IB Math HL Y1 Review of limits, derivatives, applications of derivatives Name _________________________________ Date __________ Note: The IB Formula booklet will NOT be used on this test. Topic 1 – Limits • Evaluating limits o Algebraic techniques o Estimation using a table • One-­‐sided limits • Limits and continuity o Types of discontinuities • Differentiability 1. True or False. (a) Limits can sometimes be found simply by direct substitution. 0
, then the limit does not exist. 0
0
(c) If direct substitution gives , then the limit is equal to ±∞ . 0
(d) The notation x → 2 − means “as x approaches two from the left.” (e) If lim f (x) ≠ lim f (x) , then lim f (x) does not exist and the function is discontinuous at x = a . (f) If lim f (x) = lim f (x) = L but f (a) = M , where L ≠ M , then lim f (x) does not exist. (g) If lim f (x) = lim f (x) = L but f (a) = M , where L ≠ M , then the function is (b) If direct substitution gives x→a −
x→a +
x→a −
x→a +
x→a −
x→a +
x→a
x→a
discontinuous at x = a . 2. Evaluate the limits. (
)
(a) lim x + 3 x→−1
2
x−3
x→3 x − x − 6
x2 − 1
(b) lim
x→3 x − 1
(c) lim
1 1
−
2
4 (e) lim x
x→2 x − 2
2
x 3 − 4x
(d) lim
x→−2 x + 2
(f) lim
x→4
2x + 1 − 3
x−4
3. Sketch a possible graph for each of the functions described. (a) has a removable discontinuity at x = 1 (b) approaches – 2 as x → 1− approaches 5 as x → 1+ (c) f (6) exists, lim f (x) exists, f is not continuous at x = 6 x→6
4. Multiple Choice ⎧ x 2 −1, x ≠ 1
. x =1
⎩ 4,
Let f (x) = ⎨
Which of the following statements I, II, and III, are true? I. lim f (x) exists II. f (1) exists III. f has a jump discontinuity at x = 1 (A) I only x→1
(B) II only 5. Consider the function f (x) =
(C) I and II only (D) none (E) all e x−1 − e
. x−2
Make a table of values/use a graph and estimate lim f (x) to the nearest hundredth. x→2
⎧ x + 3, x ≤ 0
⎪
6. Let f be the function defined by f (x) = ⎨ 2x + b
, where b and c are real numbers. ,
x
>
0
⎪⎩ cx −1
(a) Suppose b = c = 2 . Find and classify the points of discontinuity. (b) Suppose c = 4 . Find the value(s) of b for which f is continuous at x = 0 . (c) Find the value(s) of b and c for which f is differentiable at x = 0 . Topic 2 – Derivative basics and application • The meaning of a derivative • Estimating the value of derivative • The derivative as a limit – 2 ways • Derivative rules – power rule, product rule, quotient rule, chain rule • Tangent lines and normal lines • 2nd and higher-­‐order derivatives 7. Find the derivative. (a) f (x) =
x 2 − 2x + 5 (b) y =
4x 5 − x 3 + 3x
x2
x2
(c) g(x) =
3x + 5
2
⎛
x2 ⎞
(d) y = ⎜ 1 − ⎟ 4⎠
⎝
8. Given the following table, find the indicated values. x f (x) g(x) f ′(x) g′(x) 0 1 1 3 1 – 4 5 – 1 3 – 2 (a) h′ ( 0 ) if h(x) = f (g(x)) (b) h′ (1) if h(x) =
f (x)
g(x)
(c) h′ ( 0 ) if h(x) = x f (x) 2
9. Use the table from Question 8 to write the equation of a line tangent to the graph of y = f (x) at x = 0 . 10. The line y = 16x − 9 is a tangent to the curve y = 2x 3 + ax 2 + bx − 9 at the point (1, 7) . Find the values of a and b. 11. Given f (x) =
x5 + 2
, find the exact value of the x-­‐coordinate of the point where f ′′(x) = 0 . x
12. Find the derivative from first principles. (a) f ′(−1) where f (x) =
1
. 2x + 3
(b) f ′(x) where f (x) = 2x +
x Topic 3 – Applications of differential calculus • Kinematics – position, velocity, acceleration • First derivatives and graphical interpretation o Increasing and decreasing functions o Maximum and minimum points • Mean Value Theorem • Second derivatives and graphical interpretation o Concavity o Points of inflection • First Derivative Test/Second Derivative Test 13. Consider the function f (x) = 4 x . Find the value of c on the interval 1 < x < 4 that satisfies the Mean Value Theorem. 14. 15.
16. Use the graph from Question 15. (a) If the domain of f is [0, 7] , find the value(s) of x where f has a relative maximum or a relative minimum. Classify these extrema and justify your answers. (b) Find the intervals where f is concave up and the intervals where it is concave down. Justify your answers. (c) If f (2) = 5 , write the equation of the tangent to f at the point ( 2, 5) .