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Math 160. Test 1. (Harvey Fall 2004)
Name: (2 points)
No notes or texts allowed. You may use a TI-83 or TI-86 or equivalent calculator. Show all work.
Problems 1-5: Find the derivative f ′(x) (6 points each)
f (x) = x^3 + 3x^2 + 3x + 1
f (x) = (x + 1)(
x − 4)
f (x) = 3
x +
π √ (^3) x
f (x) = 5 + π^3
f (x) = x^1 /^2 + x^1 /^3 + x^1 /^4
Problems 6, 7: Compute the derivative dydx using the definition of the derivative. No credit will be given for any other method.
- (10 points) y = 5x^2 + 3
- (10 points)
y = 1 4 x
8-10 (6 points each) Compute the limits:
x^ lim→ 5 x
x^2 − 2 x − 15
x^ lim→ 0
x^2 + 3x + 4 2 x^2 + x + 1
x^ lim→∞^2 x
(^3) + x + 1 3 x^3 + 2x^2 + x + 2
11 (10 points). Let f (x) = x^2 + 5. What is the equation of the tangent line to f (x) at x = 2?
- (20 points)
1 2 3 4 5 6 7 8
f(x)
(a) For what values of c is limx→c f (x) undefined? (b) For what values of c is it 0? (c) For what values of c is f ′(c) undefined? (d) At which values of x does f (x) fail to be continuous?
Solutions.
f ′(x) = 3x^2 + 6x + 3
f (x) = x^3 /^2 − 4 x + x^1 /^2 − 4 =⇒ f ′(x) =
2 x
1 / 2 − 4 +^1
2 x
− 1 / 2
f (x) = x^1 /^3 + πx−^1 /^3 =⇒ f ′(x) =
3 x
− 2 / (^3) − π 3 x
− 4 / 3
f ′(x) = 0
f ′(x) =^1 2
x−^1 /^2 +^1 3
x−^2 /^3 +^1 4
x−^3 /^4
dy dx
= lim h→ 0 [5(x^ +^ h)
(^2) + 3] − [5x (^2) + 3] h = lim h→ 05 x
(^2) + 10xh + 5h (^2) + 3 − 5 x (^2) − 3 h = lim h→ 010 xh^ + 5h
2 h = lim h→ 0 10 x^ + 5h^ = 10x
