{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

finalps3

# finalps3 - Math 16A Short Calculus section 2 Practice Final...

This preview shows pages 1–4. Sign up to view the full content.

Math 16A Short Calculus, section 2 Practice Final Exam Answers 1. Continuity For each of the following functions, state: the range, the inter- val(s) on which the function is continuous, and for each disconti- nuity whether or not the discontinuity is removable. (a) y = x 2 + 4 Range: (4 , ) Continuous on: ( -∞ , ) No discontinuities (b) y = x +2 x 2 - 4 Range: ( -∞ , 0) (0 , ) Continuous on: ( -∞ , - 2) ( - 2 , 2) (2 , ) Removable discontinuity at x = - 2 Nonremovable discontinuity at x = 2 (c) y = 2 b x c Range: all even integers Continuous on: [ x, x + 1) for all integers x Nonremovable discontinuities at each integer; no other discontinuities (d) y = 4 cos( θ ) + 1 Range: (-3,5) Continuous on: ( -∞ , ) No discontinuities

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2. Slope and Derivative (a) Find an equation of the line tangent to f ( x ) = x 2 + 1 with slope - 2. We have f 0 ( x ) = 2 x , so slope - 2 occurs when - 2 = f 0 ( x ) = 2 x = x = - 1. When x = - 1, f ( - 1) = 2. The tangent line is: ( y - 2) = ( - 2)( x + 1) = y = - 2 x. (b) Use the limit definition to find the derivative of the following functions. i. f ( x ) = x 2 - 1 f 0 ( x ) = lim Δ x 0 f ( x + Δ x ) - f ( x ) Δ x = lim Δ x 0 ( x + Δ x ) 2 - 1 - x 2 + 1 Δ x = lim Δ x 0 x 2 + 2 x Δ x + (Δ x ) 2 - x 2 Δ x = lim Δ x 0 x )(2 x + Δ x ) Δ x = lim Δ x 0 (2 x + Δ x ) = 2 x. ii. h ( t ) = 2 t h 0 ( x ) = lim Δ t 0 h ( t + Δ t ) - h ( t ) Δ t = lim Δ t 0 2 t + Δ t - 2 t Δ t = lim Δ t 0 2( t + Δ t - t ) Δ t · t + Δ t + t t + Δ t + t = lim Δ t 0 2(( t + Δ t ) - t ) t )( t + Δ t + t ) = lim Δ t 0 t t )( t + Δ t + t ) = lim Δ t 0 2 t + Δ t + t = 2 2 t = 1 t .
3. Rates of Change (a) The profit P from selling x units of a product is given by P = 15 + 12 x - 81 x .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 9

finalps3 - Math 16A Short Calculus section 2 Practice Final...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online