the parabola at the endpoints of any interval [p, q] equals the slope of the tangent line at the midpoint of the interval.

Let Find the values of and that make differentiable everywhere.

Find numbers and such that the given function is differentiable at 1.

Evaluate \(\lim \limits_{x \rightarrow 1} \frac{x^{1000}-1}{x-1}\).

A tangent line is drawn to the hyperbola at a point as shown in the figure. 1. Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is . 2. Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where is located on the hyperbola.

Draw a diagram showing two perpendicular lines that intersect on the y-axis and are both tangent to the parabola . Where do these lines intersect?

Sketch the parabolas \(y=x^{2}\) and \(y=x^{2}-2 x+2\). Do you think there is a line that is tangent to both curves? If so, find its equation. If not, why not?

If , how many lines through the point are normal lines to the parabola ? What if ?