The following are some questions from a review sheet for a Calculus 1 class...

Any help would be greatly appreciated.

1) A product of functions is written as f1(x)f2(x)...fn(x), where n is greater than or equal to 2, and a quotient as f(x)/g(x).

Classify the following expressions as products, quotients or neither:

(a) x ln x cos x (b) 6x/x2 + 1 (c) x sin x/ cos x (d) ln sin x

2) Composite Functions

A composite function such as esin x2 can be written as h(g(f(x))), where f(x) = x2, g(x) = sin x and h(x) = ex.

Write the following quantities in a similar manner, in each case defining the component functions.

(a)1/x -1

(b) ln cosh x

(c) e^cos(x+pi/4)

(d)( ln tanh^ -1) (x^2 + 1)^ -3

3) Graphs of Functions

(a) For the following pairs of functions, state how the gradient and position of the graph of the first is related to that of the second.

(a) 3 sin x, sin x

(b) 7x^3 - 5, x^3

(c) 2 cos (x + pi/4 ), cos x

(d) e^x-1, e^x

4) Tangents to Graphs

(a) A straight line has the equation y = mx + c. If two points on a particular straight-line graph are (4, 5) and (8, 7), find the equation for this line and the angle that the line makes with the horizontal.

(b) If theta is the angle between the tangent to the curve y = f(x) at x and the horizontal then theta = tan^ -1 f ' (x), where f ' (x) is the derivative of f(x). (For example, if f(x) = x^3 - 1 then f ' (x) = 3x^2.)

Explain what f ' (x) tells you about the tangent to the graph at x. (It maybe useful to draw a sketch.)

5) Partial Fractions 1

Express the following in partial-fraction form.

(a) x/ (x - 1)(x + 2)

(b) 3/ x^2 - 4x + 3

6. Partial Fractions 2

Express the following in partial-fraction form.

(a) x^2 - 3/ (x + 1)(x^2 + 4)

(b) x^2 - 3/ (x + 1)(x + 4)^2

7. Trigonometric Formulae

Using the formulae for double angles, express the following in multiple-angle form.

(a) sin^2 theta

(b) cos^2 theta

(c) sin^2 3(theta)

(d) cos^2(theta/2)

8. Notation

If y = f(x) then the derivative of y is written as dy/dx or f ' (x). If s = g(t), the derivative is written as ds/ dt or g ' (t).

Write down the appropriate notation for the derivatives in the following cases.

(a) T = f(theta)

(b) theta = f(t)

(c) p = h(w)

(d) z = g(y)

9. Differentiation

Differentiate the following functions with respect to the variable, i.e., write down the derivative, using the correct notation. For example, if a function is given as s = 3t then the derivative is usually written as ds/dt= 3; if given as f(t) = 3t, then the derivative is written as f ' (t) = 3.

(a) y = x^2

(b) x = 3t^3 - 2t + 1

(c) f(theta) = sin theta

(d) y = cos t

10. Integration

Integrate the following with respect to the variable concerned.

(a) *Integral 4 d(theta)

(b) *Integral 2x^2 dx

(c) *Integral 3 sin t dt

(d) *Integral (2y^4 - 3y) dy

Not quite sure on how to use all the notations on here yet, so sorry for any confusion.

Thanks for any and all help! (Happy)