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Practice Test 1 (Differentiation) Part I: Multiple Choice: No Calculator ࢌሺ࢞ሻ ൌ ࢞ ૛െ ૢ ૜െ࢞ൌ൅૜ ࢞ This function has a discontinuity where the denominator is zero. i.... e., at ࢞ ൌ െ૜. If this discontinuity is removed, then, we have: So, ࢌሺെ૜ሻ ൌ െ૜ െ ૜ ൌ െ૟ Answer: E For this rational function, the highest power of ࢞ takes over when calculating a limit. Since the ࢞ ૟term exists in the denominator and not in the numerator, the ratio of the two polynomials gets smaller and smaller as ࢞ ,૙ → eventually approaching ૙. Alternatively, apply L’Hospital’s rule repeatedly (5 times) until you have a constant in the numerator and a linear term in the denominator. The ratio can then be seen to approach zero as ࢞.∞ → Answer: A ሻ ൅ ૛࢞૛ ܖܑܛ ૝ሺ ൌ ሻ࢞ሺࢌ ૚ ૛ൗ ᇱࢌ ሺ࢞ሻ ൌ ૚ ૚ିሻ ൅ ૛࢞૛ ܖܑܛ ૝ሺ૛ ૛ൗ ∙ ሺ૝ ܛܗ܋૛ ࢞ሻ ∙ ૛ ൌ ሻ࢞૛ ܛܗ܋ ૝ሺ ૛ ൅ ࢞૛ ܖܑܛ ૝√ ᇱࢌ ሺ૙ሻ ൌ ሺ૝ ܛܗ܋૙ ሻ ૛ ൅ ૙ ܖܑܛ ૝√ ൌ ૝ √૛ ൌ ૛√૛ Answer: D Page | 2 Use the chain rule in Lagrange (Prime) Notation. ᇱࢎ ሺ࢞ሻ ൌ ࢌᇱ ࢍ ∘ ࢌ ൌ ࢎ :where, ሻ࢞ሺ′ࢍ ∙ ሻ൯࢞ሺࢍ൫ ᇱࢎ ሺ૚ሻ ൌ ࢌᇱ ൫ࢍሺ૚ሻ൯ ∙ ࢍᇱ ሺ૚ሻ ൌ ࢌᇱ ሺ૜ሻ ∙ ࢍᇱ ሺ૚ሻ ൌ ሺെ૞ሻ ∙ ሺെ૜ሻ ൌ ૚૞ Answer: B ܕܑܔ ૙→ࢎ ቀ ܛܗ܋ ࣊ ૛ ൅ ࢎቁ െ ܛܗ܋ ቀ ࣊ ૛ቁ ࢊ ൌ ࢎ ቤ ሻ࢞ ܛܗ܋ሺ ࢞ࢊ ൌ ࢞ ࣊ ૛ This limit is the definition of a derivative. ൌ ሺെ ܛܑܖ ࢞ሻ ฬ ࣊ୀ࢞ ૛ ࣊ ܖܑܛ െ ൌ ૛ ൌ െ૚ Answer: A ࢌሺ࢞ሻ ൌ ܛܑܖ૛ሺ૜ െ ࢞ሻ ᇱࢌ ሺ࢞ሻ ൌ ૛ ∙ ܛܑܖሺ૜െ࢞ሻ ∙ ࢊ ሻሿ࢞ െ૜ ሺܖܑܛሾ ࢞ࢊ ൌ ૛ ∙ ܛܑܖሺ૜െ࢞ሻ ∙ ܛܗ܋ሺ૜െ࢞ሻ ∙ ሺെ૚ሻ ᇱࢌ ሺ૙ሻ ൌ െ૛ ∙ ܛܑܖሺ૜ሻ ∙ ܛܗ܋ሺ૜ሻ ൌ ૛ ∙ ܛܑܖሺ૜െ࢞ሻ ∙ ܛܗ܋ሺ૜ െ ࢞ሻ ∙ ࢊ ࢞ࢊ ሺ૜ െ ࢞ሻ Answer: B Page | 3 ۷. ࢌᇱ ࢇ→࢞ܕܑܔ ൌ ሻ࢞ሺ ࢌሺ࢞ሻ െ ࢌሺࢇሻ ࢇെ࢞ ᇱࢌ ሺ૛ሻ ൌ ܔܑ࢞ܕ૛→ ࢌሺ࢞ሻ െ ࢌሺ૛ሻ ૛െ࢞ ᇱࢌ ,܎ܑ ૛→࢞ ܕܑܔ ൌ ሻ૛ሺ ሻ࢞ሺࢌ ૛→࢞ ܕܑܔ ൌ૛ െ࢞ ࢌሺ࢞ሻ െ ૙ ૛െ࢞ ۳܃܀܂ ܛܑ ۷ ૙ ൌ ሻ૛ሺࢌ ܜ܉ܐܜ ܍ܝܚܜ ܍܊ ܜܛܝܕ ܜܑ Check the conditions one at a time. II. ࢌ is differentiable at ࢞ൌ૛, and so must be continuous at ࢞ ൌ ૛. II is TRUE III. The first derivative provides the slope of the tangent line. Since ࢌ′ሺ૛ሻ ൌ ૙, there must be a tangent of slope ૙ (horizontal) at ࢞ൌ૛. III is TRUE Answer: E [Show More]

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